M-matrix asymptotics for Sturm-Liouville problems on graphs

Authors:
Sonja Currie;Bruce A. Watson
Affiliations:
School of Mathematics, University of the Witwatersrand, Private Bag 3, P.O. WITS 2050, South Africa;School of Mathematics, University of the Witwatersrand, Private Bag 3, P.O. WITS 2050, South Africa
Venue:
Journal of Computational and Applied Mathematics
Year:
2008

Citing 3
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Abstract

We consider a system formulation for Sturm-Liouville operators with formally self-adjoint boundary conditions on a graph. An M-matrix associated with the boundary value problem is defined and related to the matrix Prufer angle associated with the system boundary value problem, and consequently with the boundary value problem on the graph. Asymptotics for the M-matrix are obtained as the eigenparameter tends to negative infinity. We show that the boundary conditions may be recovered, up to a unitary equivalence, from the M-matrix and that the M-matrix is a Herglotz function. This is the first in a series of papers devoted to the reconstruction of the Sturm-Liouville problem on a graph from its M-matrix.