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# $PT$-symmetry and Schrödinger operators. The double well case

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Abstract : We study a class of $PT$-symmetric semiclassical Schrodinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double-well potential. In the simple well case, two of the authors have proved in [6] that, when the potential is analytic, the eigenvalues stay real for a perturbation of size $O(1)$. We show here, in the double-well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB-analysis, leading to a fairly explicit quantization condition.
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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-01410406
Contributeur : Imb - Université de Bourgogne <>
Soumis le : mardi 6 décembre 2016 - 15:59:41
Dernière modification le : jeudi 28 janvier 2021 - 10:28:03

### Citation

Nawal Mecherout, Naima Boussekkine, Thierry Ramond, Johannes Sjöstrand. $PT$-symmetry and Schrödinger operators. The double well case. Mathematical News / Mathematische Nachrichten, Wiley-VCH Verlag, 2016, 289 (7), pp.854-887. ⟨10.1002/mana.201500075⟩. ⟨hal-01410406⟩

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