Ruedi Seiler

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.

On the Born-Oppenheimer Expansion for Polyatomic Molecules

We consider the Schrödinger operatorP(h) for a polyatomic molecule in the semiclassical limit where the mass ratioh2 of electronic to nuclear mass tends to zero. We obtain WKB-type expansions of eigenvalues and eigenfunctions ofP(h) to all orders inh. This allows to treat the splitting of the ground state energy of a non-planar molecule. Our class of potentials covers the physical case of the Coulomb interaction. We use methods ofh-pseudodifferential operators with operator valued symbols, which by use of appropriate coordinate changes in local coordinate patches covering the classically accessible region become applicable even to our class of singular potentials.

Bounds for the adiabatic approximation with applications to quantum computation

We present straightforward proofs of estimates used in the adiabatic approximation. The gap dependence is analyzed explicitly. We apply the result to interpolating Hamiltonians of interest in quantum computing.

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.

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.

The Shape Resonance

For a class of Schrödinger operatorsH:=−(ℏ2/2m)Δ+V onL2(ℝn), with potentials having minima embedded in the continuum of the spectrum and non-trapping tails, we show the existence of shape resonances exponentially close to the real axis as ℏ↘0. The resonant energies are given by a convergent perturbation expansion in powers of a parameter exhibiting the expected exponentially small behaviour for tunneling.

A Topological Look at the Quantum Hall Effect

The amazingly precise quantization of Hall conductance in a two-dimensional electron gas can be understood in terms of a topological invariant known as the Chern number.

Classical bounds and limits for energy distributions of Hamilton operators in electromagnetic fields

Abstract We discuss Hamiltonians in arbitrary C ∞ external electromagnetic fields and with positive potentials. Classical bounds are given on inverse energy moments in non-zero-temperature Gibbs states. We also prove convergence to the classical limit when ħ → 0 as well as for the expectation values of some local observables.

Krein's formula and one-dimensional multiple-well

Abstract It is shown that all the discrete eigenvalues of a one-dimensional Schrodinger operator with a multiple well potential possess an asymptotic expansions in power of h 1 2 when h → 0 . A formula for all the coefficients of these expansions is given. The method uses two main tools: Perturbation by boundary conditions and exponential decay of eigenfunctions which are developed in this article. As a by-product of this work, the exponential localization of eigenvectors when h goes to zero can be proved.

A Quantum Version of Sanov's Theorem

We present a quantum version of Sanovs theorem focussing on a hypothesis testing aspect of the theorem: There exists a sequence of typical subspaces for a given set Ψ of stationary quantum product states asymptotically separating them from another fixed stationary product state. Analogously to the classical case, the separating rate on a logarithmic scale is equal to the infimum of the quantum relative entropy with respect to the quantum reference state over the set Ψ. While in the classical case the separating subsets can be chosen universally, in the sense that they depend only on the chosen set of i.i.d. processes, in the quantum case the choice of the separating subspaces depends additionally on the reference state.