Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Dependence of the Weyl coefficient on singular interface conditions

Matthias Langera1 and Harald Woraceka2

a1 Department of Mathematics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, UK; Email: (ml@maths.strath.ac.uk)

a2 Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstrasse 8–10/101, 1040 Wien, Austria; Email: (harald.woracek@tuwien.ac.at)

Abstract

We investigate the influence of interface conditions at a singularity of an indefinite canonical system on its Weyl coefficient. An explicit formula which parametrizes all possible Weyl coefficients of indefinite canonical systems with fixed Hamiltonian function is derived. This result is illustrated with two examples: the Bessel equation, which has a singular end point, and a Sturm–Liouville equation whose potential has an inner singularity, which arises from a continuation problem for a positive definite function.

(Received June 22 2007)

Keywords

  • indefinite canonical system;
  • Weyl coefficient;
  • singular potential

2000 Mathematics subject classification

  • Primary 34B20;
  • Secondary 34B30;
  • 46C20