On a Model Graph with a Loop and Small Edges. Holomorphy Property of Resolvent

We consider the Schrodinger operator on a graph consisting of two infinite edges, a loop, and a glued (at the start and end points of the loop) graph obtained by e−1 times contraction of some fixed graph. The Kirchhoff conditions are imposed at interior vertices and the Dirichlet or Neumann conditions are imposed at boundary vertices of the graph. We show that the resolvent of the Schrodinger operator is holomorphic with respect to the small parameter e and write out the first three terms of the asymptotic expansion of the resolvent.