Schr\"odinger equation in moving domains

TL;DR

This paper develops a framework to define and analyze solutions of the Schrödinger equation on moving domains by transforming it into a fixed reference domain, revealing magnetic boundary conditions and establishing well-posedness and adiabatic results.

Contribution

It introduces a unitary transformation approach to handle Schrödinger equations on moving domains, identifying magnetic boundary conditions and extending results to include diffusion and potentials.

Findings

Established existence of weak and strong solutions.

Identified magnetic boundary conditions for moving domains.

Proved adiabatic and regularization results for domain deformations.

Abstract

We consider the Schr\''odinger equation \begin{equation}\label{eq_abstract} i\partial_t u(t)=-\Delta u(t)~~~~~\text{ on }\Omega(t) \tag{$\ast$} \end{equation}where $\Omega(t)\subset\mathbb{R}$ is a moving domain depending on the time $t\in [0,T]$. The aim of this work is to provide a meaning to the solutions of such an equation. We use the existence of a bounded reference domain $\Omega_0$ and a specific family of unitary maps $h^\sharp(t): L^2(\Omega(t),\mathbb{C})\longrightarrow L^2(\Omega_0,\mathbb{C})$. We show that the conjugation by $h^\sharp$ provides a newequation of the form \begin{equation}\label{eq_abstract2}i\partial_t v= h^\sharp(t)H(t)h_\sharp(t) v~~~~~\text{ on }\Omega_0\tag{$\ast\ast$} \end{equation} where $h_\sharp=(h^\sharp)^{-1}$. The Hamiltonian $H(t)$ is a magnetic Laplacian operator of the form$$H(t)=-(div+iA)\circ(grad+iA)-|A|^2$$where $A$ is an explicit magnetic potential depending on the deformation of the domain $\Omega(t)$. The formulation \eqref{eq_abstract2} enables to ensure the existence of weak and strong solutions of the initial problem \eqref{eq_abstract} on $\Omega(t)$ endowed with Dirichlet boundary conditions. In addition, it also indicates that the correct Neumann type boundary conditions for \eqref{eq_abstract} are not the homogeneous but the magnetic ones$$\partial_\nu u(t)+i\langle\nu| A\rangle u(t)=0,$$even though \eqref{eq_abstract} has no magnetic term. All the previous results are also studied in presence of diffusion coefficients as well as magnetic and electric potentials. Finally, we prove some associated byproducts as an adiabatic result for slow deformations of the domain and atime-dependent version of the so-called ``Moser's trick''. We use this outcome in order to simplify Equation \eqref{eq_abstract2} and to guarantee the well-posedness for slightly less regular deformations of $\Omega(t)$.