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Bender-Wu singularities. (English) Zbl 1353.81052

Summary: We consider the properties of the family of double well quantum Hamiltonians \(H_{\hbar} = -\hbar^{2} (d^{2}/dx^{2}) + i(x^{3} - x),\; x \in \mathbb{R},\; \hbar > 0\), starting from the resonances of the cubic oscillator \(H_{\epsilon} = -(d^{2}/dx^{2}) + x^{2} + \epsilon x^{3},\; \epsilon > 0\), and studying their analytic continuations obtained by generalized changes of the representation. We prove the existence of infinite crossings of the eigenvalues of \(H_{\hbar}\) together with the selection rules of the pairs of eigenvalues taking part in a crossing. This is a semiclassical localization effect. The eigenvalues at the crossings accumulate at a critical energy for some of the Stokes lines.{
©2016 American Institute of Physics}

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L05 General spectral theory of ordinary differential operators
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