Magnetic perturbations of the Robin Laplacian in the strong coupling limit

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues of the Robin Laplacian under strong magnetic fields in planar domains, revealing how boundary curvature and magnetic effects influence spectral properties.

Contribution

It provides a comparison between the Robin Laplacian spectrum and an effective boundary operator, including explicit eigenvalue asymptotics for disc domains.

Findings

Eigenvalues approximated by an effective boundary operator.

Magnetic field does not affect the eigenvalue expansion when curvature has a maximum.

Explicit magnetic contribution in eigenvalues for disc domains.

Abstract

This paper is devoted to the asymptotic analysis of the eigenvalues of the Laplace operator with a strong magnetic field and Robin boundary condition on a smooth planar domain and with a negative boundary parameter. We study the singular limit when the Robin parameter tends to infinity which is equivalent to a semi-classical limit involving a small positive semi-classical parameter. The main result is a comparison between the spectrum of the Robin Laplacian with an effective operator defined on the boundary of the domain via the Born-Oppenheimer approximation. More precisely, the low-lying eigenvalue of the Robin Laplacian is approximated by the those of the effective operator. When the curvature has a unique non-degenerate maximum, we estimate the spectral gap and find that the magnetic field does not contribute to the three-term expansion of the eigenvalues. In the case of the disc domains, the eigenvalue asymptotics displays the contribution of the magnetic field explicitly.