Absolute continuity of the free Hamiltonian and almost exponential decay of resonances in non-relativistic quantum mechanical graphs

The free Hamiltonian operator for a non-relativistic particle on a finite connected acyclic graph with one boundary point is absolutely continuous. A given Hamiltonian operator H on a connected, unbounded graph Γ, whose potential is zero on all unbounded elements, has a resonance at $z\in C$ if there is a non-normalizable solution $y$ of (H − z)$y$ = 0 satisfying given boundary conditions. For the function $4$, which is obtained by cutting off $y$ on all unbounded elements, is $Oblnb$, where .