Eigenparameter dependent Sturm–Liouville problems in boundary conditions with transmission conditions

Abstract

In this paper, we investigate the nonselfadjoint (dissipative) boundary value transmission problems in Weyl’s limit-circle case. At first using the method of operator-theoretic formulation we pass to a new operator. After showing that this new operator is a maximal dissipative operator, we construct a selfadjoint dilation of the maximal dissipative operator. Using the equivalence of the Lax–Phillips scattering function and the Sz.-Nagy-Foiaş characteristic function, we show that all eigenfunctions and associated functions are complete in the space Lw2(Ω).