Asymptotic eigenfunctions for Schrödinger operators on a vector bundle

Abstract

In the limit $\hslash \to 0$, we analyze a class of Schrödinger operators $H_{\hslash }={\hslash }^{2}L+\hslash W+V\cdot \mathrm{id}$ acting on sections of a vector bundle $\mathcal{E}h$ over a Riemannian manifold $M$ where $L$ is a Laplace type operator, $W$ is an endomorphism field and the potential energy $V$ has a non-degenerate minimum at some point $p\in M$. We construct quasimodes of WKB-type near $p$ for eigenfunctions associated with the low lying eigenvalues of $H_{\hslash }$. These are obtained from eigenfunctions of the associated harmonic oscillator $H_{p,\hslash }$ at $p$, acting on smooth functions on the tangent space.

29 pages, minor mistakes corrected

References in corpus (2)

• Asymptotic Eigenfunctions for a class of Difference Operators
• Vector Fields with a non-degenerate Source