Magnetic WKB constructions on surfaces

This article is devoted to the description of the eigenvalues and eigenfunctions of the magnetic Laplacian in the semiclassical limit via the complex WKB method. Under the assumption that the magnetic field has a unique and non-degenerate minimum, we construct the local complex WKB approximations for eigenfunctions on a general surface. Furthermore, in the case of the Euclidean plane, with a radially symmetric magnetic field, the eigenfunctions are approximated in an exponentially weighted space.

On the Level Set of a Function with Degenerate Minimum Point

Forn≥2, letMbe ann-dimensional smooth closed manifold andf:M→Ra smooth function. We setminf(M)=mand assume thatmis attained by unique pointp∈Msuch thatpis a nondegenerate critical point. Then the Morse lemma tells us that ifais slightly bigger thanm,f-1(a)is diffeomorphic toSn-1. In this paper, we relax the condition onpfrom being nondegenerate to being an isolated critical point and obtain the same consequence. Some application to the topology of polygon spaces is also included.