We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian, \$\$ H(beta)=-frac{d^2}{dx^2}+x^2+isqrt{beta}x^3, \$\$ for \$beta\$ in the cut plane \$C_c:=Cbackslash R_-\$. Moreover, we prove that the spectrum consists of the perturbative eigenvalues \${E_n(beta)}_{ngeq 0}\$ labeled by the constant number \$n\$ of nodes of the corresponding eigenfunctions. In addition, for all \$betainC_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.

### The spectrum of the cubic oscillator

#### Abstract

We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian, \$\$ H(beta)=-frac{d^2}{dx^2}+x^2+isqrt{beta}x^3, \$\$ for \$beta\$ in the cut plane \$C_c:=Cbackslash R_-\$. Moreover, we prove that the spectrum consists of the perturbative eigenvalues \${E_n(beta)}_{ngeq 0}\$ labeled by the constant number \$n\$ of nodes of the corresponding eigenfunctions. In addition, for all \$betainC_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.
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2013
V. Grecchi; A. Martinez
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/126928`
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