Related papers: Reflectionless Potentials for Difference Schr\"odi…
We present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schr\"odinger equation is solved in terms of the confluent hypergeometric functions. The potential involves an attractive inverse…
A series of conjectures is obtained as further investigation of the integral transformation I(alpha) introduced in the previous paper. A Macdonald-type difference operator D is introduced. It is conjectured that D and I(alpha) are…
Original English Summary. - A systematic method of constructing potentials, for which the one-variable Schroedinger equation can be solved in terms of the hypergeometric (HGM) function, is presented. All the potentials, obtained by…
We consider parabolic Schr\"odinger type equations associated to fractional powers of uniformly elliptic 2m-order operators with constant coefficients. Potentials and initial data are considered in suitable Morrey spaces. By means of…
We express the defining relations of the $q$-deformed Minkowski space algebra as well as that of the corresponding derivatives and differentials in the form of reflection equations. This formulation encompasses the covariance properties…
We consider a quantum above-barrier reflection of a Bose-Einstein condensate by a one-dimensional rectangular potential barrier, or by a potential well, for nonlinear Schroedinger equation (Gross-Pitaevskii equation) with a small…
The reflection and transmission of wave functions at a potential step is a well-known issue in a textbook of quantum mechanics. We studied the reflection and transmission characteristics analytically when the potential step is moving at a…
We derive dispersion estimates for solutions of a one-dimensional discrete Dirac equations with a potential. In particular, we improve our previous result, weakening the conditions on the potential. To this end we also provide new results…
The Schrodinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring in…
We introduce and study a class of analytic difference operators admitting reflectionless eigenfunctions. Our construction of the class is patterned after the Inverse Scattering Transform for the reflectionless self-adjoint Schr\"odinger and…
We construct a relationship between integral and differential representation of second-order Jordan chains. Conditions to obtain regular potentials through the confluent supersymmetry algorithm when working with the differential…
We prove that the results in scattering theory that involve resonances are still valid for non-analytic potentials, even if the notion of resonance is not defined in this setting. More precisely, we show that if the potential of a…
We introduce two potentials explicitly given by the Lambert-W function for which the exact solution of the one-dimensional stationary Schr\"odinger equation is written through the first derivative of a double-confluent Heun function. One of…
This work presents a direct and highly accurate method to solve ordinary differential equations, in particular the Schr\"odinger equation in one dimension, through the direct substitution of a power series solution to obtain a purely…
Complex Wadati-type potentials of the form $V(x)=-w^2(x) + iw_x(x)$, where $w(x)$ is a real-valued function, are known to possess a number of intriguing features, unusual for generic non-Hermitian potentials. In the present work, we…
On the base of a 1D Shr\"{o}dinger equation the non-linear first-order differential equation (Ricatti type) for a quantum wave impedance function was derived. The advantages of this approach were discussed and demonstrated for a case of a…
We propose a new method for calculating reflection and transmission coefficients for an arbitrarily polarized electromagnetic plane wave incident on a one-dimensional dielectric medium of finite thickness and with dielectric permittivity…
In this paper we develop a theory of linear differential systems analogous to the classical one for ODEs, including the obtaining of fundamental matrices, the development of a variation of parameters formula and the expression of the…
In this paper we consider the space-fractional Schr\"odinger equation with a singular potential for a wide class of fractional hypoelliptic operators. Such analysis can be conveniently realised in the setting of graded Lie groups. The paper…
We extend the notion of Dirac oscillator in two dimensions, to construct a set of potentials. These potentials becomes exactly and quasi-exactly solvable potentials of non-relativistic quantum mechanics when they are transformed into a…