Tridiagonal Substitution Hamiltonians

¹ Mathematics & Computer Science, Denison University, Granville, OH 43023-0810
² Mathematics, Rice University, 1600 Main St. MS-136, Houston, TX 77005

^⋆ Supported by the Michele T. Myers PD Account through Denison University. Part of the work presented herein was supported by DMS-0901627 (PI: A. Gorodetski)

Abstract

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the fractal structure and fractal dimensions of the spectrum, exact dimensionality of the integrated density of states, and the gap structure. We present a review of previous results, some applications, and open problems. Our investigation is based largely on the dynamics of trace maps. This work is an extension of similar results on Schrödinger operators, although some of the results that we obtain differ qualitatively and quantitatively from those for the Schrödinger operators. The nontrivialities of this extension lie in the dynamics of the associated trace map as one attempts to extend the trace map formalism from the Schrödinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are, in a sense, a test item, as many other models can be attacked via the same techniques, and we present an extensive discussion on this.