Alexander Elgart

ADIABATIC THEOREM WITHOUT A GAP CONDITION

Abstract:We prove the adiabatic theorem for quantum evolution without the traditional gap condition. All that this adiabatic theorem needs is a (piecewise) twice differentiable finite dimensional spectral projection. The result implies that the adiabatic theorem holds for the ground state of atoms in quantized radiation field. The general result we prove gives no information on the rate at which the adiabatic limit is approached. With additional spectral information one can also estimate this rate.

Moment analysis for localization in random Schrödinger operators

We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.

Optimal Quantum Pumps

We study adiabatic quantum pumps on time scales that are short relative to the cycle of the pump. In this regime the pump is characterized by the matrix of energy shift which we introduce as the dual to Wigners time delay. The energy shift determines the charge transport, the dissipation, the noise, and the entropy production. We prove a general lower bound on dissipation in a quantum channel and define optimal pumps as those that saturate the bound. We give a geometric characterization of optimal pumps and show that they are noiseless and transport integral charge in a cycle. Finally we discuss an example of an optimal pump related to the Hall effect.

Geometry, statistics, and asymptotics of quantum pumps

We give a pedestrian interpretation of a formula of Buttiker et. al. (BPT) relating the adiabatically pumped current to the S matrix and its (time) derivatives. We relate the charge in BPT to Berrys phase and the corresponding Brouwer pumping formula to curvature. As applications we derive explicit formulas for the joint probability density of pumping and conductance when the S matrix is uniformly distributed; and derive a new formula that describes hard pumping when the S matrix is periodic in the driving parameters.

Equality of the Bulk and Edge Hall Conductances in a Mobility Gap

We consider the edge and bulk conductances for 2D quantum Hall systems in which the Fermi energy falls in a band where bulk states are localized. We show that the resulting quantities are equal, when appropriately defined. An appropriate definition of the edge conductance may be obtained through a suitable time averaging procedure or by including a contribution from states in the localized band. In a further result on the Harper Hamiltonian, we show that this contribution is essential. In an appendix we establish quantized plateaus for the conductance of systems which need not be translation ergodic.

Anderson Localization for a Class of Models with a Sign-Indefinite Single-Site Potential via Fractional Moment Method

A technically convenient signature of Anderson localization is exponential decay of the fractional moments of the Green function within appropriate energy ranges. We consider a random Hamiltonian on a lattice whose randomness is generated by the sign-indefinite single-site potential, which is however sign-definite at the boundary of its support. For this class of Anderson operators, we establish a finite-volume criterion which implies that the fractional moment decay property holds. This constructive criterion is satisfied at typical perturbative regimes, e.g. at spectral boundaries which satisfy “Lifshitz tail estimates” on the density of states and for sufficiently strong disorder. We also show how the fractional moment method facilitates the proof of exponential (spectral) localization for such random potentials.

Transport and Dissipation in Quantum Pumps

This paper is about adiabatic transport in quantum pumps. The notion of “energy shift,” a self-adjoint operator dual to the Wigner time delay, plays a role in our approach: It determines the current, the dissipation, the noise and the entropy currents in quantum pumps. We discuss the geometric and topological content of adiabatic transport and show that the mechanism of Thouless and Niu for quantized transport via Chern numbers cannot be realized in quantum pumps where Chern numbers necessarily vanish.

Localisation for non-monotone Schrödinger operators

We study localisation effects of strong disorder on the spectral and dynamical properties of (matrix and scalar) Schroedinger operators with non-monotone random potentials, on the d-dimensional lattice. Our results include dynamical localisation, i.e. exponentially decaying bounds on the transition amplitude in the mean. They are derived through the study of fractional moments of the resolvent, which are finite due to resonance-diffusing effects of the disorder. One of the byproducts of the analysis is a nearly optimal Wegner estimate. A particular example of the class of systems covered by our results is the discrete alloy-type Anderson model.

A note on the switching adiabatic theorem

We derive a nearly optimal upper bound on the running time in the adiabatic theorem for a switching family of Hamiltonians. We assume the switching Hamiltonian is in the Gevrey class Gα as a function of time, and we show that the error in adiabatic approximation remains small for running times of order g−2 |ln g |6α. Here g denotes the minimal spectral gap between the eigenvalue(s) of interest and the rest of the spectrum of the instantaneous Hamiltonian.

Localization via fractional moments for models on {\bb Z} with single-site potentials of finite support

One of the fundamental results in the theory of localization for discrete Schrodinger operators with random potentials is the exponential decay of Greens function and the absence of a continuous spectrum. In this paper, we provide a new variant of these results for one-dimensional alloy-type potentials with finitely supported sign-changing single-site potentials using the fractional moment method.