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The theory of
Hahn-meromorphic functions, a holomorphic Fredholm theorem,
and its applications
Jörn Müller and Alexander Strohmaier |
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Vol. 7 (2014), No. 3, 745–770
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Abstract |
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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 and also terms containing 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. |
Keywords
Hahn series, holomorphic Fredholm theorem, scattering
theory
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Mathematical Subject Classification 2010
Primary: 47A56, 58J50
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Milestones
Received: 3 September 2013
Revised: 22 November 2013
Accepted: 22 December 2013
Published: 18 June 2014
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Authors |
| Jörn Müller | |
| Institut für Mathematik Humboldt-Universität zu Berlin Unter den Linden 6 D-10099 Berlin Germany |
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| Alexander Strohmaier | |
| Department of Mathematical
Sciences Loughborough University Loughborough, Leicestershire LE11 3TU United Kingdom |
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