Spectral conditions for admissibility and observability of Schrödinger systems: Applications to finite element discretizations

In this article, we derive uniform admissibility and observability properties for the finite element space semi-discretizations of iż=A₀z, where A₀ is an unbounded self-adjoint positive definite operator with compact resolvent. In order to address this problem, we present several spectral criteria for admissibility and observability of such systems, which will be used to derive several results for space semi-discretizations of iż=A₀z. Our approach provides very general results, which stand in any dimension and for any regular mesh (in the sense of finite elements). We also present applications to admissibility and observability for fully discrete approximation schemes, and to controllability and stabilization issues.