Braunschweig-Bremen Seminar

Fiona Gottschalk (TU Braunschweig)

Correlation Asymptotics of Classical Lattice Spin Systems in the High-Temperature Limit and the Witten Laplacian

Abstract: Subject of our considerations is a spectral theoretic application of the Witten Laplacian in classical statistical mechanical lattice spin systems in the limit of high temperatures. The involved Hamiltonian function consists of an on-site potential term given by even polynomials and a finite range interaction term, both of which are not assumed to be translation invariant. In particular, no specific spatial structure is imposed on the model.
For such systems, we establish existence and uniqueness of the associated Gibbs measure in the thermodynamic limit. Using this, we show exponential decay of the correlations and, what is more, we derive the correlation asymptotics in terms of a metric inherent to the model that is primarily governed by the underlying interaction rather than the geometry of the lattice. The spectral gap of the Witten Laplacian plays a crucial and recurring role throughout the proofs. This is joint work with V. Bach.

Konstantin Merz (TU Braunschweig)

Eigenvalue Estimates for Schrödinger Operators Using Fourier Analysis

Abstract: Estimating the location and accumulation rate of eigenvalues of Schrödinger operators is a classic problem in spectral theory and mathematical physics. For "short-range" potentials these problems can often be effectively treated using Fourier analytic methods like the Tomas-Stein restriction theorem. As an example we derive eigenvalue asymptotics for Schrödinger-type operators whose kinetic energy vanishes on a codimension one submanifold. Time permitting, we discuss another example: locating eigenvalues of ordinary Schrödinger operators with randomized, long-range, complex-valued potentials using a randomized version of the Tomas-Stein theorem by Bourgain. The talk is based on joint work with Jean-Claude Cuenin.

Jens Hoppe (TU Braunschweig)

Recent Progress on Membrane Theory

Abstract: The talk will be based on results reported in my recent arxiv-preprints 2107.03319, 2107.00569, 2103.16540, 2103.08653, 2102.03904, 2101.04495, 2101.01803.