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Title: Applications of rigged Hilbert spaces in quantum mechanics and signal processing

Abstract

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.

Authors:
 [1];  [2]; ;  [3]
  1. Dipartimento di Fisica, Università di Firenze and INFN-Sezione di Firenze, 150019 Sesto Fiorentino, Firenze (Italy)
  2. (Spain)
  3. Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, Paseo Belén 7, 47011 Valladolid (Spain)
Publication Date:
OSTI Identifier:
22596590
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 57; Journal Issue: 7; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; FOURIER TRANSFORMATION; HILBERT SPACE; QUANTUM MECHANICS; QUANTUM SYSTEMS

Citation Formats

Celeghini, E., E-mail: celeghini@fi.infn.it, Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, Paseo Belén 7, 47011 Valladolid, Gadella, M., E-mail: manuelgadella1@gmail.com, and Olmo, M. A. del, E-mail: olmo@fta.uva.es. Applications of rigged Hilbert spaces in quantum mechanics and signal processing. United States: N. p., 2016. Web. doi:10.1063/1.4958725.
Celeghini, E., E-mail: celeghini@fi.infn.it, Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, Paseo Belén 7, 47011 Valladolid, Gadella, M., E-mail: manuelgadella1@gmail.com, & Olmo, M. A. del, E-mail: olmo@fta.uva.es. Applications of rigged Hilbert spaces in quantum mechanics and signal processing. United States. doi:10.1063/1.4958725.
Celeghini, E., E-mail: celeghini@fi.infn.it, Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, Paseo Belén 7, 47011 Valladolid, Gadella, M., E-mail: manuelgadella1@gmail.com, and Olmo, M. A. del, E-mail: olmo@fta.uva.es. Fri . "Applications of rigged Hilbert spaces in quantum mechanics and signal processing". United States. doi:10.1063/1.4958725.
@article{osti_22596590,
title = {Applications of rigged Hilbert spaces in quantum mechanics and signal processing},
author = {Celeghini, E., E-mail: celeghini@fi.infn.it and Departamento de Física Teórica, Atómica y Óptica and IMUVA, Universidad de Valladolid, Paseo Belén 7, 47011 Valladolid and Gadella, M., E-mail: manuelgadella1@gmail.com and Olmo, M. A. del, E-mail: olmo@fta.uva.es},
abstractNote = {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.},
doi = {10.1063/1.4958725},
journal = {Journal of Mathematical Physics},
number = 7,
volume = 57,
place = {United States},
year = {Fri Jul 15 00:00:00 EDT 2016},
month = {Fri Jul 15 00:00:00 EDT 2016}
}
  • Within the rigged Hilbert space formulation of quantum mechanics idealized resonances (without background) are described by generalized eigenvectors of an essentially self-adjoint Hamiltonian with complex eigenvalue and a Breit--Wigner energy distribution. This establishes the link between the S matrix description of resonances by a pole and the usual description of states by vectors, overcomes theoretical problems connected with the deviation from exponential law and simplifies the calculation of the decay rate formula.
  • We consider the problem of rigging for the Koopman operators of the Renyi and the baker maps. We show that the rigged Hilbert space for the Renyi maps has some of the properties of a strict inductive limit and give a detailed description of the rigged Hilbert space for the baker maps. {copyright} {ital 1996 American Institute of Physics.}
  • The problem of decaying states and resonances is examined within the framework of scattering theory in a rigged Hilbert space formalism. The stationary free,''in,'' and ''out'' eigenvectors of formal scattering theory, which have a rigorous setting in rigged Hilbert space, are considered to be analytic functions of the energy eigenvalue. The value of these analytic functions at any point of regularity, real or complex, is an eigenvector with eigenvalue equal to the position of the point. The poles of the eigenvector families give origin to other eigenvectors of the Hamiltonian: the singularities of the ''out'' eigenvector family are the samemore » as those of the continued S matrix, so that resonances are seen as eigenvectors of the Hamiltonian with eigenvalue equal to their location in the complex energy plane. Cauchy theorem then provides for expansions in terms of ''complete'' sets of eigenvectors with complex eigenvalues of the Hamiltonian. Applying such expansions to the survival amplitude of a decaying state, one finds that resonances give discrete contributions with purely exponential time behavior; the background is of course present, but explicitly separated. The resolvent of the Hamiltonian, restricted to the nuclear space appearing in the rigged Hilbert space, can be continued across the absolutely continuous spectrum; the singularities of the continuation are the same as those of the ''out'' eigenvectors. The free, ''in'' and ''out'' eigenvectors with complex eigenvalues and those corresponding to resonances can be approximated by physical vectors in the Hilbert space, as plane waves can. The need for having some further physical information in addition to the specification of the total Hamiltonian is apparent in the proposed framework. The formalism is applied to the Lee--Friedrichs model and to the scattering of a spinless particle by a local central potential.« less