The semiclassical limit for a truncated coherent state on a star-graph

A star-graph is a metric graph constituted by N half-lines (called edges) with common origin (called vertex). It is well known that on such metric space one can construct self-adjoint realizations of the Laplacian, this is achieved by prescribing suitable boundary conditions in the vertex. We consider the dynamics of a quantum particle on a star-graph generated by a self-adjoint realization of the Laplacian. 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. The limiting classical dynamics is defined by a Liouville operator on the graph, obtained by means of Krein’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 with respect to the reference Hamiltonian with Dirichlet conditions in the vertex. This is a joint work with Andrea Posilicano and Davide Fermi.