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G. M. Graf

Publications and source records attributed to G. M. Graf.

27 records · Page 2Linked to original sources
arXiv2002

Time-Energy coherent states and adiabatic scattering

Coherent states in the time-energy plane provide a natural basis to study adiabatic scattering. We relate the (diagonal) matrix elements of the scattering matrix in this basis with the frozen on-shell scattering data. We describe an exactly solvable model, and show that the error in the frozen data cannot be estimated by the Wigner time delay alone. We introduce the notion of energy shift, a conjugate of Wigner time delay, and show that for incoming state $ρ(H_0)$ the energy shift determines the outgoing state.

math-ph↗
arXiv2001

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 Wigner's 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.

math-ph↗
arXiv2000

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 Berry's 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.

cond-mat.mes-hall↗
arXiv1999

On the extended nature of edge states of Quantum Hall Hamiltonians

Properties of eigenstates of one-particle Quantum Hall Hamiltonians localized near the boundary of a two-dimensional electron gas - so-called edge states - are studied. For finite samples it is shown that edge states with energy in an appropriate range between Landau levels remain extended along the boundary in the presence of a small amount of disorder, in the sense that they carry a non-zero chiral edge current. For a two-dimensional electron gas confined to a half-plane, the Mourre theory of positive commutators is applied to prove absolute continuity of the energy spectrum well in between Landau levels, corresponding to edge states.

math-ph↗
arXiv1998

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. The present 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.

cond-mat↗
arXiv1993

Proof of stability of the hydrogen molecule

We sketch two rigorous proofs of the stability of the hydrogen molecule in quantum mechanics. The first one is based on an extrapolation of variational estimates of the groundstate energy of a positronium molecule to arbitrary mass ratios. The second one is an extension of Heitler-London theory to nuclei of finite mass.

nucl-th↗