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

Abstract

In this paper we consider the two-dimensional Schrödinger operatorof the form:
H_V=−\frac{∂²}{∂x₁²}+(\frac{1}{i}\frac{∂}{∂x₂}−b(x₁))²+V(x₁, x₂),
where the magnetic field B(x₁)=rot(0, b(x₁)) is monotone increasing and steplike, namely the limits lim_x1→±∞ B(x₁)=B_± exist with 0<B₋<B₊<∞, and V is the slowly power-decaying electric potential. The spectrum σ(H₀) of the unperturbed operator H₀ (=H_V with V=0) has the band structure and H_V has the discrete spectrum in the gaps of the essential spectrum σ_ess(H_V)=σ(H₀). 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.