@article{Gosson2005,
  author    = {Gosson, Maurice A. de},
  title     = {Extended Weyl calculus and application to the phase-space Schrodinger equation},
  year      = {2005},
  abstract  = {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},
  language  = {en}
}
@book{Gosson2005,
  author    = {Gosson, Maurice A. de},
  title     = {On the weyl representation of metapletic operators},
  series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  journal   = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  publisher = {Univ.},
  address   = {Potsdam},
  issn      = {1437-739X},
  pages     = {15 S.},
  year      = {2005},
  language  = {en}
}
@book{Gosson2005,
  author    = {Gosson, Maurice A. de},
  title     = {Extended weyl calculus and application to the phase-space scr{\"o}dinger equation},
  series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  journal   = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  publisher = {Univ.},
  address   = {Potsdam},
  issn      = {1437-739X},
  pages     = {7 S.},
  year      = {2005},
  language  = {en}
}
@article{Gosson2005,
  author    = {Gosson, Maurice A. de},
  title     = {On the Weyl representation of metaplectic operators},
  year      = {2005},
  abstract  = {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},
  language  = {en}
}
@article{Gosson2005,
  author    = {Gosson, Maurice A. de},
  title     = {Symplectically covariant Schrodinger equation in phase space},
  year      = {2005},
  abstract  = {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},
  language  = {en}
}
@unpublished{Gosson2005,
  author    = {Gosson, Maurice A. de},
  title     = {Extended Weyl calculus and application to the phase-space Schr{\"o}dinger equation},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29879},
  year      = {2005},
  abstract  = {We show that the Schr¨odinger equation in phase space proposed by Torres-Vega and Frederick is canonical in the sense that it is a natural consequence of the extendedWeyl 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.},
  language  = {en}
}
@unpublished{Gosson2005,
  author    = {Gosson, Maurice A. de},
  title     = {On the Weyl representation of metaplectic operators},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-29865},
  year      = {2005},
  abstract  = {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.},
  language  = {en}
}
@book{Gosson2006,
  author    = {Gosson, Maurice A. de},
  title     = {Symplectic geometry, wigner-weyl-moyal calculus, and quantum mechanics, in phase space ; Part 1},
  series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  journal   = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  publisher = {Univ.},
  address   = {Potsdam},
  issn      = {1437-739X},
  pages     = {367 S.},
  year      = {2006},
  language  = {en}
}
@book{Gosson2006,
  author    = {Gosson, Maurice A. de},
  title     = {Symplectic geometry, wigner-weyl-moyal calculus, and quantum mechanics, in phase space ; Part 2},
  series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  journal   = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  publisher = {Univ.},
  address   = {Potsdam},
  issn      = {1437-739X},
  pages     = {367 S.},
  year      = {2006},
  language  = {en}
}
@book{Gosson2006,
  author    = {Gosson, Maurice A. de},
  title     = {Symplectic geometry, wigner-weyl-moyal calculus, and quantum mechanics, in phase space ; Part 3},
  series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  journal   = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Arbeitsgruppe Partiell},
  publisher = {Univ.},
  address   = {Potsdam},
  issn      = {1437-739X},
  pages     = {367 S.},
  year      = {2006},
  language  = {en}
}
@article{GossonGosson2006,
  author    = {Gosson, Maurice A. de and Gosson, Serge M. de},
  title     = {Extension of the Conley-Zehnder index, a product formula, and an application to the Weyl representation of metaplectic operators},
  series = {Journal of mathematical physics},
  volume    = {47},
  journal   = {Journal of mathematical physics},
  number    = {12},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {0022-2488},
  doi       = {10.1063/1.239066},
  pages     = {15},
  year      = {2006},
  abstract  = {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.},
  language  = {en}
}
@unpublished{Gosson2006,
  author    = {Gosson, Maurice A. de},
  title     = {Symplectic geometry, Wigner-Weyl-Moyal calculus, and quantum mechanics in phase space},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-30212},
  year      = {2006},
  abstract  = {Contents: Part I: Symplectic Geometry Chapter 1: Symplectic Spaces and Lagrangian Planes Chapter 2: The Symplectic Group Chapter 3: Multi-Oriented Symplectic Geometry Chapter 4: Intersection Indices in Lag(n) and Sp(n) Part II: Heisenberg Group, Weyl Calculus, and Metaplectic Representation Chapter 5: Lagrangian Manifolds and Quantization Chapter 6: Heisenberg Group and Weyl Operators Chapter 7: The Metaplectic Group Part III: Quantum Mechanics in Phase Space Chapter 8: The Uncertainty Principle Chapter 9: The Density Operator Chapter 10: A Phase Space Weyl Calculus},
  language  = {en}
}