The spectrum of the cubic oscillator

TL;DR

This paper proves the eigenvalues of the cubic oscillator Hamiltonian are simple, analytic, and can be computed via perturbation series, extending known results to the cubic case and confirming real spectra for positive parameters.

Contribution

It establishes the simplicity, analyticity, and perturbative computability of eigenvalues for the cubic oscillator Hamiltonian in a complex domain, extending previous results to this specific case.

Findings

Eigenvalues are simple and analytic in the complex plane cut.

Eigenvalues can be computed as Stieltjes-Padé sums of perturbation series.

Spectrum is real for positive parameter values.

Abstract

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.