Second-order supersymmetric operators and excited states

Abstract

Factorization of quantum mechanical Hamiltonians has been a useful technique for some time. This procedure has been given an elegant description by supersymmetric quantum mechanics, and the subject has become well-developed. We demonstrate that the existence of raising and lowering operators for the harmonic oscillator (and many other potentials) can be extended to their supersymmetric partners. The use of double supersymmetry (or a factorization chain) is used to obtain non-singular isospectral potentials, and the explicit expressions for the ladder operators, wave functions and probability densities are provided. This application avoids the technical complexities of the most general approaches, and requires relatively modest methods from supersymmetric quantum mechanics.PACS numbers: 03.65. Ge, 03.65.Fd, 03.65.Ca In supersymmetric quantum mechanics (SUSY QM) the form of the Schrödinger equation of a particle is determined by its Hamiltonian ( = 2m = 1):where V (x) is nonsingular on −∞ < x < ∞ and has at least one bound state. If its ground state wave function ψ 0 and energy E 0 are known, then it is always possible to factor the Hamiltonian H (0) − = H − E 0 into a product of two linear differential operators:where W 0 (x) is the superpotential related to the potential V

Interplay between Riccati, Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

“…Yet the limit ε 2 → ε 1 permits to avoid the problem in elegant form and gives rise to the confluent version of Susy QM . Additional results on 2‐step Darboux transformations can be found in, eg, Berger and Ussembayev and Midya Nevertheless, staying in the first‐order approach, the oscillation theorems satisfied by the seed function u with eigenvalue E n ≤ ε ≤ E n +1 prohibit the construction of real‐valued potentials V ( x ) that are free of singularities in Dom V 0 . The situation is different for the complex‐valued potentials V λ ( x ) since the nonlinear superposition of u p and v removes the possibility of zeros in , so the function is regular on .…”

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.…”

“…A supersymmetric quantum mechanical system is characterized by supercharges which can be differential operators of the first [1,2,3,4] or higher [5,6,7,8,9,10,11,12,13,14,15] order. At the classical level, the corresponding supercharges are linear or nonlinear functions of the momentum [9,10].…”

Singular Isotonic Oscillator, Supersymmetry and Superintegrability

“…2.1 Constrained harmonic oscillator V II For the quantum system V II given by equation (3) we have the following constraints on the solution y(z) of the equation ( The equation (5) can be solved in terms of the parabolic cylinder functions [1,44]…”

“…In recent years, many papers [2,4,5,8,13,17,19,23,38] were devoted to a nonsingular isotonic oscillator and its various generalizations. It is also referred in literature as CPRS system and also often written using the following parameter ω = 2a 2 or taking = 1.…”