Inverse Sturm–Liouville problem on equilateral regular tree

Abstract

In this article, we consider a spectral problem generated by the Sturm–Liouville equation on the edges of an equilateral regular tree. It is assumed that the Dirichlet boundary conditions are imposed at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices. The potential in the Sturm–Liouville equations, the same on each edge, is real, symmetric with respect to the middle of an edge and belongs to L ₂(0, a) where a is the length of an edge. Conditions are obtained on a sequence of real numbers necessary and sufficient to be the spectrum of the considered spectral problem.

Keywords:

• Dirichlet boundary condition
• Neumann boundary conditions
• Kirchhoff condition
• eigenvalue
• characteristic function
• multiplicity
• recurrent relations

AMS Subject Classifications::

• 34A55
• 34B45
• 81U40

Acknowledgements

The authors were partly supported by the Swiss National Science Foundation through grant IZ73Z0-128135.