Authors:David Baraglia Abstract: Reviews in Mathematical Physics, Volume 27, Issue 03, April 2015. We give a simplified definition of topological T-duality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for T-duals are shown to follow. We determine necessary and sufficient conditions for existence of a T-dual in the case of affine torus bundles. This is general enough to include all principal torus bundles as well as torus bundles with arbitrary monodromy representations. We show that isomorphisms in twisted cohomology, twisted K-theory and of Courant algebroids persist in this general setting. We also give an example where twisted K-theory groups can be computed by iterating T-duality. Citation: Reviews in Mathematical Physics PubDate: Tue, 05 May 2015 06:11:58 GMT DOI: 10.1142/S0129055X15500087

Authors:Wolfgang Bock, Maria João Oliveira, José Luís da Silva, Ludwig Streit Abstract: Reviews in Mathematical Physics, Ahead of Print. Through chaos decomposition, we improve the Varadhan estimate for the rate of convergence of the centered approximate self-intersection local time of planar Brownian motion. Citation: Reviews in Mathematical Physics PubDate: Mon, 27 Apr 2015 03:01:57 GMT DOI: 10.1142/S0129055X15500099

Authors:Karsten Leonhardt, Norbert Peyerimhoff, Martin Tautenhahn, Ivan Veselić Abstract: Reviews in Mathematical Physics, Ahead of Print. We study Schrödinger operators on L2(ℝd) and ℓ2(ℤd) with a random potential of alloy-type. The single-site potential is assumed to be exponentially decaying but not necessarily of fixed sign. In the continuum setting, we require a generalized step-function shape. Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. In the described situation, a Wegner estimate, which is polynomial in the volume of the box and linear in the size of the energy interval, holds. We apply the established Wegner estimate as an ingredient for a localization proof via multiscale analysis. Citation: Reviews in Mathematical Physics PubDate: Mon, 20 Apr 2015 07:09:38 GMT DOI: 10.1142/S0129055X15500075

Authors:Miguel Ballesteros, Ricardo Weder Abstract: Reviews in Mathematical Physics, Ahead of Print. We introduce a general class of long-range magnetic potentials and derive high velocity limits for the corresponding scattering operators in quantum mechanics, in the case of two dimensions. We analyze the high velocity limits that we obtain in the presence of an obstacle and we uniquely reconstruct from them the electric potential and the magnetic field outside the obstacle, that are accessible to the particles. We additionally reconstruct the inaccessible fluxes (magnetic fluxes produced by fields inside the obstacle) modulo 2π, which give a proof of the Aharonov–Bohm effect. For every magnetic potential A in our class, we prove that its behavior at infinity ($A_{\infty}(\hat{{\bf v}}), \hat{{\bf v}} \in {\mathbb S}^1$) can be characterized in a natural way; we call it the long-range part of the magnetic potential. Under very general assumptions, we prove that $A_{\infty}(\hat{{\bf v}}) + A_{\infty}(-\hat{{\bf v}})$ can be uniquely reconstructed for every $\hat{{\bf v}} \in {\mathbb S}^{1}$. We characterize properties of the support of the magnetic field outside the obstacle that permit us to uniquely reconstruct $A_{\infty}(\hat{{\bf v}})$ either for all $\hat{{\bf v}} \in {\mathbb S}^{1}$ or for $\hat{{\bf v}}$ in a subset of ð•Š1. We also give a wide class of magnetic fields outside the obstacle allowing us to uniquely reconstruct the total magnetic flux (and $A_{\infty}(\hat{{\bf v}})$ for all $\hat{{\bf v}} \in {\mathbb S}^{1}$). This is relevant because, as it is well-known, in general the scattering operator (even if is known for all velocities or energies) does not define uniquely the total magnetic flux (and $A_{\infty}(\hat{{\bf v}})$). We analyze additionally injectivity (i.e. uniqueness without giving a method for reconstruction) of the high velocity limits of the scattering operator with respect to $A_{\infty}(\hat{{\bf v}})$. Assuming that the magnetic field outside the obstacle is not identically zero, we provide a class of magnetic potentials for which injectivity is valid. Citation: Reviews in Mathematical Physics PubDate: Wed, 25 Mar 2015 03:12:09 GMT DOI: 10.1142/S0129055X15500063