Asymptotics for the Low-Lying Eigenstates of the Schrödinger Operator with Magnetic Field near Corners

Abstract

The Neumann realization for the Schrodinger operator with magnetic field is considered in a bounded two-dimensional domain with corners. This operator is associated with a small semi-classical parameter h or, equivalently, with a large magnetic field.We investigate the behavior of its eigenpairs as h tends to zero, like in a semi-classical limit. We prove, in the situation where the domain is a polygon and the magnetic field is constant, that the lowest eigenvalues are exponentially close to those of model problems associated with the corners. We approximate the corresponding eigenvectors by linear combinations of functions concentrated in corners at the scale $$\sqrt h .$$ If the domain has curved sides and the magnetic field is smoothly varying, we exhibit a full asymptotics for eigenpairs in powers of $$\sqrt h .$$ Communicated by Christian Gerard