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Développements semi-classiques exacts des niveaux d’énergie d’un oscillateur à une dimension. (Exact semi-classical expansions of the energy levels of a one-dimensional oscillator). (French) Zbl 0712.35071

The eigenvalues of the one dimensional Schrödinger operator \({\mathcal H}=- \hslash^ 2d^ 2/dq^ 2+V(q)\), for a real polynomial potential V, going to \(+\infty\) when \(q\to^+_ -\infty\), are investigated. These eigenvalues are the zeros of \(a^+(E)=a(E-i0)\) (or his complex conjugate), where a(E) is the Jost function associated to \({\mathcal H}.\)
Developing the ideas of A. Voros [Ann. Inst. Henri Poincaré, Sect. A 39, 211-338 (1983; Zbl 0526.34046)] and J. Ecalle [Cinq applications des fonctions résurgentes, preprint 84 T 62, Orsay] on resurgent functions and Jost symbols, the authors obtain the exact semi- classical expansions of the eigenvalues. In the case of a symmetric double well, they get the proof of a conjecture of J. Zinn-Justin [J. Math. Phys. 25, 549-555 (1984)].
Reviewer: D.Huet

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35J10 Schrödinger operator, Schrödinger equation

Citations:

Zbl 0526.34046
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