Weyl asymptotics for magnetic Schr\"odinger operators and de Gennes' boundary condition

TL;DR

This paper derives Weyl asymptotics for the discrete spectrum of a magnetic Schrödinger operator with de Gennes boundary conditions, extending understanding of eigenvalue distribution in semi-classical quantum systems.

Contribution

It provides a new asymptotic expansion for the eigenvalue count of magnetic Schrödinger operators with Robin boundary conditions, building on prior eigenvalue behavior results.

Findings

Derived Weyl-type asymptotics for eigenvalues below the essential spectrum.

Extended previous eigenvalue asymptotic results to magnetic Schrödinger operators.

Enhanced understanding of spectral properties under de Gennes boundary conditions.

Abstract

This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schr\"odinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an asymptotic expansion of the number of eigenvalues below the essential spectrum (Weyl-type asymptotics). The methods of proof relies on results concerning the asymptotic behavior of the first eigenvalue obtained in a previous work [A. Kachmar, J. Math. Phys. Vol. 47 (7) 072106 (2006)].