Vol. 7, No. 3, 2014

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

Jörn Müller and Alexander Strohmaier

Vol. 7 (2014), No. 3, 745–770

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 noninteger powers of z and also terms containing logz can appear. We show that, under natural assumptions, some important theorems from complex analysis carry over to this class of 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 in establishing analyticity of the scattering matrix and the existence of generalized eigenfunctions at points in the spectrum.

Hahn series, holomorphic Fredholm theorem, scattering theory
Mathematical Subject Classification 2010
Primary: 47A56, 58J50
Received: 3 September 2013
Revised: 22 November 2013
Accepted: 22 December 2013
Published: 18 June 2014
Jörn Müller
Institut für Mathematik
Humboldt-Universität zu Berlin
Unter den Linden 6
D-10099 Berlin
Alexander Strohmaier
Department of Mathematical Sciences
Loughborough University
Loughborough, Leicestershire
LE11 3TU
United Kingdom