Recent Research

My new book Symplectic Geometry and Quantum Mechanics was published by Bikhäuser-Basel in June 2006, in the series "Operator Theory and Applications" (subseries: partial differential equations).  See the link:

http://www.springer.com/sgw/cda/frontpage/0,11855,4-151-69-1236131-0,00.html

From the blurb:

"This book is devoted to a rather complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a very readable introduction to symplectic geometry. Many topics are also of genuine interest for pure mathematicians working in geometry and topology. TOC:Preface.- Notation.- I. Symplectic Geometry.- 1. Symplectic Spaces and Lagrangian Planes.- 2. The Symplectic Group.- 3. Multi-Oriented Symplectic Geometry.- 4. Intersection Indices.- II. Heisenberg Group, Weyl Calculus, and Metaplectic Representation.- 5. Lagrangian Manifolds and Quantization.- 6. Heisenberg Group and Weyl Operators.- 7. The Metaplectic Group.- III. Quantum Mechanics in Phase Space.- 8. The Uncertainty Principle.- 9. The Density Operator.- 10. A Phase Space Weyl Calculus.- Appendices.- Bibliography.- Index."
 
Table of contents: click here http://www.freewebs.com/cvdegosson/TOC.pdf
Preface: click here http://www.freewebs.com/cvdegosson/TOC.pdf
 
If you want to download a preliminary (uncorrected!) version of the book you can go to the University of Potsdam preprint server
http://www.math.uni-potsdam.de/prof/a_partdiff/prepr/2006.html 
(three files).


Publication list

You are welcome to upload a complete list of my publications by clicking on the link below:
http://www.freewebs.com/cvdegosson/Publications.pdf
(updated on April 5, 2007)

Publications and Preprints 2007

  1. Quantum States and Hardy's Formulation of the Uncertainty Principle: a Symplectic Approach. (With Franz Luef). Lett. Math. Phys., Published online: 4 April 2007.
  2. Abelian Gerbes as a Gauge Theory of Quantum Mechanics on Phase Space (With José Isidro). J. Phys. A: Math. Theor. 40 3549-3567, 2007.
  3. A Gauge Theory of Quantum Mechanics. Modern Phys. Lett. A. (With José Isidro). Mod. Phys. Letters A, 22(3), 191-200, 2007.
  4. Remarks on the fact that the uncertainty principle doe not characterize the quantum state. (With Franz Luef). Phys. Lett. A. 364, 453-457, 2007.


Publications and Preprints 2006

  1. An Extension of the Conley-Zehnder Index, a Product Formula, and an Application to the Weyl Representation of Metaplectic Operators. To appear in Journal of Mathematical Physics, 41 (december 2006).
    http://www.freewebs.com/cvdegosson/JMPCZ.pdf. The Conley-Zehnder index plays a fundamental role in the theory of Hamiltonian periodic orbits. Ín this paper I express that indexd in terms of the cohomological index of Leray index. This allows me to extend the Conley-Zehnder index to arbitrary Hamiltonian paths, and to prove a fundamental product formula. Applications top the theory of the metaplectic group are given.
  2. Uncertainty Principle, Phase Space Ellipsoids, and Weyl Calculus. In: Operator Theory: advances and applications, Ed. Wong, Vol. 164, 2006 Birkhäuser Verlag
    Ch9Wong.pdf
  3. The adiabatic limit for multi-dimensional Hamiltonian systems; to appear in Journ. Geom. and Symmetry in Physics, 2006
    http://www.freewebs.com/cvdegosson/Adiabatic.pdf
  4. Metaplectic Representation, Conley-Zehnder Index, and Weyl Calculus on Symplectic Phase Space
    http://www.freewebs.com/cvdegosson/MetaWeylCZ.pdf
  5. Non-Squeezing Theorems, Quantization of Integrable Systems, and Quantum Uncertainty
    http://www.freewebs.com/cvdegosson/Symplecticetc.pdf


Publications and Preprints 2005

  1. Symplectically covariant Schrdinger equation in phase space. J. Phys. A:Math. Gen. 38, 2005
    JPhysA.SymplecticCov.pdf. One of the themes of this paper is that the Schrdinger equation in phase space is natural in the sense that it is compatible with the Stone-von Neumann theorem on the uniqueness (up to an isomorphism) of the representation of the canonical commutation relations. Another theme, of interest in harmonic analysis, is the construction of a modification and extension of Weyl calculus in which operators act on functions defined on symplectic phase space.
  2. Extended Weyl calculus and application to the phase-space Schrdinger equation. J. Phys. A:Math. Gen. 38, 2005
    ExtendedWeyl.pdf This paper has been downloaded more than 500 times from thIOP site. It is devoted to a rigorous mathematical justification and study of the Schrdinger equation in phase space proposed by Frederick and Torres.Vega
  3. On the Weyl Representation of Metaplectic Operators. Lett. Math. Phys. 72, 2005
    LettMathPhysWeylMetaplectic.pdf. In this paper I study the twisted Weyl symbol of metaplectic operators and its relation to the "Mehlig-Wilkinson formula" which plays an essential role in the understanding of Gutzwiller's trace formula for quantum systems exhibiting a classical chaotic behavior. It is also intrumental in the study of phase-space quantum mechanics. The intersting point is that the Conley-Zehnder index appears in the Weyl representation instead of the usual maslov index.
  4. Cellules quantiques symplectiques et fonctions de Husimi-Wigner. Bull. sci. math.129, 2005
    BSMcelulles.pdf. This paper (written in French) is about "quantum blobs" as symplectically invariant quantum cells. I show that to every phase-space ellipsoid with symplectic capcity one-half of Planck's constant one can associate canonically a configuration space Gaussian, unique up to phase factor. To the author's great satisfaction and pride this paper was ranked 11th among the most downloaded papers of Bull. sci. Math.
  5. Schrdinger equation in phase space and deformation quantization:
    http://www.arxiv.org/pdf/math.SG/0504013. In this short paper I discuss the relationship between one form of the Schrdinger equation in phase space and deformation quantization.


Some Previous Highlights

  1. The classical and quantum evolution of Lagrangian half-forms. Ann. Inst. Henri Poincar 70(6), 1999 http://www.freewebs.com/cvdegosson/AIHP.pdf
  2. The structure of q-symplectic geometry. Jour. Math. Pures et Appl. 71(5), 1992
  3. Maslov indices on the metaplectic group Mp(n). Ann. Inst. Fourier.40(3), 1990 AIF2.pdf


Previous books

  • The Principles of Newtonian and Quantum Mechanics: the Need for Planck's Constant h; with a foreword by B. Hiley. Imperial College Press (2001), ca 380 pages. Click here to read Basil Hiley's foreword: http://www.freewebs.com/cvdegosson/Foreword.pdf
  • Maslov Classes, Metaplectic Representation and Lagrangian Quantization. Mathematical Research 95, Wiley VCH (1997), ca 190 pages.


Edited volumes

Proceedings of the Karlskrona Conference in the Honor of Jean Leray, Springer (2003), ca 580 pages