Quantum Physics and Signal Processing in Rigged Hilbert Spaces by means of Special Functions, Lie Algebras and Fourier and Fourier-like Transforms

Quantum Mechanics and Signal Processing in the line R, are strictly related to Fourier Transform and Weyl-Heisenberg algebra. We discuss here the addition of a new discrete variable that measures the degree of the Hermite functions and allows to obtain the projective algebra io(2). A Rigged Hilbert space is found and a new discrete basis in R obtained. The operators {O[R]} defined on R are shown to belong to the Universal Enveloping Algebra UEA[io(2)] allowing, in this way, their algebraic discussion. Introducing in the half-line a Fourier-like Transform, the procedure is extended to R^+ and can be easily generalized to R^n and to spherical reference systems.Comment: 12 pages, Contribution to the 30th International Colloquium on Group Theoretical Methods in Physics, July 14-18, 2014, Gent (Belgium

SU(2), Associated Laguerre Polynomials and Rigged Hilbert Spaces

We present a family of unitary irreducible representations of SU(2) realized in the plane, in terms of the Laguerre polynomials. These functions are similar to the spherical harmonics defined on the sphere. Relations with an space of square integrable functions defined on the plane, $L^2(\R^2)$, are analyzed. We have also enlarged this study using rigged Hilbert spaces that allow to work with iscrete and continuous bases like is the case here.Comment: 10 page

Intertwining Symmetry Algebras of Quantum Superintegrable Systems

We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwinig operators, that span pairs of Lie algebras like $(su(n),so(2n))$ or $(su(p,q),so(2p,2q))$. The eigenstates of the associated Hamiltonian hierarchies belong to unitary representations of these algebras. It is shown that these intertwining operators, related with separable coordinates for the system, are very useful to determine eigenvalues and eigenfunctions of the Hamiltonians in the hierarchy. An study of the corresponding superintegrable classical systems is also included for the sake of completness

Groups, Special Functions and Rigged Hilbert Spaces

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Symmetry Groups, Quantum Mechanics and Generalized Hermite Functions

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

Unitary-Preserving Holography in dSd_d

This letter introduces a unitarity-preserving holographic correspondence within a $d$-dimensional de Sitter (dS$_d$) spacetime, distinctly challenging the prevailing notion that the holographic framework of dS$_d$ falls short in maintaining unitarity. The proposed approach is rooted in the geometry of the complex dS$_d$ spacetime and leverages the inherent properties of the (global) dS$_d$ plane waves, as defined within their designated tube domains.Comment: 6 pages, no figur