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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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References:

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