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 HK= - (2 m) - 1ħ2Δ 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 Kreĭn’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 (HK,HD⊕), where HD⊕ is the Hamiltonian with Dirichlet conditions in the vertex.