Diffusion Operator On The Finite Tree Graph Of The Borg-levinson Theorem

We study boundary value problems on compact graphs without circles(i.e. on finite trees) for the quadratic differential pencil of the Schrodinger operator(i.e. diffusion op-erator). We establish the properties of spectral characteristics and investigate the inverse spectral problem of uniquely determining the potentials in the differential equations by using the so-called Dirichlet-Neumann map instead of the Titchmarsh-Weyl function (m-function) for the classical Sturm-Liouville operators, and then as an application, we discuss the inverse spectral problems of the Klein-Gordon equations on the finite tree graphs. We note that the obtained results are natural generalizations of the well-known Borg-Levinson theorem on the inverse spectral theory for the classical Sturm-Liouville operators.