Symmetry Groups, Quantum Mechanics and Generalized Hermite Functions

This is a review paper on the generalization of Euclidean as well pseudo-Euclidean groups interest in quantum mechanics. The Weyl–Heisenberg groups, Hn, together with Euclidean, En, and Ep,q, are two families particular due to their applications physics. In present manuscript, we show that, together, they give rise more general family Kp,q, that contain Hp,q Ep,q subgroups. It noteworthy properties such self-similarity invariance respect orientation axes properly included structure Kp,q. We construct generalized Hermite functions multidimensional spaces, which serve orthogonal bases Hilbert spaces supporting unitary irreducible representations type By extending these obtain Kp,q rigged (Gelfand triplets). study transformation laws under Fourier transform.