On quantum star graphs with eigenparameter dependent vertex conditions

We investigate the spectral properties of two different boundary value problems on a compact star graph in which the vertex conditions are dependent on the spectral parameter. We treat these boundary value problems as eigenvalue problems in some extended Hilbert spaces by associating them with vector-valued operators. We prove that the corresponding operators are self-adjoint. We construct the characteristic functions of these eigenvalue problems and prove that the corresponding operators have discrete spectrum. Moreover, we present some examples where we construct fundamental solutions and derive the resolvent operators.

Automated summary

We examine the spectral properties of two different boundary value problems on a compact star graph with vertex conditions dependent on the spectral parameter. Treating these problems as eigenvalue problems in extended Hilbert spaces associated with vector-valued operators, we prove that the corresponding operators are self-adjoint. We construct the characteristic functions and show that the operators have discrete spectra. Additionally, we provide examples where fundamental solutions are constructed and resolvent operators are derived.