Extending the class of solvable potentials. IV Inverse square potential with a rich spectrum

Abstract

This is the fourth article in a series where we succeed in enlarging the class of exactly solvable quantum systems. We do that by working in a complete set of square integrable basis that carries a tridiagonal matrix representation for the wave operator. Consequently, the matrix wave equation becomes a three-term recursion relation for the expansion coefficients of the wavefunction. Finding solutions of this recursion relation in terms of orthogonal polynomials is equivalent to solving the original problem. This method, called the Tridiagonal Representation Approach (TRA), gives a larger class of solvable potentials. Here, we obtain S-wave solutions for a new four-parameter 1/r2 singular but short-range radial potential with an elaborate configuration structure and rich spectral property. A particle scattered by this potential has to overcome a barrier then could be trapped within the potential valley in a resonance or bound state. Using complex scaling (complex rotation), we display the rich spectral property of the potential in the complex energy plane for non-zero angular momentum and show how this structure varies with the physical parameters.