Eigenvalues in the semiclassical limit

Question

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?

Answer 1

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").

Answer 2

Another reference is Helffer and several Coauthors (Sjöstrand, Nier, Klein, Garyard, ...) using the Witten-Laplace approach.

An introduction is given in the book Semiclassical analysis, Witten Laplacians, and statistical mechanics

Later sharp asymptotics for the low lying spectra in the case where $V$ consists of several minima were obtained. Some lecture note on this topic Low lying eigenvalues of Witten Laplacians and metastability (after Helffer-Klein-Nier and Helffer-Nier).

If you are only interessted in the Schrödinger Operator, maybe the book Semi-Classical Analysis for the Schrödinger Operator and Applications is the most interesting for you. There are also lecture notes available Semiclassical Analysis for Schrödinger Operators, Laplace Integrals and Transfer Operators in large dimension: an introduction.