Title: Resolvent expansion for the Schrödinger operator on a graph with infinite rays,
Abstract: We consider the Schrödinger operator on a combinatorial graph consisting of
a finite graph and a finite number of discrete half-lines, all jointed
together, and compute an asymptotic expansion of its resolvent around the
threshold $0$. Precise expressions are obtained for the first few coefficients
of the expansion in terms of the generalized eigenfunctions. This result
justifies the classification of threshold types solely by growth properties of
the generalized eigenfunctions. By choosing an appropriate free operator a
priori possessing no zero eigenvalue or zero resonance we can simplify the
expansion procedure as much as that on the single discrete half-line.