Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 3 (2014), 745-770.

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

Jörn Müller and Alexander Strohmaier

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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 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.

Article information

Anal. PDE, Volume 7, Number 3 (2014), 745-770.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones) 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]

Hahn series holomorphic Fredholm theorem scattering theory


Müller, Jörn; Strohmaier, Alexander. The theory of Hahn-meromorphic functions, a holomorphic Fredholm theorem, and its applications. Anal. PDE 7 (2014), no. 3, 745--770. doi:10.2140/apde.2014.7.745.

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