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This is the fourth article in a series where we succeed in enlarging theclass of exactly solvable quantum systems. We do that by working in a completeset of square integrable basis that carries a tridiagonal matrix representationfor the wave operator. Consequently, the matrix wave equation becomes athree-term recursion relation for the expansion coefficients of thewavefunction. Finding solutions of this recursion relation in terms oforthogonal polynomials is equivalent to solving the original problem. Thismethod, called the Tridiagonal Representation Approach (TRA), gives a largerclass of solvable potentials. Here, we obtain S-wave solutions for a newfour-parameter 1/r2 singular but short-range radial potential with an elaborateconfiguration structure and rich spectral property. A particle scattered bythis potential has to overcome a barrier then could be trapped within thepotential valley in a resonance or bound state. Using complex scaling (complexrotation), we display the rich spectral property of the potential in thecomplex energy plane for non-zero angular momentum and show how this structurevaries with the physical parameters.

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