Papers by Jean Bellissard
Ergodic Theory and Dynamical Systems, Jun 1, 2009
Communications in Mathematical Physics, Jun 1, 1983
We study the spectrum of the almost Mathieu hamiltonian : where θ is an irrational number and x i... more We study the spectrum of the almost Mathieu hamiltonian : where θ is an irrational number and x is in the circle ΊΓ. For a small enough coupling constant μ and any x there is a closed energy set of non-zero measure in the absolutely continuous spectrum of H. For sufficiently high μ and almost all x we prove the existence of a set of eigenvalues whose closure has positive measure. The two results are obtained for a subset of irrational numbers θ of full Lebesgue measure.
WORLD SCIENTIFIC eBooks, May 1, 1985
We studyH=-d 2/dx 2+ V (x) withV (x) limit periodic, eg V (x)= Sa n cos (x/2 n) with S| an|&a... more We studyH=-d 2/dx 2+ V (x) withV (x) limit periodic, eg V (x)= Sa n cos (x/2 n) with S| an|< 8. We prove that for a genericV (and for generica n in the explicit example), s (H) is a Cantor (= nowhere dense, perfect) set. For a dense set, the spectrum is both Cantor and purely ...
This material is based upon work supported by the National Science Foundation
D. Spehner, Contributions a la theorie du transport electronique,
We reconsider the various denitions of dynamical localization in view of Solid State Physics ana-... more We reconsider the various denitions of dynamical localization in view of Solid State Physics ana-log. To treat the semiclassical limit on a rigorous background we introduce an algebraic formalism already used for disordered crystals. We get a complete analogy between classical motion and quan-tum motion in term of the language of Non Commutative Geometry. Various rigorous results on the semiclassical limit are obtained and we discuss the main diculties in getting a rigorous approach of the Chirikov- Izrailev-Shepelyansky theory.
1.1.1. Periodic Media. In Condensed Matter Physics, the basic tool to describe models and to perf... more 1.1.1. Periodic Media. In Condensed Matter Physics, the basic tool to describe models and to perform calculations is Bloch’s Theory [13]. The main ingredient is to use the invariance of the Hamiltonian under the action of the translation group and to diagonalize simultaneously the Hamiltonian and the unitary group representing the translations. Since the translation group (either Rd or Zd, with d = 1, 2, 3 in practice) is both Abelian and locally compact, the diagonalization of the unitaries representing it can be done through its group of characters (Pontryagin dual) known as the Brillouin zone [16] in Condensed Matter theory and it will be denoted here by B. If the focus of attention is put on the one-electron motion, then for each quasi-momentum k ∈ B, there is a self-adjoint Hamiltonian H(k) = H(k)† which is mostly a matrix, either finite dimensional, when the energy is restricted to a neighborhood of the Fermi level (in the so called tight-binding representation), or infinite d...
Encyclopedia of Condensed Matter Physics, 2005
Contemporary Mathematics, 2005
... d with real Julia set E, EC£,£] T~ l:£,£ We want to get a differential equation for J. Let ... more ... d with real Julia set E, EC£,£] T~ l:£,£ We want to get a differential equation for J. Let 23 be a unitary matrix such that J< B=< BA, where A= diag {Afc}. Since we can choose Ai (x)< A2 (x)<... Aj (x) Page 68. 46 J. BELLISSARD, J. GERONIMO, A. VOLBERG, AND P ...
It is shown that the category of homogeneous self-dual cones in Hilbert spaces is isomorphic to t... more It is shown that the category of homogeneous self-dual cones in Hilbert spaces is isomorphic to the category of Jordan Banach algebras with predual (JBW-algebras).
This material is based upon work supported by the National Science Foundation
References: I. Guarneri, Spectral properties of quantum diusion on discrete lattices, Europhys. L... more References: I. Guarneri, Spectral properties of quantum diusion on discrete lattices, Europhys. Lett., 10, 95-100, (1989).On an estimate concerning quantum diusion in the presence of a fractal spectrum, Europhys. Lett., 21, 729-733, (1993). J. Bellissard, A. van Elst & H. Schulz-Baldes, The Non-Commutative Geom-
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Papers by Jean Bellissard