АN ELLIPTIC SELF-ADJOINT OPERATOR OF THE SECOND ORDER ON A GRAPH WITH SMALL EDGES

Abstract

This work is devoted to the study of a second-order elliptic self-adjoint operator on a metric graph with short edges. The underlying structure is constructed by rescaling a given graph by the factor ε^-1 and attaching it to another fixed graph, where ε > 0 is a small parameter. No substantial restrictions are imposed on the pair of graphs. On this combined structure, we define a general second-order elliptic self-adjoint operator whose differential expression involves derivatives of arbitrary order with variable coefficients and a non-constant potential. The vertex
conditions are taken in a general form as well. All coefficients, both in the differential expression and in the vertex conditions, are allowed to depend analytically on the small parameter ε. It was previously established that the components of the resolvent corresponding to the restrictions of the operator to the fixed-length edges and to the short edges are analytic in ε as operators in the
corresponding functional spaces, with the restriction on short edges additionally conjugated by dilation operators. Analyticity here means representability of these operator families by Taylor series. The first principal result of the paper is a recursive procedure, reminiscent of the method
of matched asymptotic expansions, for determining all coefficients of such Taylor series. The second main result provides a convergent expansion of the resolvent in the form of a Taylor-type series, together with effective estimates of the remainder terms.