Unstable equilibrium in semi-classical regime. I: Microlocal concentration. (Équilibre instable en régime semi-classique. I: Concentration microcale.) (French) Zbl 0819.35116

Colin de Verdière, Yves; Parisse, Bernard

Commun. Partial Differ. Equations 19, No. 9-10, 1553-1563 (1994).

The authors are interested in the study of the eigenfunctions of the Schrödinger operator in dimension 1 \[ \Biggl( - {{h^ 2} \over 2} {{d^ 2} \over {dx^ 2}}+ V(x) \Biggr) \varphi(x)= E(h) \varphi(x), \] where \(V\) is a \(C^ \infty\) potential, \(V(x)= -x^ 2/2+ o(x^ 2)\) in a neighborhood of 0 and \(\liminf_{| x| \to\infty} V(x)> 0\), \(h\) tends to 0 (semiclassical limit), \(E(h)\) tends to 0 as \(h\) tends to 0.
They analyze how these eigenfunctions are concentrated near the point \(x=0\). As a corollary, they obtain the existence of eigenfunctions of the Laplacian of a revolution surface with \(-1\) curvature which are concentrated on an unstable geodesic.

Reviewer: B.Helffer (Paris)

MSC:

35Q40	PDEs in connection with quantum mechanics
35P10	Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
81Q10	Selfadjoint operator theory in quantum theory, including spectral analysis

Keywords:

eigenfunctions of the Schrödinger operator; eigenfunctions of the Laplacian