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.

Principally inspired by the very geometrically physic characteristics of the divergence theorem (regarded as one of the fundamental theorems within mathematical physics & vector calculus), this talk will address: quasilinear relative capacity; deficits via isocapacity & isoperimeter; geometric Green/Robin function; trace principle & energy evaluation; quasilinear graph mass & mean curvature.