Generalization of the Darboux transformation and generalized harmonic oscillators

Abstract

The Darbroux transformation is generalized for time-dependent Hamiltonian systems which include a term linear in momentum and a time-dependent mass. The formalism for the N -fold application of the transformation is also established, and these formalisms are applied for a general quadratic system (a generalized harmonic oscillator) and a quadratic system with an inverse-square interaction up to N = 2. Among the new features found, it is shown, for the general quadratic system, that the shape of potential difference between the original system and the transformed system could oscillate according to a classical solution, which is related to the existence of coherent states in the system.

Supersymmetry of Generalized Linear Schrödinger Equations in (1+1) Dimensions

“…We observe that S 1 and S 2 are symmetry operators for the TDSEs (26) and (27), respectively, such that S = diag(S 1 , S 2 ) provides a symmetry operator for the matrix TDSE (30). This can be proved exactly as in the conventional case, see the calculations (16)- (18). Furthermore, the results (19)- (23) transfer to the present, generalized case without change of notation:…”

“…This TDSE resembles a system that is minimally coupled to a vector potential [18]. The form of the coefficients in the TDSE (26) will be a bit more involved than in the other cases.…”

Form-Preserving Transformations for Hamiltonians with Linear Terms in the Momentum

“…Together with the only recently established formalism of Darboux transformations (see Ref. [22]), our form-preserving transformations provide a useful tool for tracking down exact solutions of TDSEs associated to Hamiltonians of the form (3). It is known that for conventional Hamiltonians (without linear terms in p) there is a relation between form-preserving-and Darboux transformations, see Ref.…”

“…Besides the extended Darboux transformation in Ref. [22], our formpreserving transformations are the only known method for tracking down exact solutions of TDSEs associated to Hamiltonians of the form (2). This note is organized as follows: in Sec.…”

Generalized linear Schrödinger equations and their Darboux transformations

“…the Dirac equation [3,4], the nonlinear Schrödinger equation [5] (in fact here all classical nonlinear equations are considered), the Korteweg-de-Vries equation [6], the sine-Gordon equation [6] and many more. Among these equations that allow for Darboux transformations there is the class of linear generalizations of the Schrödinger equation, such as Schrödinger equations with positiondependent (effective) mass [7], Schrödinger equations coupled to a magnetic field [8] and Schrödinger equations with weighted energy [9]. All these equations are linear and of second order, and all of them allow for the construction of certain Darboux transformations.…”