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Series: Math Physics Seminar

This is part of the 2017 Quolloquium series.

Starting from the classical Berezin- and Li-Yau-bounds onthe eigenvalues of the Laplace operator with Dirichlet boundaryconditions I give a survey on various improvements of theseinequalities by remainder terms. Beside the Melas inequalitywe deal with modifications thereof for operators with and withoutmagnetic field and give bounds with (almost) classical remainders.Finally we extend these results to the Heisenberg sub-Laplacianand the Stark operator in domains.

Series: Math Physics Seminar

This is part of the 2017 Quolloquium series.

We use the weighted isoperimetric inequality of J. Ratzkin for a wedge domain in higher dimensions to prove new isoperimetric inequalities for weighted $L_p$-norms of the fundamental eigenfunction of a bounded domain in a convex cone-generalizing earlier work of Chiti, Kohler-Jobin, and Payne-Rayner. We also introduce relative torsional rigidity for such domains and prove a new Saint-Venant-type isoperimetric inequality for convex cones. Finally, we prove new inequalities relating the fundamental eigenvalue to the relative torsional rigidity of such a wedge domain thereby generalizing our earlier work to this higher dimensional setting, and show how to obtain such inequalities using the Payne interpretation in Weinstein fractional space. (Joint work with A. Hasnaoui)

Series: Math Physics Seminar

Series: Math Physics Seminar

Existence of ballistic transport for Schr ̈odinger operator with a quasi-
periodic potential in dimension two is discussed. Considerations are based on the
following properties of the operator: the spectrum of the operator contains a semiaxis
of absolutely continuous spectrum and there are generalized eigenfunctions being close
to plane waves ei⟨⃗k,⃗x⟩ (as |⃗k| → ∞) at every point of this semiaxis. The isoenergetic
curves in the space of momenta ⃗k corresponding to these eigenfunctions have a form
of slightly distorted circles with holes (Cantor type structure).

Series: Math Physics Seminar

During the last few years there has been a systematic pursuit for sharp estimates of the energy components of atomic systems in terms of their single particle density. The common feature of these estimates is that they include corrections that depend on the gradient of the density. In this talk I will review these results. The most recent result is the sharp estimate of P.T. Nam on the kinetic energy. Towards the end of my talk I will present some recent results concerning geometric estimates for generalized Poincaré inequalities obtained in collaboration with C. Vallejos and H. Van Den Bosch. These geometric estimates are a useful tool to estimate the numerical value of the constant of Nam's gradient correction term.

Series: Math Physics Seminar

We consider the two-dimensional BCS functional with a radial pair interaction. We show that the translational symmetry is not broken in a certain temperature interval below the critical temperature. For this purpose, we first introduce the full BCS functional and the translation invariant BCS functional. Our main result states that theminimizers of the full BCS functional coincide with the minimizers of the translation invariant BCS functional for temperatures in the aforementioned interval. In the case of vanishing angular momentum our results translate to the three dimensional case. Finally, we will explain the strategy and main ideas of the proof. This is joint work with Andreas Deuchert, Christian Hainzl and Michael Loss.

Series: Math Physics Seminar

We study regularity properties of weak solutions of the homogeneous Boltzmann equation. While under the so called Grad cutoff assumption the homogeneous Boltzmann equation is known to propagate smoothness and singularities, it has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplace operator. This has led to the hope that the homogenous Boltzmann equation enjoys similar smoothing properties as the heat equation with a fractional Laplacian. We prove that any weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation (for Maxwellian molecules) with initial datum $f_0$ with ﬁnite mass, energy and entropy, that is, $f_0 \in L^1_2(\R^d) \cap L \log L(\R^d)$, immediately becomes Gevrey regular for strictly positive times, i.e. it gains infinitely many derivatives and even (partial) analyticity.This is achieved by an inductive procedure based on very precise estimates of nonlinear, nonlocal commutators of the Boltzmann operator with suitable test functions involving exponentially growing Fourier multipliers.(Joint work with Jean-Marie Barbaroux, Dirk Hundertmark, and Semjon Vugalter)

Series: Math Physics Seminar

The discrete truncated moment problem considers the question whether
given a discrete subsets $K \subset \mathbb{R}$ and a sequence of real
numbers one can find a measure supported on $K$ whose (power) moments
are exactly these numbers. The truncated
moment is a challenging problem. We derive a minimal set of necessary
and sufficient conditions. This simple problem is surprisingly hard and
not treatable with known techniques. Applications to the truncated
moment problem for point processes, the so-called
relizability or representability problem are given. The relevance of
this problem for statistical mechanics in particular the theory of
classic liquids, is explained. This is a joint work with M. Infusino, J.
Lebowitz and E. Speer.

Series: Math Physics Seminar

In this talk we will consider a few different mathematical
models of gas-like systems of particles, which interact through
binary collisions that conserve momentum and mass.
The aim of the talk will be to present how one can employ ideas from
dynamical systems theory to derive macroscopic properties of such models.

Series: Math Physics Seminar

We consider the motion of a particle on the two-dimensional hexagonal lattice
whose sites are occupied by flipping rotators, which scatter the particle
according to a deterministic rule. We find that the particle's trajectory
is a self-avoiding walk between returns to its initial position. We
show that this behavior is a consequence of the deterministic scattering rule
and the particular class of initial scatterer configurations we consider. Since
self-avoiding walks are one of the main tools used to model the growth of
crystals and polymers, the particle's motion in this class of systems is potentially
important for the study of these processes.