Weyl series for Aharonov-Bohm billiards (Q2766278)

The author derives and discusses asymptotic expansion of the resolvent for two- and three-dimensional quantum billiards of regular shape (circular or spherical) with a flux line at the center. Quantum billiard systems describe a point particle of mass \(m\) and charge \(q\) confined to a flat region of \(\mathbb{R}^n\), \(n=2,3\). If, in addition, the particle is influenced by a magnetic field \(B=\nabla\times A\) the quantum eigenfunctions \(\psi\) and eigenenergies satisfy the Schrödinger equation \(\frac{1}{2m}(-i\hbar\nabla-q A)^2\psi=E\psi\) with Dirichlet boundary conditions. In the case considered in the paper the magnetic field is of the form of a single flux line: \(B=2\pi\delta(r-r_0)\). The main results of the paper consist in obtaining expressions for the coefficients of the asymptotic expansions of the resolvent of the Schrödinger equations for large values of the argument of the resolvent and exhibiting their dependence on the magnetic flux. It is shown that for a circular plane billiard only one coefficient of expansion is changed by the magnetic flux whereas in the three-dimensional case (spherical billiard) all coefficients are influenced by the flux.