Serguei Tcheremchantsev

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as

Upper bounds in quantum dynamics

\lambda \to \infty, {\rm dim} (\sigma(H_\lambda)) \cdot {\rm log} \lambda

Scaling estimates for solutions and dynamical lower bounds on wavepacket spreading

converges to an explicit constant,

Nonlinear variation of diffusion exponents in quantum dynamics

{\rm log}(1+\sqrt{2})\approx 0.88137

Transfer matrices and transport for Schrödinger operators

. We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schrödinger dynamics generated by the Fibonacci Hamiltonian.

Power-law bounds on transfer matrices and quantum dynamics in one dimension–II

Abstract: We present an approach to quantum dynamical lower bounds for discrete one-dimensional Schrödinger operators which is based on power-law bounds on transfer matrices. It suffices to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamiltonian.