Eigenvalues in the semiclassical limit - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:45:05Z http://mathoverflow.net/feeds/question/77806 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77806/eigenvalues-in-the-semiclassical-limit Eigenvalues in the semiclassical limit Kofi 2011-10-11T11:26:06Z 2011-10-11T12:32:27Z <p>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 $|x|$, ...).</p> <p>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 $\hbar(2n+1)$.</p> <p>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?</p> <p>What about if M is a (possibly compact) riemannian manifold and $\Delta$ the Laplace-Beltrami operator?</p> http://mathoverflow.net/questions/77806/eigenvalues-in-the-semiclassical-limit/77807#77807 Answer by Igor Rivin for Eigenvalues in the semiclassical limit Igor Rivin 2011-10-11T11:50:10Z 2011-10-11T11:50:10Z <p>The canonical reference is:</p> <p>Introduction to spectral theory: with applications to Schrödinger operators by Hislop and Sigal.</p> <p>Your statement about the semiclassical behavior of eigenvalues seems to be proved by Barry Simon in:</p> <p><a href="http://archive.numdam.org/ARCHIVE/AIHPA/AIHPA_1983__38_3/AIHPA_1983__38_3_295_0/AIHPA_1983__38_3_295_0.pdf" rel="nofollow">http://archive.numdam.org/ARCHIVE/AIHPA/AIHPA_1983__38_3/AIHPA_1983__38_3_295_0/AIHPA_1983__38_3_295_0.pdf</a></p> <p>(where he calls it a "folk theorem").</p> http://mathoverflow.net/questions/77806/eigenvalues-in-the-semiclassical-limit/77809#77809 Answer by André Schlichting for Eigenvalues in the semiclassical limit André Schlichting 2011-10-11T12:32:27Z 2011-10-11T12:32:27Z <p>Another reference is Helffer and several Coauthors (Sjöstrand, Nier, Klein, Garyard, ...) using the Witten-Laplace approach.</p> <p>An introduction is given in the book <a href="http://books.google.de/books?id=43vAd5KoEkUC" rel="nofollow">Semiclassical analysis, Witten Laplacians, and statistical mechanics</a></p> <p>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 <a href="http://www.math.kth.se/spect/preprints04_05/Helf3.pdf" rel="nofollow">Low lying eigenvalues of Witten Laplacians and metastability (after Helffer-Klein-Nier and Helffer-Nier)</a>. </p> <p>If you are only interessted in the Schrödinger Operator, maybe the book <a href="http://www.springerlink.com/content/978-3-540-50076-6" rel="nofollow">Semi-Classical Analysis for the Schrödinger Operator and Applications</a> is the most interesting for you. There are also lecture notes available <a href="http://www.math.u-psud.fr/~helffer/CoursDEA95main.ps" rel="nofollow">Semiclassical Analysis for Schrödinger Operators, Laplace Integrals and Transfer Operators in large dimension: an introduction.</a></p>