Gian Michele Graf

Localization bounds for an electron gas

Mathematical analysis of the Anderson localization has been facilitated by the use of suitable fractional moments of the Green function. Related methods permit now a readily accessible derivation of a number of physical manifestations of localization, in regimes of strong disorder, extreme energies, or weak disorder away from the unperturbed spectrum. This work establishes on this basis exponential decay for the modulus of the two-point function, at all temperatures as well as in the ground state, for a Fermi gas within the one-particle approximation. Different implications, in particular for the integral quantum Hall effect, are reviewed.

Asymptotic completeness forN-body short-range quantum systems: A new proof

We give an alternative geometrical proof of asymptotic completeness for an arbitrary number of quantum particles interacting through shortrange pair potentials. It relies on an estimate showing that the intercluster motion concentrates asymptotically on classical trajectories.

Bulk-Edge Correspondence for Two-Dimensional Topological Insulators

Topological insulators can be characterized alternatively in terms of bulk or edge properties. We prove the equivalence between the two descriptions for two-dimensional solids in the single-particle picture. We give a new formulation of the

Anderson localization and the space-time characteristic of continuum states

{\mathbb{Z}_{2}}

A General Resonance Theory Based on Mourre’s Inequality

Z2-invariant, which allows for a bulk index not relying on a (two-dimensional) Brillouin zone. When available though, that index is shown to agree with known formulations. The method also applies to integer quantum Hall systems. We discuss a further variant of the correspondence, based on scattering theory.

Geometry, statistics, and asymptotics of 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.

On the Extended Nature of Edge States of Quantum Hall Hamiltonians

A proof of Anderson localization is obtained by ruling out any continuous spectrum on the basis of the space-time characteristic of its states.

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

Abstract: The integral quantum Hall effect can be explained either as resulting from bulk or edge currents (or, as it occurs in real samples, as a combination of both). This leads to different definitions of Hall conductance, which agree under appropriate hypotheses, as shown by Schulz-Baldes et al. by means of K-theory. We propose an alternative proof based on a generalization of the index of a pair of projections to more general operators. The equality of conductances is an expression of the stability of that index as a flux tube is moved from within the bulk across the boundary of a sample.