arXiv subjects
Publications and source records attributed to Y. Last.
We summarize (and comment on) the research program carried out in (CMP 319, 501 (2013)), (CMP 338, 347 (2015)), (CMP 344, 959 (2016), (LMP 106, 787 (2016)). This program is devoted to the characterization of the absolutely continuous spectrum of a self-adjoint operator H in terms of the transport properties of a suitable class of open quantum systems canonically associated to H.
We derive a general upper bound on the spreading rate of wavepackets in the framework of Schrödinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time independent Schrödinger equation. We also derive a new lower bound on the spreading rate, which is strongly connected with our upper bound. We apply these new bounds to the Fibonacci Hamiltonian--the most studied one-dimensional model of quasicrystals. As a result, we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization. The bounds have the same qualitative behavior in the limit of large coupling.
We look at invariance of a.e. boundary condition spectral behavior under perturbations, $W$, of half-line, continuum or discrete Schrödinger operators. We extend the results of del Rio, Simon, Stolz from compactly supported $W$'s to suitable short-range $W$. We also discuss invariance of the local Hausdorff dimension of spectral measures under such perturbations.
We study the (zero temperature) quantum piezoelectric response of Harper-like models with broken inversion symmetry. The charge transport in these models is related to topological invariants (Chern numbers). We show that there are arbitrarily small periodic modulations of the atomic positions that lead to nonzero charge transport for the electrons.