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## The spectrum of the cubic oscillator

Vincenzo Grecchi, André Martinez

We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian,$$H(\beta)=-d^2/dx^2+x^2+i\sqrt{\beta}x^3$$,for $$\beta$$ in the cut plane $$\C_c:=\C\backslash (-\infty, 0)$$. Moreover, we prove that the spectrum consists of the perturbative eigenvalues $$\{E_n(\beta)\}_{n\geq 0}$$ labeled by the constant number $n$ of nodes of the corresponding eigenfunctions. In addition, for all $$\beta\in\C_c$$, $$E_n(\beta)$$ can be computed as the Stieltjes-Pad\’e sum of its perturbation series at $$\beta=0$$. This also gives an alternative proof of the fact that the spectrum of $$H(\beta)$$ is real when $$\beta$$ is a positive number. This way, the main results on the repulsive PT-symmetric and on the attractive quartic oscillators are extended to the cubic case.

http://arxiv.org/abs/1201.2797
Mathematical Physics (math-ph); Spectral Theory (math.SP)