J. E. Avron

Schrödinger operators with magnetic fields. III. Atoms in homogeneous magnetic field

We prove a large number of results about atoms in constant magnetic field including (i) Asymptotic formula for the ground state energy of Hydrogen in large field, (ii) Proof that the ground state of Hydrogen in an arbitrary constant field hasLz = 0 and of the monotonicity of the binding energy as a function ofB, (iii) Borel summability of Zeeman series in arbitrary atoms, (iv) Dilation analyticity for arbitrary atoms with infinite nuclear mass, and (v) Proof that every once negatively charged ion has infinitely many bound states in non-zero magnetic field with estimates of the binding energy for smallB and largeLz.

Separation of center of mass in homogeneous magnetic fields

We show that a system of particles in a homogeneous magnetic field, with translation invariant interaction, has a constant of motion analogous to the total momentum when B = 0. Next, we consider the separation of the center of mass. When the total charge of the system is zero, the situation is similar to (but more complicated than) the B = 0 case. When the total charge is nonzero, the analysis is quite different. The two-body problem is worked out in some detail and we also state and prove a version of the HVZ theorem in homogeneous magnetic field.

Spectral and scattering theory of Schrödinger operators related to the Stark effect

We analyze the spectral properties and discuss the scattering theory of operators of the formH=H0+V,H0=−Δ+E·x. Among our results are the existence of wave operators, Ω±(H, H0), the nonexistence of bound states, and a (speculative) description of resonances for certain classes of potentials.

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.

Adiabatic Theorems and Applications to the Quantum Hall Effect

We study an adiabatic evolution that approximates the physical dynamics and describes a natural parallel transport in spectral subspaces. Using this we prove two folk theorems about the adiabatic limit of quantum mechanics: 1. For slow time variation of the Hamiltonian, the time evolution reduces to spectral subspaces bordered by gaps. 2. The eventual tunneling out of such spectral subspaces is smaller than any inverse power of the time scale if the Hamiltonian varies infinitly smoothly over a finite interval. Except for the existence of gaps, no assumptions are made on the nature of the spectrum. We apply these results to charge transport in quantum Hall Hamiltonians and prove that the flux averaged charge transport is an integer in the adiabatic limit.

Singular continuous spectrum for a class of almost periodic Jacobi matrices

We consider operators (parametrized by a, O, \) on /2 with matrix ojy+i + o / ; i + fl/o/y with an = \ COS(2TTCW + 6). If ce is a Liouville number and \ > 2, we prove that for a.e. 0, the operators spectral measures are all singular continuous. We consider the operator// on l2(Z) depending upon three parameters, X, a, 0, (1) [#(X, a, 0)u] (n) = u(n + 1) + u(n 1) + X cos(2iran + 6)u(n). In this note we will sketch the proof of the following result whose detailed proof will appear elsewhere [3]. THEOREM 1. Fix a, an irrational number obeying

Pushmepullyou: an efficient micro-swimmer

The swimming of a pair of spherical bladders that change their volumes and mutual distance is superior to other models of artificial swimmers at low Reynolds numbers. The swimming resembles the wriggling motion known as metaboly of certain protozoa.

VISCOSITY OF QUANTUM HALL FLUIDS

The viscosity of quantum fluids with an energy gap at zero temperature is non-dissipative and is related to the adiabatic curvature on the space of flat background metrics (which plays the role of the parameter space). For a quantum Hall fluid on two dimensional tori this viscosity is computed. In this case the average viscosity is quantized and is proportional to the total magnetic flux through the torus.

Entanglement on demand through time reordering

Entangled photons can be generated on demand in a novel scheme involving unitary time reordering of the photons emitted in a radiative decay. This scheme can be applied to the biexciton cascade in quantum dots.

Charge deficiency, charge transport and comparison of dimensions

We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissards theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.