Tunnel effect in a shrinking shell enlacing a magnetic field

Abstract

Let $C$ be a smooth planar curve. We assume that $C$ is simple, closed, smooth, symmetric with respect to an axis and its curvature attains its minimum at exactly two points away from the axis of symmetry. In a tubular neighborhood about $C$, we study the Laplace operator with a magnetic flux and mixed boundary conditions. As the thickness of the domain tends to 0, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues (tunneling effect).