| Many commonly used time-frequency distributions are members of the Cohen class. This class is deened for continuous signals and since time-frequency distributions in the Cohen class are quadratic, the formulation for discrete signals is not straightforward. The Cohen class can be derived as the class of all quadratic time-frequency distributions that are covariant to time shifts and frequency...

We consider the problem of high-frequency asymptotics for the time-dependent onedimensional Schrödinger equation with rapidly oscillating initial data. This problem is commonly studied via the WKB method. An alternative method is based on the limit Wigner measure. This approach recovers geometrical optics, but, like the WKB method, it fails at caustics. To remedy this deficiency we employ the s...

The properties of spatial distribution of luminous matter are investigated analysing all the available three dimensional catalogues of galaxies. In standard view, galaxies are believed to have a fractal distribution at small scale with a crossover to an homogeneous one at large scale. However up to now, the quantitative determination of this presumed homogeneity scale is still lacking. Contrary...

Therefore, the potential V is subquadratic and the operator − ε 2 ∆q+V (q) is essentially self–adjoint on the Schwartz space. Such systems naturally arise in quantum chemistry when studying the dynamics of large molecules in the frame of Born-Oppenheimer approximation. The parameter ε is small because it encounters for the ratio of the mass of the electron and of the averaged mass of the nuclei...

As a technical supplement to the above mentioned paper we present 192 consecutive eigenstates for the Robnik billiard with the shape parameter λ = 0.15 from 10,001st to 10,192nd, by showing the plots in the configuration space and in the phase space. The latter is smoothed projection of the Wigner function onto the surface of section. By comparison with the classical SOS plots we thus examine a...

We prove that Wigner functions contain a symplectic connection. The latter covariantises the symplectic exterior derivative on phase space. We analyse the role played by this connection and introduce the notion of local symplectic covariance of quantum–mechanical states. This latter symmetry is at work in the Schroedinger equation on phase space.

We quantize non-commutative Maxwell theory canonically in the background field gauge for weak and slowly varying background fields. We determine the complete basis for expansion under such an approximation. As an application, we derive the Wigner function which determines the leading order high temperature behavior of the perturbative amplitudes of non-commutative Maxwell theory. To leading ord...

A unified mathematical formulation for designing and analyzing even the most general optical processor is presented. It exploits the Wigner distribution function to characterize the illumination, the input, the inherent filter, and the output results. To characterize the propagation of the light through the optical processor setup, we exploit the Wigner matrix formalism, which is appealing beca...