New Solvable Singular Potentials

Abstract

We obtain three new solvable, real, shape invariant potentials starting from the harmonic oscillator, Pöschl-Teller I and Pöschl-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 behavior at the origin. Our new systems possess underlying non-linear potential algebras, which can also be used to determine their spectra analytically.

19 pages, 4 figures

References in corpus (5)

• PT -symmetric harmonic oscillators
• sl(2, C) as a complex Lie algebra and the associated non-Hermitian Hamiltonians with real eigenvalues
• Exact solution for Morse oscillator in PT-symmetric quantum mechanics
• Harmonic oscillator well with a screened Coulombic core is quasi-exactly solvable
• Coordinate Realizations of Deformed Lie Algebras with Three Generators

Cited by in corpus (8)

• A quantum exactly solvable non-linear oscillator related with the isotonic oscillator
• Non-Hermitian supersymmetry and singular PT symmetrized oscillators
• Relativistic and nonrelativistic bound states of the isotonic oscillator by Nikiforov-Uvarov method
• SUSY Intertwining Relations of Third Order in Derivatives
• Superintegrable quantum u(3)--systems and higher rank factorizations
• Comment on "Supersymmetry and Singular Potentials" by Das and Pernice [Nucl. Phys. B 561 (1999) 357]
• Algebraic Shape Invariant Potentials as the Generalized Deformed Oscillator
• Dynamical symmetries for superintegrable quantum systems