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Unstable equilibrium in semiclassical regimes. II: Bohr-Sommerfeld conditions. (Équilibre instable en régime semi-classique. II: Conditions de Bohr-Sommerfeld.) (French) Zbl 0845.35076

The authors study the eigenvalues of a one-dimensional Schrödinger operator which are located near a local maximum of a potential. This is the continuation of a first paper [Commun. Partial Differ. Equations 19, No. 9–10, 1553–1563 (1994; Zbl 0819.35116)], where the microlocal concentration of the associated eigenfunctions was analyzed.
In the particular case of the symmetric double well potential, the authors describe the transition zone between pairs of exponentially close eigenvalues and regularly spaced eigenvalues as the energy grows. This permits to recover (with control of the remainder) the quantization conditions given by some physicists like N. Fröman [Ark. Fys. 32, 79–97 (1966)].
Reviewer: B.Helffer (Orsay)

MSC:

81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)

Citations:

Zbl 0819.35116
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References:

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