A Generalization of Mielnik’s One-parameter Isospectrality in Unbroken Susyqm

Abstract

Within unbroken SUSYQM and for zero factorization energy, I present an iterative generalization of Mielnik’s one-parameter strictly isospectral method which leads to multiple-parameter bosonic zero modes and corresponding isospectral multipleparameter Schroedinger potentials. The supersymmetric procedures are an interesting and fruitful extension of onedimensional quantum mechanics. For recent reviews see [1]. These techniques are, essentially, factorizations of one-dimensional Schroedinger operators, first discussed in the supersymmetric context by Witten in 1981 [2], and well known in the mathematical literature in the broader sense of Darboux covariance of Schroedinger equations [3]. In 1984, Mielnik [4] introduced a different factorization of the quantum harmonic oscillator based on the general Riccati solution. As a result, Mielnik obtained a one-parameter family of potentials with exactly the same spectrum as that of the harmonic oscillator. However, eventhough in the same year Nieto discussed the connection of such a factorization with the inverse scattering approach, and Fernández applied it to the hydrogen atom case, Mielnik’s result remained as a curiosity for a decade during which only a few authors payed attention to it. On the other hand, constructing families of strictly-isospectral potentials is an important possibility with many potential applications all over physics [1]. This explains the recent surge of interest in this supersymmetric issue [5]. My goal in this work is to give a multiple-parameter generalization of Mielnik’s procedure based on the ground state function of any soluble one-dimensional quantum mechanical problem. This is just a form of Crum’s iterations, i.e., repeated Darboux transformations. Some work along this line has already been done by Keung et al. [6], who performed an iterative construction for the reflectionless, solitonic, sech potentials and the attractive Coulomb potential presenting relevant plots as well. However, they first go n steps away from a given ground state and only afterwards perform the n steps backwards. On the other hand, Pappademos, Sukhatme and Pagnamenta [7], working in the continuum part of the spectrum, got oneand two-parameter supersymmetric families of potentials strictly isospectral with respect to the halfline free particle and Coulomb potentials and focused on the supersymmetric bound states in the continuum. Their procedure is closer to the method I will present in the following. More recent works belong to Bagrov and Samsonov [8], Fernández et al [9], and Rosas-Ortiz [10]. In the following, I first briefly recall the mathematical background of Mielnik’s method and next pass to a simple multiple-parameter generalization for the particular but physically relevant zero mode case. I begin with the “fermionic” Riccati (FR) equation y ′ = −y2 + V1(x) (the “bosonic” one being y ′ = y + V0(x)) for which I suppose to know a particular solution y0. Notice also that I do not put any free constant in the Riccati equations, that is I work at zero factorization energy. Let us seek the general solution in the form y1 = w1 + y0. By substituting y1 in the FR equation one gets the Bernoulli equation −w′ 1 = w 1 + (2y0)w1. Furthermore, using w2 = 1/w1, the simple firstorder linear differential equation w ′ 2 − (2y0)w2 − 1 = 0 is obtained, which can be solved by employing the integration factor F0(x) = e − ∫ x c 2y0 , leading to the solution w2(x) = (λ + ∫ x c F0(z)dz)/F0(x), where λ occurs as an integration constant. In applications the lower limit c is either −∞ or 0 depending on whether one deals with full line or half line problems, respectively. In the latter case, λ is restricted to be a positive number. Thus, the general FR solution reads

Cite this paper

@inproceedings{Rosu1999AGO, title={A Generalization of Mielnik’s One-parameter Isospectrality in Unbroken Susyqm}, author={H. C. Rosu}, year={1999} }