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

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

1999-04-30

arxiv.org/abs/math-ph/9904039v1

J. Phys. A: Math. Gen. 32 (1999) 4563-4570

Physics

Mathematical Physics

13 pages, Latex file, to appear in J. Phys. A: Math. Gen

Scientific paper

10.1088/0305-4470/32/24/318

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.

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