The spectrum of the cubic oscillator

Abstract

We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian,$H(\beta )=-d^2/d{x}^2+x^2+i\sqrt{\beta }{x}^3$,for $\beta$ in the cut plane $\C_c:=\C\backslash (-\infty, 0)$. Moreover, we prove that the spectrum consists of the perturbative eigenvalues $\{E_n(\beta ){\}}_{n\ge 0}$ labeled by the constant number $n$ of nodes of the corresponding eigenfunctions. In addition, for all $β\in\C_c$, $E_n(\beta )$ 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.

23 pages, 3 figures

References in corpus (7)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• On the reality of the eigenvalues for a class of PT-symmetric oscillators
• Numerical Evidence that the Perturbation Expansion for a Non-Hermitian $\mathcal{P}\mathcal{T}$-Symmetric Hamiltonian is Stieltjes
• Analytic continuation of eigenvalues of a quartic oscillator
• Distributional Borel Summability of Odd Anharmonic Oscillators
• Order-dependent mappings: strong coupling behaviour from weak coupling expansions in non-Hermitian theories

Cited by in corpus (6)

• Nonperturbative Quantum Physics from Low-Order Perturbation Theory
• Spectral projections of the complex cubic oscillator
• Level Crossings in a PT-symmetric Double Well
• Global solution of the cubic oscillator
• Bender-Wu singularities
• Continuation of the Stieltjes Series to the Large Regime by Finite-part Integration