ar X iv : q ua nt - p h / 01 02 09 1 v 1 1 9 Fe b 20 01 ON FIRST - ORDER SCALING INTERTWINING IN QUANTUM MECHANICS

We generalize the standard first-order intertwining relationship of supersymmetric quantum mechanics in order to include simultaneous scaling transformations in both the original Hamiltonian and the intertwining operator. It is argued that in this way one can generate potentials with more interesting spectra than those obtained by means of the standard first-order intertwining technique and, as an outcome, a simple engineering procedure is presented. The harmonic oscillator potential is used in order to illustrate the previous statements. Moreover, a matrix representation of the scaled intertwining relationship is sketched up allowing for higher-dimensional generalizations in the case of separable potentials. Resumen. Generalizamos la relación de entrelazamiento estandar de primer orden de la mecánica cuántica supersimétrica para incluir de manera simultánea transformaciones de escalamiento tanto en el Hamiltoniano original como en el operador de entrelazamiento. Se argumenta que en esta forma uno puede generar potenciales con espectros más interesantes que aquellos obtenidos por medio de la técnica de entrelazamiento estandar de primer orden y, como un resultado, un procedimiento sencillo de ingenieria cuántica es presentado. El potencial de oscilador armónico es usado para ilustrar las afirmaciones anteriores. Mas aún, una representación matricial de la relación de entrelazamiento escalado es bosquejado que permite la generalización a más de una dimensión en el caso de los potenciales separables. Factorizations of second order linear one-dimensional (1D) differential operators are common tools in Witten’s supersymmetric quantum mechanics (SUSYQM) [1], which may be considered as a form of Darboux transformations [2]. They imply either particular solutions of Riccati equations known as superpotentials or the general Riccati solution (RS), the latter case being first used in physics by Mielnik for the quantum harmonic oscillator [3]. In recent years, it became clear that SUSYQM is not only a form of Darboux transformations but also a particular case in the more general framework provided by the technique of intertwining operators [4], of extensive use in the mathematical literature. Two (Hamiltonian) operators 2 Fernández and Rosu, Rev. Mex. F́ıs. 46 S2 (Nov. 2000) 153-156 are said to be intertwined by an operator A if the following relationship is fulfilled H1A + = AH. (1) In SUSYQM A is a first order differential (factorization) operator of the form A = 1 √ 2 (