Dynamics on quantum graphs as constrained systems

We study the possibility of regarding the dynamics on a quantum graph as limit, as a small parameter ∈ → O, of a dynamics with a strong confining potential. We define a projection operator along the first eigenfunction of a transversal operator and, under suitable assumptions, we prove that the projection of the solution strongly converges along subsequences to a function that satisfies the Schrödinger equation on each open edge of the graph. Moreover the limit dynamics is unitary. If the limit is independent of the subsequence, one has a limit one-parameter group, generated by one of the self-adjoint extensions of a symmetric operator defined on the open graph (with the vertices deleted). The crucial role of the shape of the confining potential at the vertices is pointed out.