Bender-Wu singularities

We consider a family of quantum Hamiltonians $H_\hbar=-\hbar^2\,(d^2\!/dx^2) +V(x)$, $x\in\mathbb{R},$ $\hbar>0,$ where $V(x)=i(x^3-x)$ is an imaginary double well potential. We prove the existence of infinite eigenvalue crossings with the selection rules of the eigenvalue pairs 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.