The Hamiltonian H = (−1/2) d2/d x2 + x2/2 + λ/x2 reobserved

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Abstract
The Schrödinger problem for the title Hamiltonian is considered as a perturbed one‐dimensional harmonic oscillator. Exact bound state solutions can be derived from a classical differential equation in the theory of Laguerre polynomials. These solutions are valid and analytically dependent on λ only in a limited range of the perturbation strength. Within this region the oscillator Hamiltonian restricted to odd and even parity subspaces is unitary equivalent to H restricted over the respective perturbed subspaces. It is shown that due to the singular nature of the perturbation the allowed λ range is narrowed if side conditions are imposed to make the wavefunctions ’’physically interpretable.’’

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