Asymptotics of Eigenvalues and Eigenfunctionsof the Sturm-Liouville Operator with a Singular Potential on a Star Graph: I

Spectral problems on the three-edge star graph with a Sturm-Liouville operator defined on each of the edges are studied. The spectral properties of such operators are analyzed; in particular, asymptotic formulas for the eigenvalues and eigenfunctions of the operator with the Dirichlet boundary conditions at the free ends and the continuity and Kirchhoff conditions at the common vertex are obtained. The potential in the Sturm-Liouville problem is assumed to be singular; namely, it is the derivative of a square integrable function in the sense of distributions.