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Wave Dynamics on Graphs

Lionel Kameni
University of Bristol

Postgrad seminar: Mathematical Physics and beyond

Organisers: Dale Smith, Rupert Small

ABSTRACT
We examine how a wave spreads in time along the edges of a quantum graph -
a metric graph equipped with a differential operator. We find that the time
evolution of the wave can be expressed using pseudo-differential operators
and fourier integral operators in terms of a sum over the paths visited by
the components of the wave travelling in opposite directions by performing
a projection on the momentum spaces. We discuss numerical simulations of a
state propagated on the bonds of quantum graphs whose classical analogues have ergodic or mixing (chaotic) dynamics, and observe that in the high-energy (semi-classical) limit, the wave equi-distributes on
the bond-edges, which serves to illustrate a behaviour akin to quantum
ergodicity on quantum graphs. We also observe that how well the wave
equi-distribute is influenced by how strongly chaotic the graph is.