Four-parameter $1/r^2$ singular short-range potential with rich bound states and a resonance spectrum

A. D. Alhaidari

Teoretičeskaâ i matematičeskaâ fizika, Tome 195 (2018) no. 3, pp. 422-436

Résumé

We use the tridiagonal representation approach to enlarge the class of exactly solvable quantum systems. For this, we use a square-integrable basis in which the matrix representation of the wave operator is tridiagonal. In this case, the wave equation becomes a three-term recurrence relation for the expansion coefficients of the wave function with a solution in terms of orthogonal polynomials that is equivalent to a solution of the original problem. We obtain S-wave bound states for a new four-parameter potential with a $1/r^2$ singularity but short-range, which has an elaborate configuration structure and rich spectral properties. A particle scattered by this potential must overcome a barrier and can then be trapped in the potential valley in a resonance or bound state. Using complex rotation, we demonstrate the rich spectral properties of the potential in the case of a nonzero angular momentum and show how this structure varies with the parameters of the potential.

Détail

Keywords: $1/r^2$ singular potential, tridiagonal representation, recurrence relation, parameter spectrum, bound state, resonance.

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