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We consider the eigenvalues (levels) and the eigenfunctions (states) of a one-parameter family of Hamiltonians 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 the m-nodes state is PT-symmetric and the corresponding level is positive. For small there are j-nodes states localized about one of the two wells, namely one of the two stationary points of the potential which are real. The corresponding levels are non-real. We prove the existence of a crossing of each pair of levels at a parameter giving, for smaller parameters, the pair of complex levels The connection between the states is given by the instability of the imaginary node of 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. [ABSTRACT FROM AUTHOR]

We study the top resonance states of the cubic anharmonic oscillator H(\beta )=p^2+x^2+\mbox{i}\sqrt{\beta } x^3 for b on the complex plane cut on the negative semiaxis. In particular, by the semiclassical scaling and semiclassical methods, we prove that the top resonance states do not belong to L^2(\mbox{\sc R}). [ABSTRACT FROM AUTHOR]

We prove the Pade (Stieltjes) summability of the perturbation series of any energy level En,1(b), n\in {\mathbb {N}}, of the cubic anharmonic oscillator, H_1(\beta )=p^2+x^2+{\rm i}\sqrt{\beta } x^3, as suggested by the numerical studies of Bender and Weniger. At the same time, we give a simple proof of the positivity of every level of the \mathcal {PT}-symmetric Hamiltonian H1(b) for positive b (Bessis-Zinn Justin conjecture). The n zeros, of a state psn,1(b), stable at b = 0, are confined for b on the cut complex plane, and are related to the level En,1(b) by the Bohr-Sommerfeld quantization rule (semiclassical phase-integral condition). We also prove the absence of non-perturbative eigenvalues and the simplicity of the spectrum of our Hamiltonians. [ABSTRACT FROM AUTHOR]