scholarly journals Hermite functions and special Hermite functions  depending on Heisenberg group (ℍⁿ ): دوال هرميت ودوال هرميت الخاصة اعتماداً على زمرة هايزنبرغ (ℍⁿ)

2020 ◽  
Vol 4 (2) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Representation Theory ◽  
Harmonic Oscillator ◽  
Heisenberg Group ◽  
Hermite Functions ◽  
Hermite Operator ◽  
Twisted Laplacian ◽  
Special Hermite Functions ◽  
Hermite Operators ◽  
The Relationship ◽  
Hermite Expansions

In this paper, we talk about Heisenberg group, the most known example from the lie groups. After that, we talk about the representation theory of this group, and the relationship between the representation theory of the Heisenberg group and the position and momentum operator and momentum operators (ors). relationship between the representation theory of the Heisenberg group and the position and momentum, that shows how we will make the connection between the Heisenberg group and physics. Then we introduce and study some properties of the Hermite and special Hermite functions. These functions are eigenfunctions of the Hermite and special Hermite operators, respectively. The Hermite operator is often called the harmonic oscillator and the special Hermite operator is sometimes called the twisted Laplacian. As we will later see, the two operators are directly related to the sub-laplacian on the Heisenberg group. The theory of Hermite and special Hermite expansions is intimately connected to the harmonic analysis on the Heisenberg group. They play an important role in our understanding of several problems on ℍⁿ .

Heat kernels and Green functions of sub-Laplacians on Heisenberg groups with multi-dimensional center

2018 ◽  
Vol 13 (4) ◽  
pp. 38
Author(s):  
Shahla Molahajloo ◽  
M.W. Wong
Keyword(s):  
Fourier Transform ◽  
Green Function ◽  
Heisenberg Group ◽  
Inverse Fourier Transform ◽  
Heat Kernels ◽  
Hermite Functions ◽  
Green Functions ◽  
Heisenberg Groups ◽  
Explicit Formulas ◽  
Special Hermite Functions

We compute the sub-Laplacian on the Heisenberg group with multi-dimensional center. By taking the inverse Fourier transform with respect to the center, we get the parametrized twisted Laplacians. Then by means of the special Hermite functions, we find the eigenfunctions and the eigenvalues of the twisted Laplacians. The explicit formulas for the heat kernels and Green functions of the twisted Laplacians can then be obtained. Then we give an explicit formula for the heat kernal and Green function of the sub-Laplacian on the Heisenberg group with multi-dimensional center.

scholarly journals The role of Heisenberg group in Harmonic analysis: دور زمرة هايزنبرغ في التحليل التوافقي

2019 ◽  
Vol 3 (3) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Representation Theory ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Lie Groups ◽  
Mathematical Concepts ◽  
Group Fourier Transform ◽  
The Relationship

In this paper we talk about Heisenberg group, the most know example from the lie groups. After that we discuss the representation theory of this group, and the relationship between the representation theory of the Heisenberg group and the position and momentum operatorsو and momentum operators.ors. ielationship between the representation theory of the Heisenberg group and the position and momen, that shows how we will make the connection between the Heisenberg group and physics. we have considered only the Schr dinger picture. That is, all the representations we considered are realized on the Hilbert space . we define the group Fourier transform on the Heisenberg group as an operator valued function, and other facts and properties. The main aim of our research is having the formula of Schr dinger Representation that connect physics with the Heisenberg group. Depending on this Representation we will study new formulas for some mathematical concepts such us Fourier Transform and  .

scholarly journals Heisenberg–Weyl Groups and Generalized Hermite Functions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1060
Author(s):  
Enrico Celeghini ◽  
Manuel Gadella ◽  
Mariano A. del del Olmo
Keyword(s):  
Hilbert Space ◽  
Fourier Transform ◽  
Heisenberg Group ◽  
Real Line ◽  
Hermite Functions ◽  
Unitary Representations ◽  
Weyl Groups ◽  
The Real ◽  
Symmetry Properties ◽  
The Fourier Transform

We introduce a multi-parameter family of bases in the Hilbert space L2(R) that are associated to a set of Hermite functions, which also serve as a basis for L2(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 Weyl–Heisenberg group and some of their extensions.

scholarly journals Superunitary Representations of Heisenberg Supergroups

2018 ◽  
Vol 2020 (19) ◽  
pp. 5926-6006 ◽  
Author(s):  
Axel de Goursac ◽  
Jean-Philippe Michel
Keyword(s):  
Representation Theory ◽  
Heisenberg Group ◽  
Fourier Transformation ◽  
Coadjoint Orbits ◽  
New Approach ◽  
Von Neumann ◽  
Neumann Theorem ◽  
Lie Supergroups ◽  
Definition Of ◽  
Von Neumann Theorem

Abstract Numerous Lie supergroups do not admit superunitary representations (SURs) except the trivial one, for example, Heisenberg and orthosymplectic supergroups in mixed signature. To avoid this situation, we introduce in this paper a broader definition of SUR, relying on a new definition of Hilbert superspace. The latter is inspired by the notion of Krein space and was developed initially for noncommutative supergeometry. For Heisenberg supergroups, this new approach yields a smooth generalization, whatever the signature, of the unitary representation theory of the classical Heisenberg group. First, we obtain Schrödinger-like representations by quantizing generic coadjoint orbits. They satisfy the new definition of irreducible SURs and serve as ground to the main result of this paper: a generalized Stone–von Neumann theorem. Then, we obtain the superunitary dual and build a group Fourier transformation, satisfying Parseval theorem. We eventually show that metaplectic representations, which extend Schrödinger-like representations to metaplectic supergroups, also fit into this definition of SURs.

Classical Limit and Chaotic Regime in a Semi-Quantum Hamiltonian

2003 ◽  
Vol 13 (08) ◽  
pp. 2315-2325 ◽  
Author(s):  
A. M. Kowalski ◽  
M. T. Martin ◽  
A. Plastino ◽  
A. N. Proto
Keyword(s):  
Harmonic Oscillator ◽  
Classical Limit ◽  
Chaotic Regime ◽  
Classical Particle ◽  
Quantum Harmonic Oscillator ◽  
Quantum Dynamical ◽  
Quartic Term ◽  
Quantum Hamiltonian ◽  
The Relationship ◽  
Full Quantum

Based on a quantum dynamical invariant of motion, I, we study the classical limit of a semiclassical Hamiltonian composed by a full quantum harmonic oscillator plus a classical particle plus a "semiclassical" coupling quartic term. The motion-invariant is closely related to the uncertainty principle. The classical limit (CL) is determined by the relationship between I and the total energy of the system, defining an adimensional invariant Er. We find that the CL coincides with the results of a purely classical treatment. Both invariants allow to follow the transit between quantum nonchaotic to the classical chaotic regime. Particularly, with Er we define the threshold above which chaos appears, and the interval during which both regimes co-exist.

On the representation theory of quantum Heisenberg group and algebra

1994 ◽  
Vol 44 (11-12) ◽  
pp. 1019-1027
Author(s):  
Demosthenes Ellinas ◽  
Jan Sobczy
Keyword(s):  
Representation Theory ◽  
Heisenberg Group ◽  
And Algebra

scholarly journals Studying in Lee groups and most important examples of it (Heisenberg group): دراسة في زمر لي وأهم الأمثلة عنها (زمرة هايزنبرغ)

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Number Theory ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Representation Theory ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Lie Groups ◽  
General Definition ◽  
Nilpotent Lie Groups ◽  
Basic Facts

In this research, we present some basic facts about Lie algebra and Lie groups. We shall require only elementary facts about the general definition and knowledge of a few of the more basic groups, such as Euclidean groups. Then we introduce the Heisenberg group which is the most well-known example from the realm of nilpotent Lie groups and plays an important role in several branches of mathematics, such as representation theory, partial differential equations and number theory... It also offers the greatest opportunity for generalizing the remarkable results of Euclidean harmonic analysis.

scholarly journals From linear algebra to quantum information

2021 ◽  
pp. 032-047
Author(s):  
Yu LW ◽  
Wang NL ◽  
Kanemitsu S
Keyword(s):  
Functional Analysis ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Information ◽  
Linear Algebra ◽  
Smooth Transition ◽  
Quantum Computers ◽  
Hermite Operator ◽  
Hermite Matrix ◽  
Hermite Operators

Anticipating the realization of quantum computers, we propose the most reader-friendly exposition of quantum information and qubits theory. Although the latter lies within framework of linear algebra, it has some fl avor of quantum mechanics and it would be easier to get used to special symbols and terminologies. Quantum mechanics is described in the language of functional analysis: the state space (the totality of all states) of a quantum system is a Hilbert space over the complex numbers and all mechanical quantities are taken as Hermite operators. Hence some basics of functional analysis is necessary. We make a smooth transition from linear algebra to functional analysis by comparing the elements in these theories: Hilbert space vs. fi nite dimensional vector space, Hermite operator vs. linear map given by a Hermite matrix. Then from Newtonian mechanics to quantum mechanics and then to the theory of qubits. We elucidate qubits theory a bit by accommodating it into linear algebra framework under these precursors.

scholarly journals Gelfand―Tsetlin Polytopes and Feigin―Fourier―Littelmann―Vinberg Polytopes as Marked Poset Polytopes

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Thomas Bliem ◽  
Dido Salazar
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Ehrhart Polynomial ◽  
International Audience ◽  
Order Polytope ◽  
Finite Partially Ordered Set ◽  
The Relationship

International audience Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras. Stanley (1986) a montré que chaque ensemble fini partiellement ordonné permet de définir deux polyèdres, le polyèdre de l'ordre et le polyèdre des cha\^ınes. Ces polyèdres ont le même polynôme de Ehrhart, bien qu'ils soient tout à fait distincts du point de vue combinatoire. On généralise ce résultat à une famille plus générale de polyèdres, construits à partir d'un ensemble partiellement ordonné ayant des entiers attachés à certains de ses éléments. Par cette construction, on explique en termes combinatoires la relation entre les polyèdres de Gelfand-Tsetlin (1950) et ceux de Feigin-Fourier-Littelmann-Vinberg (2010, 2005), qui apparaissent dans la théorie des représentations des algèbres de Lie linéaires spéciales. On utilise les polyèdres de Gelfand-Tsetlin généralisés par Berenstein et Zelevinsky (1989) afin d'obtenir des analogues (conjecturés) des polytopes de Feigin-Fourier-Littelmann-Vinberg pour les algèbres de Lie symplectiques et orthogonales impaires.

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