
Reviews in Mathematical Physics [SJR: 1.092] [HI: 36] [0 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 0129055X  ISSN (Online) 17936659 Published by World Scientific [118 journals] 
 Superradiance initiated inside the ergoregion
 Authors: Gregory Eskin
Abstract: Reviews in Mathematical Physics, Volume 28, Issue 10, November 2016.
We consider the stationary metrics that have both the black hole and the ergoregion. The class of such metric contains, in particular, the Kerr metric. We study the Cauchy problem with highly oscillatory initial data supported in a neighborhood inside the ergoregion with some initial energy [math]. We prove that when the time variable [math] increases this solution splits into two parts: one with the negative energy [math] ending at the event horizon in a finite time, and the second part, with the energy [math], escaping, under some conditions, to the infinity when [math]. Thus we get the superradiance phenomenon. In the case of the Kerr metric the superradiance phenomenon is “shortlived”, since both the solutions with positive and negative energies cross the outer event horizon in a finite time (modulo [math]) where [math] is a large parameter. We show that these solutions end on the singularity ring in a finite time. We study also the case of naked singularity.
Citation: Reviews in Mathematical Physics
PubDate: 20161115T08:26:10Z
DOI: 10.1142/S0129055X16500252
 Authors: Gregory Eskin
 The Wasserstein geometry of nonlinear [math] models and the
Hamilton–Perelman Ricci flow Authors: Mauro Carfora
Abstract: Reviews in Mathematical Physics, Ahead of Print.
Nonlinear sigma models are quantum field theories describing, in the large deviation sense, random fluctuations of harmonic maps between a Riemann surface and a Riemannian manifold. Via their formal renormalization group analysis, they provide a framework for possible generalizations of the Hamilton–Perelman Ricci flow. By exploiting the heat kernel embedding introduced by Gigli and Mantegazza, we show that the Wasserstein geometry of the space of probability measures over Riemannian metric measure spaces provides a natural setting for discussing the relation between nonlinear sigma models and Ricci flow theory. In particular, we analyze the embedding of Ricci flow into a heat kernel renormalization group flow for dilatonic nonlinear sigma models, and characterize a nontrivial generalization of the Hamilton–Perelman version of the Ricci flow. We discuss in detail the monotonicity and gradient flow properties of this extended flow.
Citation: Reviews in Mathematical Physics
PubDate: 20161117T01:50:35Z
DOI: 10.1142/S0129055X17500015
 Authors: Mauro Carfora
 Generalized Connes–Chern characters in [math]theory with an application
to weak invariants of topological insulators Authors: Emil Prodan, Hermann SchulzBaldes
Abstract: Reviews in Mathematical Physics, Ahead of Print.
We use constructive bounded Kasparov [math]theory to investigate the numerical invariants stemming from the internal Kasparov products [math], [math], where the last morphism is provided by a tracial state. For the class of properly defined finitelysummable Kasparov [math]cycles, the invariants are given by the pairing of [math]theory of [math] with an element of the periodic cyclic cohomology of [math], which we call the generalized Connes–Chern character. When [math] is a twisted crossed product of [math] by [math], [math], we derive a local formula for the character corresponding to the fundamental class of a properly defined Dirac cycle. Furthermore, when [math], with [math] the algebra of continuous functions over a disorder configuration space, we show that the numerical invariants are connected to the weak topological invariants of the complex classes of topological insulators, defined in the physics literature. The end products are generalized index theorems for these weak invariants, which enable us to predict the range of the invariants and to identify regimes of strong disorder in which the invariants remain stable. The latter will be reported in a subsequent publication.
Citation: Reviews in Mathematical Physics
PubDate: 20161102T10:34:41Z
DOI: 10.1142/S0129055X16500240
 Authors: Emil Prodan, Hermann SchulzBaldes
 Trace formulas for a class of nonFredholm operators: A review
 Authors: Alan Carey, Fritz Gesztesy, Harald Grosse, Galina Levitina, Denis Potapov, Fedor Sukochev, Dmitriy Zanin
Abstract: Reviews in Mathematical Physics, Ahead of Print.
Take a oneparameter family of selfadjoint Fredholm operators [math] on a Hilbert space [math], joining endpoints [math]. There is a long history of work on the question of whether the spectral flow along this path is given by the index of the operator [math] acting in [math], where [math] denotes the multiplication operator [math] for [math]. Most results are about the case where the operators [math] have compact resolvent. In this article, we review what is known when these operators have some essential spectrum and describe some new results. Using the operators [math], [math], an abstract trace formula for Fredholm operators with essential spectrum was proved in [23], extending a result of Pushnitski [35], although, still under strong hypotheses on [math]: trL2(ℝ;ℋ)((H2 − zI)−1 − (H 1 − zI)−1) = 1 2ztrL2(ℋ)(gz(A+) − gz(A−)), where [math], [math], [math]. Associated to the pairs [math] and [math] are Krein spectral shift functions [math] and [math], respectively. From the trace formula, it was shown that there is a second, Pushnitskitype, formula: ξ(λ; H2,H1) = 1 π∫−λ1/2λ1/2 ξ(ν; A+,A−)dν (λ − ν2)1/2 for a.e. λ > 0. This can be employed to establish the desired equality, Fredholm index = ξ(0; A+,A−) = spectral flow. This equality was generalized to nonFredholm operators in [14] in the form Witten index = [ξR(0; A+,A−) + ξL(0; A+,A−)]/2, replacing the Fredholm index on the lefthand side by the Witten index of [math] and [math] on the righthand side by an appropriate arithmetic mean (assuming [math] is a right and left Lebesgue point for [math] denoted by [math] and [math], respectively). But this applies only under the restrictive assumption that the endpoint [math] is a relatively trace class perturbation of [math] (ruling out general differential operators). In addition to reviewing this previous work, we describe in this article some extensions using a [math]dimensional setup, where [math] are nonFredholm differential operators. By a careful analysis we prove, for a class of examples, that the preceding trace formula still holds in this more general situation. Then we prove that the Pushnitskitype formula for spectral shift functions also holds and this then gives the equality of spectral shift functions in the form ξ(λ; H2,H1) = ξ(ν; A+,A−)for a.e. λ > 0 and a.e. ν ∈ ℝ, for the [math]dimensional model operator at hand. This shows that neither the relatively trace class perturbation assumption nor the Fredholm assumption are required if one works with spectral shift functions. The results support the view that the spectral shift function should be a replacement for the spectral flow in certain nonFredholm situations and also point the way to the study of higherdimensional cases. We discuss the connection with summability questions in Fredholm modules in an appendix.
Citation: Reviews in Mathematical Physics
PubDate: 20161025T03:38:31Z
DOI: 10.1142/S0129055X16300028
 Authors: Alan Carey, Fritz Gesztesy, Harald Grosse, Galina Levitina, Denis Potapov, Fedor Sukochev, Dmitriy Zanin
 Notes on topological insulators
 Authors: Ralph M. Kaufmann, Dan Li, Birgit WehefritzKaufmann
Abstract: Reviews in Mathematical Physics, Ahead of Print.
This paper is a survey of the [math]valued invariant of topological insulators used in condensed matter physics. The [math]valued topological invariant, which was originally called the TKNN invariant in physics, has now been fully understood as the first Chern number. The [math] invariant is more mysterious; we will explain its equivalent descriptions from different points of view and provide the relations between them. These invariants provide the classification of topological insulators with different symmetries in which Ktheory plays an important role. Moreover, we establish that both invariants are realizations of index theorems which can also be understood in terms of condensed matter physics.
Citation: Reviews in Mathematical Physics
PubDate: 20161025T03:38:29Z
DOI: 10.1142/S0129055X1630003X
 Authors: Ralph M. Kaufmann, Dan Li, Birgit WehefritzKaufmann
 He–McKellar–Wilkenstype effect, quantum holonomies and
Aharonov–Bohmtype effect for bound states from the Lorentz symmetry
breaking effects Authors: A. G. de Lima, H. Belich, K. Bakke
Abstract: Reviews in Mathematical Physics, Ahead of Print.
From the effects of the Lorentz symmetry violation in the CPTeven gauge sector of the Standard Model Extension determined by a tensor background [math], we establish a possible scenario where an analogue of the He–McKellar–Wilkens effect can stem from. Besides, we build quantum holonomies associated with the analogue of the He–McKellar–Wilkens effect and discuss a possible analogy with the geometric quantum computation. Finally, we investigate the dependence of the energy levels on the He–McKellar–Wilkens geometric phase induced by Lorentz symmetry breaking effects when the particle is confined to a hardwall confining potential.
Citation: Reviews in Mathematical Physics
PubDate: 20161007T03:58:38Z
DOI: 10.1142/S0129055X16500239
 Authors: A. G. de Lima, H. Belich, K. Bakke
 He–McKellar–Wilkenstype effect, quantum holonomies and
Aharonov–Bohmtype effect for bound states from the Lorentz symmetry
breaking effects Authors: A. G. de Lima, H. Belich, K. Bakke
Abstract: Reviews in Mathematical Physics, Ahead of Print.
From the effects of the Lorentz symmetry violation in the CPTeven gauge sector of the Standard Model Extension determined by a tensor background [math], we establish a possible scenario where an analogue of the He–McKellar–Wilkens effect can stem from. Besides, we build quantum holonomies associated with the analogue of the He–McKellar–Wilkens effect and discuss a possible analogy with the geometric quantum computation. Finally, we investigate the dependence of the energy levels on the He–McKellar–Wilkens geometric phase induced by Lorentz symmetry breaking effects when the particle is confined to a hardwall confining potential.
Citation: Reviews in Mathematical Physics
PubDate: 20161007T03:58:38Z
DOI: 10.1142/S0129055X16500239
 Authors: A. G. de Lima, H. Belich, K. Bakke
 Stability of gas measures under perturbations and discretizations
 Authors: Roberto Fernández, Pablo Groisman, Santiago Saglietti
Abstract: Reviews in Mathematical Physics, Ahead of Print.
For a general class of gas models — which includes discrete and continuous Gibbsian models as well as contour or polymer ensembles — we determine a diluteness condition that implies: (1) uniqueness of the infinitevolume equilibrium measure; (2) stability of this measure under perturbations of parameters and discretization schemes, and (3) existence of a coupled perfectsimulation scheme for the infinitevolume measure together with its perturbations and discretizations. Some of these results have previously been obtained through methods based on cluster expansions. In contrast, our treatment is purely probabilistic and its diluteness condition is weaker than existing convergence conditions for cluster expansions.
Citation: Reviews in Mathematical Physics
PubDate: 20161004T09:28:48Z
DOI: 10.1142/S0129055X16500227
 Authors: Roberto Fernández, Pablo Groisman, Santiago Saglietti