Harmonic oscillator well with a screened Coulombic core is quasi-exactly solvable

TL;DR

This paper explores a quasi-exactly solvable quantum model combining harmonic and Coulomb potentials with PT symmetry, revealing multiple elementary eigenstates at specific energy levels and eigencharges.

Contribution

It introduces a PT-symmetric Hamiltonian with a superposition of harmonic and Coulomb potentials that admits a set of elementary eigenstates at each excited energy level.

Findings

Existence of elementary Sturmian eigenstates at specific energies and charges.

Model is quasi-exactly solvable with multiple eigenstates beyond the ground state.

Perturbative methods are recommended for higher multiplicities.

Abstract

In the quantization scheme which weakens the hermiticity of a Hamiltonian to its mere PT invariance the superposition V(x) = x^2+ Ze^2/x of the harmonic and Coulomb potentials is defined at the purely imaginary effective charges (Ze^2=if) and regularized by a purely imaginary shift of x. This model is quasi-exactly solvable: We show that at each excited, (N+1)-st harmonic-oscillator energy E=2N+3 there exists not only the well known harmonic oscillator bound state (at the vanishing charge f=0) but also a normalizable (N+1)-plet of the further elementary Sturmian eigenstates \psi_n(x) at eigencharges f=f_n > 0, n = 0, 1, ..., N. Beyond the first few smallest multiplicities N we recommend their perturbative construction.