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Mathematische Nachrichten

‐symmetry and Schrödinger operators. The double well case

Nawal Mecherout, Naima Boussekkine, Thierry Ramond, Johannes Sjöstrand

Published Nov 24, 2015• 4 citations
Publisher

Abstract

We study a class of ‐symmetric semiclassical Schrödinger 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 that, when the potential is analytic, the eigenvalues stay real for a perturbation of size . 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.

DOI
10.1002/mana.201500075

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We study a class of ‐symmetric semiclassical Schrödinger 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 that, when the potential is analytic, the eigenvalues stay real for a perturbation of size . 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.
We study a class of PT-symmetric semiclassical Schrödinger 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 \cite{BoMe14} 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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WKB analysis of P\mathcal PPT\mathcal TT-symmetric Sturm–Liouville problems
Most studies of -symmetric quantum-mechanical Hamiltonians have considered the Schrödinger eigenvalue problem on an infinite domain. This paper examines the consequences of imposing the boundary conditions on a finite domain. As is the case with regular Hermitian Sturm–Liouville problems, the eigenvalues of the -symmetric Sturm–Liouville problem grow like n 2 for large n . However, the novelty is that a eigenvalue problem on a finite domain typically exhibits a sequence of critical points at which pairs of eigenvalues cease to be real and become complex conjugates of one another. For the potentials considered here this sequence of critical points is associated with a turning point on the imaginary axis in the complex plane. WKB analysis is used to calculate the asymptotic behaviours of the real eigenvalues and the locations of the critical points. The method turns out to be surprisingly accurate even at low energies. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.
In this work we consider PT-symmetric perturbations of a self-adjoint semi-classical Schrödinger operator on the real axis in the case of a simple potential well. We assume that the potential is analytic and show that the eigenvalues remain real under the perturbation. Dans ce travail, on considère des perturbations PT -symétriques d'un opérateur de Schrödinger semi-classique auto-adjoint sur la droite réel dans le cas d'un puits de potentiel simple. On suppose que le potentiel soit analytique et on montre que les valeurs propres restent réelles sous la perturbation.