Level Crossings in a PT-symmetric Double Well

TL;DR

This paper investigates the spectral properties of a PT-symmetric cubic oscillator with an imaginary double well, proving the existence of infinite level crossings and analyzing their semiclassical behaviors and node structures.

Contribution

It establishes the existence of infinitely many level crossings with a specific selection rule in a PT-symmetric double well and analyzes their semiclassical and node behaviors.

Findings

Infinite level crossings with a selection rule are proven.

Level energies exhibit different semiclassical behaviors along various paths.

Crossing parameters accumulate at zero, indicating an instability point.

Abstract

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.