Eigenvalue Asymptotics for the Schrödinger Operator with Steplike Magnetic Field and Slowly Decreasing Electric Potential

Abstract

In this paper we consider the two-dimensional Schrödinger operator of the form: where the magnetic field B(\_x\_1) = rot(0, b(\_x\_1)) is monotone increasing and steplike, namely the limits lim \_x\_1 →±∞ B(\_x\_1) = \_B\_± exist with 0 < \_B\_− < B+ < ∞, and V is the slowly power-decaying electric potential. The spectrum σ(\_H\_0) of the unperturbed operator H\_0 (= HV with V = 0) has the band structure and HV has the discrete spectrum in the gaps of the essential spectrum σ\_ess (HV) = σ(\_H\_0). The aim of this paper is to study the asymptotic distribution of the eigenvalues near the edges of the spectral gaps. Using the min-max argument, we prove that the classical Weyl-type asymptotic formula is satisfied under suitable assumptions on B and V.