Andreas Knauf

Number Theoretic Background

This paper is an expanded version of some of the lectures given at the summer school in Bologna. In these lectures we gave an introduction to very basic number theory, assuming practically no background. The lectures were intended for graduate students in Math and Physics and while the material is completely standard, we tried to make the presentation as elementary as possible.

Ergodic and topological properties of Coulombic periodic potentials

The motion of a classical pointlike particle in a two-dimensional periodic potential with negative coulombic singularities is examined. This motion is shown to be Bernoullian for many potentials and high enough energies. Then the motion on the plane is a diffusion process. All such motions are topologically conjugate and the periodic orbits can be analysed with the help of a group.

Entropy distance: New quantum phenomena

We study a curve of Gibbsian families of complex 3 × 3-matrices and point out new features, absent in commutative finite-dimensional algebras: a discontinuous maximum-entropy inference, a discontinuous entropy distance, and non-exposed faces of the mean value set. We analyze these problems from various aspects including convex geometry, topology, and information geometry. This research is motivated by a theory of infomax principles, where we contribute by computing first order optimality conditions of the entropy distance.

Coulombic periodic potentials: The quantum case

Abstract The motion of a quantum mechanical particle in a 2-dimensional crystal with attracting nuclei is considered. For a large class of potentials the particle is shown to be slow not only for low but also for high energies. The remarkable high energy behaviour is a manifestation of “quantum chaos.”

Semiclassical resolvent estimates for Schrodinger operators with Coulomb singularities.

Abstract.Consider the Schrödinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator’s resolvent at a positive energy λ are bounded by O(h−1) if and only if the associated Hamilton flow is non-trapping at energy λ. In the present paper, we extend this result to the case where the potential may possess Coulomb singularities. Since the Hamilton flow then is not complete in general, our analysis requires the use of an appropriate regularization.

The Low Activity Phase of Some Dirichlet Series

We show that a rigorous statistical mechanics description of some Dirichlet series is possible. Using the abstract polymer model language of statistical mechanics and the polymer expansion theory we characterize the low activity phase by the suitable exponential decay of the truncated correlation functions.

On a ferromagnetic spin chain. II. Thermodynamic limit

The existence of the thermodynamic limit of the free energy for the ferromagnetic spin chain connected with the Riemann zeta function is proven.

Maximizing the Divergence from a Hierarchical Model of Quantum States

We study many-party correlations quantified in terms of the Umegaki relative entropy (divergence) from a Gibbs family known as a hierarchical model. We derive these quantities from the maximum-entropy principle which was used earlier to define the closely related irreducible correlation. We point out the differences between quantum states and probability vectors which exist in hierarchical models, in the divergence from a hierarchical model and in local maximizers of this divergence. The differences are, respectively, missing factorization, discontinuity and reduction of uncertainty. We discuss global maximizers of the mutual information of separable qubit states.

The non-trapping degree of scattering

We consider classical potential scattering. If at energy E no orbit is trapped, the Hamiltonian dynamics define an integer-valued topological degree deg(E) ≤ 1. This is calculated explicitly for all potentials, and exactly integers ≤1 are shown to occur for suitable potentials.The non-trapping condition is restrictive in the sense that for a bounded potential it is shown to imply that boundary of Hills region in configuration space is either empty or homeomorphic to a sphere.However, in many situations one can decompose a potential into a sum of non-trapping potentials with a non-trivial degree and embed symbolic dynamics of multi-obstacle scattering. This comprises a large number of earlier results, obtained by different authors on multi-obstacle scattering.

Stochastically stable quenched measures

We analyze a class of stochastically stable quenched measures.We prove that stochastic stability is fully characterized by an infinite family of zero average polynomials in the covariance matrix entries.