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.
Mathematical Physics (math-ph); Spectral Theory (math.SP)

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