Resolvent expansions on hybrid manifolds

We study Laplace-type operators on hybrid manifolds, i.e. on configurations consisting of closed two-dimensional manifolds and one-dimensional segments. Such an operator can be constructed by using the Laplace-Beltrami operators on each component with some boundary conditions at the points of gluing. The large spectral parameter expansion of the trace of the second power of the resolvent is obtained. Some questions of the inverse spectral theory are adressed.