Energy of surface states for 3D magnetic Schrodinger operators

TL;DR

This paper derives a semi-classical formula for the sum of boundary-localized eigenvalues of 3D magnetic Schrödinger operators, linking them to model operators in simplified geometries.

Contribution

It introduces a new semi-classical formula for surface state energies of 3D magnetic Schrödinger operators with boundary conditions.

Findings

Established a semi-classical eigenvalue sum formula

Connected boundary eigenvalues to model operators in half-space

Provided a method to analyze surface states in 3D domains

Abstract

We establish a semi-classical formula for the sum of eigenvalues of a magnetic Schrodinger operator in a three-dimensional domain with compact smooth boundary and Neumann boundary conditions. The eigenvalues we consider have eigenfunctions localized near the boundary of the domain, hence they correspond to surface states. Using relevant coordinates that straighten out the boundary, the leading order term of the energy is described in terms of the eigenvalues of model operators in the half-axis and the half-plane.