Wilhelm Schlag

Sixty Years of Bernoulli Convolutions

The distribution νλ of the random series random series Σ±λn is the infinite convolution product of These measures have been studied since the 1930’s, revealing connections with harmonic analysis, the theory of algebraic numbers, dynamical systems, and Hausdorff dimension estimation. In this survey we describe some of these connections, and the progress that has been made so far on the fundamental open problem: For which λ∈ is νλ, absolutely continuous?

Dispersive Estimates for Schrödinger Operators in Dimensions One and Three

We consider L1→L∞ estimates for the time evolution of Hamiltonians H=−Δ+V in dimensions d=1 and d=3 with bound We require decay of the potentials but no regularity. In d=1 the decay assumption is ∫(1+|x|)|V(x)|dx<∞, whereas in d=3 it is |V(x)|≤C(1+|x|)−3−.

Renormalization and blow up for charge one equivariant critical wave maps

We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ2+1→S2 of the form

Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions

u(t,r)=Q(\lambda(t)r)+\mathcal{R}(t,r)

Dispersive analysis of charge transfer models

where u is the polar angle on the sphere,

On the focusing critical semi-linear wave equation

Q(r)=2\arctan r

Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension

is the ground state harmonic map, λ(t)=t-1-ν, and

Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions

\mathcal{R}(t,r)

Spectral theory and mathematical physics : a festschrift in honor of Barry Simon's 60th birthday

is a radiative error with local energy going to zero as t→0. The number

Global dynamics away from the ground state for the energy-critical nonlinear wave equation

\nu>\frac{1}{2}