On a dynamical symmetry group of the relativistic linear singular oscillator

Abstract

Abstract. -An exact approach for the factorization of the relativistic linear singular oscillator is proposed. This model is expressed by the finite-difference Schrödinger-like equation. We have found finite-difference raising and lowering operators, which are with the Hamiltonian operator form the close Lie algebra of the SU (1, 1) group.Introduction. -The singular harmonic oscillator is one of the rare exactly solvable problems in non-relativistic quantum mechanics [1,2]. This model is useful to explain many phenomena, such as description of interacting many-body systems [3], diatomic [4] and polyatomic [5] molecules, spin chains [6], quantum Hall effect [7], fractional statistics and anyons [8]. In spite of many interesting papers devoted to the study of the non-relativistic singular harmonic oscillator model [9], the number of works studying relativistic approachs to singular oscillator exact solution is still rather few [10].Recently, we constructed the exactly solvable relativistic model of the quantum linear singular oscillator [11]. Later, this model was generalized for 3D case in [12]. Our model was formulated in the framework of the finite-difference version of the relativistic quantum mechanics, developed in [13][14][15][16][17]. As we noted in [11], unlike the case of the Coulomb potential, the relativistic generalization of the oscillator or singular oscillator potentials are not uniquely defined. Therefore, one of the main requirements for proposed relativistic models is existence of their dynamical symmetry. In this Letter, we present the simplest way for construction of the close Lie algebra of the SU (1, 1) group for a relativistic model of the linear singular oscillator. In spite of expression of the problem under consideration by the finite-difference equation, it is also exactly factorizable and is in complete analogy with relevant non-relativistic problem. This factorization will allow us to obtain the exact finite-difference expression of the generators of SU (1, 1) group.

A relativistic model of theN-dimensional singular oscillator

“…It is necessary to note the recent published work [32], where the dynamical symmetry and factorization scheme is proposed for Hamiltonian leading to eigenfunctions expressed by continuous dual Hahn polynomials. The finite-difference Hamiltonian is formulated in the framework of Ruijsenaars-Schneider approach [26][27][28] and therefore, it is different than factorization scheme used here and in [33]. Taking into account that, both of these approaches to solve explicitly the finite-difference equations lead to the close results, we aim to show in detail possible relations and transformations between them as a further step of investigations.…”

Polyanalytic relativistic second Bargmann transforms

“…Remark 4.1. We should note that the eigenstates {ψ ν k } in (4.3) can also be obtained by the k-fold action of a finite-difference raising operator to the ground state, see [21] where the authors have established an exact factorization of the RIO in a complete analogy with the non-relativistic problem. In particular, they concluded that eigenfunctions {ψ ν k } constitute the basis of the irreducible representation D + ν+α 2 of the SU(1, 1) Lie group.…”

Oscillating and rotating boundary conditions in quantum mechanics

“…Time-varying potentials and moving boundary conditions are time-dependent problems in quantum mechanics. There are a small number of exact analytic solutions for time-dependent potentials [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The Schrödinger equation endowed with time-dependent boundary conditions (0, t) = 0, (L(t), t) = 0,…”