Power-Law Bounds on Transfer Matrices and Quantum Dynamics in One Dimension

Abstract

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.

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

“…The other approach is based on complex energy methods and the Plancherel Theorem; see [11] (and also [12] for a way to combine the two approaches). For θ = 0, the paper [11] has (6) with f # replaced by 1/6.…”

“…For θ = 0, the paper [11] has (6) with f # replaced by 1/6. It is possible to treat general θ along the same lines using [10], but the dynamical lower bound has a somewhat smaller constant in the general case.…”

Lower Transport Bounds for One-dimensional Continuum Schrödinger Operators

“…For discrete one-dimensional Schrödinger operators on ℓ 2 (N) and ℓ 2 (Z), respectively, Damanik and Tcheremchantsev [7] have developed a general method which allows one to derive lower bounds on diffusion exponents from upper bounds on the growth of norms of transfer matrices. In Section 2 we will present an extension of their method to continuum operators.…”

“…The paper [7] only considers the case f = δ 1 , a discrete unit mass. While this is somewhat natural in discrete space, there is no corresponding choice in the continuum (at least if one wants to stay in Hilbert space).…”

Localization for the Random Displacement Model

“…Thus, the transfer matrix also grows linearly. By Corollary 2.1 in [6], which is based on work in [8], this implies that the time-averaged moments |X| p of suitable solutions of the timedependent Schrödinger equation are bounded below by CT (p−5)/2 . This rules out dynamical…”