Boundary observability of time discrete Schrodinger equations

Abstract: In this paper we study the boundary observability estimate of time discrete Schrödinger equations in a bounded domain. By means of a time discrete version of the classical multiplier technique, we prove the uniform observability inequality of the solutions in an appropriate filtered space in which the high frequency components have been filtered. In this way, the well-known boundary observability property of the Schödinger equation can be reproduced as the limit, as , h → 0 of the observability of the time discrete one. Better than the existing result in Ervedoza et al. (2008), our alterative proof shows the rigorous relationship between the filtering parameter and the optimal observation time T. Moreover, the latter one tends to zero as the time scale tends to zero. Finally, the optimality of the order of the filtering parameter is also established for lower dimensional case.