  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.
