Singular Isotonic Oscillator, Supersymmetry and Superintegrability

Abstract

Abstract. In the case of a one-dimensional nonsingular Hamiltonian H and a singular supersymmetric partner H a , the Darboux and factorization relations of supersymmetric quantum mechanics can be only formal relations. It was shown how we can construct an adequate partner by using infinite barriers placed where are located the singularities on the real axis and recover isospectrality. This method was applied to superpartners of the harmonic oscillator with one singularity. In this paper, we apply this method to the singular isotonic oscillator with two singularities on the real axis. We also applied these results to four 2D superintegrable systems with second and third-order integrals of motion obtained by Gravel for which polynomial algebras approach does not allow to obtain the energy spectrum of square integrable wavefunctions. We obtain solutions involving parabolic cylinder functions.

Schwarzian derivative treatment of the quantum second-order supersymmetry anomaly, and coupling-constant metamorphosis

“…x 2 − γ , the same mutually shifted superpartner harmonic oscillators given by potentials V ∓ (x) = γ 2 x 2 ∓ 2 γ are intertwined through the isotonic oscillator system, see [14,51,52,53]. It could seem that there is no sense to consider such alternative singular (at x = 0 here) factorizing operators for regular on all the real line R superpartner Hamiltonians.…”

SUSY method for the three-dimensional Schrödinger equation with effective mass

“…Different realizations of these relations exist: they depend both on the form of Hamiltonians (scalar, matrix, multidimensional) and on the order of differential operators Q ± (linear or higher order in derivatives). Independently on realization, intertwining relations (3), (4) with nonsingular operators Q ± lead to the isospectrality of Hamiltonians H 1,2 , but up to possible zero modes of Q ± (see [74] as example of model with singular superpotential). Correspondingly, the wave functions are obtained from each other (up to the constant multipliers):…”

Coherent states of the two-dimensional non-separable supersymmetric Morse potential

“…When two systems are related through supersymmetry, they share nearly identical spectra and states from one system may be transformed into states of the other system by the action of differential operators known as supercharges. In one-dimensional quantum mechanics, supersymmetry has been used to study a plethora of new potentials including partners of: the Rosen-Morse potentials [3,4], the truncated oscillator [5][6][7], and the singular oscillator [8]. In higherdimensional quantum systems, supersymmetry is less explored, though coherent states for the two-dimensional infinite well and its coordinate separable supersymmetric partners have been discussed [9].…”