Level crossings in a PT-symmetric double well

We consider the eigenvalues (levels) and the eigenfunctions (states) of a one-parameter family of Hamiltonians H (h) with a PT-symmetric double well. We call nodes the zeros of the states that are stable in the free limit of an associated perturbation theory. For large positive parameter h, the m-nodes state ym (h) is PT-symmetric and the corresponding level Em (h) is positive. For small h there are j-nodes states ypmj (h) , localized about one of the two wells, namely one of the two stationary points of the potential which are real. The corresponding levels E j pm(h) are non-real. We prove the existence of a crossing of each pair of levels (E2n + 1 (h) , E 2n (h)) at a parameter h n > 0, giving, for smaller parameters, the pair of complex levels (En + (h ), En - (h)). The connection between the states (y 2n + 1 (h), y2n (h)) is given by the instability of the imaginary node of y2n + 1 (h). We extend the analysis of the level crossings to the complex plane of the parameter and we propose a through understanding of the full process by considering the Stokes complex and the nodes.