Dirichlet and Neumann Eigenvalues for Half-Plane Magnetic Hamiltonians

TL;DR

This paper compares the eigenvalue distributions of magnetic Schrödinger operators with Dirichlet and Neumann boundary conditions on a half-plane, revealing how the discrete spectra behave near the essential spectrum's infimum.

Contribution

It introduces effective Hamiltonians to analyze the asymptotic distribution of eigenvalues and demonstrates differing spectral behaviors for Dirichlet and Neumann cases based on potential decay.

Findings

The Dirichlet operator has infinitely many eigenvalues below the essential spectrum.

The Neumann operator's discrete spectrum can be finite or infinite depending on potential decay.

Effective Hamiltonians accurately describe the asymptotic eigenvalue distribution.

Abstract

Let $H_{0, D}$ (resp., $H_{0,N}$) be the Schroedinger operator in constant magnetic field on the half-plane with Dirichlet (resp., Neumann) boundary conditions, and let $H_\ell : = H_{0, \ell} - V$, $\ell =D,N$, where the scalar potential $V$ is non negative, bounded, does not vanish identically, and decays at infinity. We compare the distribution of the eigenvalues of $H_D$ and $H_N$ below the respective infima of the essential spectra. To this end, we construct effective Hamiltonians which govern the asymptotic behaviour of the discrete spectrum of $H_\ell$ near $\inf \sigma_{ess}(H_\ell) = \inf \sigma(H_{0,\ell})$, $\ell = D,N$. Applying these Hamiltonians, we show that $\sigma_{disc}(H_D)$ is infinite even if $V$ has a compact support, while $\sigma_{disc}(H_N)$ could be finite or infinite depending on the decay rate of $V$.