Semi-classical trace asymptotics for magnetic Schrodinger operators with Robin condition

TL;DR

This paper derives semi-classical asymptotic formulas for the eigenvalues of magnetic Schrödinger operators with Robin boundary conditions, emphasizing boundary effects and using boundary reduction, coherent states, and Lieb-Thirring inequalities.

Contribution

It provides new semi-classical trace asymptotics for magnetic Schrödinger operators with Robin boundary conditions, accounting for boundary influence under weak regularity assumptions.

Findings

Derived explicit formulas for eigenvalue sums and counts near the boundary.

Showed boundary conditions significantly affect spectral asymptotics.

Validated formulas using Lieb-Thirring inequalities and boundary coherent states.

Abstract

We compute the sum and number of eigenvalues for a certain class of magnetic Schrodinger operators in a domain with boundary. Functions in the domain of the operator satisfy a (magnetic) Robin condition. The calculations are valid in the semi-classical asymptotic limit and the eigenvalues concerned correspond to eigenstates localized near the boundary of the domain. The formulas we derive display the influence of the boundary and the boundary condition and are valid under a weak regularity assumption of the boundary function. Our approach relies on three main points: reduction to the boundary; construction of boundary coherent states; handling the boundary term as a surface electric potential and controlling the errors by various Lieb-Thirring inequalities.