Spectral asymptotics for magnetic Schrödinger operators in domains with corners

Abstract

This paper is on magnetic Schrodinger operators in two dimensional domains with corners. Semiclassical formulas are obtained for the sum and number of eigenvalues. The obtained results extend former formulas for smooth domains in (11, 10) to piecewise smooth domains. The spectral analysis of magnetic Schrodinger operators in domains with boundary has been the subject of many research papers in the last two decades. Apart from the mathematical interest behind the study of their spectra, magnetic Schrodinger operators in interior/exterior domains with various boundary conditions arise in several models of condensed matter physics, as superconductivity (14, 9, 15, 16), liquid crystals (12, 18) and Fermi gases (6). The present paper is devoted to the study of magnetic Schrodinger operators in domains with corners (piecewise smooth domains). The presence of corners in the domain has a strong effect on the spectrum of the operator. In particular, it is shown in (2, 13) that the presence of corners decreases the value of the ground state energy of the operator compared with the case of smooth domains. Discussion of this effect in the framework of superconductivity is given in (3). We give in a simple particular case, a brief presentation of the semiclassical results proved in this paper. Suppose for simplicity that is a simply connected bounded domain in R 2 and that A is a vector field such that b = curlA is constant. Let Ph, = −(h∇ − iA) 2 be the magnetic Laplacian in L 2 () with magnetic Neumann boundary condition, N(λh) and E(λh) be the number and sum of negative eigenvalues of Ph, − λId. If the boundary of is smooth, it is proved in (10, 11) that, as h → 0, N(λh) ∼ c1(b,λ)|∂|h −1/2 , E(λh) ∼ c2(b,λ)|∂|h 1/2 , (1.1)