Asymptotic eigenfunctions for Schrödinger operators on a vector bundle

Abstract

In the limit [Formula: see text], we analyze a class of Schrödinger operators [Formula: see text] acting on sections of a vector bundle [Formula: see text] over a Riemannian manifold [Formula: see text] where [Formula: see text] is a Laplace type operator, [Formula: see text] is an endomorphism field and the potential energy [Formula: see text] has a non-degenerate minimum at some point [Formula: see text]. We construct quasimodes of WKB-type near [Formula: see text] for eigenfunctions associated with the low-lying eigenvalues of [Formula: see text]. These are obtained from eigenfunctions of the associated harmonic oscillator [Formula: see text] at [Formula: see text], acting on smooth functions on the tangent space.