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 $B(x_1)=\mathrm{rot}(0,b(x_1))$ is monotone increasing and steplike, namely the limits ${\lim }_{x_1\to \pm \infty }B(x_1)=B_{\pm }$ exist with $0<B_{-}<B_{+}<\infty$, and $V$ is the slowly power-decaying electric potential. The spectrum $\sigma (H_0)$ of the unperturbed operator $H_0$ ($=H_V$ with $V=0$) has the band structure and $H_V$ has the discrete spectrum in the gaps of the essential spectrum ${\sigma }_{\text{ess}}(H_V)=\sigma (H_0)$. 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 $V$.