Consider the Schrödinger operator $H\_\hbar = -\hbar^2\Delta + V$ on $M=\mathbb{R}^n$, where $V$ is a potential that behaves well in a certain sense ($C^\infty$, bounded from below, going to infinity for large $|x|$, ...).


In the case $V = x^2$, $n=1$, the eigenfunctions are $\psi\_n(x/\sqrt{\hbar})$ where the $\psi\_n$ are hermite functions, and the corresponding eigenvalues are $\hbar(2n+1)$.


My question is: Is it a well-known theorem, that in the semiclassical limit $\hbar \rightarrow 0$, the eigenvalues tend to the minimum or minima of $V$ and the corresponding eigenvectors behave asymptotically like delta peaks? Can you give me references?


What about if M is a (possibly compact) riemannian manifold and $\Delta$ the Laplace-Beltrami operator?



Eigenvalues in the semiclassical limit
The canonical reference is:


Introduction to spectral theory: with applications to Schrödinger operators by Hislop and Sigal.


Your statement about the semiclassical behavior of eigenvalues seems to be proved by Barry Simon in:


<http://archive.numdam.org/ARCHIVE/AIHPA/AIHPA_1983__38_3/AIHPA_1983__38_3_295_0/AIHPA_1983__38_3_295_0.pdf>


(where he calls it a "folk theorem").


