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

In the quantization scheme which weakens the hermiticity of a Hamiltonian to its mere T 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 = i f ) 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 = 2 N +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 } <0, n = 0,1, ... , N . Beyond the smallest multiplicities N we recommend perturbative methods for their construction.