On the energy of bound states for magnetic Schr\"odinger operators

TL;DR

This paper derives precise semiclassical asymptotics for the energy of bound states in magnetic Schr"odinger operators with Neumann boundary conditions, revealing boundary and bulk contributions near the spectrum's bottom.

Contribution

It introduces boundary coherent states and magnetic Lieb-Thirring estimates to analyze the energy asymptotics of magnetic Schr"odinger operators in two-dimensional domains.

Findings

Asymptotic formula valid up to the essential spectrum's bottom

Identification of boundary and bulk energy components

Development of boundary coherent states for analysis

Abstract

We provide a leading order semiclassical asymptotics of the energy of bound states for magnetic Neumann Schr\"odinger operators in two dimensional (exterior) domains with smooth boundaries. The asymptotics is valid all the way up to the bottom of the essential spectrum. When the spectral parameter is varied near the value where bound states become allowed in the interior of the domain, we show that the energy has a boundary and a bulk component. The estimates rely on coherent states, in particular on the construction of `boundary coherent states', and magnetic Lieb-Thirring estimates.