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19 pages, 351 KB

Open AccessArticle

Octonion Special Affine Fourier Transform: Pitt’s Inequality and the Uncertainty Principles

by Mohammad Younus Bhat, Aamir Hamid Dar, Mohra Zayed and Serkan Araci

Fractal Fract. 2023, 7(5), 356; https://doi.org/10.3390/fractalfract7050356 - 27 Apr 2023

Cited by 4 | Viewed by 1735

The special affine Fourier transform (SAFT) is an extended version of the classical Fourier transform and incorporates various signal processing tools which include the Fourier transforms, the fractional Fourier transform, the linear canonical transform, and other related transforms. This paper aims to introduce [...] Read more.

The special affine Fourier transform (SAFT) is an extended version of the classical Fourier transform and incorporates various signal processing tools which include the Fourier transforms, the fractional Fourier transform, the linear canonical transform, and other related transforms. This paper aims to introduce a novel octonion special affine Fourier transform (

O -

SAFT) and establish several classes of uncertainty inequalities for the proposed transform. We begin by studying the norm split and energy conservation properties of the proposed (

O -

SAFT). Afterwards, we generalize several uncertainty relations for the (

O -

SAFT) which include Pitt’s inequality, Heisenberg–Weyl inequality, logarithmic uncertainty inequality, Hausdorff–Young inequality, and local uncertainty inequalities. Finally, we provide an illustrative example and some possible applications of the proposed transform. Full article

(This article belongs to the Special Issue New Insights into Nonlinear Coupled Differential Equations with Its Applications)

8 pages, 283 KB

Open AccessArticle

Energy Transition Density of Driven Chaotic Systems: A Compound Trace Formula

by Alfredo M. Ozorio de Almeida

Quantum Rep. 2022, 4(4), 558-565; https://doi.org/10.3390/quantum4040040 - 30 Nov 2022

Cited by 1 | Viewed by 1914

Abstract

Oscillations in the probability density of quantum transitions of the eigenstates of a chaotic Hamiltonian within classically narrow energy ranges have been shown to depend on closed compound orbits. These are formed by a pair of orbit segments, one in the energy shell [...] Read more.

Oscillations in the probability density of quantum transitions of the eigenstates of a chaotic Hamiltonian within classically narrow energy ranges have been shown to depend on closed compound orbits. These are formed by a pair of orbit segments, one in the energy shell of the original Hamiltonian and the other in the energy shell of the driven Hamiltonian, with endpoints that coincide. Viewed in the time domain, the same pair of trajectory segments arises in the semiclassical evaluation of the trace of a compound propagator: the product of the complex exponentials of the original Hamiltonian and of its driven image. It is shown here that the probability density is the double Fourier transform of this trace, and that the closed compound orbits emulate the role played by the periodic orbits in Gutzwiller’s trace formula in its semiclassical evaluation. The phase of the oscillations with the energies or the evolution parameters agree with those previously obtained, whereas the amplitude of the contribution of each closed compound orbit is more compact and independent of any feature of the Weyl–Wigner representation in which the calculation was carried out. Full article

(This article belongs to the Special Issue Continuous and Discrete Phase-Space Methods and Their Applications)

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21 pages, 372 KB

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 2902

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)

25 pages, 13254 KB

Open AccessArticle

Quantum Particle on Dual Weight Lattice in Weyl Alcove

by Adam Brus, Jiří Hrivnák and Lenka Motlochová

Symmetry 2021, 13(8), 1338; https://doi.org/10.3390/sym13081338 - 24 Jul 2021

Cited by 4 | Viewed by 2391

Abstract

Families of discrete quantum models that describe a free non-relativistic quantum particle propagating on rescaled and shifted dual weight lattices inside closures of Weyl alcoves are developed. The boundary conditions of the presented discrete quantum billiards are enforced by precisely positioned Dirichlet and [...] Read more.

Families of discrete quantum models that describe a free non-relativistic quantum particle propagating on rescaled and shifted dual weight lattices inside closures of Weyl alcoves are developed. The boundary conditions of the presented discrete quantum billiards are enforced by precisely positioned Dirichlet and Neumann walls on the borders of the Weyl alcoves. The amplitudes of the particle’s propagation to neighbouring positions are determined by a complex-valued dual-weight hopping function of finite support. The discrete dual-weight Hamiltonians are obtained as the sum of specifically constructed dual-weight hopping operators. By utilising the generalised dual-weight Fourier–Weyl transforms, the solutions of the time-independent Schrödinger equation together with the eigenenergies of the quantum systems are exactly resolved. The matrix Hamiltonians, stationary states and eigenenergies of the discrete models are exemplified for the rank two cases

C_{2}

and

G_{2}

. Full article

(This article belongs to the Section Physics)

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21 pages, 363 KB

Open AccessArticle

Heisenberg–Weyl Groups and Generalized Hermite Functions

by Enrico Celeghini, Manuel Gadella and Mariano A. del Olmo

Symmetry 2021, 13(6), 1060; https://doi.org/10.3390/sym13061060 - 12 Jun 2021

Cited by 4 | Viewed by 2988

Abstract

We introduce a multi-parameter family of bases in the Hilbert space

L^{2} (R)

that are associated to a set of Hermite functions, which also serve as a basis for

L^{2} (R)

. The Hermite functions are eigenfunctions [...] Read more.

We introduce a multi-parameter family of bases in the Hilbert space

L^{2} (R)

that are associated to a set of Hermite functions, which also serve as a basis for

L^{2} (R)

. The Hermite functions are eigenfunctions of the Fourier transform, a property that is, in some sense, shared by these “generalized Hermite functions”. The construction of these new bases is grounded on some symmetry properties of the real line under translations, dilations and reflexions as well as certain properties of the Fourier transform. We show how these generalized Hermite functions are transformed under the unitary representations of a series of groups, including the Heisenberg–Weyl group and some of their extensions. Full article

(This article belongs to the Special Issue Recent Advances in the Application of Symmetry Group)

28 pages, 417 KB

Open AccessArticle

Connecting (Anti)Symmetric Trigonometric Transforms to Dual-Root Lattice Fourier–Weyl Transforms

by Adam Brus, Jiří Hrivnák and Lenka Motlochová

Symmetry 2021, 13(1), 61; https://doi.org/10.3390/sym13010061 - 31 Dec 2020

Cited by 1 | Viewed by 2760

Abstract

Explicit links of the multivariate discrete (anti)symmetric cosine and sine transforms with the generalized dual-root lattice Fourier–Weyl transforms are constructed. Exact identities between the (anti)symmetric trigonometric functions and Weyl orbit functions of the crystallographic root systems

A_{1}

and

C_{n}

are utilized [...] Read more.

Explicit links of the multivariate discrete (anti)symmetric cosine and sine transforms with the generalized dual-root lattice Fourier–Weyl transforms are constructed. Exact identities between the (anti)symmetric trigonometric functions and Weyl orbit functions of the crystallographic root systems

A_{1}

and

C_{n}

are utilized to connect the kernels of the discrete transforms. The point and label sets of the 32 discrete (anti)symmetric trigonometric transforms are expressed as fragments of the rescaled dual root and weight lattices inside the closures of Weyl alcoves. A case-by-case analysis of the inherent extended Coxeter–Dynkin diagrams specifically relates the weight and normalization functions of the discrete transforms. The resulting unique coupling of the transforms is achieved by detailing a common form of the associated unitary transform matrices. The direct evaluation of the corresponding unitary transform matrices is exemplified for several cases of the bivariate transforms. Full article

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26 pages, 6117 KB

Open AccessFeature PaperArticle

Central Splitting of A₂ Discrete Fourier–Weyl Transforms

by Jiří Hrivnák, Mariia Myronova and Jiří Patera

Symmetry 2020, 12(11), 1828; https://doi.org/10.3390/sym12111828 - 4 Nov 2020

Cited by 3 | Viewed by 2550

Abstract

Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group

A_{2}

constitute the kernels of the considered transforms. The central splitting of any [...] Read more.

Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group

A_{2}

constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of

A_{2}

is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified. Full article

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38 pages, 532 KB

Open AccessArticle

Generalized Dual-Root Lattice Transforms of Affine Weyl Groups

by Tomasz Czyżycki, Jiří Hrivnák and Lenka Motlochová

Symmetry 2020, 12(6), 1018; https://doi.org/10.3390/sym12061018 - 16 Jun 2020

Cited by 8 | Viewed by 2864

Abstract

Discrete transforms of Weyl orbit functions on finite fragments of shifted dual root lattices are established. The congruence classes of the dual weight lattices intersected with the fundamental domains of the affine Weyl groups constitute the point sets of the transforms. The shifted [...] Read more.

Discrete transforms of Weyl orbit functions on finite fragments of shifted dual root lattices are established. The congruence classes of the dual weight lattices intersected with the fundamental domains of the affine Weyl groups constitute the point sets of the transforms. The shifted weight lattices intersected with the fundamental domains of the extended dual affine Weyl groups form the sets of labels of Weyl orbit functions. The coinciding cardinality of the point and label sets and corresponding discrete orthogonality relations of Weyl orbit functions are demonstrated. The explicit counting formulas for the numbers of elements contained in the point and label sets are calculated. The forward and backward discrete Fourier-Weyl transforms, together with the associated interpolation and Plancherel formulas, are presented. The unitary transform matrices of the discrete transforms are exemplified for the case

A_{2}

. Full article

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26 pages, 2457 KB

Open AccessArticle

Discrete Orthogonality of Bivariate Polynomials of A₂, C₂ and G₂

by Jiří Hrivnák, Jiří Patera and Marzena Szajewska

Symmetry 2019, 11(6), 751; https://doi.org/10.3390/sym11060751 - 3 Jun 2019

Viewed by 2444

Abstract

We develop discrete orthogonality relations on the finite sets of the generalized Chebyshev nodes related to the root systems

A_{2}

,

C_{2}

and

G_{2}

. The orthogonality relations are consequences of orthogonality of four types of Weyl orbit functions on [...] Read more.

We develop discrete orthogonality relations on the finite sets of the generalized Chebyshev nodes related to the root systems

A_{2}

,

C_{2}

and

G_{2}

. The orthogonality relations are consequences of orthogonality of four types of Weyl orbit functions on the fragments of the dual weight lattices. A uniform recursive construction of the polynomials as well as explicit presentation of all data needed for the discrete orthogonality relations allow practical implementation of the related Fourier methods. The polynomial interpolation method is developed and exemplified. Full article

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16 pages, 314 KB

Open AccessArticle

Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane

by Hervé Bergeron and Jean-Pierre Gazeau

Entropy 2018, 20(10), 787; https://doi.org/10.3390/e20100787 - 13 Oct 2018

Cited by 8 | Viewed by 4048

Abstract

Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary [...] Read more.

Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all resources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. We first review this quantization scheme before revisiting the cases where symmetry covariance is described by the Weyl-Heisenberg group and the affine group respectively, and we emphasize the fundamental role played by Fourier transform in both cases. As an original outcome of our generalisations of the Wigner-Weyl transform, we show that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

26 pages, 314 KB

Open AccessArticle

The Fractional Orthogonal Derivative

by Enno Diekema

Mathematics 2015, 3(2), 273-298; https://doi.org/10.3390/math3020273 - 22 Apr 2015

Cited by 7 | Viewed by 4801

Abstract

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder [...] Read more.

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder in 2012. Here, an approximation of the Weyl or Riemann–Liouville fractional derivative is considered by replacing the n-th derivative by its approximation in the formula for the fractional derivative. In the case of, for instance, Jacobi polynomials, an explicit formula for the kernel of this approximate fractional derivative can be given. Next, we consider the fractional derivative as a filter and compute the frequency response in the continuous case for the Jacobi polynomials and in the discrete case for the Hahn polynomials. The frequency response in this case is a confluent hypergeometric function. A different approach is discussed, which starts with this explicit frequency response and then obtains the approximate fractional derivative by taking the inverse Fourier transform. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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