virtuell

In the talk we discuss an arbitrary metric graph, to which we glue a graph with edges of lengths proportional to a small parameter . On such graph, we consider a general self-adjoint second order differential operator with varying coefficients subject to general vertex conditions; all coefficients in differential expression and vertex conditions are supposed to be holomorphic in . We introduce a special operator on a special graph obtained by rescaling the aforementioned small edges and assume that it has no embedded eigenvalues at the threshold of its essential spectrum. Under such assumption, we show that that certain parts of the resolvent of the original operator are holomorphic in and we show how to find effectively all coefficients in their Taylor series. This allows us to represent the resolvent of by an uniformly converging Taylor-like series and its partial sums can be used for approximating the resolvent up to an arbitrary power of . In particular, the zero-order approximation reproduces recent convergence results by G. Berkolaiko, Yu. Latushkin, S. Sukhtaiev and by C. Cacciapuoti, but we additionally show that next-to-leading terms in -expansions of the coefficients in the differential expression and vertex conditions can contribute to the limiting operator producing the Robin part at the vertices, to which small edges are incident.

Denis Borisov

Institute of Mathematics UFRC RAS