Applications of Functional Analysis in Quantum Physics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 15405

Special Issue Editor


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Guest Editor
Departamento de Física Teórica, Atómica y Optica and IMUVA, Universidad de Valladolid, 47011 Valladolid, Spain
Interests: functional analytic methods in theoretical physics; rigged Hilbert spaces; operator algebras; self adjoint extensions of symmetric operators and point potentials; scattering theory; quantum resonances; time asymmetry in quantum mechanics;

Special Issue Information

Dear Colleagues,

Functional analytic methods have been an essential mathematical tool for quantum mechanics since the very first years of the development of quantum theory, starting with the seminal work by von Neumann. Since then, an enormous effort has been made in order to give order and mathematical rigor to quantum physics. Great developments in the theory of operators in Hilbert spaces have helped very much in the understanding of quantum theory through a large variety of models.

We may mention a variety of very successful applications of functional analysis to quantum mechanics, mostly through the theory of operators on Hilbert spaces. In fact, self-adjoint operators and their spectral decomposition have played an important role, since states and observables are traditionally described by self-adjoint operators. This has driven impressive developments like the Kato perturbation theory, the formal scattering theory, the theory of extensions of symmetric non-self-adjoint operators or the renormalization techniques often necessary to properly define point potentials. In addition, we have the outstanding field of group and algebra representations as operators on Hilbert spaces. The above represent just a selection of the applications to date.

The objective of this Special Issue is to foster the extension of investigation in this field. In addition to the traditional fields of research mentioned above, we welcome contributions of algebras of operators in quantum mechanics, as well as quantum field theory, Banach spaces, locally convex spaces, Gelfand triplets, and theory of operators on all these structures with possible applications in quantum mechanics and quantum field theory. Additionally, rigorous mathematical developments of PT symmetries and non-Hermitian quantum mechanics would be most welcome. Finally, other rigorous developments not mentioned here would also be considered, provided that they could be included in this field. This Special Issue will accept high-quality papers including original research results with illustrative applications, as well as survey articles of exceptional merit.

Prof. Dr. Manuel Gadella
Guest Editor

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Keywords

  • Theory of operators on Hilbert and Banach spaces: applications of quantum mechanics and quantum field theory
  • Scattering theory
  • Locally convex spaces, Gelfand triplets, frames, and the theory of operators on these structures
  • Self-adjoint extensions of symmetric operators and point potentials
  • Regularization theory and point potentials
  • Rigorous theory of quantum resonances
  • Groups and algebra representations as operators on Hilbert, Banach, locally convex spaces, Gelfand triplets, frames, nets, etc
  • PT symmetries and non-Hermitian Hamiltonians in quantum theory

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Published Papers (9 papers)

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Research

13 pages, 261 KiB  
Article
Toeplitz Operators on Harmonic Fock Spaces with Radial Symbols
by Zhi-Ling Sun, Wei-Shih Du and Feng Qi
Mathematics 2024, 12(4), 565; https://doi.org/10.3390/math12040565 - 13 Feb 2024
Viewed by 1051
Abstract
The main aim of this paper is to study new features and specific properties of the Toeplitz operator with radial symbols in harmonic Fock spaces. A new spectral decomposition of a Toeplitz operator with Wick symbols is also established. Full article
(This article belongs to the Special Issue Applications of Functional Analysis in Quantum Physics)
11 pages, 292 KiB  
Article
Operators in Rigged Hilbert Spaces, Gel’fand Bases and Generalized Eigenvalues
by Jean-Pierre Antoine and Camillo Trapani
Mathematics 2023, 11(1), 195; https://doi.org/10.3390/math11010195 - 30 Dec 2022
Cited by 1 | Viewed by 1825
Abstract
Given a self-adjoint operator A in a Hilbert space H, we analyze its spectral behavior when it is expressed in terms of generalized eigenvectors. Using the formalism of Gel’fand distribution bases, we explore the conditions for the generalized eigenspaces to be one-dimensional, [...] Read more.
Given a self-adjoint operator A in a Hilbert space H, we analyze its spectral behavior when it is expressed in terms of generalized eigenvectors. Using the formalism of Gel’fand distribution bases, we explore the conditions for the generalized eigenspaces to be one-dimensional, i.e., for A to have a simple spectrum. Full article
(This article belongs to the Special Issue Applications of Functional Analysis in Quantum Physics)
21 pages, 370 KiB  
Article
An Algebraic Model for Quantum Unstable States
by Sebastian Fortin, Manuel Gadella, Federico Holik, Juan Pablo Jorge and Marcelo Losada
Mathematics 2022, 10(23), 4562; https://doi.org/10.3390/math10234562 - 1 Dec 2022
Viewed by 1376
Abstract
In this review, we present a rigorous construction of an algebraic method for quantum unstable states, also called Gamow states. A traditional picture associates these states to vectors states called Gamow vectors. However, this has some difficulties. In particular, there is no consistent [...] Read more.
In this review, we present a rigorous construction of an algebraic method for quantum unstable states, also called Gamow states. A traditional picture associates these states to vectors states called Gamow vectors. However, this has some difficulties. In particular, there is no consistent definition of mean values of observables on Gamow vectors. In this work, we present Gamow states as functionals on algebras in a consistent way. We show that Gamow states are not pure states, in spite of their representation as Gamow vectors. We propose a possible way out to the construction of averages of observables on Gamow states. The formalism is intended to be presented with sufficient mathematical rigor. Full article
(This article belongs to the Special Issue Applications of Functional Analysis in Quantum Physics)
13 pages, 648 KiB  
Article
A Fully Pseudo-Bosonic Swanson Model
by Fabio Bagarello
Mathematics 2022, 10(21), 3954; https://doi.org/10.3390/math10213954 - 25 Oct 2022
Cited by 1 | Viewed by 1185
Abstract
We consider a fully pseudo-bosonic Swanson model and we show how its Hamiltonian H can be diagonalized. We also deduce the eigensystem of H, using the general framework and results deduced in the context of pseudo-bosons. We also construct, using different [...] Read more.
We consider a fully pseudo-bosonic Swanson model and we show how its Hamiltonian H can be diagonalized. We also deduce the eigensystem of H, using the general framework and results deduced in the context of pseudo-bosons. We also construct, using different approaches, the bi-coherent states for the model, study some of their properties, and compare the various constructions. Full article
(This article belongs to the Special Issue Applications of Functional Analysis in Quantum Physics)
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23 pages, 438 KiB  
Article
Interference of Non-Hermiticity with Hermiticity at Exceptional Points
by Miloslav Znojil
Mathematics 2022, 10(20), 3721; https://doi.org/10.3390/math10203721 - 11 Oct 2022
Cited by 2 | Viewed by 1065
Abstract
The recent growth in popularity of the non-Hermitian quantum Hamiltonians H(λ) with real spectra is strongly motivated by the phenomenologically innovative possibility of an access to the non-Hermitian degeneracies called exceptional points (EPs). What is actually presented in the present [...] Read more.
The recent growth in popularity of the non-Hermitian quantum Hamiltonians H(λ) with real spectra is strongly motivated by the phenomenologically innovative possibility of an access to the non-Hermitian degeneracies called exceptional points (EPs). What is actually presented in the present paper is a perturbation-theory-based demonstration of a fine-tuned nature of this access. This result is complemented by a toy-model-based analysis of the related details of quantum dynamics in the almost degenerate regime with λλ(EP). In similar studies, naturally, one of the decisive obstacles is the highly nontrivial form of the underlying mathematics. Here, many of these obstacles are circumvented via several drastic simplifications of our toy models—i.a., our N by N matrices H(λ)=H(N)(λ) are assumed real, tridiagonal and PT-symmetric, and our H(N)(λ) is assumed to be split into its Hermitian and non-Hermitian components staying in interaction. This is shown to lead to several remarkable spectral features of the model. Up to N=8, their description is even shown tractable non-numerically. In particular, it is shown that under generic perturbation, the “unfolding” removal of the spontaneous breakdown of PT-symmetry proceeds via intervals of λ with complex energy spectra. Full article
(This article belongs to the Special Issue Applications of Functional Analysis in Quantum Physics)
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11 pages, 400 KiB  
Article
On Hermite Functions, Integral Kernels, and Quantum Wires
by Silvestro Fassari, Manuel Gadella, Luis M. Nieto and Fabio Rinaldi
Mathematics 2022, 10(16), 3012; https://doi.org/10.3390/math10163012 - 21 Aug 2022
Viewed by 1824
Abstract
In this note, we first evaluate and subsequently achieve a rather accurate approximation of a scalar product, the calculation of which is essential in order to determine the ground state energy in a two-dimensional quantum model. This scalar product involves an integral operator [...] Read more.
In this note, we first evaluate and subsequently achieve a rather accurate approximation of a scalar product, the calculation of which is essential in order to determine the ground state energy in a two-dimensional quantum model. This scalar product involves an integral operator defined in terms of the eigenfunctions of the harmonic oscillator, expressed in terms of the well-known Hermite polynomials, so that some rather sophisticated mathematical tools are required. Full article
(This article belongs to the Special Issue Applications of Functional Analysis in Quantum Physics)
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11 pages, 290 KiB  
Article
Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals
by Jing Zhang, Lixia Zhang and Caishi Wang
Mathematics 2022, 10(15), 2635; https://doi.org/10.3390/math10152635 - 27 Jul 2022
Viewed by 1146
Abstract
Let N be the number operator in the space H of real-valued square-integrable Bernoulli functionals. In this paper, we further pursue properties of N from a probabilistic perspective. We first construct a nuclear space G, which is also a dense linear subspace [...] Read more.
Let N be the number operator in the space H of real-valued square-integrable Bernoulli functionals. In this paper, we further pursue properties of N from a probabilistic perspective. We first construct a nuclear space G, which is also a dense linear subspace of H, and then by taking its dual G*, we obtain a real Gel’fand triple GHG*. Using the well-known Minlos theorem, we prove that there exists a unique Gauss measure γN on G* such that its covariance operator coincides with N. We examine the properties of γN, and, among others, we show that γN can be represented as a convolution of a sequence of Borel probability measures on G*. Some other results are also obtained. Full article
(This article belongs to the Special Issue Applications of Functional Analysis in Quantum Physics)
21 pages, 372 KiB  
Article
Symmetry Groups, Quantum Mechanics and Generalized Hermite Functions
by Enrico Celeghini, Manuel Gadella and Mariano A. del Olmo
Mathematics 2022, 10(9), 1448; https://doi.org/10.3390/math10091448 - 26 Apr 2022
Cited by 4 | Viewed by 2237
Abstract
This is a review paper on the generalization of Euclidean as well as pseudo-Euclidean groups of interest in quantum mechanics. The Weyl–Heisenberg groups, Hn, together with the Euclidean, En, and pseudo-Euclidean Ep,q, groups are two [...] Read more.
This is a review paper on the generalization of Euclidean as well as pseudo-Euclidean groups of interest in quantum mechanics. The Weyl–Heisenberg groups, Hn, together with the Euclidean, En, and pseudo-Euclidean Ep,q, groups are two families of groups with a particular interest due to their applications in quantum physics. In the present manuscript, we show that, together, they give rise to a more general family of groups, Kp,q, that contain Hp,q and Ep,q as subgroups. It is noteworthy that properties such as self-similarity and invariance with respect to the orientation of the axes are properly included in the structure of Kp,q. We construct generalized Hermite functions on multidimensional spaces, which serve as orthogonal bases of Hilbert spaces supporting unitary irreducible representations of groups of the type Kp,q. By extending these Hilbert spaces, we obtain representations of Kp,q on rigged Hilbert spaces (Gelfand triplets). We study the transformation laws of these generalized Hermite functions under Fourier transform. Full article
(This article belongs to the Special Issue Applications of Functional Analysis in Quantum Physics)
20 pages, 352 KiB  
Article
On the Schrödinger Equation for Time-Dependent Hamiltonians with a Constant Form Domain
by Aitor Balmaseda, Davide Lonigro and Juan Manuel Pérez-Pardo
Mathematics 2022, 10(2), 218; https://doi.org/10.3390/math10020218 - 11 Jan 2022
Cited by 4 | Viewed by 1680
Abstract
We study two seminal approaches, developed by B. Simon and J. Kisyński, to the well-posedness of the Schrödinger equation with a time-dependent Hamiltonian. In both cases, the Hamiltonian is assumed to be semibounded from below and to have a constant form domain, but [...] Read more.
We study two seminal approaches, developed by B. Simon and J. Kisyński, to the well-posedness of the Schrödinger equation with a time-dependent Hamiltonian. In both cases, the Hamiltonian is assumed to be semibounded from below and to have a constant form domain, but a possibly non-constant operator domain. The problem is addressed in the abstract setting, without assuming any specific functional expression for the Hamiltonian. The connection between the two approaches is the relation between sesquilinear forms and the bounded linear operators representing them. We provide a characterisation of the continuity and differentiability properties of form-valued and operator-valued functions, which enables an extensive comparison between the two approaches and their technical assumptions. Full article
(This article belongs to the Special Issue Applications of Functional Analysis in Quantum Physics)
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