Time evolution and the Schrödinger equation on time dependent quantum graphs * *Dedicated to Michael Berry on his forthcoming anniversary.

Abstract

The purpose of the present paper is to discuss the time dependent Schrödinger equation on a metric graph with time-dependent edge lengths, and the proper way to pose the problem so that the corresponding time evolution is unitary. We show that the well posedness of the Schrödinger equation can be guaranteed by replacing the standard Kirchhoff Laplacian with a magnetic Schrödinger operator with a harmonic potential. We then generalize the result to time dependent families of vertex conditions. We also apply the theory to show the existence of a geometric phase associated with a slowly changing quantum graph.