3.10 Quantum Mechanics

Organizers A. Laptev (Stokholm), B. Simon (Pasadena)

3.10.1 Recent Progress in the Spectral Theory of Quasi-Periodic Operators

David Damanik

CALTECH damanik@caltech.edu

I will describe recent results and technique used in analysing random walks (and diffusions) in random environment in the perturbative regime, when the environment satisfies certain isotropy conditions.

3.10.2 Recent Results on Localization for Random Schrödinger Operators

Francois Germinet

Universitéé de Cergy-Pontoise germinet@math.u-cergy.fr

Since Fröhlich and Spencer in 1983, localization of random Schrödinger operators can be studied with a so called multiscale analysis. We shall review some recent developments of this technique and of the kind of localization it implies. It will include the Anderson Bernoulli model as well as the Schrödinger operator with Poisson random potential.

3.10.3 Quantum Dynamics and Enhanced Diffusion for Passive Scalar

Alexander Kiselev

University of Wisconsin kiselev@math.wisc.edu

Consider a dissipative evolution equation ψt = iLψ − Γ ψ , where Γ , L are selfadjoint operators, Γ > 0, small. Can the presence of unitary evolution corresponding to L significantly speed up dissipation due to Γ ? The question has a long history in the particular case of the elliptic operators, and has been studied using probabilistic and PDE tools. We prove a sharp result describing the operators L that have this property in the general setting. The methods employ ideas from quantum dynamics. Applications include the classical passive scalar equation and reaction-diffusion equations.

3.10.4 Lieb-Thirring Inequalities, Recent Results

Ari Laptev

KTH, Stockholm laptev@math.kth.se

Some new recent results concerning Lieb-Thirring inequalities will be discussed. In particular, inequalities are derived for power sums of the real-part and modulus of the eigenvalues of a Schrödinger operator with a complex-valued potential. This is my recent joint paper with Rupert Frank, Elliott Lieb and Robert Seiringer.

3.10.5Exponential Decay Laws in Perturbation Theory of Threshold and Embedded Eigenvalues

Gheorghe Nenciu

Univ. of Bucharest

Gheorghe.Nenciu@imar.ro

Exponential decay laws for the metastable states resulting from perturbation of unstable eigenvalues are discussed. Eigenvalues embedded in the continuum as well as threshold eigenvalues are considered. Stationary methods are used, i.e. the evolution group is written in terms of the resolvent via Stone’s formula and Schur-Feschbach partition technique is used to localize the essential terms. No analytic continuation of the resolvent is required. The main result is about threshold case: for Schrödinger operators in odd dimensions the leading term of the decay rate in the perturbation strength, ε, is of order εν/2 where ν is an odd integer, ν ≥ 3.

This is joint work with Arne Jensen.

3.10.6Homogenization of Periodic Operators of Mathematical Physics as a Spectral Threshold Effect

Tatiana A. Suslina

St. Petersburg State University suslina@list.ru

In L2(Rd ), we consider matrix periodic elliptic second order differential operators A admitting a factorization of the form A = X X. Here X is a homogeneous first order differential operator. Many operators of mathematical physics have such structure. We study a homogenization problem in the small period limit. Namely, for the operator A with rapidly oscillating coefficients (depending on x/ ), we study the behavior of the resolvent (A + I )−1 as tends to zero. We find approximation for this resolvent in the (L2 → L2)-operator norm in terms of the resolvent of the effective operator. For the norm of the difference of the resolvents, we obtain the sharp-order estimate (by C ). The constant in this estimate is controlled explicitly.