The heat kernel for two Aharonov–Bohm solenoids in a uniform magnetic field

Abstract

A non-relativistic quantum model is considered with a point particle carrying a charge e and moving on the plane pierced by two infinitesimally thin AharonovBohm solenoids and subjected to a perpendicular uniform magnetic field of magnitude B. Relying on a technique due to Schulman and Sunada which is applicable to Schrödinger operators on multiply connected configuration manifolds a formula is derived for the corresponding heat kernel. As an application of the heat kernel formula, an approximate asymptotic expressions are derived for the lowest eigenvalue lying above the first Landau level and for the corresponding eigenfunction while assuming that |eB|R 2 /( c) is large where R is the distance between the two solenoids.

Improved time-decay for a class of scaling critical electromagnetic Schrödinger flows

“…The spectrum of the Hamiltonian H α,B 0 consists of pure discrete eigenvalues and the corresponding (L 2 -normalized) eigenfunctions form a complete orthonormal basis for L 2 (R 2 ) (see e.g. [27]).…”