Scale Invariant Effective Hamiltonians for a Graph with a Small Compact Core

Abstract

We consider a compact metric graph of size $\epsilon$, and attach to it several edges (leads) of length of order one (or of infinite length). As $\epsilon$ goes to zero, the graph ${\mathcal{G}}^{\epsilon }$ obtained in this way looks like the star-graph formed by the leads joined in a central vertex. On ${\mathcal{G}}^{\epsilon }$ we define an Hamiltonian $H^{\epsilon }$, properly scaled with the parameter $\epsilon$. We prove that there exists a scale invariant effective Hamiltonian on the star-graph that approximates $H^{\epsilon }$ (in a suitable norm resolvent sense) as $\epsilon \to 0$. The effective Hamiltonian depends on the spectral properties of an auxiliary $\epsilon$-independent Hamiltonian defined on the compact graph obtained by setting $\epsilon =1$. If zero is not an eigenvalue of the auxiliary Hamiltonian, in the limit $\epsilon \to 0$, the leads are decoupled.

23 pages

References in corpus (4)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Limits of Quantum Graph Operators With Shrinking Edges
• Approximations of quantum-graph vertex couplings by singularly scaled rank-one operators
• Schrödinger operators on star graphs with singularly scaled potentials supported near the vertices