The semiclassical limit on a star-graph with Kirchhoff conditions

Abstract

AbstractWe consider the dynamics of a quantum particle of mass m on a n-edges star-graph with Hamiltonian $$H_K=-(2m)^{-1}\hbar ^2 \Delta $$ H K = - ( 2 m ) - 1 ħ 2 Δ 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 $$(H_K,H_{D}^{\oplus })$$ ( H K , H D ⊕ ) , where $$H_{D}^{\oplus }$$ H D ⊕ is the Hamiltonian with Dirichlet conditions in the vertex.