Abstract
We consider a compact metric graph of size ε and attach to it several edges (leads) of length of order one (or of infinite length). As ε goes to zero, the graph G ε obtained in this way looks like the star-graph formed by the leads joined in a central vertex. On G ε we define an Hamiltonian H ε , properly scaled with the parameter ε . We prove that there exists a scale invariant effective Hamiltonian on the star-graph that approximates H ε (in a suitable norm resolvent sense) as ε → 0 . The effective Hamiltonian depends on the spectral properties of an auxiliary ε -independent Hamiltonian defined on the compact graph obtained by setting ε = 1 . If zero is not an eigenvalue of the auxiliary Hamiltonian, in the limit ε → 0 , the leads are decoupled.
Highlights
One nice feature of quantum graphs is that they are simple objects
For example in the framework of the analysis of self-adjoint realizations of the Laplacian, it is possible to write down explicit formulae for the relevant quantities, such as the resolvent or the scattering matrix
One may be interested in a simpler, effective model which captures only the most essential features of a complex quantum graph
Summary
One nice feature of quantum graphs (metric graphs equipped with differential operators) is that they are simple objects. In the domain of b out , the boundary conditions in v0 are associated to the values of these the effective Hamiltonian H eigenfunctions in the connecting vertices, see Definition 7 In this case, the boundary conditions in the vertex v0 are scale invariant but, in general, not of decoupling type. In [20] (see references therein), it is shown that all the possible self-adjoint boundary conditions at the central vertex of a star-graph, can be obtained as the limit of Hamiltonians with δ-interactions and magnetic field terms on a graph with a shrinking inner part. We conclude the paper with two appendices: in Appendix A we briefly discuss the proofs of the Kreın resolvent formulae from Section 3; in Appendix B we prove some useful bounds on the eigenvalues and eigenfunctions of Hin. For the convenience of the reader we recall here the notation for the Hamiltonians used in our analysis.
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