Scattering theory on Riemann surfaces I: Schiffer operators, cohomology, and index theorems

TL;DR

This paper studies conformally invariant integral operators on Riemann surfaces with boundary, developing a calculus for Schiffer operators, and establishing index theorems linking conformal and topological invariants.

Contribution

It introduces a comprehensive calculus for Schiffer and Cauchy operators on Riemann surfaces with boundary, including jump formulas and index theorems, extending previous results to quasicircles.

Findings

Derived a Plemelj-Sokhotski jump formula for quasicircles.

Proved Schiffer operators are isomorphisms for quasicircles.

Established index theorems connecting conformal and topological invariants.

Abstract

We consider a compact Riemann surface $\mathscr{R}$ with a complex of non-intersecting Jordan curves, whose complement is a pair of Riemann surfaces with boundary, each of which may be possibly disconnected. We investigate conformally invariant integral operators of Schiffer, which act on $L^{2}$ anti-holomorphic one-forms on one of these surfaces with boundary and produce holomorphic one-forms on the disjoint union. These operators arise in potential theory, boundary value problems, approximation theory, and conformal field theory, and are closely related to a kind of Cauchy operator. We develop an extensive calculus for the Schiffer and Cauchy operators, including a number of adjoint identities for the Schiffer operators. In the case that the Jordan curves are quasicircles, we derive a Plemelj-Sokhotski jump formula for Dirichlet-bounded functions. We generalize a theorem of Napalkov and Yulmukhametov, which shows that a certain Schiffer operator is an isomorphism for quasicircles. Finally, we characterize the kernels and images, and derive index theorems for the Schiffer operators, which will in turn connect conformal invariants to topological invariants.