Mathematics

Research

12 pages, 261 KiB

Open AccessArticle

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 1288

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

Open AccessArticle

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 2 | Viewed by 1946

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

Open AccessArticle

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 1537

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

Open AccessArticle

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 1308

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)

► Show Figures

Figure 1

23 pages, 438 KiB

Open AccessArticle

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 1170

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

λ \approx λ^{(E P)}

. 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)

► Show Figures

Figure 1

11 pages, 400 KiB

Open AccessFeature PaperArticle

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 1977

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)

► Show Figures

Figure 1

11 pages, 290 KiB

Open AccessArticle

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 1294

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

G \subset H \subset G^{*}

. Using the well-known Minlos theorem, we prove that there exists a unique Gauss measure

γ_{N}

on

G^{*}

such that its covariance operator coincides with N. We examine the properties of

γ_{N}

, and, among others, we show that

γ_{N}

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

Open AccessFeature PaperArticle

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 5 | Viewed by 2429

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,

H_{n}

, together with the Euclidean,

E_{n}

, and pseudo-Euclidean

E_{p, 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,

H_{n}

, together with the Euclidean,

E_{n}

, and pseudo-Euclidean

E_{p, 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,

K_{p, q}

, that contain

H_{p, q}

and

E_{p, 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

K_{p, 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

K_{p, q}

. By extending these Hilbert spaces, we obtain representations of

K_{p, 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

Open AccessArticle

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 1899

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)

Journal Menu

Journal Browser

Applications of Functional Analysis in Quantum Physics

Share This Special Issue

Special Issue Editor

Special Issue Information

Keywords

Benefits of Publishing in a Special Issue

Published Papers (9 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI