Existence of a tree of Stieltjes strings corresponding to two given spectra

Abstract

Transversal vibrations of a plane rooted tree of Stieltjes strings are considered with Dirichlet boundary conditions at each pendant vertex. Continuity and Kirchhoff conditions are imposed at each interior vertex except the root. The Dirichlet problem is that with the Dirichlet condition at the root, and the Neumann problem is that with continuity and Kirchhoff conditions at the root. It is shown that strict interlacing is a sufficient condition for two sequences of real numbers to be the spectra of Neumann and Dirichlet problems generated by Stieltjes string recurrent relations on any prescribed tree.