# The semiclassical limit on a star-graph with Kirchhoff conditions

May 08, 2020

We consider the dynamics of a quantum particle of mass $m$ on a $n$-edges
star-graph with Hamiltonian $H_K=-(2m)^{-1}\hbar^2 \Delta$ 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.

Keywords:

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