Level Crossings in a PT-symmetric Double Well

We consider a \textit{PT}-symmetric cubic oscillator with an imaginary double well. We prove the existence of an infinite number of level crossings with a definite selection rule. Decreasing the positive parameter $\hbar$ from large values, at a value $\hbar_n$ we find the crossing of the pair of levels $(E_{2n+1}(\hbar),E_{2n}(\hbar))$ becoming the pair of levels $(E_n^+(\hbar),E_n^-(\hbar))$. For large parameters, a level is a holomorphic function $E_m(\hbar)$ with different semiclassical behaviors, $E_j^\pm(\hbar),$ along different paths. The corresponding $m$-nodes delocalized state $\psi_m(\hbar)$ behaves along the same paths as the semiclassical $j$-nodes states $\psi_j^\pm(\hbar),$ localized at one of the wells $x_\pm$ respectively. In particular, if the crossing parameter $\hbar_n$ is by-passed from above, the levels $E_{2n+(1/2)\pm(1/2)}(\hbar)$ have respectively the semiclassical behaviors of the levels $E_n^\mp(\hbar)$ along the real axis. These results are obtained by the control of the nodes. There is evidence that the parameters $\hbar_n$ accumulate at zero and the accumulation point of the corresponding energies is aninstability point of a subset of the Stokes complex called the monochord, consisting of the vibrating string and the sound board.