The semiclassical limit on a star-graph with Kirchhoff conditions

Abstract

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

arXiv comment31 pages

References in corpus (8)

• Kirchhoff's Rule for Quantum Wires
• Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics
• Chaotic Scattering on Graphs
• Fast solitons on star graphs
• Transport and dynamics on open quantum graphs
• Classical dynamics on graphs
• The properties of the distribution of Gaussian packets on a spatial network
• The semi-classical limit with delta potentials

Cited by in corpus (1)

• The semi-classical limit with a delta-prime potential