# Controllability of periodic bilinear quantum systems on infinite graphs

## TL;DR

This paper investigates the controllability of bilinear Schrödinger equations on infinite graphs, establishing well-posedness and controllability results for quantum states on complex network structures.

## Contribution

It introduces a framework for analyzing controllability of quantum systems on infinite graphs, including well-posedness and examples like tadpole and star graphs.

## Key findings

- Well-posedness in subspaces of $D(| riangle|^{3/2})$
- Global exact controllability results
- Applications to tadpole and star graphs with infinite spokes

## Abstract

In this work, we study the controllability of the bilinear Schr\"odinger equation on infinite graphs for periodic quantum states. We consider the bilinear Schr\"odinger equation $i\partial_t\psi=-\Delta\psi+u(t)B\psi$ in the Hilbert space $L^2_p$ composed by functions defined on an infinite graph $\mathscr{G}$ verifying periodic boundary conditions on the infinite edges. The Laplacian $-\Delta$ is equipped with specific boundary conditions, $B$ is a bounded symmetric operator and $u\in L^2((0,T),\mathbb{R})$ with $T>0$. We present the well-posedness of the system in suitable subspaces of $D(|\Delta|^{3/2})$ . In such spaces, we study the global exact controllability and we provide examples involving for instance tadpole graphs and star graphs with infinite spokes.