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

## TL;DR

This paper develops a scale-invariant effective Hamiltonian for a graph with a small core connected to leads, showing it approximates the original Hamiltonian as the core shrinks, with implications for decoupling in the limit.

## Contribution

It introduces a novel scale-invariant effective Hamiltonian that approximates the scaled Hamiltonian on shrinking graphs, depending on spectral properties of an auxiliary Hamiltonian.

## Key findings

- Effective Hamiltonian approximates original as core shrinks
- Leads decouple if zero is not an eigenvalue of auxiliary Hamiltonian
- Provides a norm resolvent convergence result

## Abstract

We consider a compact metric graph of size $\varepsilon$, and attach to it several edges (leads) of length of order one (or of infinite length). As $\varepsilon$ goes to zero, the graph $\mathcal{G}^\varepsilon$ obtained in this way looks like the star-graph formed by the leads joined in a central vertex. On $\mathcal{G}^\varepsilon$ we define an Hamiltonian $H^\varepsilon$, properly scaled with the parameter $\varepsilon$. We prove that there exists a scale invariant effective Hamiltonian on the star-graph that approximates $H^\varepsilon$ (in a suitable norm resolvent sense) as $\varepsilon\to0$. The effective Hamiltonian depends on the spectral properties of an auxiliary $\varepsilon$-independent Hamiltonian defined on the compact graph obtained by setting $\varepsilon = 1$. If zero is not an eigenvalue of the auxiliary Hamiltonian, in the limit $\varepsilon\to0$, the leads are decoupled.