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 H1 0(Ω and the right hand side f can be decomposed in f=g+h where g ∞ L1(0,T;H10(Ω) and hεL 2(0,T;L 2(Ω)), Δh=0 and h /Γ εL 2(0,T;L 2(Γ)), then the solution is in C([0,T];H10(Ω). This obviously contains the case fεL 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.