# 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:

- Dipartimento di Fisica, Università di Firenze and INFN-Sezione di Firenze, 150019 Sesto Fiorentino, Firenze (Italy)
- (Spain)
- 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}

}