The theory of Hahn meromorphic functions, a holomorphic Fredholm theorem and its applications

TL;DR

This paper develops a new class of functions with generalized power series expansions, extending complex analysis tools like the Fredholm theorem, and applies these to spectral theory and scattering problems involving Bessel and Laplace-Beltrami operators.

Contribution

It introduces a class of Hahn meromorphic functions with generalized expansions and extends the Fredholm theorem to this class, enabling new spectral analysis applications.

Findings

Established a field of generalized meromorphic functions.

Proved a version of the Fredholm theorem for these functions.

Applied results to spectral theory of differential operators.

Abstract

We introduce a class of functions near zero on the logarithmic cover of the complex plane that have convergent expansions into generalized power series. The construction covers cases where non-integer powers of $z$ and also terms containing $\log z$ can appear. We show that under natural assumptions some important theorems from complex analysis carry over to the class of these functions. In particular it is possible to define a field of functions that generalize meromorphic functions and one can formulate an analytic Fredholm theorem in this class. We show that this modified analytic Fredholm theorem can be applied in spectral theory to prove convergent expansions of the resolvent for Bessel type operators and Laplace-Beltrami operators for manifolds that are Euclidean at infinity. These results are important in scattering theory as they are the key step to establish analyticity of the scattering matrix and the existence of generalized eigenfunctions at points in the spectrum.