## The spectrum of the cubic oscillator.(English)Zbl 1268.81073

Summary: We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian, $H(\beta)=-\frac{d^2}{dx^2}+x^2+i\sqrt{\beta}x^3,$ for $$\beta$$ in the cut plane $${\mathcal{C}_c := \mathcal{C}\backslash \mathcal{R}_-}$$. 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 $${\beta \in \mathcal{C}_c, E_n(\beta)}$$ can be computed as the Stieltjes-Padé 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.

### MSC:

 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 81Q15 Perturbation theories for operators and differential equations in quantum theory 41A21 Padé approximation
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