## 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.{