## 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

**Quantum States and Hardy's Formulation of the Uncertainty Principle: a Symplectic Approach. (With Franz Luef). Lett. Math. Phys., Published online: 4 April 2007.****Abelian Gerbes as a Gauge Theory of Quantum Mechanics on Phase Space (With José Isidro). J. Phys. A: Math. Theor. 40 3549-3567, 2007.****A Gauge Theory of Quantum Mechanics. Modern Phys. Lett. A. (With José Isidro). Mod. Phys. Letters A, 22(3), 191-200, 2007.****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

**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.**Uncertainty Principle, Phase Space Ellipsoids, and Weyl Calculus. In: Operator Theory: advances and applications, Ed. Wong, Vol. 164, 2006 Birkhäuser Verlag**

Ch9Wong.pdf**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**Metaplectic Representation, Conley-Zehnder Index, and Weyl Calculus on Symplectic Phase Space**

http://www.freewebs.com/cvdegosson/MetaWeylCZ.pdf**Non-Squeezing Theorems, Quantization of Integrable Systems, and Quantum Uncertainty**

http://www.freewebs.com/cvdegosson/Symplecticetc.pdf

## Publications and Preprints 2005

**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.**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**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.**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.**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

- The classical and quantum evolution of Lagrangian half-forms. Ann. Inst. Henri Poincar 70(6), 1999 http://www.freewebs.com/cvdegosson/AIHP.pdf
- The structure of q-symplectic geometry. Jour. Math. Pures et Appl. 71(5), 1992
- 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.