Schrödinger Equation in Moving Domains

We consider the Schrodinger equation where $$\Omega (t)\subset \mathbb {R}^N$$ 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 new equation of the form where $$h_\sharp =(h^\sharp )^{-1}$$ . The Hamiltonian H(t) is a magnetic Laplacian operator of the form $$\begin{aligned} H(t)=-({\text {div}}_x+iA)\circ (\nabla _x+iA)-|A|^2 \end{aligned}$$ where A is an explicit magnetic potential depending on the deformation of the domain $$\Omega (t)$$ . The formulation ( $$**$$ ) enables to ensure the existence of weak and strong solutions of the initial problem ( $$*$$ ) on $$\Omega (t)$$ endowed with Dirichlet boundary conditions. In addition, it also indicates that the correct Neumann-type boundary conditions for ( $$*$$ ) are not the homogeneous but the magnetic ones $$\begin{aligned} \partial _\nu u(t)+i\langle \nu | A\rangle u(t)=0, \end{aligned}$$ even though ( $$*$$ ) has no magnetic term. All the previous results are also studied in the presence of diffusion coefficients as well as magnetic and electric potentials. Finally, we prove some associated by-products as an adiabatic result for slow deformations of the domain and a time-dependent version of the so-called Moser’s trick. We use this outcome in order to simplify Eq. ( $$**$$ ) and to guarantee the well-posedness for slightly less regular deformations of $$\Omega (t)$$ .