The one-dimensional pseudoharmonic oscillator potential has an inverse square term (with dimensionless coefficient α) and a harmonic oscillator term. For α −1/4, the spectrum consists of normalizable even- and odd-parity states that are doubly degenerate. Here we study a regularized version of this potential (with a cutoff δ) in the regime α −1/4. In the case of the inverse-square potential, in this regime a particle “falls” into the origin. The regularization procedure enables one to study the eigenstates and eigenvalues as a function of δ, where δ → 0 corresponds to the case where a particle falls into the origin. As δ → 0, the energy eigenvalues behave as ∼−1/δ2; also, there is a single ground state with even parity and a probability distribution which limits to a Dirac delta function, whereas the excited energy eigenstates are doubly degenerate and described by a universal wave function in terms of a modified Bessel function of the second kind.
Coulam et al. (Sun,) studied this question.