New Solvable Singular Potentials

We obtain three new solvable, real, shape-invariant potentials starting from the harmonic oscillator, Poschl-Teller I and Poschl-Teller II potentials on the half-axis and extending their domain to the full line, while taking special care to regularize the inverse-square singularity at the origin. The regularization procedure gives rise to a delta-function behaviour at the origin. Our new systems possess underlying nonlinear potential algebras, which can also be used to determine their spectra analytically.