Institute of Mathematics Jena

Summer School

Spectral Theory
Schrödinger Operators

Jena, July 31 - August 3, 2018

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David Borthwick (Emory University)
Resonances in Geometric Scattering Theory

Hyperbolic manifolds (i.e., manifolds of constant negative sectional curvature) provide a "laboratory" for conjectures about the distribution of scattering resonances in the complex plane and their relationship with classical dynamics, i.e., geodesic flow. In these lectures we'll discuss the geometric scattering theory of a distinguished class of hyperbolic manifolds, the convex co-compact manifolds. We'll begin by developing scattering theory on convex co-compact surfaces and study the associated Selberg zeta function that encodes the relationship between geodesic flow (the length spectrum of closed orbits) and quantum flow (the scattering resonances of the Laplacian). We'll then discuss higher-dimensional analogues and more detailed questions such as spectral gaps and the relationship between the distribution of resonances and the Hausdorff dimension of the trapped set of geodesics.