Time dependent quantum graphs, the geometric phase and knots on closed Lissajous figures
The purpose of the present paper is threefold. The first is to discuss the time dependent Schrodinger equation on a metric graph with time-dependent edges, and the proper way to pose the problem so that the corresponding time evolution is unitary. The second is to compute the geometric phase acquired by an eigenstate as it adiabatically evolves in time as the edge lengths change. The third is to study the connection between the geometric phase and the topology of the cycles in parameter space when they are taken to be Lissajous knots. We show that the frequencies of the Lissajous knot can be deduced from the geometric phase.