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?