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2014 The theory of Hahn-meromorphic functions, a holomorphic Fredholm theorem, and its applications
Jörn Müller, Alexander Strohmaier
Anal. PDE 7(3): 745-770 (2014). DOI: 10.2140/apde.2014.7.745

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 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.

Citation

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Jörn Müller. Alexander Strohmaier. "The theory of Hahn-meromorphic functions, a holomorphic Fredholm theorem, and its applications." Anal. PDE 7 (3) 745 - 770, 2014. https://doi.org/10.2140/apde.2014.7.745

Information

Received: 3 September 2013; Revised: 22 November 2013; Accepted: 22 December 2013; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1319.30038
MathSciNet: MR3227433
Digital Object Identifier: 10.2140/apde.2014.7.745

Subjects:
Primary: 47A56, 58J50

Rights: Copyright © 2014 Mathematical Sciences Publishers

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