Study of Second-Order Differential Operators on Graphs with Small Edges

In this work, we investigate a Schrödinger operator defined on a model graph containing small loops, under the assumption that the standard nonresonance condition—typically ensuring the holomorphic dependence of the resolvent for elliptic operators on graphs with short edges—is violated. Our analysis focuses on the behavior of those components of the resolvent that correspond to finite edges and small loops. It is shown that these components retain their holomorphic dependence on a small parameter characterizing the length of the loops. In contrast to the nonresonant case, however, the part of the resolvent associated with the small loops develops an additional contribution in the leading term of its Taylor expansion, which results in a certain localization of the resolvent on these loops.