Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

STURM–LIOUVILLE PROBLEMS WITH REDUCIBLE BOUNDARY CONDITIONS

Paul A. Bindinga1, Patrick J. Brownea2 and Bruce A. Watsona3

a1 Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada (binding@ucalgary.ca)

a2 Mathematical Sciences Group, Department of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E6, Canada (browne@admin.usask.ca)

a3 School of Mathematics, University of the Witwatersrand, Private Bag 3, PO WITS 2050, South Africa (bwatson@maths.wits.ac.za)

Abstract

The regular Sturm–Liouville problem

$$ \tau y:=-y''+qy=\lambda y\quad\text{on }[0,1],\ \lambda\in\CC, $$

is studied subject to boundary conditions

$$ P_j(\lambda)y'(j)=Q_j(\lambda)y(j),\quad j=0,1, $$

where $q\in L^1(0,1)$ and $P_j$ and $Q_j$ are polynomials with real coefficients. A comparison is made between this problem and the corresponding ‘reduced’ one where all common factors are removed from the boundary conditions. Topics treated include Jordan chain structure, eigenvalue asymptotics and eigenfunction oscillation.

(Online publication January 25 2007)

(Received February 10 2005)

Keywords

  • Primary 34B25;
  • 46D05;
  • 47E05;
  • Sturm–Liouville problem;
  • eigenparameter-dependent boundary conditions;
  • Jordan chain