Time evolution and the Schr\"odinger equation on time dependent quantum graphs

TL;DR

This paper studies the time-dependent Schr"odinger equation on quantum graphs with evolving edge lengths and vertex conditions, ensuring well-posedness through magnetic operators and exploring geometric phases.

Contribution

It introduces a framework for well-posed quantum dynamics on time-dependent graphs using magnetic Schr"odinger operators and extends the theory to variable vertex conditions.

Findings

Well-posedness guaranteed by magnetic Schr"odinger operators

Extension to time-dependent vertex conditions

Existence of geometric phase in slowly changing quantum graphs

Abstract

The purpose of the present paper is to discuss the time dependent Schr\"odinger 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 Schr\"odinger equation can be guaranteed by replacing the standard Kirchhoff Laplacian with a magnetic Schr\"odinger 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.