Related papers: An $su(1,1)$ algebraic approach for the relativist…
We evaluate the matrix elements <Or^{p}>, where O ={1, \beta, i\alpha n \beta} are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, in terms of the…
We study here the behavior of the solutions to a $2\times 2$ semi-linear cooperative system involving Schr\" odinger operators (considered in its variational form): $$LU:=(-\Delta + q(x))U = AU+\mu U + F(x,U) \quad{\rm in}\ \mathbb{R}^N$$…
A Schroedinger type equation on the superspace R^{D|2n} is studied, which involves a potential inversely proportional to the negative of the osp(D|2n) invariant "distance" away from the origin. An osp(2,D+1|2n) dynamical supersymmetry for…
We compare two recent approaches of quasi-exactly solvable Schr\" odinger equations, the first one being related to finite-dimensional representations of $sl(2,R)$ while the second one is based on supersymmetric developments. Our results…
A $r$-parameter ${u}_{\{\kappa_1, \kappa_2, \cdots, \kappa_r\}}(2)$ algebra is introduced. Finite unitary representations are investigated. This polynomial algebra reduces via a contraction procedure to the generalized Weyl-Heisenberg…
In this paper we use different techniques from the fractional and pseudo-operators calculus to solve partial differential equations involving operators with non integer exponents. We apply the method to equations resembling generalizations…
In this paper we consider the problem of deformation quantization of the algebra of polynomial functions on coadjoint orbits of semisimple lie groups. The deformation of an orbit is realized by taking the quotient of the universal…
We revise the symmetries of the Zernike polynomials that determine the Lie algebra su(1,1) + su(1,1). We show how they induce discrete as well continuous bases that coexist in the framework of rigged Hilbert spaces. We also discuss some…
We show Rellich's theorem, the limiting absorption principle, and a Sommerfeld uniqueness result for a wide class of one-body Schr\"odinger operators with long-range potentials, extending and refining previously known results. Our general…
Quantum-mechanical system -- spin 1 particle in external Coulomb field is studied on the base of the matrix Duffin-Kemmer-Petiau formalism with the use of the tetrad technique. Separation of the variables is performed with the help of…
From the algebraic treatment of the quasi-solvable systems, and a q-deformation of the associated $su(2)$ algebra, we obtain exact solutions for the q-deformed Schrodinger equation with a 3-dimensional q-deformed harmonic oscillator…
The quantum group analogue of the normalizer of SU(1,1) in SL(2,C) is an important and non-trivial example of a non-compact quantum group. The general theory of locally compact quantum groups in the operator algebra setting implies the…
A class of quantum superintegrable Hamiltonians defined on a two-dimensional hyperboloid is considered together with a set of intertwining operators connecting them. It is shown that such intertwining operators close a su(2,1) Lie algebra…
We analyse the $n$-dimensional superintegrable Kepler-Coulomb system with non-central terms. We find a novel underlying chain structure of quadratic algebras formed by the integrals of motion. We identify the elements for each sub-structure…
A class of Schr\"odinger-type second-order linear differential equations with a large parameter $u$ is considered. Analytic solutions of this type of equations can be described via (divergent) formal series in descending powers of $u$.…
We present the general ideas on SuperSymmetric Quantum Mechanics (SUSY-QM) using different representations for the operators in question, which are defined by the corresponding bosonic Hamiltonian as part of SUSY Hamiltonian and its…
We establish a simple algebraic relationship between the energy eigenstates of the rational Calogero-Sutherland model with harmonic oscillator and Coulomb-like potentials. We show that there is an underlying SU(1,1) algebra in both of these…
We consider two analytic representations of the SU(1,1) Lie group: the representation in the unit disk based on the SU(1,1) Perelomov coherent states and the Barut-Girardello representation based on the eigenstates of the SU(1,1) lowering…
The second order N-dimensional Schrodinger equation with Mie-type potentials is reduced to a first order differential equation by using the Laplace transformation. Exact bound state solutions are obtained using convolution or Faltungs…
In this paper we introduce the bosonic generators of the $sp(4,R)$ algebra and study some of their properties, based on the $SU(1,1)$ and $SU(2)$ group theory. With the developed theory of the $Sp(4,R)$ group, we solve the interaction part…