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

Abstract

In the quantization scheme which weakens the hermiticity of a Hamiltonian to its mere T 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)th 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 _{n}(x) at eigencharges f = f_{n}n = 0,1, ... ,N. Beyond the smallest multiplicities N we recommend perturbative methods for their construction.