Multiple Parameter Structure of Mielnik’s Isospectrality in Unbroken Susyqm

Within unbroken SUSYQM and for zero factorization energy, I present an iterative generalization of Mielnik’s isospectral method by employing a Schrödinger true zero mode in the first-step general Riccati solution and imposing the physical condition of normalization at each iterative step. This procedure leads to a well-defined multipleparameter structure within Mielnik’s construction for both zero modes and potentials. Int. J. Theor. Phys. 39, 105-114 (January 2000) 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 Schrödinger 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, even though 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 a curiosity for a decade during which only a few authors paid attention to it. On the other hand, constructing families of strictly isospectral potentials is an important possibility with many potential applications in 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 et al. [7], working in the continuum part of the spectrum, got oneand two-parameter supersymmetric families of potentials strictly isospectral with respect to the half-line 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. For more recent works see Bagrov and Samsonov [8], Fernández et al [9], Junker and Roy [10], and Rosas-Ortiz [11]. 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 ′ = −y + 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, we obtain the simple first-order linear differential equation w ′ 2 − (2y0)w2 − 1 = 0, 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