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arXiv · 1309.4178

Asymptotic eigenfunctions for Schr\"odinger operators on a vector bundle

Matthias Ludewig·Elke Rosenberger
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Abstract

In the limit $\hbar\to 0$, we analyze a class of Schr\"odinger operators $H_\hbar = \hbar^2 L + \hbar W + V\cdot \mathrm{id}$ acting on sections of a vector bundle $\mathcal{Eh}$ 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.

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Matthias Ludewig, Elke Rosenberger. 2013-09-17. Asymptotic eigenfunctions for Schr\"odinger operators on a vector bundle. https://doi.org/10.1142/s0129055x20500208

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