, Volume 26, Issue 1, pp 183-192

A regularity property for Schrödinger equations on bounded domains

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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 hL 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 fL 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.