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Authors:Matteo Gallone, Alessandro Michelangeli, Eugenio Pozzoli Abstract: Reviews in Mathematical Physics, Ahead of Print. We classify the self-adjoint realizations of the Laplace–Beltrami operator minimally defined on an infinite cylinder equipped with an incomplete Riemannian metric of Grushin type, in the class of metrics yielding an infinite deficiency index. Such realizations are naturally interpreted as Hamiltonians governing the geometric confinement of a Schrödinger quantum particle away from the singularity, or the dynamical transmission across the singularity. In particular, we characterize all physically meaningful extensions qualified by explicit local boundary conditions at the singularity. Within our general classification we retrieve those distinguished extensions previously identified in the recent literature, namely the most confining and the most transmitting one. Citation: Reviews in Mathematical Physics PubDate: 2022-04-30T07:00:00Z DOI: 10.1142/S0129055X22500180

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Authors:Fei Han, Varghese Mathai Abstract: Reviews in Mathematical Physics, Ahead of Print. This paper is a step towards realizing T-duality and Hori formulae for loop spaces. Here, we prove T-duality and Hori formulae for winding [math]-loop spaces, which are infinite dimensional subspaces of loop spaces. Citation: Reviews in Mathematical Physics PubDate: 2022-04-18T07:00:00Z DOI: 10.1142/S0129055X22500192

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Authors:Wojciech Dybalski, Alessandro Pizzo Abstract: Reviews in Mathematical Physics, Ahead of Print. We consider the massless Nelson model with two types of massive particles which we call atoms and electrons. The atoms interact with photons via an infrared regular form-factor and thus they are Wigner-type particles with sharp mass-shells. The electrons have an infrared singular form-factor and thus they are infraparticles accompanied by soft-photon clouds correlated with their velocities. In the weak coupling regime, we construct scattering states of one atom and one electron, and demonstrate their asymptotic clustering into individual particles. The proof relies on the Cook’s argument, clustering estimates, and the non-stationary phase method. The latter technique requires sharp estimates on derivatives of the ground state wave functions of the fiber Hamiltonians of the model, which were proven in the earlier papers of this series. Although we rely on earlier studies of the atom–atom and electron–photon scattering in the Nelson model, the paper is written in a self-contained manner. A perspective on the open problem of the electron–electron scattering in this model is also given. Citation: Reviews in Mathematical Physics PubDate: 2022-04-08T07:00:00Z DOI: 10.1142/S0129055X22500143

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Authors:Domenico Fiorenza, Hisham Sati, Urs Schreiber Abstract: Reviews in Mathematical Physics, Ahead of Print. We characterize the integral cohomology and the rational homotopy type of the maximal Borel-equivariantization of the combined Hopf/twistor fibration, and find that subtle relations satisfied by the cohomology generators are just those that govern Hořava–Witten’s proposal for the extension of the Green–Schwarz mechanism from heterotic string theory to heterotic M-theory. We discuss how this squares with the Hypothesis H that the elusive mathematical foundation of M-theory is based on charge quantization in tangentially twisted unstable Cohomotopy theory. Citation: Reviews in Mathematical Physics PubDate: 2022-03-31T07:00:00Z DOI: 10.1142/S0129055X22500131

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Authors:Claudio Cacciapuoti, Davide Fermi, Andrea Posilicano Abstract: Reviews in Mathematical Physics, Ahead of Print. We consider the quantum evolution [math] of a Gaussian coherent state [math] localized close to the classical state [math], where [math] denotes a self-adjoint realization of the formal Hamiltonian [math], with [math] the derivative of Dirac’s delta distribution at [math] and [math] a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (with respect to the [math]-norm, uniformly for any [math] away from the collision time) by [math], where [math], [math] and [math] is a suitable self-adjoint extension of the restriction to [math], [math], of ([math] times) the generator of the free classical dynamics. While the operator [math] here utilized is similar to the one appearing in our previous work [C. Cacciapuoti, D. Fermi and A. Posilicano, The semi-classical limit with a delta potential, Ann. Mat. Pura Appl. 200 (2021) 453–489], in the present case the approximation gives a smaller error: it is of order [math], [math], whereas it turns out to be of order [math], [math], for the delta potential. We also provide similar approximation results for both the wave and scattering operators. Citation: Reviews in Mathematical Physics PubDate: 2022-03-31T07:00:00Z DOI: 10.1142/S0129055X22500155

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Authors:Masaya Maeda, Akito Suzuki, Kazuyuki Wada Abstract: Reviews in Mathematical Physics, Ahead of Print. This paper is a continuation of work by the third author, which studied quantum walks with special long-range perturbations of the coin operator. In this paper, we consider general long-range perturbations of the coin operator and prove the non-existence of a singular continuous spectrum and embedded eigenvalues. The proof relies on the gauge transformation and construction of generalized eigenfunctions (Jost solutions) which was studied in the short-range case. Citation: Reviews in Mathematical Physics PubDate: 2022-03-31T07:00:00Z DOI: 10.1142/S0129055X22500167

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Authors:L. Amour, J. Nourrigat Abstract: Reviews in Mathematical Physics, Ahead of Print. The purpose of this paper is to give different interpretations of the first non-vanishing term (quadratic) of the ground state asymptotic expansion for a spin system in quantum electrodynamics, as the spin magnetic moments go to [math]. One of the interpretations makes a direct link with some classical physics laws. A central role is played by an operator [math] acting only in the finite-dimensional spin state space and making the connections with the different interpretations, and also being in close relation with the multiplicity of the ground state. Citation: Reviews in Mathematical Physics PubDate: 2022-03-29T07:00:00Z DOI: 10.1142/S0129055X22500179

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Authors:Alan Carey, Galina Levitina, Denis Potapov, Fedor Sukochev Abstract: Reviews in Mathematical Physics, Ahead of Print. In [Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99] Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd-dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin–Salamon [The spectral flow and the Maslov index, Bull. London Math. Soc. 27(1) (1995) 1–33, MR 1331677]. In [F. Gesztesy, Y. Latushkin, K. Makarov, F. Sukochev and Y. Tomilov, The index formula and the spectral shift function for relatively trace class perturbations, Adv. Math. 227(1) (2011) 319–420, MR 2782197], a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper [A. Pushnitski, The spectral flow, the Fredholm index, and the spectral shift function, in Spectral Theory of Differential Operators, American Mathematical Society Translations: Series 2, Vol. 225 (American Mathematical Society, Providence, RI, 2008), pp. 141–155, MR 2509781]. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and its perturbation by a relatively trace-class operator. In this paper, we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher [math]th Schatten class condition for [math], thus allowing differential operators on manifolds of any dimension [math]. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by [M.-T. Benameur, A. Carey, J. Phillips, A. Rennie, F. Sukochev and K. Wojciechowski, An analytic approach to spectral flow in von Neumann algebras, in Analysis, Geometry and Topology of Elliptic Operators (World Scientific Publisher, Hackensack, NJ, 2006), pp. 297–352, MR 2246773; A. Carey, H. Grosse and J. Kaad, On a spectral flow formula for the homological index, Adv. Math. 289 (2016) 1106–1156, MR 3439708]. We illustrate our results using Dirac type operators on [math] for arbitrary [math] (see Sec. 8). In this setting Theorem 6.4 substantially extends Theorem 3.5 of [A. Carey, F. Gesztesy, H. Grosse, G. Levitina, D. Potapov, F. Sukochev and D. Zanin, Trace formulas for a class of non-Fredholm operators: a review, Rev. Math. Phys. 28(10) (2016) 1630002, MR 3572626], where the case [math] was treated. Citation: Reviews in Mathematical Physics PubDate: 2022-02-17T08:00:00Z DOI: 10.1142/S0129055X22500118

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Authors:Manseob Lee, Jumi Oh, Junmi Park Abstract: Reviews in Mathematical Physics, Ahead of Print. Expansiveness has been used to study dynamic systems and has been developed for various forms of expansiveness. In this paper, we introduce the concept of kinematic [math]-expansiveness for flows on a [math] compact connected manifold [math], which is an extension of [math]-expansive homeomorphisms. We prove that if a vector field [math] on [math] is [math] robustly kinematic [math]-expansive, then it is quasi-Anosov. Furthermore, we consider the divergence-free vector fields and Hamiltonian systems with the kinematic [math]-expansive property; then, we study their robustness. Citation: Reviews in Mathematical Physics PubDate: 2022-02-17T08:00:00Z DOI: 10.1142/S0129055X2250012X

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Authors:Christian Gérard, Théo Stoskopf Abstract: Reviews in Mathematical Physics, Ahead of Print. We consider Dirac equations on even-dimensional Lorentzian manifolds of bounded geometry with a spin structure. For the associated free quantum field theory, we construct pure Hadamard states using global pseudodifferential calculus on a Cauchy surface. We also give two constructions of Hadamard states for Dirac fields for arbitrary spacetimes with a spin structure. Citation: Reviews in Mathematical Physics PubDate: 2022-01-31T08:00:00Z DOI: 10.1142/S0129055X22500088

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Authors:Xianguo Geng, Jiao Wei Abstract: Reviews in Mathematical Physics, Ahead of Print. The Itoh–Narita–Bogoyavlensky lattice hierarchy associated with a discrete [math] matrix spectral problem is derived by using Lenard recursion equations. Resorting to the characteristic polynomial of Lax matrix for the lattice hierarchy, we introduce a three-sheeted Riemann surface [math] of arithmetic genus [math] and construct the corresponding Baker–Akhiezer function and meromorphic function on it. On the basis of the theory of Riemann surface, the continuous flow and discrete flow related to the lattice hierarchy are straightened with the help of the Abel map. Quasi-periodic solutions of the lattice hierarchy in terms of the Riemann theta function are constructed by using the asymptotic properties and the algebro-geometric characters of the meromorphic function and Riemann surface. Citation: Reviews in Mathematical Physics PubDate: 2022-01-22T08:00:00Z DOI: 10.1142/S0129055X2250009X

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Authors:S. Richard, N. Tsuzu Abstract: Reviews in Mathematical Physics, Ahead of Print. In this paper, we investigate the spectral and scattering theory for operators acting on topological crystals and on their perturbations. Special attention is paid to perturbations obtained by the addition of an infinite number of edges, and/or by the removal of a finite number of them, but perturbations of the underlying measures and perturbations by the addition of a multiplication operator are also considered. The description of the nature of the spectrum of the resulting operators and the existence and completeness of the wave operators are standard outcomes for these investigations. Citation: Reviews in Mathematical Physics PubDate: 2022-01-22T08:00:00Z DOI: 10.1142/S0129055X22500106

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Authors:Konstantin Wernli Abstract: Reviews in Mathematical Physics, Ahead of Print. We give a detailed introduction to the classical Chern–Simons gauge theory, including the mathematical preliminaries. We then explain the perturbative quantization of gauge theories via the Batalin–Vilkovisky (BV) formalism. We then define the perturbative Chern–Simons partition function at any (possibly non-acylic) reference flat connection using the BV formalism, using a Riemannian metric for gauge fixing. We show that it exhibits an anomaly known as the “framing anomaly” when the Riemannian metric is changed, that is, it fails to be gauge invariant. We explain how one can deal with this anomaly to obtain a topological invariant of framed manifolds. Citation: Reviews in Mathematical Physics PubDate: 2022-01-21T08:00:00Z DOI: 10.1142/S0129055X22300035

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Authors:Martin Gebert, Alvin Moon, Bruno Nachtergaele Abstract: Reviews in Mathematical Physics, Ahead of Print. We consider a quantum spin chain with nearest neighbor interactions and sparsely distributed on-site impurities. We prove commutator bounds for its Heisenberg dynamics which incorporate the coupling strengths of the impurities. The impurities are assumed to satisfy a minimum spacing, and each impurity has a non-degenerate spectrum. Our results are proven in a broadly applicable setting, both in finite volume and in thermodynamic limit. We apply our results to improve Lieb–Robinson bounds for the Heisenberg spin chain with a random, sparse transverse field drawn from a heavy-tailed distribution. Citation: Reviews in Mathematical Physics PubDate: 2022-01-15T08:00:00Z DOI: 10.1142/S0129055X22500076

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Authors:Aldo Procacci Abstract: Reviews in Mathematical Physics, Ahead of Print. In this paper, we revisit the recent and less recent developments concerning rigorous results on the Mayer series and the virial series of a continuous system of classical particles interacting via a stable and tempered pair potential and we provide new lower bounds for the convergence radius of the virial series when the potential has a strictly positive stability constant. As an application we obtain a new estimate for the convergence radius of the virial series of the Lennard-Jones gas which improves sensibly previous estimates available in the literature. Citation: Reviews in Mathematical Physics PubDate: 2021-12-23T08:00:00Z DOI: 10.1142/S0129055X22500064

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Authors:Nuno Costa Dias, Maurice de Gosson, João Nuno Prata Abstract: Reviews in Mathematical Physics, Ahead of Print. The notion of partial trace of a density operator is essential for the understanding of the entanglement and separability properties of quantum states. In this paper, we investigate these notions putting an emphasis on the geometrical properties of the covariance ellipsoids of the reduced states. We thereafter focus on Gaussian states and we give new and easily numerically implementable sufficient conditions for the separability of all Gaussian states. Unlike the positive partial transposition criterion, none of these conditions is however necessary. Citation: Reviews in Mathematical Physics PubDate: 2021-11-09T08:00:00Z DOI: 10.1142/S0129055X22500052

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Authors:A. Zuevsky Abstract: Reviews in Mathematical Physics, Ahead of Print. A way to characterize the space of leaves of a foliation in terms of connections is proposed. A particular example of vertex algebra cohomology of codimension one foliations on complex curves is considered. Multiple applications in Mathematical Physics are revealed. Citation: Reviews in Mathematical Physics PubDate: 2021-11-06T07:00:00Z DOI: 10.1142/S0129055X22300023