# Seminars & Events for Mathematical Physics Seminar

##### Generalized eigenfunctions and spectrum for Dirichlet forms

How existence of certain solutions determines the spectrum is a classical issue for Schrödinger operators. We will discuss such results in the context of Dirichlet forms. The framework of Dirichlet forms covers in particular rather general elliptic operators on manifolds as well as (suitable) quantum graphs.

The talk is based on joint works with Anne Boutet de Monvel, Peter Stollmann and Ivan Veselic.

##### The Resolvent Algebra: A Novel Approach to Canonical Quantum Systems

The standard C*-algebraic version of the algebra of canonical commutation relations, the Weyl algebra, frequently causes difficulties in applications since it neither admits the automorphic action of physically interesting dynamics nor does it incorporate pertinent physical observables such as (bounded functions of) the Hamiltonian. In this talk a novel C*-algebra of the canonical commutation relations is presented which is based on the resolvents of the canonical operators. It has many desirable analytic properties and the regularity structure of its representations is surprisingly simple. Moreover, it provides a convenient framework for the study of (infinite) interacting quantum systems and of constraints, as will be illustrated by several examples.

##### Minimizing the ground state energy of an electron in a randomly deformed lattice

We provide a characterization of the spectral minimum for a random Schrödinger operator of the form $H=-\Delta + \sum_i {\in Z^d} q(x-i-\omega_i)$ in $L^2(R^d)$, where the single site potential $q$ is reflection symmetric, compactly supported in the unit cube centered at $0$, and the displacement parameters $\omega_i$ are restricted so that adjacent single site potentials do not overlap. In particular, we show that a minimizing configuration of the displacements is given by a periodic pattern of densest possible $2^d$-clusters of single site potentials. This is joint work with Günter Stolz and Jeff Baker.

##### Localisation in the Anderson tight binding model with several particles

The Anderson model (which will celebrate its 50th anniversary in 2008) is among most popular topics in the random matrix and operator theory. However, so far the attention here was concentrated on single-particle models, where the random external potential is either IID or has a rapid decay of spatial correlations. Multi-particle models remained out of scope in mathematical (and, surprisingly, physical) literature. Recently, Chulaevsky and Suhov (2007) proposed a version of the multi-scale analysis (MSA) scheme tackling the multi-particle case. I'll discuss one of results in this direction: localisation in the lattice (tight binding) multi-particle models for large values of the amplitude (coupling) constant.

##### Photon localization and Dicke superradiance : a crossover to small world networks

We study photon localization in a gas of cold atoms, using a Dicke Hamiltonian that accounts for photon mediated atomic dipolar interactions. The photon escape rates are obtained from a new class of random matrices. A scaling behavior is observed for photons escape rates as a function of disorder and system size. Photon localization is described using statistical properties of random networks which display a "small world" cross-over. Those results are compared to the Anderson photon localization transition.

##### Non-Hermitian Anderson model: Lyapunov exponents, eigenvalues, and eigenfunctions

The Non-Hermitian Anderson model was introduce in 1996 by N. Hatano and D. Nelson. Their numerical studies reveled very interesting and unusual spectral properties of this model. The aim of my talk is to explain how the theory of Lyapunov exponents allows one to:

(a) obtain the equations for the curves on which the non-real eigenvalues lie

(b) study the unusually regular behavior of these eigenvalues

(c) show that the eigenfunctions corresponding to the non-real eigenvalues are $exp(-\sqrt{n})$-localized in a finite box of size $N$ but become de-localized as $N\rightarrow \infty$.

##### Nontrivial coupling at quantum graph vertices obtained through squeezing of Dirichlet networks

The problem discussed in this talk is motivated by efforts to understand approximation of quantum graph Hamiltonians by Laplacians on families of "fat graphs." The emphasis is on new results in the Dirichlet case, however, first we review the background and explain the importance of vertex boundary conditions using a lattice graph example, and mention known result in both the Neumann and Dirichlet setting. After that we suggest a way how a wider classes of vertex couplings can be obtained from squeezed Dirichlet networks. To illustrate the proposed strategy we work out the simplest nontrivial example, a family of bent tubes giving a graph of one vertex and two edges, or a two-parameter family of generalized point interactions on the line.

##### Long range order for lattice dipoles

We consider a system of classical Heisenberg spins on a cubic lattice in dimensions three or more, interacting via the dipole-dipole interaction. We prove that at low enough temperature the system displays orientational long range order, as expected by spin wave theory. The proof is based on reflection positivity methods. In particular, we demonstrate a previously unproven conjecture on the dispersion relation of the spin waves, first proposed by Froehlich and Spencer, which allows one to apply infrared bounds for estimating the long distance behavior of the spin-spin correlation functions.