Recent zbMATH articles in MSC 81Qhttps://www.zbmath.org/atom/cc/81Q2021-11-25T18:46:10.358925ZWerkzeugEndowing evolution algebras with properties of discrete structureshttps://www.zbmath.org/1472.170992021-11-25T18:46:10.358925Z"González-López, Rafael"https://www.zbmath.org/authors/?q=ai:gonzalez-lopez.rafael"Núñez, Juan"https://www.zbmath.org/authors/?q=ai:nunez-valdes.juanIn this paper, the authors delve into the basic properties characterizing the directed graph that is uniquely associated to an evolution algebra, and which was introduced in [\textit{J. P. Tian}, Evolution algebras and their applications. Berlin: Springer (2008; Zbl 1136.17001)]. These properties are conveniently translated to algebraic concepts and results concerning the evolution algebra under consideration. In particular, the authors focus on the adjacency of graphs, whose immediate translation into algebraic language enables them to introduce the concepts of adjacency, walk, trail, circuit, path and cycle of an evolution algebra. Also the notions of strongly and weakly connected evolution algebras are introduced as the algebraic equivalences of the same concepts in graph theory. It enables the authors to introduce the notions of distance, girth, circumference, eccentricity, center, radio, diameter and geodesic of an evolution algebra, together with the concepts of Eulerian and Hamiltonian evolution algebras. Some basic results on these topics are then described. The relationship among all of these notions and their analogous in graph theory are visually illustrated throughout the paper.On a class of sharp multiplicative Hardy inequalitieshttps://www.zbmath.org/1472.350132021-11-25T18:46:10.358925Z"Guzu, D."https://www.zbmath.org/authors/?q=ai:guzu.dorian"Hoffmann-Ostenhof, T."https://www.zbmath.org/authors/?q=ai:hoffmann-ostenhof.thomas"Laptev, A."https://www.zbmath.org/authors/?q=ai:laptev.ariSummary: A class of weighted Hardy inequalities is treated. The sharp constants depend on the lowest eigenvalues of auxiliary Schrödinger operators on a sphere. In particular, for some block radial weights these sharp constants are given in terms of the lowest eigenvalue of a Legendre type equation.Functional difference equations and eigenfunctions of a Schrödinger operator with \(\delta' -\) interaction on a circular conical surfacehttps://www.zbmath.org/1472.351052021-11-25T18:46:10.358925Z"Lyalinov, Mikhail A."https://www.zbmath.org/authors/?q=ai:lyalinov.mikhail-anatolievichSummary: Eigenfunctions and their asymptotic behaviour at large distances for the Laplace operator with singular potential, the support of which is on a circular conical surface in three-dimensional space, are studied. Within the framework of incomplete separation of variables an integral representation of the Kontorovich-Lebedev (KL) type for the eigenfunctions is obtained in terms of solution of an auxiliary functional difference equation with a meromorphic potential. Solutions of the functional difference equation are studied by reducing it to an integral equation with a bounded self-adjoint integral operator. To calculate the leading term of the asymptotics of eigenfunctions, the KL integral representation is transformed to a Sommerfeld-type integral which is well adapted to application of the saddle point technique. Outside a small angular vicinity of the so-called singular directions the asymptotic expression takes on an elementary form of exponent decreasing in distance. However, in an asymptotically small neighbourhood of singular directions, the leading term of the asymptotics also depends on a special function closely related to the function of parabolic cylinder (Weber function).Subharmonic functions in scattering theoryhttps://www.zbmath.org/1472.352602021-11-25T18:46:10.358925Z"Denisov, Sergey A."https://www.zbmath.org/authors/?q=ai:denisov.sergey-aSummary: We present a method that uses the properties of subharmonic functions to control spatial asymptotics of Green's kernel of multidimensional Schrödinger operator with rough potential.On the one dimensional Dirac equation with potentialhttps://www.zbmath.org/1472.353222021-11-25T18:46:10.358925Z"Erdoğan, M. Burak"https://www.zbmath.org/authors/?q=ai:erdogan.mehmet-burak"Green, William R."https://www.zbmath.org/authors/?q=ai:green.william-rSummary: We investigate \(L^1\to L^\infty\) dispersive estimates for the one dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies the natural \(t^{-\frac{1}{2}}\) decay rate, which may be improved to \(t^{-\frac{3}{2}}\) at the cost of spatial weights when the thresholds are regular. We classify the structure of threshold obstructions, showing that there is at most a one dimensional space at each threshold. We show that, in the presence of a threshold resonance, the Dirac evolution satisfies the natural decay rate, and satisfies the faster weighted bound except for a piece of rank at most two, one per threshold. Further, we prove high energy dispersive bounds that are near optimal with respect to the required smoothness of the initial data. To do so we use a variant of a high energy argument that was originally developed to study Kato smoothing estimates for magnetic Schrödinger operators. This method has never been used before to obtain \(L^1\to L^\infty\) estimates. As a consequence of our analysis we prove a uniform limiting absorption principle, Strichartz estimates, and prove the existence of an eigenvalue free region for the one dimensional Dirac operator with a non-self-adjoint potential.Semiclassical evolution with low regularityhttps://www.zbmath.org/1472.353232021-11-25T18:46:10.358925Z"Golse, François"https://www.zbmath.org/authors/?q=ai:golse.francois"Paul, Thierry"https://www.zbmath.org/authors/?q=ai:paul.thierrySummary: We prove semiclassical estimates for the Schrödinger-von Neumann evolution with \(C^{1,1}\) potentials and density matrices whose square root have either Wigner functions with low regularity independent of the dimension, or matrix elements between Hermite functions having long range decay. The estimates are settled in different weak topologies and apply to initial density operators whose square root have Wigner functions 7 times differentiable, independently of the dimension. They also apply to the \(N\)-body quantum dynamics uniformly in \(N\) and to concentrating pure and mixed states without any regularity assumption. In an appendix, we finally estimate the dependence in the dimension of the constant appearing on the Calderón-Vaillancourt Theorem.Restrictions on the existence of a canonical system flow hierarchyhttps://www.zbmath.org/1472.353342021-11-25T18:46:10.358925Z"Hur, Injo"https://www.zbmath.org/authors/?q=ai:hur.injo"Ong, Darren C."https://www.zbmath.org/authors/?q=ai:ong.darren-cSummary: The KdV hierarchy is a family of evolutions on a Schrödinger operator that preserves its spectrum. Canonical systems are a generalization of Schrödinger operators, that nevertheless share many features with Schrödinger operators. Since this is a very natural generalization, one would expect that it would also be straightforward to build a hierarchy of isospectral evolutions on canonical systems analogous to the KdV hierarchy. Surprisingly, we show that there are many obstructions to constructing a hierarchy of flows on canonical systems that obeys the standard assumptions of the KdV hierarchy. This suggests that we need a more sophisticated approach to develop such a hierarchy, if it is indeed possible to do so.Patterns of water in lighthttps://www.zbmath.org/1472.353572021-11-25T18:46:10.358925Z"Horikis, Theodoros P."https://www.zbmath.org/authors/?q=ai:horikis.theodoros-p"Frantzeskakis, Dimitrios J."https://www.zbmath.org/authors/?q=ai:frantzeskakis.dimitri-jSummary: The intricate patterns emerging from the interactions between soliton stripes of a two-dimensional defocusing nonlinear Schrödinger (NLS) model with a non-local nonlinearity are considered. We show that, for sufficiently strong non-locality, the model is asymptotically reduced to a Kadomtsev-Petviashvilli-II (KPII) equation, which is a common model arising in the description of shallow water waves, as such patterns of water may indeed exist in light (this non-local NLS finds applications in nonlinear optics, modelling beam propagation in media featuring thermal nonlinearities, in plasmas, and in nematic liquid crystals). This way, approximate antidark soliton solutions of the NLS model are constructed from the stable KPII line solitons. By means of direct numerical simulations, we demonstrate that non-resonant and resonant two- and three-antidark NLS stripe soliton interactions give rise to wave configurations that are found in the context of the KPII equation. Thus, our study indicates that patterns which are usually observed in water can also be found in optics.Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphshttps://www.zbmath.org/1472.354182021-11-25T18:46:10.358925Z"Hofmann, Matthias"https://www.zbmath.org/authors/?q=ai:hofmann.matthias"Kennedy, James B."https://www.zbmath.org/authors/?q=ai:kennedy.james-bSummary: We prove interlacing inequalities between spectral minimal energies of metric graphs built on Dirichlet and standard Laplacian eigenvalues, as recently introduced in [the second author et al., Calc. Var. Partial Differ. Equ. 60, No. 2, Paper No. 61, 63 p. (2021; Zbl 1462.35222)]. These inequalities, which involve the first Betti number and the number of degree one vertices of the graph, recall both interlacing and other inequalities for the Laplacian eigenvalues of the whole graph, as well as estimates on the difference between the number of nodal and Neumann domains of the whole graph eigenfunctions. To this end we study carefully the principle of \textit{cutting} a graph, in particular quantifying the size of a cut as a perturbation of the original graph via the notion of its \textit{rank}. As a corollary we obtain an inequality between these energies and the actual Dirichlet and standard Laplacian eigenvalues, valid for all compact graphs, which complements a version for tree graphs of Friedlander's inequalities between Dirichlet and Neumann eigenvalues of a domain. In some cases this results in better Laplacian eigenvalue estimates than those obtained previously via more direct methods.Weighted Hurwitz numbers and topological recursionhttps://www.zbmath.org/1472.370782021-11-25T18:46:10.358925Z"Alexandrov, A."https://www.zbmath.org/authors/?q=ai:aleksandrov.a-i|alexandrov.artem|alexandrov.alexey|alexandrov.a-d|aleksandrov.alexandr-g|aleksandrov.a-yu|alexandrov.alexander-s|aleksandrov.a-k|aleksandrov.a-a|aleksandrov.a-t|alexandrov.a-n|aleksandrov.aleksei-b|aleksandrov.avraam-ya|alexandrov.a-s|aleksandrov.albert-georgievich|aleksandrov.aleksandr-yurevich|aleksandrov.a-l|alexandrov.andrey|alexandrov.anatoli-v|aleksandrov.a-b|aleksandrov.a-m|aleksandrov.a-v.1"Chapuy, G."https://www.zbmath.org/authors/?q=ai:chapuy.guillaume"Eynard, B."https://www.zbmath.org/authors/?q=ai:eynard.bertrand"Harnad, J."https://www.zbmath.org/authors/?q=ai:harnad.johnSummary: The KP and 2D Toda \(\tau \)-functions of hypergeometric type that serve as generating functions for weighted single and double Hurwitz numbers are related to the topological recursion programme. A graphical representation of such weighted Hurwitz numbers is given in terms of weighted constellations. The associated classical and quantum spectral curves are derived, and these are interpreted combinatorially in terms of the graphical model. The pair correlators are given a finite Christoffel-Darboux representation and determinantal expressions are obtained for the multipair correlators. The genus expansion of the multicurrent correlators is shown to provide generating series for weighted Hurwitz numbers of fixed ramification profile lengths. The WKB series for the Baker function is derived and used to deduce the loop equations and the topological recursion relations in the case of polynomial weight functions.Spectral enclosures for non-self-adjoint extensions of symmetric operatorshttps://www.zbmath.org/1472.470182021-11-25T18:46:10.358925Z"Behrndt, Jussi"https://www.zbmath.org/authors/?q=ai:behrndt.jussi"Langer, Matthias"https://www.zbmath.org/authors/?q=ai:langer.matthias"Lotoreichik, Vladimir"https://www.zbmath.org/authors/?q=ai:lotoreichik.vladimir"Rohleder, Jonathan"https://www.zbmath.org/authors/?q=ai:rohleder.jonathanIn the description of many quantum mechanical systems, operators appear as a consequence of heuristic arguments which suggest in a first step a formal expression for the Hamiltonian or Schrödinger operator describing the model. These operators \(S\) are typically unbounded and symmetric on a domain \(\operatorname{dom}{S}\) which is a dense subspace of a Hilbert space \(\mathcal{H}\). In a second crucial step for the description of the quantum mechanical system, one has to choose a closed (in many cases self-adjoint) extension of \(S\) in order to start the analysis of the model. Typically, fixing an extension means to specify the relevant boundary conditions for the system. The paper under review focuses on the description of closed non-selfadjoint extensions of \(S\) which appear as restrictions of the adjoint operator \(S^*\) and on the analysis of some of their spectral properties. The article also presents in its final part several applications of their results to elliptic operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with \(\delta\)-interactions, and to quantum graphs with non-self-adjoint vertex couplings.
In the first part of the article, the authors use an abstract and systematic approach to the description of extensions in terms of so-called boundary triples \((\Gamma_0,\Gamma_1,\mathcal{G})\), where \(\Gamma_{0,1}: \operatorname{dom}S^*\to \mathcal{G}\) satisfy a Green identity on the auxiliary Hilbert space \(\mathcal{G}\). Boundary triples (or their generalizations called quasi boundary triples) provide a useful technique to describe extensions encoding abstractly the boundary data of the problem. To formulate their main results in this part (e.g., Theorem 3.1), the authors use the Weyl function, which is an operator-valued function on the auxiliary Hilbert space defined in terms of the boundary triples. In addition, they introduce also a boundary operator \(B\) (in general non-symmetric) which serves to label the different extensions.
The article has an informative and well-written introduction to the topic describing their methods and results, but also introducing the reader to the literature and alternative approaches in this very active field. The bibliography list contains more than 130 references.Some properties of Dirac-Einstein bubbleshttps://www.zbmath.org/1472.530582021-11-25T18:46:10.358925Z"Borrelli, William"https://www.zbmath.org/authors/?q=ai:borrelli.william"Maalaoui, Ali"https://www.zbmath.org/authors/?q=ai:maalaoui.aliSummary: We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac-Einstein equations on \(\mathbb{R}^3\), which appear in the bubbling analysis of conformal Dirac-Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin-Talenti functions, while the spinorial part is the conformal image of \(- \frac{1}{2}\)-Killing spinors on the round sphere \(\mathbb{S}^3\).From classical mechanics to quantum field theory. A tutorialhttps://www.zbmath.org/1472.810042021-11-25T18:46:10.358925Z"Asorey, Manuel"https://www.zbmath.org/authors/?q=ai:asorey.manuel"Ercolessi, Elisa"https://www.zbmath.org/authors/?q=ai:ercolessi.elisa"Moretti, Valter"https://www.zbmath.org/authors/?q=ai:moretti.valterAfter the preface this book is organized in 3 chapters, and an index.
The preface informs the reader that this Nook grew out from mini courses the authors delivered st the at the Fall workshops on geometry and physics at Granada, Zaragoza, and Madrid for advanced graduate students, PhD students, and young researchers.
Chapter 1: ``A short course on quantum mechanics and methods of quantization,'' is written by Elisa Ercolessi. In her introduction she begins with a short overview of the developments in the latter centuries. Geometric methods became more and more important at the end of the latter and beginning of the present century, symplectic manifolds and geometric quantization are mentioned. In section 1.2 an overview of quantum mechanics is given. Whereas a pure state in classical mechanics is a point in a real symplectic \(\mathbb{R}^{2n}\) such state in quantum mechanics is a unit ray in a complex Hilbert space \(\mathcal{H}\) for to describe interferences, probabilities etc.. As an example the two level case is considered. Observables with pure discrete spectra and their expectation values are explained. Then the position and linear momentum operators are introduced. Then ladder operators for two level Fermionic and general Bosonic systems as well as composite systems are considered. The Schrödinger equation is introduced and applied to the harmonic oscillator. The description of quantum measurement is shortly described. The first remark about geometric quantum mechanics is that \(T\mathcal{H} = \mathcal{H}\times\mathcal{H}\), s. t. a vector field \(\Gamma\) is given by \(\mathcal{H}\ni\psi\mapsto (\psi,\phi) = \Gamma(\psi),\, \phi\in\mathcal{H}\). The Lie derivative along \(\Gamma\), the dilation and further concepts are discussed. It is shown how the Hermitean scalar product is related to the Riemann metric and the symplectic form when \(\mathcal{H}\)is understood as a manifold. The ray representation of states leads to the projective Hilbert space whose relation to a Kähler manifold is shown up. Several notations for vector fields, tensor fields, forms, and brackets are introduced. The section ``Methods of Quantization'' begins introducing the Heisenberg-Weyl algebra. Then coherent states and the Bargmann-Fock representation are considered. Distributions of the number of particles, quasi-classical states, the property of minimizing the uncertainty relation, and generalized coherent states are described. Feynman's appoach to quantize field theories from the principle of least action to Feyman integrals is considered. Examples of application are given for the free particle. the harmonic oscillator, partition function and imaginary time integrals, the use of coherent states,and particle exchange symmeties in case of many particles. Then the Weyl quantization omitting the problem of domains for unbounded operators and relating quantization with phase space representations and Wigner functions is considered. The Weyl map is introduced exemplified for the free particle. Linear transformations preserving symplectic structures and the Fourier transform lead to the free particle evolution. The Wigner map is introduced as a kind of inversion to the Weyl map and discussed to some extend including the Moyal product. Several properties are considered ending with Ehrenfest's theorem.
Chapter 2: ``Mathematical Formulations of Quantum Mechanics. An advanced short Course,''is written by Valter Moretti. Whereas the previous chapter points out the mathematical methods used in quantum mechanics supposing the reader to know well mathematics and being costumed to think in respective abstract categories, the present one is a concrete introduction to their applications in the foundation of quantum mechanics. At first the author states that quantum systems are characterized by the occurrence of actions of the order of Planck's constant. This is demonstrated comparing orbital periods in hydrogen atoms with those of mechanical pendulums. The randomness of measurement outcomes, state collapses, eigenstates of compatible and incompatible observables with rrespect to successive measurements, and uncertainty relations as well as time evolution are described preliminary in finite dimensional unitary space. Then the more complicated description in Hilbert space is considered. After clearing up the peculiarities of the countably infinite dimensions. For unbounded operators domains and ranges have to be respected. Types of operator algebras are considered. For densely defined operators the adjoint is introduced. The properties closed, Hermitean , selfadjoint, essential selfadjoint, unitary, and normal are defined. These properties are properly explained. Spectral decompositions, spectral measures, and functional analysis are considered. The orthocomplemented lattice of projection operators, posets, quantum logics, and the respective access to the Hilbert space representation is described. The representation of states as measures and Gleason's theorem are explained. Contextuality and the theorem of Kochen and Specker follow. Symmetry transformations, one parameter groups of unitary operators, projective representations, and conservation laws are considered until von Neumann algebras, and the existence of superselection rules are pointed on. All this is extensively explained and formulated with mathematical rigor.
Chapter 3: ``Concise Introduction to Quantum Field Theory'' is written by Manuel Asorey. After a short overview on the history of development two points are stated: (1) It provides a framework to unify quantum theory with relativity. (2) The handling of arising divergences requires the use of special mathematical methods. The problem to replace the Galileian space-time symmetry transfprmations by Poincaré transformations is exemplified considering the multidimensional harmonic oscillator and its quantization. The light cone, lightlike and spacelike distances of events are introduced and explained. Quantum field operators are introduced by the requirement of fundamental principles P1 through P7. It is shown how canonical quantization of scalar fields leads to ultraviolet and infrared divergences. As a way around the torus compactification of \(\mathbb{R}^3\) is applied which discretizes the wavelengths of plane waves. But the vacuum energy of the quantum field is still divergent because of an infinite series the positive ground state energies of harmonic oscillations. It lasted two decades until suitable renormalization procedures were found. Two previous steps are mentioned, the regularization of field operators including cut offs, and the introduction of parameters like masses, charges,coupling constants etc.. Short examles of such procedures are given. The Casimir effect is described. Arguments of field versus particle interpretations are given before the Fock space is introduced and discussed together with creation, destruction and number operators. The Wick theorem is stated which is useful in the treatment of interacting quantum fields. The interaction term of the Hamiltonian causes an extra divergence in the vacuum energy determined by the Wick theorem. Its subtraction leads to the finite vacuum energy. The renormalization treatment for excited states are described too. The validity of the covariance principles P5 through P7 for he Heisenberg time translation of the field operators is considered. The Euklidean approach is motivated by the difficulty to handle unbounded field operators in the Fock space. Wightman functions and Schwinger functions being their analytic continuations are mentioned. Their evolution in the imaginary time is a semigroup instead a group. The necessity of time orderings is founded. Five Euklidean principle which follow from P1 through P7 are formulated. Since the relativistic quantum field can be fully reconstructed this approach is also called constructive. Conformal field theories are invariant under an extension of the Poincaré group including dilations and special conformal transformations. Difficulties with two-point functions are described The \(O(2)\) sigma model is dicussed. A functional integral approach is described and refines the Euclidean approach.After a look on possibly to expect further creations three appendices deepening the descriptions of the Casimir effect, Gaussian measures on Hilbert spaces, and Peierls brackets close the chapter.
This book is really suited for to extend the insights to the title problems for interested people who have previous knowledge in mathematics and physics.Temporal relationalismhttps://www.zbmath.org/1472.810082021-11-25T18:46:10.358925Z"Smolin, Lee"https://www.zbmath.org/authors/?q=ai:smolin.leeSummary: I review the hypothesis that neither space nor quantum mechanics is fundamental, and both are emergent from a more fundamental description that is neither. This fundamental description is a completion of quantum mechanics based on relational hidden variables. Here, relational means that they give a fuller description, not of an individual particle but of a network of relations among particles. This completion of quantum mechanics does not live in space, rather space is an emergent description of an underlying network of relations. Since locality is, in this sense, emergent, locality can be disordered, and one of the effects of this is quantum nonlocality. This summarizes a line of thought that weaves through many of my papers on quantum foundations, from the early 1980s to the present.
For the entire collection see [Zbl 1460.81001].Revisiting the admissibility of non-contextual hidden variable models in quantum mechanicshttps://www.zbmath.org/1472.810102021-11-25T18:46:10.358925Z"Arora, Atul Singh"https://www.zbmath.org/authors/?q=ai:arora.atul-singh"Bharti, Kishor"https://www.zbmath.org/authors/?q=ai:bharti.kishor"Arvind"https://www.zbmath.org/authors/?q=ai:arvind.n|arvind.m-t|arvind.b|arvind.kim-p-gostelow|arvind.vikraman|arvind.d-k|arvind.aSummary: We construct a non-contextual hidden variable model consistent with all the kinematic predictions of quantum mechanics (QM). The famous Bell-KS theorem shows that non-contextual models which satisfy a further reasonable restriction are inconsistent with QM. In our construction, we define a weaker variant of this restriction which captures its essence while still allowing a non-contextual description of QM. This is in contrast to the contextual hidden variable toy models, such as the one by Bell, and brings out an interesting alternate way of looking at QM. The results also relate to the Bohmian model, where it is harder to pin down such features.The appearance of particle tracks in detectorshttps://www.zbmath.org/1472.810112021-11-25T18:46:10.358925Z"Ballesteros, Miguel"https://www.zbmath.org/authors/?q=ai:ballesteros.miguel"Benoist, Tristan"https://www.zbmath.org/authors/?q=ai:benoist.tristan"Fraas, Martin"https://www.zbmath.org/authors/?q=ai:fraas.martin"Fröhlich, Jürg"https://www.zbmath.org/authors/?q=ai:frohlich.jurg-martinEver since its observation the seemingly innocuous appearance of the atomic particle tracks in detectors has puzzled the physicists as one of the most striking manifestations of the weird properties of the quantum world. To make a long story short, in the quoted words of M.\ Born at the 1927 Solvay conference: ``If one associates a spherical wave with each emission process, how can one understand that the track of each \(\alpha\)-particle appears as a (very nearly) straight line? In other words: how can the corpuscular character of the phenomenon be reconciled here with the representation by waves?'' In particular, the \(S\)-wave of the outgoing \(\alpha\)-particle is spherically symmetric, but the particle tracks are not. How is, then, that this initial symmetry is broken? In a sense the typical answer -- given in a celebrated W.\ Heisenberg 1927 thought experiment -- is well known: it is an effect of the collapse of the wave packet due to the (repeated) measurements of the particle position. But this is famously an answer that begs a lot of explanation and that, ever since its formulation, has elicited heated discussions not to be summarized here.
Admittedly however the present paper deals neither with a physical model for the atom ionization and the subsequent drop formation in a cloud chamber (along the lines, for example, of the quoted N.F.\ Mott 1929 paper), nor with an explanation of the strange nature of the quantum measurements, so that in particular no new insight is to be found about the wave packet collapse, or the \textit{Heisenberg cut}, notions that are simply accepted and used along the paper. For instance, in the words of the authors (page 438): ``To describe the effect of an instantaneous measurement of the approximate position of the particle on its state we follow the conventional wisdom of quantum mechanics: In the course of such a measurement whose result is given by some vector \(\mathbf q\in\mathbb{R}^d\), the state \(\rho\) of the particle changes according to'' the usual quantum rule of the wave packet collapse summarized in the subsequent equation (29). The focus of the discussion instead is to ``present a mathematically rigorous analysis of the appearance of particle tracks'' within the framework of the said ``conventional wisdom of quantum mechanics,'' namely to show that the track appearance is well accounted for with a scrupulous application of the quantum formalism. This is in any case a commendable task, and not a very easy one to carry out as the complexity of the subsequent discussion shows.
More precisely the authors want to ``present a theoretical analysis of a gedanken experiment of the sort Heisenberg had in mind in 1927,'' with repeated position measurements every \(\tau\) seconds. In their discussion however they do not deal with \textit{idealized} quantum measurements, but they take instead the considerable trouble of discussing the case of \textit{approximate} measurements. The general states of the charged particle of mass \(M\) are here density operators \(\rho\) in a Hilbert space \(\mathcal H\) where \(\mathbf X, \mathbf P\) are the position and momentum operators, while the state vectors \(\Omega\) of the electromagnetic (EM) field plus photomultipliers live in another Hilbert space \(\mathfrak H\). The values \(\mathbf q\in\mathbb{R}^d\) (representing the approximate position measurements) of suitable operators \(\mathbf Q\) in \(\mathfrak H\) are then ``supposed to be tightly correlated with the positions, \(\mathbf x\in\mathbb{R}^d\), of the charged particle.''
Before each measurement the EM field and photomultipliers always are in the state \(\Omega_{in}\); then during the light-scattering the state changes according to a propagator \(U_t(\mathbf x)\) (\(\mathbf x\) being the position of the charged particle during the scattering process, and \(t\ll\tau\) the scattering time span) and quickly relaxes back to \(\Omega_{in}\) long before the subsequent measurement is performed. It is therefore possible to define the transition amplitude \(V_{\mathbf q}(\mathbf x)\) of equation (7) representing the probability density amplitude of finding \(\mathbf q\) when the particle is in \(\mathbf x\). The operators \(V_{\mathbf q}(\mathbf X)\) will then constitute a positive-operator-valued measure (POVM) that turns out to be instrumental to implement the wave packet reduction of every approximate position measurement. On the other hand the evolution between two measurements of the position-momentum pair \(\mathbf X,\mathbf P\) of the freely moving particle is described by a propagator \(U_S\) associated to a symplectic matrix \(S\) in the phase space \(\Gamma\)
With the Gaussian \textit{ansatz} of equation (9) for the transition amplitude, a sequence of approximate measurements \(\mathbf q_0,\dots,\mathbf q_n\) falling in the subsets \(\Delta_0,\dots,\Delta_n\) entails a change in the initial density matrix \(\rho\) produced by a total operator \(W_n(\mathbf q_0,\dots,\mathbf q_n)= U_{S^{n+1}}V_{\mathbf q_n} (\mathbf X_{n\tau}) \ldots V_{\mathbf q_0} (\mathbf X)\) given as a repeated combination of \(V_{\mathbf q}\) and \(S\). Finally this gives rise to a probability measure \(\mathbb P_\rho\) on the process of the sequences \(\mathbf q_n\) that can be used to calculate the probability that the position of the particle at the times \(n\tau\) is within \(\Delta_n\). The aim of the paper now is to show that (page 435) ``with high probability, the cells \(\Delta_0,\dots,\Delta_n\) which indicate the positions of the particle at times \(0, \tau,\dots , n\tau\) , are centered in points ``close'' to \(\mathbf x(0), \mathbf x(\tau), \dots , \mathbf x(n\tau)\), respectively, where \(\mathbf x(t) =\mathbf x+t\mathbf v, t \in [0, n\tau]\), is the trajectory of a freely moving classical particle.'' Here \(\mathbf v\) is a value of \(\mathbf V=\mathbf P/M\)
Without retracing in this short review all the details of their exhaustive discussion we will only recall next that the authors on the one hand study (by means of a suitable family of coherent states \(|W,\zeta\rangle \) centered around phase space points \(\zeta\in\Gamma\)) ``the stochastic dynamics of a (quasi-) freely moving quantum particle subjected to repeated measurements of its approximate position;'' and on the other they ``introduce a stochastic process [equation (39)] with values in the classical phase space of the particle that indexes the trajectory of coherent states occupied by the particle under the forward dynamics.'' Their first main result is then summarized in the Theorem 2.2 that ``relates the sequence of measurement data of approximate particle positions to the sequence of phase space points determined by the stochastic process in Eq. (39)'' by establishing an equality in law between the classical positions \(\xi_n\) of the particle -- plus an independent Gaussian noise \(\eta_n\) -- and the measurement results \(\mathbf Q_n\). In the Theorem 2.4 they next ``determine the best guess of the initial condition of a phase space trajectory of the stochastic process introduced in (39) from its tail,'' and finally the Theorem 2.8 ``relates the positive operator-valued measure (POVM) induced by sequences of approximate particle position measurements to a POVM taking values in the space of coherent states.'' The bulk of the paper is thereafter devoted to a long and rather convoluted sequence of technical arguments peppered with a great deal of lemmas and propositions needed to prove the said results, and ends finally with a few examples of free particles, harmonic oscillators and particles in a constant magnetic field to show the efficacity of the method.Classicalization of quantum state of detector by amplification processhttps://www.zbmath.org/1472.810182021-11-25T18:46:10.358925Z"Tirandaz, Arash"https://www.zbmath.org/authors/?q=ai:tirandaz.arash"Taher Ghahramani, Farhad"https://www.zbmath.org/authors/?q=ai:taher-ghahramani.farhad"Asadian, Ali"https://www.zbmath.org/authors/?q=ai:asadian.ali"Golshani, Mehdi"https://www.zbmath.org/authors/?q=ai:golshani.mehdiSummary: It has been shown that a macroscopic system being in a high-temperature thermal coherent state can be, in principle, driven into a non-classical state by coupling to a microscopic system. Therefore, thermal coherent states do not truly represent the classical limit of quantum description. Here, we study the classical limit of quantum state of a more relevant macroscopic system, namely the pointer of a detector, after the phase-preserving linear amplification process. In particular, we examine to what extent it is possible to find the corresponding amplified state in a superposition state, by coupling the pointer to a qubit system. We demonstrate quantitatively that the amplification process is able to produce the classical limit of quantum state of the pointer, offering a route for a classical state in a sense of not to be projected into a quantum superposition state.On the role of Hermite-like polynomials in the Fock representations of Gaussian stateshttps://www.zbmath.org/1472.810212021-11-25T18:46:10.358925Z"Pierobon, Gianfranco"https://www.zbmath.org/authors/?q=ai:pierobon.gianfranco-l"Cariolaro, Gianfranco"https://www.zbmath.org/authors/?q=ai:cariolaro.gianfranco-l"Dattoli, Giuseppe"https://www.zbmath.org/authors/?q=ai:dattoli.giuseppeSummary: The expansion of quantum states and operators in terms of Fock states plays a fundamental role in the field of continuous-variable quantum mechanics. In particular, for general single-mode Gaussian operators and Gaussian noisy states, many different approaches have been used in the evaluation of their Fock representation. In this paper, a natural approach has been applied using exclusively the operational properties of the Hermite and Hermite-like polynomials and showing their fundamental role in this field. Closed-form results in terms of polynomials, exponentials, and simple algebraic functions are the major contribution of the paper.
{\copyright 2021 American Institute of Physics}Quantum information measures of the Aharonov-Bohm ring in uniform magnetic fieldshttps://www.zbmath.org/1472.810232021-11-25T18:46:10.358925Z"Olendski, O."https://www.zbmath.org/authors/?q=ai:olendski.olegSummary: Shannon quantum information entropies \(S_{\rho, \gamma}\), Fisher informations \(I_{\rho, \gamma}\), Onicescu energies \(O_{\rho, \gamma}\) and complexities \(e^S O\) are calculated both in the position (subscript \(\rho\)) and momentum (\(\gamma\)) spaces for the azimuthally symmetric two-dimensional nanoring that is placed into the combination of the transverse uniform magnetic field \(\mathbf{B}\) and the Aharonov-Bohm (AB) flux \(\phi_{AB}\) and whose potential profile is modelled by the superposition of the quadratic and inverse quadratic dependencies on the radius \(r\). The increasing intensity \(B\) flattens momentum waveforms \(\Phi_{nm}(\mathbf{k})\) and in the limit of the infinitely large fields they turn to zero: \(\Phi_{n m}(\mathbf{k}) \to 0\) at \(B \to \infty \), what means that the position wave functions \(\Psi_{n m}(\mathbf{r})\), which are their Fourier counterparts, tend in this limit to the \(\delta\)-functions. Position (momentum) Shannon entropy depends on the field \(B\) as a negative (positive) logarithm of \(\omega_{eff} \equiv (\omega_0^2 + \omega_c^2 / 4)^{1/2}\), where \(\omega_0\) determines the quadratic steepness of the confining potential and \(\omega_c\) is a cyclotron frequency. This makes the sum \(S_{\rho_{nm}} + S_{\gamma_{nm}}\) a field-independent quantity that increases with the principal \(n\) and azimuthal \(m\) quantum numbers and does satisfy entropic uncertainty relation. Position Fisher information does not depend on \(m\), linearly increases with \(n\) and varies as \(\omega_{eff}\) whereas its \(n\) and \(m\) dependent Onicescu counterpart \(O_{\rho_{nm}}\) changes as \(\omega_{eff}^{-1}\). The products \(I_{\rho_{nm}} I_{\gamma_{nm}}\) and \(O_{\rho_{nm}} O_{\gamma_{nm}}\) are \(B\)-independent quantities. A dependence of the measures on the ring geometry is discussed. It is argued that a variation of the position Shannon entropy or Onicescu energy with the AB field uniquely determines an associated persistent current as a function of \(\phi_{AB}\) at \(B = 0\). An inverse statement is correct too.Multipartite entanglement transfer in spin chainshttps://www.zbmath.org/1472.810262021-11-25T18:46:10.358925Z"Apollaro, Tony J. G."https://www.zbmath.org/authors/?q=ai:apollaro.tony-john-george"Sanavio, Claudio"https://www.zbmath.org/authors/?q=ai:sanavio.claudio"Chetcuti, Wayne Jordan"https://www.zbmath.org/authors/?q=ai:chetcuti.wayne-jordan"Lorenzo, Salvatore"https://www.zbmath.org/authors/?q=ai:lorenzo.salvatoreSummary: We investigate the transfer of genuine multipartite entanglement across a spin-\(\frac{1}{2}\) chain with nearest-neighbour \(XX\)-type interaction. We focus on the perturbative regime, where a block of spins is weakly coupled at each edge of a quantum wire, embodying the role of a multiqubit sender and receiver, respectively. We find that high-quality multipartite entanglement transfer is achieved at the same time that three excitations are transferred to the opposite edge of the chain. Moreover, we find that both a finite concurrence and tripartite negativity is attained at much shorter time, making \textit{GHZ}-distillation protocols feasible. Finally, we investigate the robustness of our protocol with respect to non-perturbative couplings and increasing lengths of the quantum wire.Entanglement of free fermions on Hadamard graphshttps://www.zbmath.org/1472.810272021-11-25T18:46:10.358925Z"Crampé, Nicolas"https://www.zbmath.org/authors/?q=ai:crampe.nicolas"Guo, Krystal"https://www.zbmath.org/authors/?q=ai:guo.krystal"Vinet, Luc"https://www.zbmath.org/authors/?q=ai:vinet.lucSummary: Free Fermions on vertices of distance-regular graphs are considered. Bipartitions are defined by taking as one part all vertices at a given distance from a reference vertex. The ground state is constructed by filling all states below a certain energy. Borrowing concepts from time and band limiting problems, algebraic Heun operators and Terwilliger algebras, it is shown how to obtain, quite generally, a block tridiagonal matrix that commutes with the entanglement Hamiltonian. The case of the Hadamard graphs is studied in detail within that framework and the existence of the commuting matrix is shown to allow for an analytic diagonalization of the restricted two-point correlation matrix and hence for an explicit determination of the entanglement entropy.Controlled quantum state transfer in \textit{XX} spin chains at the quantum speed limithttps://www.zbmath.org/1472.810372021-11-25T18:46:10.358925Z"Acosta Coden, D. S."https://www.zbmath.org/authors/?q=ai:acosta-coden.d-s"Gómez, S. S."https://www.zbmath.org/authors/?q=ai:gomez.s-s"Ferrón, A."https://www.zbmath.org/authors/?q=ai:ferron.a"Osenda, O."https://www.zbmath.org/authors/?q=ai:osenda.omarSummary: The Quantum Speed Limit (QSL) can be found in many different situations, in particular in the propagation of information through quantum spin chains. In homogeneous chains it implies that taking information from one extreme of the chain to the other will take a time \(O(N/2)\), where \(N\) is the chain length. Using Optimal Control Theory we design control pulses that achieve near perfect population transfer between the extremes of the chain at times on the order of \(N/2\). Our results show that the control pulses that govern the dynamical behavior of chains with different lengths are closely related. The pulses were constructed for control schemes involving one or two actuators in chains with exchange couplings without static disorder. Our results also show that the two actuator scheme is considerably more robust against the presence of static disorder than the scheme that uses just a single one.Noise-induced multilevel Landau-Zener transitions: density matrix investigationhttps://www.zbmath.org/1472.810422021-11-25T18:46:10.358925Z"Nyisomeh, I. F."https://www.zbmath.org/authors/?q=ai:nyisomeh.i-f"Ateuafack, M. E."https://www.zbmath.org/authors/?q=ai:ateuafack.m-e"Fai, L. C."https://www.zbmath.org/authors/?q=ai:fai.lukong-corneliusSummary: The generalised multilevel Landau-Zener problem is solved by applying the density matrix technique within the framework of nonstationary perturbation theory. The exact survival probability is achieved as a proof of the Brundobler-Elzer hypothesis [\textit{V. Brundobler}, and \textit{J. Elzer}, ``S-matrix for generalized Landau-Zener problem'', J. Phys. A, Math. Gen. 26, No. 5, 1211--1227 (1993; \url{doi:10.1088/0305-4470/26/5/037})]. The effect of classical Gaussian noise is investigated by averaging the solution over the noise realisation. A generalised formula for slow noise-induced transition probability is obtained and found to agree exactly with all known results. Exact results are reported for the Demkov-Osherov model in the slow and fast noise limits. Thermal transition probabilities are obtained via the activation Arrhenius law and observed to tailor a qubit from thermal decoherence.Comment on ``Deterministic CNOT gate and entanglement swapping for photonic qubits using a quantum-dot spin in a double-sided optical microcavity''https://www.zbmath.org/1472.810522021-11-25T18:46:10.358925Z"Gueddana, Amor"https://www.zbmath.org/authors/?q=ai:gueddana.amor"Lakshminarayanan, Vasudevan"https://www.zbmath.org/authors/?q=ai:lakshminarayanan.vasudevanA comment on [\textit{H.-F. Wang} et al., ibid. 377, No. 40, 2870--2876 (2013; Zbl 1301.81049)].Solving the Schrödinger equation by reduction to a first-order differential operator through a coherent states transformhttps://www.zbmath.org/1472.810652021-11-25T18:46:10.358925Z"Almalki, Fadhel"https://www.zbmath.org/authors/?q=ai:almalki.fadhel"Kisil, Vladimir V."https://www.zbmath.org/authors/?q=ai:kisil.vladimir-vSummary: The Legendre transform expresses dynamics of a classical system through first-order Hamiltonian equations. We consider coherent state transforms with a similar effect in quantum mechanics: they reduce certain quantum Hamiltonians to first-order partial differential operators. Therefore, the respective dynamics can be explicitly solved through a flow of points in extensions of the phase space. This generalises the geometric dynamics of a harmonic oscillator in the Fock space. We describe all Hamiltonians which are geometrised (in the above sense) by Gaussian and Airy beams and write down explicit solutions for such systems.Laplace transform approach for the dynamics of \(N\) qubits coupled to a resonatorhttps://www.zbmath.org/1472.810662021-11-25T18:46:10.358925Z"Amico, Mirko"https://www.zbmath.org/authors/?q=ai:amico.mirko"Berman, Oleg L."https://www.zbmath.org/authors/?q=ai:berman.oleg-l"Kezerashvili, Roman Ya."https://www.zbmath.org/authors/?q=ai:kezerashvili.roman-yaSummary: An approach to use the method of Laplace transform for the perturbative solution of the Schrödinger equation at any order of the perturbation for a system of \(N\) qubits coupled to a cavity with \(n\) photons is suggested. We investigate the dynamics of a system of \(N\) superconducting qubits coupled to a common resonator with time-dependent coupling. To account for the contribution of the dynamical Lamb effect to the probability of excitation of the qubit, we consider counter-rotating terms in the qubit-photon interaction Hamiltonian. As an example, we illustrate the method for the case of two qubits coupled to a common cavity. The perturbative solutions for the probability of excitation of the qubit show excellent agreement with the numerical calculations.Riccati-type pseudo-potentials, conservation laws and solitons of deformed sine-Gordon modelshttps://www.zbmath.org/1472.810672021-11-25T18:46:10.358925Z"Blas, H."https://www.zbmath.org/authors/?q=ai:blas.harold"Callisaya, H. F."https://www.zbmath.org/authors/?q=ai:callisaya.hector-flores"Campos, J. P. R."https://www.zbmath.org/authors/?q=ai:campos.j-p-rSummary: Deformed sine-Gordon (DSG) models \(\partial_\xi \partial_\eta w + \frac{d}{d w} V(w) = 0\), with \(V(w)\) being the deformed potential, are considered in the context of the Riccati-type pseudo-potential approach. A compatibility condition of the deformed system of Riccati-type equations reproduces the equation of motion of the DSG models. Then, we provide a pair of linear systems of equations for the DSG model and an associated infinite tower of non-local conservation laws. Through a direct construction and supported by numerical simulations of soliton scatterings, we show that the DSG models, which have recently been defined as quasi-integrable in the anomalous zero-curvature approach [\textit{L. A. Ferreira} and \textit{W. J. Zakrzewski}, J. High Energy Phys. 2011, No. 5, Paper No. 130, 39 p. (2011; Zbl 1296.81035)], possess new towers of infinite number of quasi-conservation laws. We compute numerically the first sets of non-trivial and independent charges (beyond energy and momentum) of the DSG model: the two third order conserved charges and the two fifth order asymptotically conserved charges in thepseudo-potential approach, and the first four anomalies of the new towers of charges, resp ectively. We consider kink-kink, kink-antikink and breather configurations for the Bazeia et al. potential \(V_q(w) = \frac{64}{q^2} \tan^2 \frac{w}{2}(1 - | \sin \frac{w}{2} |^q)^2\) \((q \in \mathbb{R})\), which contains the usual SG potential \(V_2(w) = 2 [1 - \cos(2w)]\). The numerical simulations are performed using the 4th order Runge-Kutta method supplied with non-reflecting boundary conditions.Lax connection and conserved quantities of quadratic mean field gameshttps://www.zbmath.org/1472.810682021-11-25T18:46:10.358925Z"Bonnemain, Thibault"https://www.zbmath.org/authors/?q=ai:bonnemain.thibault"Gobron, Thierry"https://www.zbmath.org/authors/?q=ai:gobron.thierry"Ullmo, Denis"https://www.zbmath.org/authors/?q=ai:ullmo.denisSummary: Mean field games is a new field developed simultaneously in applied mathematics and engineering in order to deal with the dynamics of a large number of controlled agents or objects in interaction. For a large class of these models, there exists a deep relationship between the associated system of equations and the non-linear Schrödinger equation, which allows us to get new insights into the structure of their solutions. In this work, we deal with the related aspects of integrability for such systems, exhibiting in some cases a full hierarchy of conserved quantities and bringing some new questions that arise in this specific context.
{\copyright 2021 American Institute of Physics}On discrete spectrum of a model graph with loop and small edgeshttps://www.zbmath.org/1472.810692021-11-25T18:46:10.358925Z"Borisov, D. I."https://www.zbmath.org/authors/?q=ai:borisov.denis-i"Konyrkulzhaeva, M. N."https://www.zbmath.org/authors/?q=ai:konyrkulzhaeva.maral-nurlanovna"Mukhametrakhimova, A. I."https://www.zbmath.org/authors/?q=ai:mukhametrakhimova.a-iSummary: We consider a perturbed graph consisting of two infinite edges, a loop, and a glued arbitrary finite graph \(\gamma \epsilon\) with small edges, where \(\gamma \epsilon\) is obtained by \(\epsilon^{-1}\) times contraction of some fixed graph and \(\epsilon\) is a small parameter. On the perturbed graph, we consider the Schrödinger operator whose potential on small edges can singularly depend on \(\epsilon\) with the Kirchhoff condition at internal vertices and the Dirichlet or Neumann condition at the boundary vertices. For the perturbed eigenvalue and the corresponding eigenfunction we prove the holomorphy with respect to \(\epsilon\) and propose a recurrent algorithm for finding all coefficients of their Taylor series.The thermal properties of the one-dimensional boson particles in Rindler spacetimehttps://www.zbmath.org/1472.810702021-11-25T18:46:10.358925Z"Boumali, Abdelmalek"https://www.zbmath.org/authors/?q=ai:boumali.abdelmalek"Rouabhia, Tarek Imad"https://www.zbmath.org/authors/?q=ai:rouabhia.tarek-imadSummary: In this paper we study the one-dimension Klein-Gordon (KG) equation in the Rindler spacetime. The solutions of the wave equation in an accelerated reference frame are obtained. As a result, (i) we derive a compact expression for the energy spectrum associated in an accelerated reference, and (ii) we show that the non-inertial effect of the accelerated reference frame mimics an external potential in the Klein-Gordon equation and, moreover, allows the formation of bound states. In addition, the thermal properties of the Klein-Gordon from the partition function, have been investigated, and the effect of the accelerated reference frame parameter \(a\) on these properties has been tested. This study is extended to the case of the one-dimensional Dirac equation where the spectrum of energy is well determined and has an exact form. As a result, we will see that the behavior of the thermal quantities of the fermion particles is similar to the case of boson particles.On the spectrum and eigenfunctions of the equivariant general boundary value problem outside the sphere for the Schrödinger operator with Coulomb potentialhttps://www.zbmath.org/1472.810712021-11-25T18:46:10.358925Z"Burskii, V. P."https://www.zbmath.org/authors/?q=ai:burskii.vladimir-p"Zaretskaya, A. A."https://www.zbmath.org/authors/?q=ai:zaretskaya.a-aSummary: We consider the Schrödinger equation of hydrogen-type atom with Coulomb potential. The eigenvalue and eigenfunction are found in the case of swing-invariant boundary value problem.Exact solutions of the harmonic oscillator plus non-polynomial interactionhttps://www.zbmath.org/1472.810722021-11-25T18:46:10.358925Z"Dong, Qian"https://www.zbmath.org/authors/?q=ai:dong.qian"Hernández, H. Iván García"https://www.zbmath.org/authors/?q=ai:hernandez.h-ivan-garcia"Sun, Guo-Hua"https://www.zbmath.org/authors/?q=ai:sun.guohua"Toutounji, Mohamad"https://www.zbmath.org/authors/?q=ai:toutounji.mohamad"Dong, Shi-Hai"https://www.zbmath.org/authors/?q=ai:dong.shihaiSummary: The exact solutions to a one-dimensional harmonic oscillator plus a non-polynomial interaction \(a x^2 + b x^2/(1 + c x^2) (a > 0, c > 0)\) are given by the confluent Heun functions \(H_{}c ( \alpha, \beta, \gamma, \delta, \eta ;z)\). The minimum value of the potential well is calculated as \(V_{\text{min}}(x) = - ( a + | b | - 2 \sqrt{a |b |} ) / c\) at \(x = \pm [ ( \sqrt{ |b | /a} - 1 ) / c ]^{1 / 2} (|b\)| > \(a)\) for the double-well case \((b < 0)\). We illustrate the wave functions through varying the potential parameters \(a, b, c\) and show that they are pulled back to the origin when the potential parameter \(b\) increases for given values of \(a\) and \(c\). However, we find that the wave peaks are concave to the origin as the parameter |\(b\)| is increased.Dynamics of solutions of a fractional NLS system with quadratic interactionhttps://www.zbmath.org/1472.810732021-11-25T18:46:10.358925Z"Esfahani, Amin"https://www.zbmath.org/authors/?q=ai:esfahani.aminSummary: The purpose of this article is twofold. First, we study the existence and asymptotic behavior of ground states of a fractional Schrodinger system with quadratic interaction. Second, we give some conditions, in terms of the mass and energy of the ground states, under which the solutions of the associated initial value problem have the uniform bound or may blow up in finite time. As a corollary, we show the strong instability of the ground states.
{\copyright 2021 American Institute of Physics}Trajectory construction of Dirac evolutionhttps://www.zbmath.org/1472.810742021-11-25T18:46:10.358925Z"Holland, Peter"https://www.zbmath.org/authors/?q=ai:holland.peter-rSummary: We extend our programme of representing the quantum state through exact stand-alone trajectory models to the Dirac equation. We show that the free Dirac equation in the angular coordinate representation is a continuity equation for which the real and imaginary parts of the wave function, angular versions of Majorana spinors, define conserved densities. We hence deduce an exact formula for the propagation of the Dirac spinor derived from the self-contained first-order dynamics of two sets of trajectories in 3-space together with a mass-dependent evolution operator. The Lorentz covariance of the trajectory equations is established by invoking the `relativity of the trajectory label'. We show how these results extend to the inclusion of external potentials. We further show that the angular version of Dirac's equation implies continuity equations for currents with non-negative densities, for which the Dirac current defines the mean flow. This provides an alternative trajectory construction of free evolution. Finally, we examine the polar representation of the Dirac equation, which also implies a non-negative conserved density but does not map into a stand-alone trajectory theory. It reveals how the quantum potential is tacit in the Dirac equation.Limiting absorption principle and radiation condition for repulsive Hamiltonianshttps://www.zbmath.org/1472.810752021-11-25T18:46:10.358925Z"Itakura, Kyohei"https://www.zbmath.org/authors/?q=ai:itakura.kyoheiSummary: For spherically symmetric repulsive Hamiltonians we prove the limiting absorption principle bound, the radiation condition bounds and the limiting absorption principle. The Sommerfeld uniqueness result also follows as a corollary of these. In particular, the Hamiltonians considered in this paper cover the case of inverted harmonic oscillator. In the proofs of our theorems, we mainly use a commutator argument invented recently by Ito and Skibsted. This argument is simple and elementary, and dose not employ energy cut-offs or the microlocal analysis.Measuring space deformation via graphene under constraintshttps://www.zbmath.org/1472.810762021-11-25T18:46:10.358925Z"Jellal, Ahmed"https://www.zbmath.org/authors/?q=ai:jellal.ahmedSummary: We describe the lattice deformation in graphene under strain effect by considering the spacial-momenta coordinates do not commute. This later can be realized by introducing the star product to end up with a generalized Heisenberg algebra. Within such framework, we build a new model describing Dirac fermions interacting with an external source that is noncommutative parameter \(\kappa\) dependent. The solutions of energy spectrum are showing Landau levels in similar way to the case of a real magnetic field applied to graphene. We show that some strain configurations can be used to explicitly evaluate \(\kappa\) and then offer a piste toward its measurement.Four-dimensional Fano quiver flag zero locihttps://www.zbmath.org/1472.810772021-11-25T18:46:10.358925Z"Kalashnikov, Elana"https://www.zbmath.org/authors/?q=ai:kalashnikov.elanaSummary: Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of representation theoretic vector bundles. We prove the Abelian/non-Abelian correspondence in this context: this allows us to compute genus zero Gromov-Witten invariants of quiver flag zero loci. We determine the ample cone of a quiver flag variety, and disprove a conjecture of Craw. In the appendices (which can be found in the electronic supplementary material), which are joint work with Tom Coates and Alexander Kasprzyk, we use these results to find four-dimensional Fano manifolds that occur as quiver flag zero loci in ambient spaces of dimension up to 8, and compute their quantum periods. In this way, we find at least 141 new four-dimensional Fano manifolds.A class of time-energy uncertainty relations for time-dependent Hamiltonianshttps://www.zbmath.org/1472.810782021-11-25T18:46:10.358925Z"Kieu, Tien D."https://www.zbmath.org/authors/?q=ai:kieu.tien-dSummary: A new class of time-energy uncertainty relations is directly derived from the Schrödinger equations for time-dependent Hamiltonians. Only the initial states and the Hamiltonians, but neither the instantaneous eigenstates nor the full time-dependent wave functions, which would demand a full solution for a time-dependent Hamiltonian, are required for our time-energy relations. Explicit results are then presented for particular subcases of interest for time-independent Hamiltonians and also for time-varying Hamiltonians employed in adiabatic quantum computation. Some estimates of the lower bounds on computational time are given for general adiabatic quantum algorithms, with Grover's search as an illustration. We particularly emphasize the role of required energy resources, besides the space and time complexity, for the physical process of (quantum) computation, in general.An optimized multistage complete in phase P-stable algorithmhttps://www.zbmath.org/1472.810792021-11-25T18:46:10.358925Z"Kovalnogov, Vladislav N."https://www.zbmath.org/authors/?q=ai:kovalnogov.vladislav-n"Fedorov, Ruslan V."https://www.zbmath.org/authors/?q=ai:fedorov.ruslan-v"Bondarenko, Aleksandr A."https://www.zbmath.org/authors/?q=ai:bondarenko.aleksandr-adamovich"Simos, Theodore E."https://www.zbmath.org/authors/?q=ai:simos.theodore-eSummary: A fourteen algebraic order P-stable symmetric four-stages two-step scheme with expunged phase-lag and its first and second derivatives, is developed, for the first time in the literature, in this paper. The new four-stages method is developed based on the following steps: \begin{itemize}\item Contentment of the necessary and sufficient conditions for P-stability. \item Contentment of the condition of the expunging of the phase-lag. \item Contentment of the junctures of the expunging of the first and second derivatives of the phase-lag. \end{itemize} The result of the above methodology is the development, for the first time in the literature, of a four-stages P-stable fourteen algebraic order symmetric two-step method with expunged phase-lag and its derivatives up to order two. \par We present also a full numerical and theoretical analysis for the new algorithm which contains the following steps: \begin{itemize}\item the development of the new four-stages method, \item the achievement of its local truncation error (LTE), \item the foundation of the asymptotic form of the LTE of the new four-stages method, \item the foundation of the stability and interval of periodicity of the new four-stages method, \item the achievement of an embedded algorithm and the determination of the variable step technique for the changing of the step sizes, \item the evaluation of the computational efficiency of the new four-stages method with its application on: \begin{itemize}\item the resonance problem of the radial Schrödinger equation and on \item the system of the coupled differential equations of the Schrödinger type. \end{itemize}\end{itemize} The above study leads to the conclusion that the new four-stages method is more efficient than the existed ones.Ground states for a linearly coupled indefinite Schrödinger system with steep potential wellhttps://www.zbmath.org/1472.810802021-11-25T18:46:10.358925Z"Lin, Ying-Chieh"https://www.zbmath.org/authors/?q=ai:lin.ying-chieh"Wang, Kuan-Hsiang"https://www.zbmath.org/authors/?q=ai:wang.kuan-hsiang"Wu, Tsung-fang"https://www.zbmath.org/authors/?q=ai:wu.tsungfangSummary: In this paper, we study a class of linearly coupled Schrödinger systems with steep potential wells, which arises from Bose-Einstein condensates. The existence of positive ground states is investigated by exploiting the relation between the Nehari manifold and fiberring maps. Some interesting phenomena are that we do not need the weight functions in the nonlinear terms to be integrable or bounded and we can relax the upper control condition of the coupling function. Moreover, the decay rate and concentration phenomenon of positive ground states are also studied.
{\copyright 2021 American Institute of Physics}Markovian embedding procedures for non-Markovian stochastic Schrödinger equationshttps://www.zbmath.org/1472.810812021-11-25T18:46:10.358925Z"Li, Xiantao"https://www.zbmath.org/authors/?q=ai:li.xiantaoSummary: We present embedding procedures for the non-Markovian stochastic Schrödinger equations, arising from studies of quantum systems coupled with bath environments. By introducing auxiliary wave functions, it is demonstrated that the non-Markovian dynamics can be embedded in extended, but Markovian, stochastic models. Two embedding procedures are presented. The first method leads to nonlinear stochastic equations, the implementation of which is much more efficient than the non-Markovian stochastic Schrödinger equations. The stochastic Schrödinger equations obtained from the second procedure involve more auxiliary wave functions, but the equations are linear, and a closed-form generalized quantum master equation for the density-matrix can be obtained. The accuracy of the embedded models is ensured by fitting to the power spectrum. The stochastic force is represented using a linear superposition of Ornstein-Uhlenbeck processes, which are incorporated as multiplicative noise in the auxiliary Schrödinger equations. It is shown that the asymptotic behavior of the spectral density in the low frequency regime, which is responsible for the long-time behavior of the quantum dynamics, can be preserved by using correlated stochastic processes. The approximations are verified by using a spin-boson system as a test example.Gauge transformations of spectral triples with twisted real structureshttps://www.zbmath.org/1472.810822021-11-25T18:46:10.358925Z"Magee, Adam M."https://www.zbmath.org/authors/?q=ai:magee.adam-m"Dąbrowski, Ludwik"https://www.zbmath.org/authors/?q=ai:dabrowski.ludwikSummary: Twisted real structures are well-motivated as a way to implement the conformal transformation of a Dirac operator for a real spectral triple without needing to twist the noncommutative one-forms. We study the coupling of spectral triples with twisted real structures to gauge fields, adopting Morita equivalence via modules and bimodules as a guiding principle and paying special attention to modifications to the inner fluctuations of the Dirac operator. In particular, we analyze the twisted first-order condition as a possible alternative to abandoning the first-order condition in order to go beyond the standard model and elaborate upon the special case of gauge transformations accordingly. Applying the formalism to a toy model, we argue that under certain physically motivated assumptions, the spectral triple based on the left-right symmetric algebra should reduce to that of the standard model of fundamental particles and interactions, as in the untwisted case.
{\copyright 2021 American Institute of Physics}Justification of the discrete nonlinear Schrödinger equation from a parametrically driven damped nonlinear Klein-Gordon equation and numerical comparisonshttps://www.zbmath.org/1472.810832021-11-25T18:46:10.358925Z"Muda, Y."https://www.zbmath.org/authors/?q=ai:muda.yuslenita"Akbar, F. T."https://www.zbmath.org/authors/?q=ai:akbar.fiki-taufik"Kusdiantara, R."https://www.zbmath.org/authors/?q=ai:kusdiantara.rudy"Gunara, B. E."https://www.zbmath.org/authors/?q=ai:gunara.bobby-eka"Susanto, H."https://www.zbmath.org/authors/?q=ai:susanto.hadiSummary: We consider a damped, parametrically driven discrete nonlinear Klein-Gordon equation, that models coupled pendula and micromechanical arrays, among others. To study the equation, one usually uses a small-amplitude wave ansatz, that reduces the equation into a discrete nonlinear Schrödinger equation with damping and parametric drive. Here, we justify the approximation by looking for the error bound with the method of energy estimates. Furthermore, we prove the local and global existence of solutions to the discrete nonlinear Schrödinger equation. To illustrate the main results, we consider numerical simulations showing the dynamics of errors made by the discrete nonlinear equation. We consider two types of initial conditions, with one of them being a discrete soliton of the nonlinear Schrödinger equation, that is expectedly approximate discrete breathers of the nonlinear Klein-Gordon equation.Symmetries of the Schrödinger-Pauli equation for neutral particleshttps://www.zbmath.org/1472.810842021-11-25T18:46:10.358925Z"Nikitin, A. G."https://www.zbmath.org/authors/?q=ai:nikitin.anatoly-gSummary: By using the algebraic approach, the Lie symmetries of Schrödinger equations with matrix potentials are classified. Thirty three inequivalent equations of such type together with the related symmetry groups are specified, and the admissible equivalence relations are clearly indicated. In particular, the Boyer results concerning kinematical invariance groups for arbitrary potentials [\textit{C. P. Boyer}, ``The maximal 'kinematical' invariance group for an arbitrary potential'', Helv. Phys. Acta 47, 589--605 ((1974; \url{doi:10.5169/seals-114583})] are clarified and corrected.
{\copyright 2021 American Institute of Physics}Characterization of Darboux transformations for quantum systems with quadratically energy-dependent potentialshttps://www.zbmath.org/1472.810852021-11-25T18:46:10.358925Z"Schulze-Halberg, Axel"https://www.zbmath.org/authors/?q=ai:schulze-halberg.axelSummary: We construct three classes of higher-order Darboux transformations for Schrödinger equations with quadratically energy-dependent potentials by means of generalized Wronskian determinants. Particular even-order cases reduce to the Darboux transformation for conventional (energy-independent) potentials. Our construction is based on an adaptation of the results for coupled Korteweg-de Vries equations [\textit{S. B. Leble} and \textit{N. V. Ustinov}, J. Math. Phys. 34, No. 4, 1421--1428 (1993; Zbl 0774.35075)].
{\copyright 2021 American Institute of Physics}A multistage full in phase P-stable scheme with optimized propertieshttps://www.zbmath.org/1472.810862021-11-25T18:46:10.358925Z"Shi, Xin"https://www.zbmath.org/authors/?q=ai:shi.xin"Simos, Theodore E."https://www.zbmath.org/authors/?q=ai:simos.theodore-eSummary: A fourteen algebraic order P-stable symmetric four-stages two-step scheme with zeroing phase-lag and it's derivatives up to order three, is constructed, for the first time in the literature, in this paper. The new four-stages method is built based on the following procedure: \begin{itemize} \item Gratification of the conditions for the characteristic of the P-stability (necessary and sufficient). \item Gratification of the condition of the zeroing of the phase-lag.\item Gratification of the junctures of the zeroing of the derivatives of the phase-lag up to order three. \end{itemize} The above procedure leads to the construction, for the first time in the literature, of a four-stages P-stable fourteen algebraic order symmetric two-step method with phase-lag and its first, second and third derivatives equal to zero. \par For the newly introduced method, a numerical and theoretical analysis is presented, which consists of the following stages: \begin{itemize} \item the construction of the newly introduced four-stages method, \item the feat of the determination of its local truncation error (LTE), \item the development of the asymptotic form of the LTE of the newly introduced four-stages method, \item the determination of the stability and interval of periodicity of the newly introduced four-stages method, \item the determination of an embedded algorithm and the definition of the variable step methodology for the foundation of the step sizes, \item the estimation of the computational effectiveness of the newly introduced four-stages method which consists of its application on: \begin{itemize} \item the resonance problem of the radial Schrödinger equation and on \item the system of the coupled differential equations arising from the Schrödinger equation. \end{itemize}\end{itemize} Based on the above research, we conclude that the newly introduced four-stages method is more efficient than the existed ones.Dirac particle with memory: proper time non-localityhttps://www.zbmath.org/1472.810872021-11-25T18:46:10.358925Z"Tarasov, Vasily E."https://www.zbmath.org/authors/?q=ai:tarasov.vasily-eSummary: A generalization of the standard model of Dirac particle in external electromagnetic field is proposed. In the generalization we take into account interactions of this particle with environment, which is described by the memory function. This function takes into account that the behavior of the particle at proper time can depend not only at the present time, but also on the history of changes on finite time interval. In this case the Dirac particle can be considered an open quantum system with non-Markovian dynamics. The violation of the semigroup property of dynamic maps is a characteristic property of dynamics with memory. We use the Fock-Schwinger proper time method and derivatives of non-integer orders with respect to proper time. The fractional differential equation, which describes the Dirac particle with memory, and the expression of its exact solution are suggested. The asymptotic behavior of the proposed solutions is described.On the derivative nonlinear Schrödinger equation on the half line with Robin boundary conditionhttps://www.zbmath.org/1472.810882021-11-25T18:46:10.358925Z"Van Tin, Phan"https://www.zbmath.org/authors/?q=ai:van-tin.phanSummary: We consider the Schrödinger equation with a nonlinear derivative term on [0, +\(\infty)\) under the Robin boundary condition at 0. Using a virial argument, we obtain the existence of blowing up solutions, and using variational techniques, we obtain stability and instability by blow-up results for standing waves.
{\copyright 2021 American Institute of Physics}Solving forward and inverse problems of the logarithmic nonlinear Schrödinger equation with \(\mathcal{PT}\)-symmetric harmonic potential via deep learninghttps://www.zbmath.org/1472.810892021-11-25T18:46:10.358925Z"Zhou, Zijian"https://www.zbmath.org/authors/?q=ai:zhou.zijian"Yan, Zhenya"https://www.zbmath.org/authors/?q=ai:yan.zhenyaSummary: In this paper, we investigate the logarithmic nonlinear Schrödinger (LNLS) equation with the parity-time \((\mathcal{PT})\)-symmetric harmonic potential, which is an important physical model in many fields such as nuclear physics, quantum optics, magma transport phenomena, and effective quantum gravity. Three types of initial value conditions and periodic boundary conditions are chosen to solve the LNLS equation with \(\mathcal{PT}\)-symmetric harmonic potential via the physics-informed neural networks (PINNs) deep learning method, and these obtained results are compared with ones deduced from the Fourier spectral method. Moreover, we also investigate the effectiveness of the PINNs deep learning for the LNLS equation with \(\mathcal{PT}\) symmetric potential by choosing the distinct space widths or distinct optimized steps. Finally, we use the PINNs deep learning method to effectively tackle the data-driven discovery of the LNLS equation with \(\mathcal{PT} \)-symmetric harmonic potential such that the coefficients of dispersion and nonlinear terms or the amplitudes of \(\mathcal{PT}\)-symmetric harmonic potential can be approximately found.A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalitieshttps://www.zbmath.org/1472.810902021-11-25T18:46:10.358925Z"Antunes, Pedro R. S."https://www.zbmath.org/authors/?q=ai:antunes.pedro-ricardo-simao"Benguria, Rafael D."https://www.zbmath.org/authors/?q=ai:benguria.rafael-d"Lotoreichik, Vladimir"https://www.zbmath.org/authors/?q=ai:lotoreichik.vladimir"Ourmières-Bonafos, Thomas"https://www.zbmath.org/authors/?q=ai:ourmieres-bonafos.thomaslet \(\Omega \subset {\mathbb R}^2\) be a \(C^\infty\) simply connected domain and let \(n = (n_1,n_2)^\top\) be the outward pointing normal field on \(\partial\Omega\). The Dirac operator with infinite mass boundary conditions in \(L^2(\Omega,{\mathbb C}^2)\) is defined as \[D^\Omega := \begin{pmatrix} 0 & -2\mathrm{i}\partial_z\\
-2\mathrm{i}\partial_{\bar z} & 0 \end{pmatrix}, \] with domain \(\{ u = (u_1,u_2)^\top \in H^1(\Omega,{\mathbb C}^2) : u_2 = \mathrm{i} \mathbf{n}u_1 \text{ on }\partial\Omega \},\) where \(\mathbf{n} := n_1 + \mathrm{i} n_2\) and \(\partial_z, \partial_{\bar{z}}\) are the Wirtinger operators. The spectrum of \(D^\Omega\) is symmetric with respect to the origin and constituted of eigenvalues of finite multiplicity \[ \cdots \leq -E_k(\Omega) \leq\cdots \leq-E_{1}(\Omega) < 0 < E_{1}(\Omega) \leq \cdots \leq E_k(\Omega) \leq \cdots.\] The authors prove the following estimate \[E_1(\Omega) \leq \frac{|\partial\Omega|}{(\pi r_i^2 + |\Omega|)}E_1({\mathbb D}) \] with equality if and only if \(\Omega\) is a disk, where \(r_i\) is the inradius of \(\Omega\) and \(\mathbb D\) is the unit disk. \par The second main result of this paper is the following non-linear variational characterization of \(E_1(\Omega)\). \(E > 0\) is the first non-negative eigenvalue of \(D^\Omega\) if and only if \(\mu^\Omega(E) = 0\), where \[\mu^\Omega(E) := \inf\limits_{u} \frac{4 \int_\Omega |\partial_{\bar z} u|^2 dx - E^2 \int_{\Omega}|u|^2dx + E \int_{\partial\Omega} |u|^2 ds}{\int_\Omega |u|^2 dx}.\] \par The authors propose the following conjecture \[\mu^\Omega(E) \geq \frac{\pi}{|\Omega|}\mu^{\mathbb D}\Big(\sqrt{\frac{|\Omega|}{\pi}}E\Big), \forall E>0\] and provide numerical evidences supporting it. This conjecture implies the validity of the Faber-Krahn-type inequality \(E_1(\Omega) \geq \sqrt{\frac{\pi}{|\Omega|}} E_1({\mathbb D})\) (it is still an open question).Heun operator of Lie type and the modified algebraic Bethe ansatzhttps://www.zbmath.org/1472.810912021-11-25T18:46:10.358925Z"Bernard, Pierre-Antoine"https://www.zbmath.org/authors/?q=ai:bernard.pierre-antoine"Crampé, Nicolas"https://www.zbmath.org/authors/?q=ai:crampe.nicolas"Shaaban Kabakibo, Dounia"https://www.zbmath.org/authors/?q=ai:shaaban-kabakibo.dounia"Vinet, Luc"https://www.zbmath.org/authors/?q=ai:vinet.lucSummary: The generic Heun operator of Lie type is identified as a certain \textit{BC}-Gaudin magnet Hamiltonian in a magnetic field. By using the modified algebraic Bethe ansatz introduced to diagonalize such Gaudin models, we obtain the spectrum of the generic Heun operator of Lie type in terms of the Bethe roots of inhomogeneous Bethe equations. We also show that these Bethe roots are intimately associated with the roots of polynomial solutions of the differential Heun equation. We illustrate the use of this approach in two contexts: the representation theory of \(O(3)\) and the computation of the entanglement entropy for free Fermions on the Krawtchouk chain.
{\copyright 2021 American Institute of Physics}A new spectral analysis of stationary random Schrödinger operatorshttps://www.zbmath.org/1472.810922021-11-25T18:46:10.358925Z"Duerinckx, Mitia"https://www.zbmath.org/authors/?q=ai:duerinckx.mitia"Shirley, Christopher"https://www.zbmath.org/authors/?q=ai:shirley.christopherThe authors consider random Schrödinger operators of the form
\[
-\Delta + \lambda V_{\omega}
\]
and the associated Schrödinger equation, where \(V_{\omega}\) is a realization of a stationary random potential \(V\). The regime under consideration here is \(0<\lambda \ll 1\). The main goal of the authors is to develop a spectral approach to describe the long time behavior of the system beyond perturbative timescales by using ideas from Malliavin calculus, leading to rigorous Mourre type results. In particular, the authors describe the dynamics by a fibered family of spectral perturbation problems. They then state a number of exact resonance conjectures which would require that Bloch waves exist as resonant modes. An approximate resonance result is obtained and the first spectral proof of the decay of time correlations on the kinetic timescale is also provided.Note about covariant Hamiltonian formalism for strings, p-branes and unstable Dp-braneshttps://www.zbmath.org/1472.810932021-11-25T18:46:10.358925Z"Klusoň, Josef"https://www.zbmath.org/authors/?q=ai:kluson.josefSummary: In this short note we formulate Covariant Hamiltonian formalism for strings, p-branes and Non-BPS Dp-branes. We also analyse the vacuum tachyon condensation in case of unstable D1-brane.Tosio Kato's work on non-relativistic quantum mechanics. IIhttps://www.zbmath.org/1472.810942021-11-25T18:46:10.358925Z"Simon, Barry"https://www.zbmath.org/authors/?q=ai:simon.barry.1The work is the second part of a review to Kato's work on nonrelativistic quantum mechanics. It focuses on bounds on the number of eigenvalues of the helium atom, on the absence of embedded bound states, on scattering theory under a trace class condition, Kato smoothness, the adiabatic theorem, and the Trotter product formula.
The author is known for the clarity of his presentation which is reflected in this work as well. The review can also serve as an introduction of the subject, since the results are not merely reviewed but put in a current perspective of the field. An example of this is the appendix where the inequality \(|p|>2/(\pi |x|)\) in \(d=3\), known as Kato's inequality or Herbst inequality, is treated. It is put in the context of the groundstate transform which [\textit{R. L. Frank} et al., J. Am. Math. Soc. 21, No. 4, 925--950 (2008; Zbl 1202.35146)] used to prove a generalization for fractional powers of \(p:=-i\nabla\).
For Part I see the author [Bull. Math. Sci. 8, No. 1, 121--232 (2018; Zbl 1416.81063)]A quantummechanical derivation of the eigenvalues of the quadratic Casimir operator of the algebra \(SU (n)\) in Young tableau representationhttps://www.zbmath.org/1472.810952021-11-25T18:46:10.358925Z"Szőke, Éva"https://www.zbmath.org/authors/?q=ai:szoke.evaSummary: The paper concerns the eigenvalues and eigenfunctions of the quadratic Casimir operator of the algebra \(SU(n)\) in the Young tableau representation. Our starting point is a concrete physical problem in second quantized formalism. We use quantummechanical and group theoretical vehicles to determine the aimed quantities. The tableaux are organized according to their eigenvalue. We also investigate the modification rules.Passage through exceptional point: case studyhttps://www.zbmath.org/1472.810962021-11-25T18:46:10.358925Z"Znojil, Miloslav"https://www.zbmath.org/authors/?q=ai:znojil.miloslavSummary: The description of unitary evolution using non-Hermitian but `hermitizable' Hamiltonians \(H\) is feasible via an \textit{ad hoc} metric \(\Theta = \Theta (H)\) and a (non-unique) amendment \(\langle \psi_1| \psi_2\rangle \rightarrow \langle \psi_1| \Theta |\psi_2\rangle\) of the inner product in Hilbert space. Via a proper fine-tuning of \(\Theta (H)\) this opens the possibility of reaching the boundaries of stability (i.e. exceptional points) in many quantum systems sampled here by the fairly realistic Bose-Hubbard (BH) and discrete anharmonic oscillator (AO) models. In such a setting, it is conjectured that the EP singularity can play the role of a quantum phase-transition interface between different dynamical regimes. Three alternative `AO \(\leftrightarrow\) BH' implementations of such an EP-mediated dynamical transmutation scenario are proposed and shown, at an arbitrary finite Hilbert-space dimension \(N\), exact and non-numerical.Unitary unfoldings of a Bose-Hubbard exceptional point with and without particle number conservationhttps://www.zbmath.org/1472.810972021-11-25T18:46:10.358925Z"Znojil, Miloslav"https://www.zbmath.org/authors/?q=ai:znojil.miloslavSummary: The conventional non-Hermitian but \(\mathcal{P} T\)-symmetric three-parametric Bose-Hubbard Hamiltonian \(H( \gamma , v, c)\) represents a quantum system of \(N\) bosons, unitary only for parameters \(\gamma , v\) and \(c\) in a domain \(\mathcal{D} \). Its boundary \(\partial\mathcal{D}\) contains an exceptional point of order \(K\) (EPK; \(K = N + 1)\) at \(c = 0\) and \(\gamma = v\), but even at the smallest non-vanishing parameter \(c \neq \) ~0 the spectrum of \(H(v, v, c)\) ceases to be real, i.e. the system ceases to be observable. In this paper, the question is inverted: all of the stable, unitary and observable Bose-Hubbard quantum systems are sought which would lie close to the phenomenologically most interesting EPK-related dynamical regime. Two different families of such systems are found. Both of them are characterized by the perturbed Hamiltonians \(\mathfrak{H}(\lambda) = H(v, v, 0) + \lambda \mathcal{V}\) for which the unitarity and stability of the system is guaranteed. In the first family the number \(N\) of bosons is assumed conserved while in the second family such an assumption is relaxed. Attention is paid mainly to an anisotropy of the physical Hilbert space near the EPK extreme. We show that it is reflected by a specific, operationally realizable structure of perturbations \(\lambda \mathcal{V}\) which can be considered small.Perturbation theory for Bose-Einstein condensates on bounded space domainshttps://www.zbmath.org/1472.810982021-11-25T18:46:10.358925Z"van Gorder, Robert A."https://www.zbmath.org/authors/?q=ai:van-gorder.robert-aSummary: Bose-Einstein condensates (BECs), first predicted theoretically by Bose and Einstein and finally discovered experimentally in the 1990s, continue to motivate theoretical and experimental physics work. Although experiments on BECs are carried out in bounded space domains, theoretical work in the modelling of BECs often involves solving the Gross-Pitaevskii equation on unbounded domains, as the combination of bounded domains and spatial heterogeneity render most existing analytical approaches ineffective. Motivated by a lack of theory for BECs on bounded domains, we first derive a perturbation theory for both ground and excited stationary states on a given bounded space domain, allowing us to explore the role various forms of the self-interaction, external potential and space domain have on BECs. We are able to show that the shape and curvature of a space domain strongly influence BEC structure, and may be used as control mechanisms in experiments. We next derive a non-autonomous perturbation theory to predict BEC response to temporal changes in an external potential. In certain cases, our approach can be extended to unbounded domains, and we conclude by constructing a perturbation theory for bright solitons within external potentials on unbounded domains.Fock quantization of canonical transformations and semiclassical asymptotics for degenerate problemshttps://www.zbmath.org/1472.810992021-11-25T18:46:10.358925Z"Dobrokhotov, Sergei"https://www.zbmath.org/authors/?q=ai:dobrokhotov.sergei-yu"Nazaikinskii, Vladimir"https://www.zbmath.org/authors/?q=ai:nazaikinskii.vladimir-eSummary: The aim of this work is to explain the role played by the Fock quantization of canonical transformations in the construction of the global semiclassical (high-frequency) asymptotic approximation. This role may well pass unnoticed as long as one deals with nondegenerate differential equations. However, the situation is different for some classes of equations with degeneration, where the Fock quantization of canonical transformations becomes instrumental in the construction of asymptotic solutions.
For the entire collection see [Zbl 1472.53006].Remarks on semiclassical wavefront sethttps://www.zbmath.org/1472.811002021-11-25T18:46:10.358925Z"Kameoka, Kentaro"https://www.zbmath.org/authors/?q=ai:kameoka.kentaroSummary: The essential support of the symbol of a semiclassical pseudodifferential operator is characterized by semiclassical wavefront sets of distributions. The proof employs a coherent state whose center in phase space is dependent on Planck's constant.Reduction and coherent stateshttps://www.zbmath.org/1472.811012021-11-25T18:46:10.358925Z"Rousseva, Jenia"https://www.zbmath.org/authors/?q=ai:rousseva.jenia"Uribe, Alejandro"https://www.zbmath.org/authors/?q=ai:uribe.alejandroThe authors apply a quantum version of dimensional reduction to Gaussian coherent states in Bargmann space to obtain squezed states on complex projective spaces (Definition 1.13) which lay perfectly in semiclassical approximation. Besides they are governed by a symbol calculus. The authors prove semiclassical norm estimates and a propagation result. The article includes the following topics: reduction of Gausian coherent states, squezed spin coherent states, covariance and Gaussian states, symbols, quantized Kahler manifolds, Bargmann spaces, reduction, classical propagation, quantum propagation, examples.How to choose master integralshttps://www.zbmath.org/1472.811022021-11-25T18:46:10.358925Z"Smirnov, A. V."https://www.zbmath.org/authors/?q=ai:smirnov.aleksandr-viktorovich.1|smirnov.andrei-v|smirnov.aleksandr-vladimirovich|smirnov.aleksandr-valeriyanovich|smirnov.aleksandr-viktorovich|smirnov.alexey-v|smirnov.andrey-v"Smirnov, V. A."https://www.zbmath.org/authors/?q=ai:smirnov.v-a|smirnov.vladimir-alekseevich|smirnov.vladimir-aSummary: The standard procedure when evaluating integrals of a given family of Feynman integrals, corresponding to some Feynman graph, is to construct an algorithm which provides the possibility to write any particular integral as a linear combination of so-called master integrals. To do this, public (\texttt{AIR}, \texttt{FIRE}, \texttt{REDUZE}, \texttt{LiteRed}, \texttt{KIRA}) and private codes based on solving integration by parts relations are used. However, the choice of the master integrals provided by these codes is not always optimal. We present an algorithm to improve a given basis of the master integrals, as well as its computer implementation; see also a competitive variant [\textit{J. Usovitsch}, ``Factorization of denominators in integration-by-parts reductions'', Preprint, \url{arXiv:2002.08173}].Scattering theory for Dirac fields inside a Reissner-Nordström-type black holehttps://www.zbmath.org/1472.811032021-11-25T18:46:10.358925Z"Häfner, Dietrich"https://www.zbmath.org/authors/?q=ai:hafner.dietrich"Mokdad, Mokdad"https://www.zbmath.org/authors/?q=ai:mokdad.mokdad"Nicolas, Jean-Philippe"https://www.zbmath.org/authors/?q=ai:nicolas.jean-philippeSummary: We show asymptotic completeness for the massive charged Dirac equation between the black hole and Cauchy horizons of a sub-extremal [(anti-)de Sitter]-Reissner-Nordström black hole.
{\copyright 2021 American Institute of Physics}Dependence of the density of states outer measure on the potential for deterministic Schrödinger operators on graphs with applications to ergodic and random modelshttps://www.zbmath.org/1472.811042021-11-25T18:46:10.358925Z"Hislop, Peter D."https://www.zbmath.org/authors/?q=ai:hislop.peter-d"Marx, Christoph A."https://www.zbmath.org/authors/?q=ai:marx.christoph-aSummary: We prove quantitative bounds on the dependence of the density of states on the potential function for discrete, deterministic Schrödinger operators on infinite graphs. While previous results were limited to random Schrödinger operators with independent, identically distributed potentials, this paper develops a deterministic framework, which is applicable to Schrödinger operators independent of the specific nature of the potential. Following ideas by Bourgain and Klein, we consider the density of states outer measure (DOSoM), which is well defined for all (deterministic) Schrödinger operators. We explicitly quantify the dependence of the DOSoM on the potential by proving a modulus of continuity in the \(\ell^\infty \)-norm. The specific modulus of continuity so obtained reflects the geometry of the underlying graph at infinity. For the special case of Schrödinger operators on \(\mathbb{Z}^d\), this implies the Lipschitz continuity of the DOSoM with respect to the potential. For Schrödinger operators on the Bethe lattice, we obtain log-Hölder dependence of the DOSoM on the potential. As an important consequence of our deterministic framework, we obtain a modulus of continuity for the density of states measure (DOSm) of ergodic Schrödinger operators in the underlying potential sampling function. Finally, we recover previous results for random Schrödinger operators on the dependence of the DOSm on the single-site probability measure by formulating this problem in the ergodic framework using the quantile function associated with the random potential.A classification of invertible phases of bosonic quantum lattice systems in one dimensionhttps://www.zbmath.org/1472.811052021-11-25T18:46:10.358925Z"Kapustin, Anton"https://www.zbmath.org/authors/?q=ai:kapustin.anton"Sopenko, Nikita"https://www.zbmath.org/authors/?q=ai:sopenko.nikita"Yang, Bowen"https://www.zbmath.org/authors/?q=ai:yang.bowenSummary: We study invertible states of 1D bosonic quantum lattice systems. We show that every invertible 1D state is in a trivial phase: after tensoring with some unentangled ancillas, it can be disentangled by a fuzzy analog of a finite-depth quantum circuit. If an invertible state has symmetries, it may be impossible to disentangle it in a way that preserves the symmetries, even after adding unentagled ancillas. We show that in the case of a finite unitary symmetry \(G\), the only obstruction is an index valued in degree-2 cohomology of \(G\). We show that two invertible \(G\)-invariant states are in the same phase if and only if their indices coincide.
{\copyright 2021 American Institute of Physics}Configurational complexity of nonautonomous discrete one-soliton and rogue waves in Ablowitz-Ladik-Hirota waveguidehttps://www.zbmath.org/1472.811062021-11-25T18:46:10.358925Z"Thakur, Pooja"https://www.zbmath.org/authors/?q=ai:thakur.pooja"Gleiser, Marcelo"https://www.zbmath.org/authors/?q=ai:gleiser.marcelo"Kumar, Anil"https://www.zbmath.org/authors/?q=ai:kumar.anil"Gupta, Rama"https://www.zbmath.org/authors/?q=ai:gupta.ramaSummary: We compute the configurational complexity (CC) for discrete soliton and rogue waves traveling along an Ablowitz-Ladik-Hirota (ALH) waveguide and modeled by a discrete nonlinear Schrödinger equation. We show that for a specific range of the soliton transverse direction \(\kappa\) propagating along the parametric time \(\zeta(t)\), CC reaches an evolving series of global minima. These minima represent maximum compressibility of information in the momentum modes along the Ablowitz-Ladik-Hirota waveguide. Computing the CC for rogue waves as a function of background amplitude modulation \(\mu\), we show that it displays two essential features: a maximum representing the optimal value for the rogue wave inception (the ``gradient catastrophe'') and saturation representing the rogue wave dispersion into constituent wave modes. We show that saturation is achieved earlier for higher values of modulation amplitude as the discrete rogue wave evolves along time \(\zeta(t)\).Stability of kinklike structures in generalized modelshttps://www.zbmath.org/1472.811072021-11-25T18:46:10.358925Z"Andrade, I."https://www.zbmath.org/authors/?q=ai:andrade.ivan|andrade.ivo-h-p"Marques, M. A."https://www.zbmath.org/authors/?q=ai:marques.miguel-a-l"Menezes, R."https://www.zbmath.org/authors/?q=ai:menezes.robertoSummary: We study the stability of topological structures in generalized models with a single real scalar field. We show that it is driven by a Sturm-Liouville equation and investigate the conditions that lead to the existence of explicit supersymmetric operators that factorize the stability equation and allow us to construct partner potentials. In this context, we discuss the property of shape invariance as a possible manner to calculate the discrete states and their respective eigenvalues.Supertime and Pauli's principlehttps://www.zbmath.org/1472.811082021-11-25T18:46:10.358925Z"Musin, Y. R."https://www.zbmath.org/authors/?q=ai:musin.yu-rSummary: Connection of Pauli's principle with the nontrivial structure of the fermion supertime is shown. When supersymmetry is localized as supergravitation, fields of gravitational and exchange interaction carriers arise. The exchange interaction quantum of free fermions, being a superpartner of graviton (gravitino), is interpreted as a paulino -- the particle responsible for the effect of mutual avoidance of identical fermions.On the construction of non-Hermitian Hamiltonians with all-real spectra through supersymmetric algorithmshttps://www.zbmath.org/1472.811092021-11-25T18:46:10.358925Z"Zelaya, Kevin"https://www.zbmath.org/authors/?q=ai:zelaya.kevin"Cruz y Cruz, Sara"https://www.zbmath.org/authors/?q=ai:cruz-y-cruz.sara"Rosas-Ortiz, Oscar"https://www.zbmath.org/authors/?q=ai:rosas-ortiz.oscarSummary: The energy spectra of two different quantum systems are paired through supersymmetric algorithms. One of the systems is Hermitian and the other is characterized by a complex-valued potential, both of them with only real eigenvalues in their spectrum. The superpotential that links these systems is complex-valued, parameterized by the solutions of the Ermakov equation, and may be expressed either in nonlinear form or as the logarithmic derivative of a properly chosen complex-valued function. The non-Hermitian systems can be constructed to be either parity-time-symmetric or non-parity-time-symmetric.
For the entire collection see [Zbl 1472.53006].Hyperspace fermions, Möbius transformations, Krein space, fermion doubling, dark matterhttps://www.zbmath.org/1472.811102021-11-25T18:46:10.358925Z"Jaroszkiewicz, George"https://www.zbmath.org/authors/?q=ai:jaroszkiewicz.georgeSummary: We develop an approach to classical and quantum mechanics where continuous time is extended by an infinitesimal parameter \(T\) and equations of motion converted into difference equations. These equations are solved and the physical limit \(T \rightarrow 0\) then taken. In principle, this strategy should recover all standard solutions to the original continuous time differential equations. We find this is valid for bosonic variables whereas with fermions, additional solutions occur. For both bosons and fermions, the difference equations of motion can be related to Möbius transformations in projective geometry. Quantization via Schwinger's action principle recovers standard particle-antiparticle modes for bosons but in the case of fermions, Hilbert space has to be replaced by Krein space. We discuss possible links with the fermion doubling problem and with dark matter.Superdeterministic hidden-variables models. I: Non-equilibrium and signallinghttps://www.zbmath.org/1472.811112021-11-25T18:46:10.358925Z"Sen, Indrajit"https://www.zbmath.org/authors/?q=ai:sen.indrajit"Valentini, Antony"https://www.zbmath.org/authors/?q=ai:valentini.antonySummary: This is the first of two papers that attempt to comprehensively analyse superdeterministic hidden-variables models of Bell correlations. We first give an overview of superdeterminism and discuss various criticisms of it raised in the literature. We argue that the most common criticism, the violation of `free-will', is incorrect. We take up Bell's intuitive criticism that these models are `conspiratorial'. To develop this further, we introduce non-equilibrium extensions of superdeterministic models. We show that the measurement statistics of these extended models depend on the physical system used to determine the measurement settings. This suggests a fine-tuning in order to eliminate this dependence from experimental observation. We also study the signalling properties of these extended models. We show that although they generally violate the formal no-signalling constraints, this violation cannot be equated to an actual signal. We therefore suggest that the so-called no-signalling constraints be more appropriately named the marginal-independence constraints. We discuss the mechanism by which marginal-independence is violated in superdeterministic models. Lastly, we consider a hypothetical scenario where two experimenters use the apparent-signalling of a superdeterministic model to communicate with each other. This scenario suggests another conspiratorial feature peculiar to superdeterminism. These suggestions are quantitatively developed in the second paper [the authors, Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 476, No. 2243, Article ID 20200214, 14 p. (2020; Zbl 1472.81112)].Superdeterministic hidden-variables models. II: Conspiracyhttps://www.zbmath.org/1472.811122021-11-25T18:46:10.358925Z"Sen, Indrajit"https://www.zbmath.org/authors/?q=ai:sen.indrajit"Valentini, Antony"https://www.zbmath.org/authors/?q=ai:valentini.antonySummary: We prove that superdeterministic models of quantum mechanics are conspiratorial in a mathematically well-defined sense, by further development of the ideas presented in a previous article
[the authors, Proc. R. Soc. Lond., A, Math. Phys. Eng. Sci. 476, No. 2243, Article ID 20200212, 14 p. (2020; Zbl 1472.81111)]. We consider a Bell scenario where, in each run and at each wing, the experimenter chooses one of \(N\) devices to determine the local measurement setting. We prove, without assuming any features of quantum statistics, that superdeterministic models of this scenario must have a finely tuned distribution of hidden variables. Specifically, fine-tuning is required so that the measurement statistics depend on the measurement settings but not on the details of how the settings are chosen. We quantify this as the overhead fine-tuning \(F\) of the model, and show that \(F > 0\) (corresponding to `fine-tuned') for any \(N > 1\). The notion of fine-tuning assumes that arbitrary (`non-equilibrium') hidden-variables distributions are possible in principle. We also show how to quantify superdeterministic conspiracy without using non-equilibrium. This second approach is based on the fact that superdeterministic correlations can mimic actual signalling. We argue that an analogous situation occurs in equilibrium where, for every run, the devices that the hidden variables are correlated with are coincidentally the same as the devices in fact used. This results in extremely large superdeterministic correlations, which we quantify as a drop of an appropriately defined formal entropy. Non-local and retrocausal models turn out to be non-conspiratorial according to both approaches.Diffeomorphism groups in quantum theory and statistical physicshttps://www.zbmath.org/1472.811142021-11-25T18:46:10.358925Z"Goldin, Gerald A."https://www.zbmath.org/authors/?q=ai:goldin.gerald-aSummary: Symmetry groups describe invariances or partial invariances in physical systems under transformations. Locality refers to the association between physical effects and spatial or spacetime regions, with ``action at a distance'' forbidden. Local symmetry joins these ideas mathematically in the theory of certain infinite-dimensional groups and their representations. This chapter is an extended abstract of lectures by the author, surveying how unitary representations of diffeomorphism groups and the corresponding current algebras provide a unifying framework for understanding or predicting a wide variety of different quantum and statistical systems.
For the entire collection see [Zbl 1472.53006].Axial momentum for the relativistic Majorana particlehttps://www.zbmath.org/1472.811182021-11-25T18:46:10.358925Z"Arodź, H."https://www.zbmath.org/authors/?q=ai:arodz.henrykSummary: The Hilbert space of states of the relativistic Majorana particle consists of normalizable bispinors with real components, hence the usual momentum operator \(- i\nabla\) can not be defined in this space. For this reason, we introduce the axial momentum operator, \(-i \gamma_5 \nabla\) as a new observable for this particle. In the Heisenberg picture, the axial momentum contains a component which oscillates with the amplitude proportional to \(m/E\), where \(E\) is the energy and \(m\) the mass of the particle. The presence of the oscillations discriminates between the massive and massless Majorana particle. Furthermore, we show how the eigenvectors of the axial momentum, called the axial plane waves, can be used as a basis for obtaining the general solution of the evolution equation, also in the case of free Majorana field. Here a novel feature is a coupling of modes with the opposite momenta, again present only in the case of massive particle or field.Spinorial \(R\) operator and algebraic Bethe ansatzhttps://www.zbmath.org/1472.811202021-11-25T18:46:10.358925Z"Karakhanyan, D."https://www.zbmath.org/authors/?q=ai:karakhanyan.david"Kirschner, R."https://www.zbmath.org/authors/?q=ai:kirschner.rolandSummary: We propose a new approach to the spinor-spinor R-matrix with orthogonal and symplectic symmetry. Based on this approach and the fusion method we relate the spinor-vector and vector-vector monodromy matrices for quantum spin chains. We consider the explicit spinor \(R\) matrices of low rank orthogonal algebras and the corresponding \(RTT\) algebras. Coincidences with fundamental \(R\) matrices allow to relate the Algebraic Bethe Ansatz for spinor and vector monodromy matrices.Periodic coherent states decomposition and quantum dynamics on the flat torushttps://www.zbmath.org/1472.811212021-11-25T18:46:10.358925Z"Zanelli, Lorenzo"https://www.zbmath.org/authors/?q=ai:zanelli.lorenzoSummary: We provide a result on the coherent states decomposition for functions in \(L^2 (\mathbb{T}^n)\) where \(\mathbb{T}^n := (\mathbb{R} / 2\pi \mathbb{Z})^n\). We study such a decomposition with respect to the quantum dynamics related to semiclassical elliptic pseudodifferential operators, and we prove a related invariance result.
For the entire collection see [Zbl 1471.47002].Tighter uncertainty relations based on Wigner-Yanase skew information for observables and channelshttps://www.zbmath.org/1472.811272021-11-25T18:46:10.358925Z"Zhang, Li-Mei"https://www.zbmath.org/authors/?q=ai:zhang.limei"Gao, Ting"https://www.zbmath.org/authors/?q=ai:gao.ting"Yan, Feng-Li"https://www.zbmath.org/authors/?q=ai:yan.fengliSummary: Uncertainty principle is the basis of quantum mechanics. It reflects the basic law of the movement of microscopic particles. Wigner-Yanase skew information, as a measure of quantum uncertainties, is used to characterize the intrinsic features of the state and the observable. In this paper, we mainly investigate the sum uncertainty relations for both quantum mechanical observables and quantum channels based on skew information. We establish a new uncertainty relation in terms of skew information for \(n\) observables, which is saturated (thus it holds as equality) for two incompatible observables. We also present two uncertainty relations for arbitrary finite \(N\) quantum channels by using skew information. Our uncertainty relations have tighter lower bounds than the existing ones. Detailed examples are provided.Multiple-scale analysis of open quantum systemshttps://www.zbmath.org/1472.811292021-11-25T18:46:10.358925Z"Bernal-García, D. N."https://www.zbmath.org/authors/?q=ai:bernal-garcia.d-n"Rodríguez, B. A."https://www.zbmath.org/authors/?q=ai:rodriguez.boris-a"Vinck-Posada, H."https://www.zbmath.org/authors/?q=ai:vinck-posada.herbertSummary: In this work, we present a multiple-scale perturbation technique suitable for the study of open quantum systems, which is easy to implement and in few iterative steps allows us to find excellent approximate solutions. For any time-local quantum master equation, whether Markovian or non-Markovian, in Lindblad form or not, we give a general procedure to construct analytical approximations to the corresponding dynamical map and, consequently, to the temporal evolution of the density matrix. As a simple illustrative example of the implementation of the method, we study an atom-cavity system described by a dissipative Jaynes-Cummings model. Performing a multiple-scale analysis we obtain approximate analytical expressions for the strong and weak coupling regimes that allow us to identify characteristic time scales in the state of the physical system.Path integrals with discarded degrees of freedomhttps://www.zbmath.org/1472.811342021-11-25T18:46:10.358925Z"Butcher, Luke M."https://www.zbmath.org/authors/?q=ai:butcher.luke-mSummary: Whenever variables \(\phi = (\phi^1, \phi^2, \dots)\) are discarded from a system, and the discarded information capacity \(\mathcal{S}(x)\) depends on the value of an observable \(x\), a quantum correction \(\Delta V_{\mathrm{eff}}(x)\) appears in the effective potential [the author, Phys. Lett., A 382, No. 36, 2555--2560 (2018; Zbl 1404.81110)]. Here I examine the origins and implications of \(\Delta V_{\mathrm{eff}}\) within the path integral, which I construct using Synge's world function. I show that the \(\phi\) variables can be `integrated out' of the path integral, reducing the propagator to a sum of integrals over observable paths \(x(t)\) alone. The phase of each path is equal to the semiclassical action (divided by \(\hbar\)) including the same correction \(\Delta V_{\mathrm{eff}}\) as previously derived. This generalises the prior results beyond the limits of the Schrödinger equation; in particular, it allows us to consider discarded variables with a history-dependent information capacity \(\mathcal{S} = \mathcal{S}(x, \int^t f(x(t^\prime)) \mathrm{d}t^\prime)\). History dependence does not alter the formula for \(\Delta V_{\mathrm{eff}}\).Wobbling double sine-Gordon kinkshttps://www.zbmath.org/1472.811382021-11-25T18:46:10.358925Z"Campos, João G. F."https://www.zbmath.org/authors/?q=ai:campos.joao-g-f"Mohammadi, Azadeh"https://www.zbmath.org/authors/?q=ai:mohammadi.azadehSummary: We study the collision of a kink and an antikink in the double sine-Gordon model with and without the excited vibrational mode. In the latter case, we find that there is a limited range of the parameters where the resonance windows exist, despite the existence of a vibrational mode. Still, when the vibrational mode is initially excited, its energy can turn into translational energy after the collision. This creates one-bounce as well as a rich structure of higher-bounce resonance windows that depend on the wobbling phase being in or out of phase at the collision and the wobbling amplitude being sufficiently large. When the vibrational mode is excited, the modified structure of one-bounce windows is observed in the whole range of the model's parameters, and the resonant interval with higher-bounce windows gradually increases with the wobbling amplitude. We estimated the center of the one-bounce windows using a simple analytical approximation for the wobbling evolution. The kinks' final wobbling frequency is Lorentz contracted, which is simply derived from our equations. We also report that the maximum energy density value always has a smooth behavior in the resonance windows.Goldstone bosons in different PT-regimes of non-Hermitian scalar quantum field theorieshttps://www.zbmath.org/1472.811432021-11-25T18:46:10.358925Z"Fring, Andreas"https://www.zbmath.org/authors/?q=ai:fring.andreas"Taira, Takanobu"https://www.zbmath.org/authors/?q=ai:taira.takanobuSummary: We study the interplay between spontaneously breaking global continuous and discrete antilinear symmetries in a newly proposed general class of non-Hermitian quantum field theories containing a mixture of complex and real scalar fields. We analyse the model for different types of global symmetry preserving and breaking vacua. In addition, the models are symmetric under various types of discrete antilinear symmetries composed out of nonstandard simultaneous charge conjugations, time-reversals and parity transformations; CPT. While the global symmetry governs the existence of massless Goldstone bosons, the discrete one controls the precise expression of the Goldstone bosons in terms of the original fields in the model and its physical regimes. We show that even when the CPT-symmetries are broken on the level of the action expanded around different types of vacua, the mass spectra might still be real when the symmetry is preserved at the tree approximation and the breaking only occurs at higher order. We discuss the parameter space of some of the models in the proposed class and identify physical regimes in which massless Goldstone bosons emerge when the vacuum spontaneously breaks the global symmetry or equivalently when the corresponding Noether currents are conserved. The physical regions are bounded by exceptional points in different ways. There exist special points in parameter space for which massless bosons may occur already before breaking the global symmetry. However, when the global symmetry is broken at these points they can no longer be distinguished from genuine Goldstone bosons.Holographic unitary renormalization group for correlated electrons. II: Insights on fermionic criticalityhttps://www.zbmath.org/1472.811762021-11-25T18:46:10.358925Z"Mukherjee, Anirban"https://www.zbmath.org/authors/?q=ai:mukherjee.anirban"Lal, Siddhartha"https://www.zbmath.org/authors/?q=ai:lal.siddharthaSummary: Capturing the interplay between electronic correlations and many-particle entanglement requires a unified framework for Hamiltonian and eigenbasis renormalization. In this work, we apply the unitary renormalization group (URG) scheme developed in a companion work [the authors, ibid. 960, Article ID 115170, 72 p. (2020; Zbl 1472.81177)] to the study of two archetypal models of strongly correlated lattice electrons, one with translation invariance and one without. We obtain detailed insight into the emergence of various gapless and gapped phases of quantum electronic matter by computing effective Hamiltonians from numerical evaluation of the various RG equations, as well as their entanglement signatures through their respective tensor network descriptions. For the translationally invariant model of a single-band of interacting electrons, this includes results on gapless metallic phases such as the Fermi liquid and Marginal Fermi liquid, as well as gapped phases such as the reduced Bardeen-Cooper-Schrieffer, pair density-wave and Mott liquid phases. Additionally, a study of a generalised Sachdev-Ye model with disordered four-fermion interactions offers detailed results on many-body localised phases, as well as thermalised phase. We emphasise the distinctions between the various phases based on a combined analysis of their dynamical (obtained from the effective Hamiltonian) and entanglement properties. Importantly, the RG flow of the Hamiltonian vertex tensor network is shown to lead to emergent gauge theories for the gapped phases. Taken together with results on the holographic spacetime generated from the RG of the many-particle eigenstate (seen through, for instance, the holographic upper bound of the one-particle entanglement entropy), our analysis offer an ab-initio perspective of the gauge-gravity duality for quantum liquids that are emergent in systems of correlated electrons.Holographic unitary renormalization group for correlated electrons. I: A tensor network approachhttps://www.zbmath.org/1472.811772021-11-25T18:46:10.358925Z"Mukherjee, Anirban"https://www.zbmath.org/authors/?q=ai:mukherjee.anirban"Lal, Siddhartha"https://www.zbmath.org/authors/?q=ai:lal.siddharthaSummary: We present a unified framework for the renormalisation of the Hamiltonian and eigenbasis of a system of correlated electrons, unveiling thereby the interplay between electronic correlations and many-particle entanglement. For this, we extend substantially the unitary renormalization group (URG) scheme introduced in [the authors, ``Scaling theory for Mott-Hubbard transitions: I. \(T = 0\) phase diagram of the 1/2-filled Hubbard model'', New J. Phys. 22, No. 6, Article ID 063007, 26 p. (2020; \url{doi:10.1088/1367-2630/ab8831}); ``Scaling theory for Mott-Hubbard transitions. II: Quantum criticality of the doped Mott insulator'', ibid. 22, No. 6, Article ID 063008, 25 p. (2020; \url{doi:10.1088/1367-2630/ab890c}); ``Holographic entanglement renormalisation of topological order in a quantum liquid'', Preprint, \url{arXiv:2003.06118}]. We recast the RG as a discrete flow of the Hamiltonian tensor network, i.e., the collection of various \(2n\)-point scattering vertex tensors comprising the Hamiltonian. The renormalisation progresses via unitary transformations that block diagonalizes the Hamiltonian iteratively via the disentanglement of single-particle eigenstates. This procedure incorporates naturally the role of quantum fluctuations. The RG flow equations possess a non-trivial structure, displaying a feedback mechanism through frequency-dependent dynamical self-energies and correlation energies. The interplay between various UV energy scales enables the coupled RG equations to flow towards a stable fixed point in the IR. The effective Hamiltonian at the IR fixed point generically has a reduced parameter space, as well as number of degrees of freedom, compared to the microscopic Hamiltonian. Importantly, the vertex RG flows are observed to govern the RG flow of the tensor network that denotes the coefficients of the many-particle eigenstates. The RG evolution of various many-particle entanglement features of the eigenbasis are, in turn, quantified through the coefficient tensor network. In this way, we show that the URG framework provides a microscopic understanding of holographic renormalisation: the RG flow of the vertex tensor network generates a eigenstate coefficient tensor network possessing a many-particle entanglement metric. We find that the eigenstate tensor network accommodates sign factors arising from fermion exchanges, and that the IR fixed point reached generically involves a trivialisation of the fermion sign factor. Several results are presented for the emergence of composite excitations in the neighbourhood of a gapless Fermi surface, as well as for the condensation phenomenon involving the gapping of the Fermi surface.
For Part II, see the authors [Nucl. Phys., B 960, Article ID 115163, 63 p. (2020; Zbl 1472.81176)].Nonrelativistic strings on \(R \times S^2\) and integrable systemshttps://www.zbmath.org/1472.811952021-11-25T18:46:10.358925Z"Roychowdhury, Dibakar"https://www.zbmath.org/authors/?q=ai:roychowdhury.dibakarSummary: We show that the (torsional) nonrelativistic string sigma models on \(R \times S^2\) can be mapped into \textit{deformed} Rosochatius like integrable models in one dimension. We also explore the associated Hamiltonian constrained structure by introducing appropriate Dirac brackets. These results show some solid evidence of the underlying integrable structure in the nonrelativistic sector of the gauge/string duality.On precision holography for the circular Wilson loop in \(AdS_5 \times S^5\)https://www.zbmath.org/1472.812032021-11-25T18:46:10.358925Z"Botao, Li"https://www.zbmath.org/authors/?q=ai:botao.li"Medina-Rincon, Daniel"https://www.zbmath.org/authors/?q=ai:medina-rincon.danielSummary: The string theory calculation of the \(\frac{1}{2}\)-BPS circular Wilson loop of \(\mathcal{N} = 4\) SYM in the planar limit at next to leading order at strong coupling is revisited in the ratio of its semiclassical string partition function and the one dual to a latitude Wilson loop with trivial expectation value. After applying a conformal transformation from the disk to the cylinder, this problem can be approached by means of the Gel'fand-Yaglom formalism. Using results from the literature and the exclusion of zero modes from a modified Gel'fand-Yaglom formula, we obtain matching with the known field theory result. As seen in the phaseshift method computation, non-zero mode contributions cancel and the end result comes from the zero mode degeneracies of the latitude Wilson loop.On some (integrable) structures in low-dimensional holographyhttps://www.zbmath.org/1472.812112021-11-25T18:46:10.358925Z"Rashkov, R. C."https://www.zbmath.org/authors/?q=ai:rashkov.r-ch|rashkov.radoslav-cSummary: Recent progress in holographic correspondence uncovered remarkable relations between key characteristics of the theories on both sides of duality and certain integrable models.
In this note we revisit the problem of the role of certain invariants in low-dimensional holography. As motivating example we consider first the entanglement entropy in 2d CFT and projective invariants. Next we consider higher projective invariants and suggest generalization to higher spin theories. Quadratic in invariants deformations is considered and conjectured to play role in low-dimensional higher spin holography.Integrability as duality: the gauge/YBE correspondencehttps://www.zbmath.org/1472.812132021-11-25T18:46:10.358925Z"Yamazaki, Masahito"https://www.zbmath.org/authors/?q=ai:yamazaki.masahitoSummary: The Gauge/YBE correspondence states a surprising connection between solutions to the Yang-Baxter equation with spectral parameters and partition functions of supersymmetric quiver gauge theories. This correspondence has lead to systematic discoveries of new integrable models based on quantum-field-theory methods. We provide pedagogical introduction to the subject and summarizes many recent developments. This is a write-up of the lecture at the String-Math 2018 conference.Qubit construction in 6D SCFTshttps://www.zbmath.org/1472.812202021-11-25T18:46:10.358925Z"Heckman, Jonathan J."https://www.zbmath.org/authors/?q=ai:heckman.jonathan-jSummary: We consider a class of 6D superconformal field theories (SCFTs) which have a large \(N\) limit and a semi-classical gravity dual description. Using the quiver-like structure of 6D SCFTs we study a subsector of operators protected from large operator mixing. These operators are characterized by degrees of freedom in a one-dimensional spin chain, and the associated states are generically highly entangled. This provides a concrete realization of qubit-like states in a strongly coupled quantum field theory. Renormalization group flows triggered by deformations of 6D UV fixed points translate to specific deformations of these one-dimensional spin chains. We also present a conjectural spin chain Hamiltonian which tracks the evolution of these states as a function of renormalization group flow, and study qubit manipulation in this setting. Similar considerations hold for theories without \(AdS\)duals, such as 6D little string theories and 4D SCFTs obtained from compactification of the partial tensor branch theory on a \(T^2\).Universality of ultra-relativistic gravitational scatteringhttps://www.zbmath.org/1472.812482021-11-25T18:46:10.358925Z"Di Vecchia, Paolo"https://www.zbmath.org/authors/?q=ai:di-vecchia.paolo"Heissenberg, Carlo"https://www.zbmath.org/authors/?q=ai:heissenberg.carlo"Russo, Rodolfo"https://www.zbmath.org/authors/?q=ai:russo.rodolfo"Veneziano, Gabriele"https://www.zbmath.org/authors/?q=ai:veneziano.gabrieleSummary: We discuss the ultra-relativistic gravitational scattering of two massive particles at two-loop (3PM) level. We find that in this limit the real part of the eikonal, determining the deflection angle, is universal for gravitational theories in the two derivative approximation. This means that, regardless of the number of supersymmetries or the nature of the probes, the result connects smoothly with the massless case discussed since the late eighties by Amati, Ciafaloni and Veneziano. We analyse the problem both by using the analyticity and crossing properties of the scattering amplitudes and, in the case of the maximally supersymmetric theory, by explicit evaluation of the 4-point 2-loop amplitude using the results for the integrals in the full soft region. The first approach shows that the observable we are interested in is determined by the inelastic tree-level amplitude describing the emission of a graviton in the high-energy double-Regge limit, which is the origin of the universality property mentioned above. The second approach strongly suggests that the inclusion of the whole soft region is a necessary (and possibly sufficient) ingredient for recovering ultra relativistic finiteness and universality at the 3PM level. We conjecture that this universality persists at all orders in the PM expansion.Scattering, spectrum and resonance states completeness for a quantum graph with Rashba Hamiltonianhttps://www.zbmath.org/1472.812512021-11-25T18:46:10.358925Z"Blinova, Irina V."https://www.zbmath.org/authors/?q=ai:blinova.irina-v"Popov, Igor Y."https://www.zbmath.org/authors/?q=ai:popov.igor-yu"Smolkina, Maria O."https://www.zbmath.org/authors/?q=ai:smolkina.maria-oSummary: Quantum graphs consisting of a ring with two semi-infinite edges attached to the same point of the ring is considered. We deal with the Rashba spin-orbit Hamiltonian on the graph. A theorem concerning to completeness of the resonance states on the ring is proved. Due to use of a functional model, the problem reduces to factorization of the characteristic matrix-function. The result is compared with the corresponding completeness theorem for the Schrödinger, Dirac and Landau quantum graphs.
For the entire collection see [Zbl 1471.47002].Tunneling through bridges: Bohmian non-locality from higher-derivative gravityhttps://www.zbmath.org/1472.812552021-11-25T18:46:10.358925Z"Duane, Gregory S."https://www.zbmath.org/authors/?q=ai:duane.gregory-sSummary: A classical origin for the Bohmian quantum potential, as that potential term arises in the quantum mechanical treatment of black holes and Einstein-Rosen (ER) bridges, can be based on 4th-order extensions of Einstein's equations. The required 4th-order extension of general relativity is given by adding quadratic curvature terms with coefficients that maintain a fixed ratio, as their magnitudes approach zero, with classical general relativity as a singular limit. If entangled particles are connected by a Planck-width ER bridge, as conjectured by Maldacena and Susskind, then a connection by a traversable Planck-scale wormhole, allowed in 4th-order gravity, describes such entanglement in the ontological interpretation. It is hypothesized that higher-derivative gravity can account for the nonlocal part of the quantum potential generally.Non-relativistic neutrinos and the weak equivalence principle apparent violationhttps://www.zbmath.org/1472.812842021-11-25T18:46:10.358925Z"Blasone, M."https://www.zbmath.org/authors/?q=ai:blasone.massimo"Jizba, P."https://www.zbmath.org/authors/?q=ai:jizba.petr"Lambiase, G."https://www.zbmath.org/authors/?q=ai:lambiase.gaetano"Petruzziello, L."https://www.zbmath.org/authors/?q=ai:petruzziello.lucianoSummary: We study the non-relativistic limit of Dirac equation for mixed neutrinos. We demonstrate that such a procedure inevitably leads to a redefinition of the inertial mass. This happens because, in contrast to the case when mixing is absent, the antiparticle sector contribution cannot be neglected for neutrinos with definite flavor. We then show that, when a gravitational interaction is switched on, in the weak-field approximation the mass parameter which couples to gravity (gravitational mass) does not undergo the same reformulation as the inertial mass, thus leading to an apparent breakdown of the weak equivalence principle.A new class of mass dimension one fermionshttps://www.zbmath.org/1472.813052021-11-25T18:46:10.358925Z"Ahluwalia, Dharam Vir"https://www.zbmath.org/authors/?q=ai:vir-ahluwalia.dharamSummary: These are notes on the square root of a \(4 \times 4\) identity matrix and associated quantum fields of spin one half. The method is illustrated by constructing a new mass dimension one fermionic field. The presented field is local. The field energy is bounded from below. It is argued that these fermions are a first-principle candidate for dark matter with an unsuppressed quartic self-interaction.Exchange interactions, Yang-Baxter relations and transparent particleshttps://www.zbmath.org/1472.813062021-11-25T18:46:10.358925Z"Polychronakos, Alexios P."https://www.zbmath.org/authors/?q=ai:polychronakos.alexios-pSummary: We introduce a class of particle models in one dimension involving exchange interactions that have scattering properties satisfying the Yang-Baxter consistency condition. A subclass of these models exhibits reflectionless scattering, in which particles are ``transparent'' to each other, generalizing a property hitherto only known for the exchange Calogero model. The thermodynamics of these systems can be derived using the asymptotic Bethe-Ansatz method.Two anyons on the sphere: nonlinear states and spectrumhttps://www.zbmath.org/1472.813072021-11-25T18:46:10.358925Z"Polychronakos, Alexios P."https://www.zbmath.org/authors/?q=ai:polychronakos.alexios-p"Ouvry, Stéphane"https://www.zbmath.org/authors/?q=ai:ouvry.stephaneSummary: We study the energy spectrum of two anyons on the sphere in a constant magnetic field. Making use of rotational invariance we reduce the energy eigenvalue equation to a system of linear differential equations for functions of a single variable, a reduction analogous to separating center of mass and relative coordinates on the plane. We solve these equations by a generalization of the Frobenius method and derive numerical results for the energies of non-analytically derivable states.Spin texture and Berry phase for heavy-mass holes confined in SiGe mixed-alloy two-dimensional system: intersubband interaction via the coexistence of Rashba and Dresselhaus spin-orbit interactionshttps://www.zbmath.org/1472.813172021-11-25T18:46:10.358925Z"Tojo, Tatsuki"https://www.zbmath.org/authors/?q=ai:tojo.tatsuki"Takeda, Kyozaburo"https://www.zbmath.org/authors/?q=ai:takeda.kyozaburoSummary: By extending the \(\boldsymbol{k} \cdot \boldsymbol{p}\) approach, we study the spin texture and Berry phase of heavy-mass holes (HHs) confined in the SiGe two-dimensional (2D) quantum well system, focusing on the intersubband interaction via the coexistence of the Rashba and Dresselhaus spin-orbit interactions (SOIs). The coexistence of both SOIs generates spin-stabilized(+)/destabilized(-) HHs. The strong intersubband interaction causes \textit{quasi}-degenerate states resembling the 2D massive Dirac fermion. Consequently, the Berry phases of HHs have unique energy dependence understood by counting the \textit{quasi}-degenerate points with the signs of the Berry phases. Thermal averaging of the Berry phase demonstrates that HH\(+/-\) has a peculiar plateau of \(+\pi/-\pi\) at less than 30 K and then changes its sign at approximately 200 K.Diffraction pattern degradation driven by intense ultrafast X-ray pulse for \(H_2^+\)https://www.zbmath.org/1472.813182021-11-25T18:46:10.358925Z"Borovykh, S. V."https://www.zbmath.org/authors/?q=ai:borovykh.s-v"Mityureva, A. A."https://www.zbmath.org/authors/?q=ai:mityureva.a-a"Smirnov, V. V."https://www.zbmath.org/authors/?q=ai:smirnov.valerii-valentinovich|smirnov.valeri-v|smirnov.vitalii-vSummary: The drastic evolution of molecular systems exposed to ultrashort intense X-ray pulse is a fundamental obstacle for single-particle imaging (SPI) by means of X-ray free electron lasers (XFEL). Here we tackle the simplest molecule \(H_2^+\) and its diffraction pattern degradations in the strong ultrashort X-ray beam. The semiclassical method of the problem solution and its advantages are described in detail. We apply the method to calculate the electron density autocorrelation functions (ACF) for a few internuclear distances and then discuss numeric simulation data.Operator growth bounds from graph theoryhttps://www.zbmath.org/1472.813212021-11-25T18:46:10.358925Z"Chen, Chi-Fang"https://www.zbmath.org/authors/?q=ai:chen.chifang"Lucas, Andrew"https://www.zbmath.org/authors/?q=ai:lucas.andrewThe article provides a connection between combinatoric quantity arising from graph theory, and the bounds of the norm of the commutator between two observables of interest. The latter is motivated by the notion of the scrambling time -- the time at which a product state is transformed into a maximally entangled one. Such quantity is described via the commutators between two observables at different times, e.g. \(A_i\left( t \right)\) and \(B_j\), so-called out-of-time-order correlators. It is then transcribed as projected operator of \(A_i\left( t \right)\) onto a local space associated with \(B_j.\) The combinatorics and graph structures arise here from the expansion of the evolution operators in \(A_i\left( t \right),\) yielding the interplay between the aforementioned projection and the products of generators, in the expansion, associated with different paths on the graph. The authors analysed the problem for both non-random and random systems, where in the first scenario they recover the Lieb-Robinson bound as a special case. For the models with randomness, the scrambling time for some classes of graphs and the evidence for the validity of the scrambling conjecture -- the scrambling time does not grow logarithmically with the number of degrees of freedom -- are provided (yet the scrambling conjecture is not formally proven.)
The analyses in this article are interesting in several aspects of quantum many-body physics. On one hand, they provide the extension of the understanding on the scrambling or the process through which the information is lost in the many-body system. On the other hand, they exemplify the connection between combinatorics and geometric pictures with the study of the dynamical process. It is interesting that similar analyses can be adapted to other similar problems such as the study of the growth of some quantity or the causal structure on quantum networks.Some properties of the potential-to-ground state map in quantum mechanicshttps://www.zbmath.org/1472.813222021-11-25T18:46:10.358925Z"Garrigue, Louis"https://www.zbmath.org/authors/?q=ai:garrigue.louisThe author considers properties of the map from potential to the ground state in many-body quantum mechanics. External potentials \(v\in L^p + L^\infty\) and interaction potentials \(w \in L^p + L^\infty\) for \(p> \max(2d/3, 2)\) where \(d\) is the dimension of the underlying space are considered. The first result is that the space of binding potentials is path-connected. Then the author shows that the map from potentials to the ground state is locally weak-strong continuous and that its differential is compact. This implies that the Kohn-Sham inverse problem in Density Functional Theory is ill-posed on a bounded set.On the effect of fractional statistics on quantum ion acoustic waveshttps://www.zbmath.org/1472.813272021-11-25T18:46:10.358925Z"Ourabah, Kamel"https://www.zbmath.org/authors/?q=ai:ourabah.kamelSummary: In this paper, I study the effect of a small deviation from the Fermi-Dirac statistics on the quantum ion acoustic waves. For this purpose, a quantum hydrodynamic model is developed based on the Polychronakos statistics, which allows for a smooth interpolation between the Fermi and Bose limits, passing through the case of classical particles. The model includes the effect of pressure as well as quantum diffraction effects through the Bohm potential. The equation of state for electrons obeying fractional statistics is obtained and the effect of fractional statistics on the kinetic energy and the coupling parameter is analyzed. Through the model, the effect of fractional statistics on the quantum ion acoustic waves is highlighted, exploring both linear and weakly nonlinear regimes. It is found that fractional statistics enhance the amplitude and diminish the width of the quantum ion acoustic waves. Furthermore, it is shown that a small deviation from the Fermi-Dirac statistics can modify the type structures, from bright to dark soliton. All known results of fully degenerate and non-degenerate cases are reproduced in the proper limits.From short-range to contact interactions in the 1d Bose gashttps://www.zbmath.org/1472.813282021-11-25T18:46:10.358925Z"Griesemer, Marcel"https://www.zbmath.org/authors/?q=ai:griesemer.marcel"Hofacker, Michael"https://www.zbmath.org/authors/?q=ai:hofacker.michael"Linden, Ulrich"https://www.zbmath.org/authors/?q=ai:linden.ulrichIn the paper under review, the authors consider a system of finite bosons in dimension \(1\) with point interactions described by a Hamiltonian \(H\). They introduce a family of operators \(H_\epsilon\) (\(\epsilon>0\)) that are obtained from \(H\) by a regularization procedure of the point interactions. The main result of the paper is the norm resolvent convergence of \(H_\epsilon\) to \(H\), as \(\epsilon\) goes to \(0\). Under a stronger requirement on the regularized potentials, they also get a bound on the convergence rate. As a consequence, the time-evolution group associated to \(H_\epsilon\) provides a good approximation of the one associated to \(H\).
As explained in the paper, the restriction to the dimension \(1\) avoids complicated difficulties. The proof is based on an appropriate representation of the Hamiltonian \(H_\epsilon\) and on the use of the Konno-Kuroda formula. The later provides a good expression of the difference between the resolvent of \(H_\epsilon\) and the one of the free Hamiltonian. Further important ingredients of the proof are the notion of \(\Gamma\)-convergence and an appropriate use of the Green's function of the multidimensional Laplace operator. This yields, together with the above convergence, a formula for the difference between the resolvent of \(H\) and the one of the free Hamiltonian and it can be shown that it only depends on the \(\mathrm{L}^1\)-norm of the regularized potentials.
We refer to the Introduction of the paper for more details. We point out, that, in the definition of the regularized potentials, it seems that they are implicitly considered as real functions.The linear dynamics of wave functions in causal fermion systemshttps://www.zbmath.org/1472.813342021-11-25T18:46:10.358925Z"Finster, Felix"https://www.zbmath.org/authors/?q=ai:finster.felix"Kamran, Niky"https://www.zbmath.org/authors/?q=ai:kamran.niky"Oppio, Marco"https://www.zbmath.org/authors/?q=ai:oppio.marcoThe goal of the paper is to study the dynamics of the physical wave functions of a causal fermion system. Due to nonlinearity of the causal action all the wave functions interact with each other. The authors interpretes this in a manner similar to back reaction of a quantum spinor field in a curved spacetime on the metric. The authors derive dynamical wave function. They show that its solution form a Hilbert space, whose salar product is represented by a conserved layer integral. By few theorems the authors prove that the initial value problem for the dynamical wave equation admits a unique global solution. Subsequently they construct causal Green' s operators and investigate their properties. As a particular example the authors investigate the regularized Minkowski vacuum. The article includes 6 sections from which one has a preliminary character. Two apendices are included in order to clarify the role of comutator jets to which a previous section was concerned.Graphene wormhole trapped by external magnetic fieldhttps://www.zbmath.org/1472.813352021-11-25T18:46:10.358925Z"Garcia, G. Q."https://www.zbmath.org/authors/?q=ai:garcia.gabriel-queiroz"Porfírio, P. J."https://www.zbmath.org/authors/?q=ai:porfirio.paulo-j"Moreira, D. C."https://www.zbmath.org/authors/?q=ai:moreira.d-c"Furtado, C."https://www.zbmath.org/authors/?q=ai:furtado.claudioSummary: In this work we study the behavior of massless fermions in a graphene wormhole and in the presence of an external magnetic field. The graphene wormhole is made from two sheets of graphene which play the roles of asymptotically flat spaces connected through a carbon nanotube with a zig-zag boundary. We solve the massless Dirac equation within this geometry, analyze the corresponding wave function, and show that the energy spectra of these solutions exhibit behavior similar to Landau levels.