JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 20, Number 3, July 2007, Pages 799–827 S 0894-0347(06)00554-6 Article electronically published on November 3, 2006
UPPER BOUNDS IN QUANTUM DYNAMICS
DAVID DAMANIK AND SERGUEI TCHEREMCHANTSEV
1. Introduction If H is a self-adjoint operator on a separable Hilbert space H, the time-dependent Schr¨odinger equation of quantum mechanics, i∂tψ = Hψ, leads to a unitary dy- namical evolution in H, (1) ψ(t)=e−itH ψ(0). Of course, the case of main interest is where H is given by L2(Rd)or2(Zd), H is aSchr¨odinger operator of the form −∆+V ,andψ(0) is a localized wavepacket. Under the time evolution (1), the wavepacket ψ(t) will in general spread out with time. Finding quantitative bounds for this spreading in concrete cases and developing methods for proving such bounds has been the objective of a great number of papers in recent years. One is often interested in the growth behavior of the moments of the position operator, that is, the function t →|X|p/2ψ(t)2. These questions are particularly complicated and interesting in situations where singular continuous spectra occur. The discovery of the occurrence of such spectra for many models of physical relevance has triggered some activity on the quantum dynamical side. For models with absolutely continuous spectra, on the other hand, quantum dynamical questions have been investigated much earlier in the context of scattering theory and have been answered to a great degree of satisfaction. Models with point spectrum, or more restrictively, models exhibiting Anderson localization, have also been investigated from a quantum dynamical point of view. While it is not true in general that spectral localization implies dynamical localization [26, 73], it has been shown that the three main approaches to spectral localization, the fractional moment method [3, 4, 5], multi-scale analysis [28, 27, 75], and the Bourgain-Goldstein-Schlag method [12, 37], yield dynamical localization as a by- product [2, 13, 22, 31, 34]. Quantum dynamics for Schr¨odinger operators with singular continuous spectra is much less understood. The two primary examples motivated by physics are the Harper model, also known as the almost Mathieu operator at critical cou- pling, which describes a Bloch electron in a constant magnetic field with irrational flux through a unit cell, and the Fibonacci model, which describes a standard one-dimensional quasicrystal. Both models have been studied heavily in both the physics and mathematics communities.
Received by the editors October 26, 2005. 2000 Mathematics Subject Classification. Primary 81Q10; Secondary 47B36. Key words and phrases. Schr¨odinger operators, quantum dynamics, anomalous transport.