Generalization of the Darboux transform

This article presents a generalization of the standard Darboux transform ap plied to Sturm-Liouville differential equations. This is achieved with the aid of an ansatz as a particular solution for the Riccati relationship invo lved, which in turn led us to obtain its generalized Darboux solution that contains, as a particular case, the standard Darboux transform. The propose d generalized Darboux transform (GDT), applied to the quantum mechanical fi eld, gives the opportunity to prove the existence of standard and generaliz ed Darboux potentials that match with the so-called isospectral potentials. This is exemplified by obtaining, through the GDT, a set of standard and g eneralized Darboux potentials that form the partner of the one-dimensional harmonic oscillator model for any quantum principal number. The worked exam ple indicates how the GDT can be used to obtain the isospectral potentials associated to any known specific potential. We consider also the applicatio n of our method as proposed to the theory of solitons in order to show why the GDT will be important in other fields of application where the standard Darboux transform is usually concerned. (C) 2001 American Institute of Phy sics.