Applications of rigged Hilbert spaces in quantum mechanics and signal processing

Simultaneous use of discrete and continuous bases in quantum systems is not possible in the context of Hilbert spaces, but only in the more general structure of rigged Hilbert spaces (RHS). In addition, the relevant operators in RHS (but not in Hilbert space) are a realization of elements of a Lie e...

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Bibliographic Details
Published in:Journal of mathematical physics 2016-07, Vol.57 (7), p.1
Main Authors: Celeghini, E., Gadella, M., del Olmo, M. A.
Format: Article
Language:English
Subjects:
ALGEBRA
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
FOURIER TRANSFORMATION
Fourier transforms
HILBERT SPACE
Operators (mathematics)
Physics
QUANTUM MECHANICS
Quantum physics
QUANTUM SYSTEMS
Quantum theory
Signal processing
Subspaces
Variation
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Summary:Simultaneous use of discrete and continuous bases in quantum systems is not possible in the context of Hilbert spaces, but only in the more general structure of rigged Hilbert spaces (RHS). In addition, the relevant operators in RHS (but not in Hilbert space) are a realization of elements of a Lie enveloping algebra and support representations of semigroups. We explicitly construct here basis dependent RHS of the line and half-line and relate them to the universal enveloping algebras of the Weyl-Heisenberg algebra and su(1, 1), respectively. The complete sub-structure of both RHS and of the operators acting on them is obtained from their algebraic structures or from the related fractional Fourier transforms. This allows us to describe both quantum and signal processing states and their dynamics. Two relevant improvements are introduced: (i) new kinds of filters related to restrictions to subspaces and/or the elimination of high frequency fluctuations and (ii) an operatorial structure that, starting from fix objects, describes their time evolution.
ISSN:0022-2488
1089-7658
DOI:10.1063/1.4958725