Authors:Gregory Eskin Abstract: Reviews in Mathematical Physics, Volume 27, Issue 02, March 2015. Aharonov–Bohm effect is a quantum mechanical phenomenon that attracted the attention of many physicists and mathematicians since the publication of the seminal paper of Aharonov and Bohm [1] in 1959. We consider different types of Aharonov–Bohm effects such as the magnetic AB effect, electric AB effect, combined electromagnetic AB effect, AB effect for the Schrödinger equations with Yang–Mills potentials, and the gravitational analog of AB effect. We shall describe different approaches to prove the AB effect based on the inverse scattering problems, the inverse boundary value problems in the presence of obstacles, spectral asymptotics, and the direct proofs of the AB effect. Citation: Reviews in Mathematical Physics PubDate: Tue, 14 Apr 2015 02:12:22 GMT DOI: 10.1142/S0129055X15300010

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

Authors:D. Wellig Abstract: Reviews in Mathematical Physics, Ahead of Print. In this paper, estimates on the ground state energy of Fröhlich N-polarons in electromagnetic fields in the strong coupling limit, α → ∞, are derived. It is shown that the ground state energy is given by α2 multiplied by the minimal energy of the corresponding Pekar–Tomasevich functional for N particles, up to an error term of order α42/23N3. The potentials A, V are suitably rescaled in α. As a corollary, binding of N-polarons for strong magnetic fields for large coupling constants is established. Citation: Reviews in Mathematical Physics PubDate: Mon, 16 Mar 2015 09:22:53 GMT DOI: 10.1142/S0129055X15500051

Authors:Andrey Mudrov Abstract: Reviews in Mathematical Physics, Ahead of Print. Let U be either the classical or quantized universal enveloping algebra of the Lie algebra $\mathfrak{sl}(n + 1)$ extended over the field of fractions of the Cartan subalgebra. We suggest a PBW basis in U over the extended Cartan subalgebra diagonalizing the contravariant Shapovalov form on generic Verma module. The matrix coefficients of the form are calculated and the inverse form is explicitly constructed. Citation: Reviews in Mathematical Physics PubDate: Fri, 06 Mar 2015 01:42:31 GMT DOI: 10.1142/S0129055X1550004X