Title: Applications of the Ky Fan inequality to random (and almost periodic) operators
Abstract: We shall discuss the Ky Fan inequality for the eigenvalues of the sum of two Hermitian matrices. As an application, we shall derive a sharp version of a recent result of Hislop and Marx pertaining to the dependence of the integrated density of states of random Schroedinger operators on the distribution of the potential. Time permitting, we shall also discuss an application to quasiperiodic operators.
Title: The (in)compatibility of 3 and 5 dimensional Heisenberg geometry with Lebesgue spaces
Abstract: The 3-dimensional (discrete) Heisenberg geometry is the shortest-path metric on the infinite graph whose vertex set is the integer grid $\Z^3$ and the neighbors of each integer vector $(a,b,c)$ are the four integer vectors $$(a+ 1,b,c), (a- 1,b,c), (a,b+ 1,c+ a), (a,b- 1,c- a).$$
Fluctuations of linear statistics for Schroedinger operators with a random
Linear statistics provide a tool for the analysis of fluctuations of random
measures and have been extensively studied for various models in random
matrix theory. In this talk we discuss the application of the same
philosophy to the analysis of the finite volume eigenvalue counting measure
of one dimensional Schroedinger operators and demonstrate it with some
interesting results in the case of a random decaying potential.
This is joint work with Jonathan Breuer and Moshe White.
Path integral representations for magnetic Schroedinger operators on graphs
We consider the semigroup and the unitary group of magnetic Schrödinger operators on graphs. Using the ideas of the Feynman Kac formula, we develop a representation of the semigroup and the unitary group in terms of the stochastic process associated with the free Laplacian. As a consequence we derive Kato-Simon estimates for the unitary group. This is joint work with Batu Güneysu (Bonn).
Title: On tiling the real line by translates of a function
Abstract: If f is a function on the real line, then a system
of translates of f is said to be a << tiling >> if it constitutes
a partition of unity. Which functions can tile the line by
translations, and what can be said about the structure of the
tiling? I will give some background on the problem and present
our results obtained in joint work with Mihail Kolountzakis.
Title: Cwikel's bound reloaded
Abstract: There are a couple of proofs by now for the famous Cwikel--Lieb--Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schr\"odinger operator, proven in the 1970s.
Title: Nodal statistics of graph eigenfunctions
Abstract: Understanding statistical properties of zeros of Laplacian
eigenfunctions is a program which is attracting much attention from
mathematicians and physicists. We will discuss this program in the
setting of "quantum graphs", self-adjoint differential operators
acting on functions living on a metric graph.
Numerical studies of quantum graphs motivated a conjecture that the
distribution of nodal surplus (a suitably rescaled number of zeros of
the n-th eigenfunction) has a universal form: it approaches Gaussian
Dilations of q-commuting unitaries
Let (u,v) be a pair of unitary operators on a Hilbert space H such that vu=quv for a complex number q of modulus 1. For q' another complex number of modulus 1, we determine the smallest constant c>0 for which there exists a pair of q' commuting unitaries (U,V) on a larger Hilbert space K containing H such that (u,v) is the compression of (cU,cV) to H.