Compactness property of Lie polynomials in the creation and annihilation operators of the q-oscillator

College

College of Science

Department/Unit

Mathematics and Statistics Department

Document Type

Article

Source Title

Letters in Mathematical Physics

Volume

110

Issue

10

First Page

2639

Last Page

2657

Publication Date

10-1-2020

Abstract

Given a real number q such that 0 < q< 1 , the natural setting for the mathematics of a q-oscillator is an infinite-dimensional, separable Hilbert space that is said to provide an interpolation between the Bargmann–Segal space of holomorphic functions and the Hardy–Lebesgue space of analytic functions. The traditional basis states are interrelated by the creation and annihilation operators. Since the commutation relation is q-deformed, the commutator algebra for the creation and annihilation operators is not a low-dimensional Lie algebra like that for the canonical commutation relation. In this study, a characterization of the elements of the said commutator algebra is obtained using spectral properties of the creation and annihilation operators as these faithfully represent the generators of a q-deformed Heisenberg algebra. The derived algebra of the commutator algebra is precisely the set of all compact operators, and the resulting Calkin algebra is algebraically isomorphic to the complex algebra of Laurent polynomials in one indeterminate. As for any operator that is not in the commutator algebra, the action of such an operator on an arbitrary basis state can be approximated by a Lie series of elements from the commutator algebra. © 2020, Springer Nature B.V.

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Digitial Object Identifier (DOI)

10.1007/s11005-020-01304-x

Disciplines

Mathematics

Keywords

Lie algebras; Commutation relations (Quantum mechanics); Commutative algebra; Compact operators; Fredholm operators

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