The spectrum of the cubic oscillator

We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian,$H(β)=-d^2/dx^2+x^2+i\sqrtβx^3$,for $β$ in the cut plane $\C_c:=\C\backslash (-\infty, 0)$. Moreover, we prove that the spectrum consists of the perturbative eigenvalues $\{E_n(β)\}_{n\geq 0}$ labeled by the constant number $n$ of nodes of the corresponding eigenfunctions. In addition, for all $β\in\C_c$, $E_n(β)$ can be computed as the Stieltjes-Padé sum of its perturbation series at $β=0$. This also gives an alternative proof of the fact that the spectrum of $H(β)$ is real when $β$ 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.

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