Harmonic oscillator well with a screened Coulombic core is quasi-exactly solvable
Miloslav Znojil
Abstract
In the quantization scheme which weakens the hermiticity of a Hamiltonian to its mere PT invariance the superposition V(x) = x2+ Ze2/x of the harmonic and Coulomb potentials is defined at the purely imaginary effective charges (Ze2=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 ψn(x) at eigencharges f=fn > 0, n = 0, 1, ..., N. Beyond the first few smallest multiplicities N we recommend their perturbative construction.
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