Controllability of a 2D quantum particle in a time-varying disc with radial data

TL;DR

This paper demonstrates local controllability of a 2D quantum particle in a disc by deforming its radius, using spectral analysis of the Laplacian and solving a moment problem with Bessel functions.

Contribution

It establishes local controllability for a quantum particle in a deformable disc via radial boundary control, employing spectral analysis and moment problem techniques.

Findings

Controllability achieved around certain radial eigenfunction combinations.

Linearisation reduces the problem to a solvable moment problem.

Analysis involves properties of Bessel functions and nonharmonic Fourier series.

Abstract

In this article we consider a 2-D quantum particle confined a disc whose radius can be deformed continuously in time. We study the problem of controllability of such a quantum particle via deformations of the initial disc, i.e., when we set the time-dependent radius of the disc to be control variable. We prove that the resulting system is locally controllable around some radial trajectories which are linear combinations of the first three radial eigenfunc-tions of the Laplacian in the unit disc with Dirichlet boundary conditions. We prove this result, thanks to the linearisation principle, by studying the linearised system, which leads to a moment problem that can be solved using some results from Nonharmonic Fourier series. In particular, we have to deal with fine properties of Bessel functions.