Eigenvalue Asymptotics for a Schr\"odinger Operator with Non-Constant Magnetic Field Along One Direction

TL;DR

This paper analyzes the asymptotic distribution of eigenvalues for a 2D Schrödinger operator with a variable magnetic field and decaying potential, revealing semiclassical and non-semiclassical behaviors.

Contribution

It introduces effective Hamiltonians to describe eigenvalue asymptotics for operators with non-constant magnetic fields depending on one variable.

Findings

Eigenvalue counting function asymptotics derived for power-like decaying potentials.

Semiclassical eigenvalue behavior identified for certain potential decay rates.

Criterion established for finiteness of eigenvalues in spectral gaps.

Abstract

We consider the discrete spectrum of the two-dimensional Hamiltonian $H=H_0+V$, where $H_0$ is a Schr\"odinger operator with a non-constant magnetic field $B$ that depends only on one of the spatial variables, and $V$ is an electric potential that decays at infinity. We study the accumulation rate of the eigenvalues of H in the gaps of its essential spectrum. First, under some general conditions on $B$ and $V$, we introduce effective Hamiltonians that govern the main asymptotic term of the eigenvalue counting function. Further, we use the effective Hamiltonians to find the asymptotic behavior of the eigenvalues in the case where the potential V is a power-like decaying function and in the case where it is a compactly supported function, showing a semiclassical behavior of the eigenvalues in the first case and a non-semiclassical behavior in the second one. We also provide a criterion for the finiteness of the number of eigenvalues in the gaps of the essential spectrum of $H$