Spectra of Fibonacci Hamiltonians

Abstract: The Fibonacci sequence is a prominent model of a 1D quasicrystal. We will talk about some properties of continuum Schr\"odinger operators with potentials that are determined by the Fibonacci sequence. We show that the spectrum is an (unbounded) Cantor set of zero Lebesgue measure and that the local Hausdorff dimension of the spectrum tends to one in the regimes of high energy and small coupling. We also show that multidimensional Schr\"odinger operators patterned on the Fibonacci sequence can exhibit the coexistence of two phenomena: (1) Cantor structure near the bottom of the spectrum and (2) an absence of gaps in the spectrum at high energies. To prove (2), we develop an "abstract" Bethe--Sommerfeld criterion for sums of extended Cantor sets, which may be of independent interest. [Based on joint projects with David Damanik, Anton Gorodetski, and May Mei]

mathematical physics

Audience: researchers in the topic

TAMU: Mathematical Physics and Harmonic Analysis Seminar