TY  - JOUR
A1  - Gosson, Maurice A. de
T1  - Extended Weyl calculus and application to the phase-space Schrodinger equation
N2  - We show that the Schrodinger equation in phase space proposed by Torres-Vega and Frederick is canonical in the sense that it is a natural consequence of the extended Weyl calculus obtained by letting the Heisenberg group act on functions (or half-densities) defined on phase space. This allows us, in passing, to solve rigorously the TF equation for all quadratic Hamiltonians
Y1  - 2005
ER  - 
TY  - JOUR
A1  - Gosson, Maurice A. de
A1  - Gosson, Serge M. de
T1  - Extension of the Conley-Zehnder index, a product formula, and an application to the Weyl representation of metaplectic operators
JF  - Journal of mathematical physics
N2  - The aim of this paper is to express the Conley-Zehnder index of a symplectic path in terms of an index due to Leray and which has been studied by one of us in a previous work. This will allow us to prove a formula for the Conley-Zehnder index of the product of two symplectic paths in terms of a symplectic Cayley transform. We apply our results to a rigorous study of the Weyl representation of metaplectic operators, which plays a crucial role in the understanding of semiclassical quantization of Hamiltonian systems exhibiting chaotic behavior.
Y1  - 2006
U6  - https://doi.org/10.1063/1.239066
SN  - 0022-2488
VL  - 47
IS  - 12
PB  - American Institute of Physics
CY  - Melville
ER  - 
TY  - JOUR
A1  - Gosson, Maurice A. de
T1  - On the Weyl representation of metaplectic operators
N2  - We study the Weyl representation of metaplectic operators associated to a symplectic matrix having no non- trivial fixed point, and justify a formula suggested in earlier work of Mehlig and Wilkinson. We give precise calculations of the associated Maslov-type indices; these indices intervene in a crucial way in Gutzwiller's formula of semiclassical mechanics, and are simply related to an index defined by Conley and Zehnder
Y1  - 2005
ER  - 
TY  - JOUR
A1  - Gosson, Maurice A. de
T1  - Symplectically covariant Schrodinger equation in phase space
N2  - A classical theorem of Stone and von Neumann states that the Schrodinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform, we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase-space Schrodinger equation
Y1  - 2005
ER  -