Jean Bellissard

Professor

Education

Ph.D., Universite de Provence, Marseille, France 1974

Research Interests

Mathematical Theory of Aperiodic Solids

Mathematical Theory of Aperiodic Solids:

- Structure: the Hull, transversal, equilibrium Gibbs state, diffraction measure.
- Application to: Bloch electrons in magnetic field, quasycrystals, semiconductors at low temperature.
- Noncommutative Geometry: the noncommutative Brillouin zone, electrons & phonons.
- Topological invariants:K-theory and Gap Labeling theorem
- Electronic and spectral properties of aperiodic media: Cantor and Singular spectra for electron or phonons. Anderson localization, weak localization and Random Matrix Theory.
- Coherent transport: spectral and transport exponents
- Dissipative transport: the Greene-Kubo formulae, models for dissipation, Mott's variable range hopping.
- Noncommutative Geometry: cyclic cohomology, application to the theory of the Quantum Hall effect.

- "Noncommutative Geometry of Aperiodic Solids ", in "Geometric and Topological Methods for Quantum Field Theory", (Villa de Leyva, 2001), pp. 86-156, World Sci. Publishing, River Edge, NJ, (2003).
- "Coherent and dissipative transport in aperiodic solids", Published in Dynamics of Dissipation P. Garbaczewski, R. Olkiewicz (Eds.), Lecture Notes in Physics,
__597__, Springer (2003), pp. 413-486.

Proceedings of the 38th Winter School of Theoretical Physics, Ladek, Poland, 6-15 Feb 2002. - "Spaces of Tilings, Finite Telescopic Approximations and Gap-labelling", with R. Benedetti, J.-M. Gambaudo, to appear in Commun. Math. Phys., (2004).
- "A Kinetic Model for Quantum Jumps", with D. Spehner, J. Stat. Phys.,
__104__, 525-572, (2001). - "Hull of Aperiodic Solids and Gap Labelling Theorems", with D. Hermmann, M. Zarrouati, in "Directions in Mathematical Quasicrystals", CRM Monograph Series, Volume 13, 207-259, (2000),

M.B. Baake and R.V. Moody Eds., AMS Providence. - "A Kinetic Theory for Quantum Transport in Aperiodic Media", with H. Schulz-Baldes, J. Stat. Phys.,
__91__, 991-1026, (1998). - "de Haas-van Alphen oscillations and magnetic breakdown: semiclassical calculation of multi-band orbits", with Jean-Yves Fortin, Miguel Gusmao, Timothy Ziman, Phys. Rev., B
__57__, (3), 1484-1497, (1998). - "The Non Commutative Geometry of the Quantum Hall Effect" with H. Schulz-Baldes, A. van Elst, J. Math. Phys.,
__35__, 5373-5471, (1994). - "Non Commutative Methods in Semiclassical Analysis", a course given at the CIME (1991). Published in "Transition to Chaos in Classical and Quantum Mechanics", Lecture Notes in Mathematics, 1589, Springer, (1994).
- "Gap Labelling Theorems for Schr?dinger's Operators", in "From Number Theory to Physics" pp.538-630, Les Houches March 89, Springer, J.M. Luck, P. Moussa, M. Waldschmidt Eds., (1993).
- "Renormalization Group Analysis and Quasicrystals", in "Ideas and Methods in Analysis, Stochastics, and Applications (in memory of Raphael H?egh-Krohn)", S. Albeverio, J. E. Fenstad, H. Holden, T. Lindstr?m, editors, Cambridge University Press, Cambridge, 1992.
- "Spectral Properties of a tight binding Hamiltonian with Period Doubling Potential'' with A. Bovier, J.M. Ghez, Comm. Math. Phys.,
__135__, (1991), 379-399. - "Heisenberg's Picture and Non Commutative Geometry of the Semiclassical Limit'' with M. Vittot, Ann. Inst. H. Poincare,
__52__, (1990),175-235. - "An Algebraic Semi-Classical Approach to Bloch Electrons in a Magnetic Field'', with R. Rammal, J. de Phys. (France),
__51__, (1990), 1803-1830. - "Spectral Properties of One-Dimensional Quasicrystals'' with B. Iochum, E. Scoppola, D. Testard, Comm. Math. Phys.,
__125__,(1989), 527-543. - "K-Theory of C*-algebras in Solid State Physics'' in "Statistical Mechanics and Field Theory, Mathematical Aspects'', T.C. Dorlas, M.N. Hugenholtz and M. Winnink, Lecture Notes in Physics,
__257__, (1986), 99-156. - "Stability and Instability in Quantum Mechanics'' in "Trends and developments in the Eighties", S. Albeverio \& Ph. Blanchard Eds, World Scientific Pub. Co., Singapore (1985).
- "Cantor Spectrum for the Almost Mathieu Equation'' with B. Simon, J.Funct. Anal.,
__48__, 408-419, (1982).