Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part II: L[sub 2]([...])-estimates.

We consider a general non-conservative Schrödinger equation defined on an open bounded domain Ω in Rn, with C²-boundary ..., subject to (Dirichlet and, as a main focus, to) Neumann boundary conditions on the entire boundary Γ. Here, Γ0 and Γ1 are the unobserved (or uncontrolled) and observed (or controlled) parts of the boundary, respectively, both being relatively open in Γ. The Schrödinger equation includes energy-level (H¹(Ω)-level) terms, which accordingly may be viewed as unbounded perturbations. The first goal of the paper is to provide Carleman-type inequalities at the H¹-level, which do not contain lower-order terms; this is a distinguishing feature over most of the literature. This goal is accomplished in a few steps: the paper obtains first pointwise Carleman estimates for C²-solutions; and next, it turns these pointwise estimates into integral-type Carleman estimates with no lower-order terms, originally for H²-solutions, and ultimately for H¹-solutions, The passage from H² to H¹-solutions is readily accomplished in the case of Dirichlet B.C., but it requires a delicate regularization argument in the case of Neumann B.C. This is so since finite energy solutions are known to have L2-normal traces in the case of Dirichlet B.C., but by contrast do not produce H¹-traces in the case of Neumann B.C. From Carleman-type inequalities with no lower-order terms, one then obtains the sought-after benefits. These consist of deducing, in one shot, as a part of the same flow of arguments, two important implications: (i) global uniqueness results for H1-solutions satisfying over-determined boundary conditions, and--above all--(ii) continuous observability (or stabilization) inequalities with an explicit constant. The more demanding purely Neumann boundary conditions requires the same geometrical conditions... [ABSTRACT FROM AUTHOR]