A regularity property for Schrödinger equations on bounded domains

We give a regularity result for the free Schrödinger equations set in a bounded domain of ℝN which extends the 1-dimensional result proved in Beauchard and Laurent (J. Math. Pures Appl. 94(5):520–554, 2010) with different arguments. We also give other equivalent results, for example, for the free Schrödinger equation, if the initial value is in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H^{1}_{0}(\varOmega)$\end{document} and the right hand side f can be decomposed in f=g+h where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g\in L^{1}(0,T;H^{1}_{0}(\varOmega))$\end{document} and h∈L2(0,T;L2(Ω)), Δh=0 and h/Γ∈L2(0,T;L2(Γ)), then the solution is in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C([0,T];H^{1}_{0}(\varOmega))$\end{document}. This obviously contains the case f∈L2(0,T;H1(Ω)). This result is essential for controllability purposes in the 1-dimensional case as shown in Beauchard and Laurent (J. Math. Pures Appl. 94(5):520–554, 2010) and might be interesting for the N-dimensional case where the controllability problem is open.