Electronic Research Announcements in Mathematical Sciences (ERA-MS)

The spectrum of the weakly coupled Fibonacci Hamiltonian

Pages: 23 - 29, January 2009      doi:10.3934/era.2009.16.23

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David Damanik - Department of Mathematics, Rice University, Houston, TX 77005, United States (email)
Anton Gorodetski - Department of Mathematics, University of California, Irvine CA 92697, United States (email)

Abstract: We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We announce the following results and explain some key ideas that go into their proofs. The thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. Moreover, the length of every gap tends to zero linearly. Finally, for sufficiently small coupling, the sum of the spectrum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz.

Keywords:  Spectrum, Fibonacci Hamiltonian, trace map, dynamically defined Cantor set, Hausdorff dimension, thickness.
Mathematics Subject Classification:  Primary: 82B44; Secondary: 37D99.

Received: January 2009;      Revised: March 2009;      Available Online: May 2009.