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

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 f...

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Main Author: Cantuba, Rafael Reno S.
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Published: Animo Repository 2020
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spelling oai:animorepository.dlsu.edu.ph:faculty_research-27362021-07-19T07:52:12Z Compactness property of Lie polynomials in the creation and annihilation operators of the q-oscillator Cantuba, Rafael Reno S. 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. 2020-10-01T07:00:00Z text text/html https://animorepository.dlsu.edu.ph/faculty_research/1737 https://animorepository.dlsu.edu.ph/context/faculty_research/article/2736/type/native/viewcontent Faculty Research Work Animo Repository Lie algebras Commutation relations (Quantum mechanics) Commutative algebra Compact operators Fredholm operators Mathematics
institution De La Salle University
building De La Salle University Library
continent Asia
country Philippines
Philippines
content_provider De La Salle University Library
collection DLSU Institutional Repository
topic Lie algebras
Commutation relations (Quantum mechanics)
Commutative algebra
Compact operators
Fredholm operators
Mathematics
spellingShingle Lie algebras
Commutation relations (Quantum mechanics)
Commutative algebra
Compact operators
Fredholm operators
Mathematics
Cantuba, Rafael Reno S.
Compactness property of Lie polynomials in the creation and annihilation operators of the q-oscillator
description 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.
format text
author Cantuba, Rafael Reno S.
author_facet Cantuba, Rafael Reno S.
author_sort Cantuba, Rafael Reno S.
title Compactness property of Lie polynomials in the creation and annihilation operators of the q-oscillator
title_short Compactness property of Lie polynomials in the creation and annihilation operators of the q-oscillator
title_full Compactness property of Lie polynomials in the creation and annihilation operators of the q-oscillator
title_fullStr Compactness property of Lie polynomials in the creation and annihilation operators of the q-oscillator
title_full_unstemmed Compactness property of Lie polynomials in the creation and annihilation operators of the q-oscillator
title_sort compactness property of lie polynomials in the creation and annihilation operators of the q-oscillator
publisher Animo Repository
publishDate 2020
url https://animorepository.dlsu.edu.ph/faculty_research/1737
https://animorepository.dlsu.edu.ph/context/faculty_research/article/2736/type/native/viewcontent
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