The semiclassical limit on a star-graph with Kirchhoff conditions

We consider the dynamics of a quantum particle of mass $m$ on a $n$-edges star-graph with Hamiltonian $H_K=-(2m{)}^{-1}{\mathrm{\hslash }}^2\mathrm{\Delta }$ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kre\u{\i}n’s theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple $(H_K,H_D^{\oplus })$, where $H_D^{\oplus }$ is the free Hamiltonian with Dirichlet conditions in the vertex.