Generalization of the Darboux transform

Abstract

This article presents a generalization of the standard Darboux transform applied to Sturm–Liouville differential equations. This is achieved with the aid of an ansatz as a particular solution for the Riccati relationship involved, which in turn led us to obtain its generalized Darboux solution that contains, as a particular case, the standard Darboux transform. The proposed generalized Darboux transform (GDT), applied to the quantum mechanical field, gives the opportunity to prove the existence of standard and generalized Darboux potentials that match with the so-called isospectral potentials. This is exemplified by obtaining, through the GDT, a set of standard and generalized Darboux potentials that form the partner of the one-dimensional harmonic oscillator model for any quantum principal number. The worked example indicates how the GDT can be used to obtain the isospectral potentials associated to any known specific potential. We consider also the application 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.

Solvable quartic potentials and their isospectral partners

“…We note that according to the standard and generalized Darboux transform [6] these potentials are isospectral partners having the same energy spectra of the specific potential under study.…”

“…where, as before, U GD (x) is used to indicate that this potential comes from the generalized Darboux transform [6], given by…”

“…On the other hand, recently Mielnik et al [4] have pointed out that one of the simpler ways of obtaining new exactly solvable spectral problems consists of looking for pairs of quantum Hamiltonians coupled supersymmetrically by an intertwining operator. Also, Morales et al [5] have shown that the standard Darboux transform leads to the isospectral potential associated with one of the aforementioned Hamiltonians and the generalized Darboux transform [6] to the operators that factorize the Hamiltonians both of the former potential and of the generalized partner isospectral potential. Taking these elements into consideration, in this work we propose an alternative method for finding solvable potentials, making use of the fact that the general solution of the Ricatti equations always incorporates a particular solution.…”

Isospectral orthogonal polynomials from the Darboux transforms

“…with ␥ Ϯ as the integration constant and where the subscript g indicates generalized. Consequently, the generalized wave functions gn Ϯ (u) are related to the standard wave functions n Ϯ (u) by means of the generalized Darboux transforms [14] gn…”

“…This transform leads to the isospectral potential associated with one of the aforementioned Hamiltonians [13]. On the other hand, in relation to the generalized Darboux transform [14], this was recently used in the search for new isospectral generalized potentials [15].…”