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