Algebraic properties of Rogers–Szegö functions: I. Applications in quantum optics

Abstract

Abstract. By means of a well-established algebraic framework, Rogers-Szegö functions associated with a circular geometry in the complex plane are introduced in the context of q-special functions, and their properties are discussed in details. The eigenfunctions related to the coherent and phase states emerge from this formalism as infinite expansions of Rogers-Szegö functions, the coefficients being determined through proper eigenvalue equations in each situation. Furthermore, a complementary study on the Robertson-Schrödinger and symmetrical uncertainty relations for the cosine, sine and nondeformed number operators is also conducted, corroborating, in this way, certain features of q-deformed coherent states.

Weak- and strong-expansions of coherent states eigenfunctions for the generalized q-deformed harmonic oscillator and their completeness relations

“…Quantum groups (QG), considered as a generalization of the fundamental symmetry concepts of classical Lie groups, have been the subject of intensive research and several standard textbooks have been devoted to this exciting field [1][2][3][4][5]. The idea to associate quantum groups to the classical groups leads to develop the concept of q-deformed quantum mechanics [6][7][8][9][10][11][12][13] and a special attention is devoted to the q-deformed Weyl-Heisenberg algebra (q-WH) of raising and lowering operators (cf, for instance, [6,7]).…”

Weak and strong expansions of the generalized q-deformed coherent states approximate eigenfunctions and its resolution of unity

“…Quantum groups (QG), considered to be a generalization of the fundamental symmetry concepts of classical Lie groups, have been the subject of intensive research and several standard textbooks have been devoted to this exciting field [1,2,3,4,5]. The idea to associate quantum groups to the classical groups leads to develop the concept of q-deformed quantum mechanics [6,7,8,9,10,11,12,13] and a special attention is devoted to the q-deformed Weyl-Heisenberg algebra (q-WH) of raising and lowering operators (see for example, [6,7]).…”