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.


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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