A regularity property for Schrödinger equations on bounded domains

Abstract

We give a regularity result for the free Schrödinger equations set in a bounded domain of ℝ^N 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 $H^{1}_{0}(\varOmega)$ and the right hand side f can be decomposed in f=g+h where $g\in L^{1}(0,T;H^{1}_{0}(\varOmega))$ and h∈L ²(0,T;L ²(Ω)), Δh=0 and h _/Γ ∈L ²(0,T;L ²(Γ)), then the solution is in $C([0,T];H^{1}_{0}(\varOmega))$. This obviously contains the case f∈L ²(0,T;H ¹(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.