Magnetic perturbations of the Robin Laplacian in the strong coupling limit

Abstract

This paper is devoted to the asymptotic analysis of the eigenvalues of the Laplace operator with a uniform magnetic field and Robin boundary condition on a smooth planar domain. We show how the singular limit when the Robin parameter tends to infinity is equivalent to a semi-classical limit involving a small positive parameter h (the 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 n-th eigenvalue of the Robin Laplacian is approximated, modulo O(h 2 ), by the n-th eigenvalue of the effective operator. When the curvature has a unique non-degenerate maximum, the eigenvalue asymptotics displays the contribution of the magnetic field explicitly.