A STURM-LIOUVILLE PROBLEM DEPENDING RATIONALLY ON THE EIGENVALUE PARAMETER

MATTHIAS LANGER

mlanger@math.uni-bremen.de

FB 3 - Mathematik
Universität Bremen
Bibliothekstrasse 1
D-28359 Bremen, Germany

We consider eigenvalues of the following Sturm-Liouville problem

$$-y''+(q-\lambda-\frac{w}{u-\lambda})y=0$$

on the interval $[0,1]$ with Dirichlet boundary conditions, where $q$, $u$, and $w$ are real functions. If $u$ is piece-wise continuous, then in every gap of the range of $u$ the eigenvalues can be characterised by a varitional principle with an index shift. The behaviour at the end-points of the gaps and the number of eigenvules are investigated.