For the study of infinite discrete systems on phase space, the three-dimensional Lorentz algebra and group, and , provide a discrete model of the repulsive oscillator. Its eigenfunctions are found in the principal irreducible representation series, where the compact generator—that we identify with the position operator—has the infinite discrete spectrum of the integers , while the spectrum of energies is a double continuum. The right- and left-moving wavefunctions are given by hypergeometric functions that form a Dirac basis for . Under contraction, the discrete system limits to the well-known quantum repulsive oscillator. Numerical computations of finite approximations raise further questions on the use of Dirac bases for infinite discrete systems.
We examine the evolution in phase space of an N-point signal, produced and sensed at finite arrays transverse to a planar waveguide within the framework of the finite quantization of geometric optics. We use the Kravchuk coherent states provided by the finite oscillator model to evince the nonlinear transformations that elliptic-profile waveguides produce on phase space by means of the SO(3) Wigner function.
The discrete Fourier transform (DFT) matrix has a manifold of fractionalizations that depend on the choice of its eigenbases. One prominent basis is that of Mehta functions; here we examine a family of fractionalizations of the DFT stemming from q-extensions of this basis. Although closed expressions are given, many results of our analysis derive from numerical computation and display. Thus we suggest that the account of fractional Fourier transformations applied on signals as presented by other authors—typically of a centred rectangle function—may be biased because the support of the function lies in the central part of the domain only. The phase and amplitude of the whole fractional DFT matrices reveal the location of departures from the continuous kernel of the fractional Fourier integral transform, whose phase and constant amplitude are well known.
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