Asymptotic Eigenfunctions for Schrödinger operators on a vector bundle

In the limit \(\hbar \to 0\), we analyze a class of Schrödinger operators \( H_\hbar = \hbar^2 L + \hbar W + V\cdot id_{\mathscr{E}}\) acting on sections of a vector bundle \(\mathscr{E}\) 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 \(H_\hbar\). These are obtained from eigenfunctions of the associated harmonic oscillator \(H_{p,\hbar}\) at \(p\), acting on smooth functions on the tangent space.