2,314 research outputs found
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
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 or
. 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
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