## Mathematical Physics and Harmonic Analysis Seminar

**Date: ** September 9, 2022

**Time: ** 1:50PM - 2:50PM

**Location: ** Zoom

**Speaker: **Konstantin Merz, Technische Universität Braunschweig

**Title: ***Random Schrödinger operators with complex decaying potentials*

**Abstract: **Estimating the location and accumulation rate of eigenvalues of
Schrödinger operators is a classical problem in spectral theory and
mathematical physics. The pioneering work of R. Frank (Bull. Lond.
Math. Soc., 2011) illustrated the power of Fourier analytic methods —
like the uniform Sobolev inequality by Kenig, Ruiz, and Sogge, or the
Stein–Tomas restriction theorem — in this quest, when the potential
is non-real and has “short range”.
Recently S. Bögli and J.-C. Cuenin (arXiv:2109.06135) showed that
Frank’s “short-range” condition is in fact optimal, thereby disproving
a conjecture by A. Laptev and O. Safronov (Comm. Math. Phys., 2009)
concerning Keller-Lieb-Thirring-type estimates for eigenvalues of
Schrödinger operators with complex potentials.
In this talk, we estimate complex eigenvalues of continuum random
Schrödinger operators of Anderson type. Our analysis relies on methods
of J. Bourgain (Discrete Contin. Dyn. Syst., 2002, Lecture Notes in
Math., 2003) related to almost sure scattering for random lattice
Schrödinger operators, and allows us to consider potentials which
decay almost twice as slowly as in the deterministic case.
The talk is based on joint work with Jean-Claude Cuenin.