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

Miloslav Znojil
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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 ψ_n(x) at eigencharges f=f_n > 0, n = 0, 1, ..., N. Beyond the first few smallest multiplicities N we recommend their perturbative construction.

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Miloslav Znojil. 1999-04-30. Harmonic oscillator well with a screened Coulombic core is quasi-exactly solvable. https://doi.org/10.1088/0305-4470%2F32%2F24%2F318

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