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RESEARCH PRODUCT
$PT$-symmetry and Schrödinger operators. The double well case
Nawal Mecherout Naima Boussekkine Thierry Ramond Johannes Sjöstrandsubject
[ MATH.MATH-SP ] Mathematics [math]/Spectral Theory [math.SP]MSC: 35P20 81Q12 81Q20 35Q40Complex WKB analysis[ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph]EigenvaluesMathematics::Spectral TheoryPT-symmetryMathematics - Spectral Theory[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]35P20 35Q40 81Q12 81Q20Quantization conditonSchrödinger operatorsMathematical Physics[MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]description
International audience; 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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2016-05-01 |