Related papers: Distortion of a Phase Space Under the Darboux Tran…
Elementary Darboux--Laplace transformations for semidiscrete and discrete second order hyperbolic operators are classified. It is proved that in the (semi)-discrete case there are two types of elementary Darboux--Laplace transformations as…
A new kind of deformed calculus (the D-deformed calculus) that takes place in fractional-dimensional spaces is presented. The D-deformed calculus is shown to be an appropriate tool for treating fractional-dimensional systems in a simple way…
Discrete interaction models for the classical harmonic oscillator are used for introducing new mathematical generalizations in the usual continuous formalism. The inverted harmonic potential and generalized discrete hyperbolic and…
A selective chronological survey of Darboux transformations as related to supersymmetric quantum mechanics, intertwining operators and inverse scattering techniques is presented. Short comments are appended to each quotation and basic…
We present a detailed numerical study of a chaotic classical system and its quantum counterpart. The system is a special case of a kicked rotor and for certain parameter values possesses cantori dividing chaotic regions of the classical…
Darboux transformations of the singular harmonic oscillator are considered. Analytical expressions for the propagators are obtained, using the image method applied to formal singular propagators. Two-well and three-well families of…
The plane wave solutions of the three-wave resonant interaction in the plane are considered. It is shown that rank-one constraints over the right derivatives of invertible operators on an arbitrary linear space gives solutions of the…
We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its…
The singular parametric oscillators obtained from the one-parameter Darboux deformation/transformation effected upon the classical harmonic oscillator are introduced and discussed in some detail using sin(omega_0 t) and cos(omega_0 t) as…
The use of effective Darboux transformations for general classes Lax pairs is discussed. The general construction of ``binary'' Darboux transformations preserving certain properties of the operator, such as self-adjointness, is given. The…
We study Darboux transformations associated with the focusing nonlinear Schr\"odinger equation (NLS_-) and their effect on spectral properties of the underlying Lax operator. The latter is a formally J-self-adjoint (but non-self-adjoint)…
It is shown how the Canonical Function approach can be used to obtain accurate solutions for the distorted wave problem taking account of direct static and polarisation potentials and exact non-local exchange. Calculations are made for…
In this paper we point out a close connection between the Darboux transformation and the group of point transformations which preserve the form of the time-dependent Schr\"odinger equation (TDSE). In our main result, we prove that any pair…
We study perturbation theory in certain quantum mechanics problems in which the perturbing potential diverges at some points, even though the energy eigenvalues are smooth functions of the coefficient of the potential. We discuss some of…
We study the Darboux transformation (DT) for Dirac equations with (1+1) potentials. Exact solutions for the adiabatic external field are constructed. The connection between the exactly soluble Dirac (1+1) potentials and the soliton…
Darboux transformation is constructed for superfields of the super sine-Gordon equation and the superfields of the associated linear problem. The Darboux transformation is shown to be related to the super B\"{a}cklund transformation and is…
Using a regularised construction of the phase space path integral due to Ingrid Daubechies and John Klauder which involves a time scale ultimately taken to vanish, and motivated by the general programme towards a noncommutative space(time)…
The known approaches of number-phase problem (for a quantum oscillator) are mutually contradictory. All of them are subsequent in respect with the Robertson-Schr\"{o}dinger uncertainty relation (RSUR). In oposition here it is proposed a new…
The three-dimensional potential equation, motivated by representations of quantum mechanics, is investigated in four different scenarios: (i) In the usual Euclidean space $\mathbb{E}_{3}$ where the potential is singular but invariant under…
Macroscopic ensembles of radiating dipoles are ubiquitous in the physical and natural sciences. In the classical limit the dipoles can be described as damped-driven oscillators, which are able to spontaneously synchronize and collectively…