Inverse/observability estimates for Schrodinger equations with variable coefficients

We consider a general Schrodinger equation defined on an open bounded domai n Omega subset of R-n with variable coefficients in both the elliptic princ ipal past and in the first-order terms as well. At first, no boundary condi tions (B.C.) are imposed. Our main result (Theorem 3.5) is a reconstruction , or inverse, estimate for solutions w: under checkable conditions on the c oefficients of the principal part, the H-1(Omega)-energy at time t = T, or at time t = 0, is dominated by the L-2(Sigma)-norms of the boundary traces partial derivative w/partial derivative v(A) and w(t), module an interior l ower-order term. Once homogeneous B.C. are imposed, our results yield - und er a uniqueness theorem, needed to absorb the lower order term - continuous observability estimates for both the Dirichlet and Neumann case, with an a rbitrarily short observability time; hence, by duality, exact controllabili ty results. Moreover, no artificial geometrical conditions are imposed on t he controlled part of the boundary in the Neumann case. In contrast to exis ting literature, the first step of our method employs a Riemann geometry ap proach to reduce the original variable coefficient principal part problem i n Omega subset of R-n to a problem on an appropriate Riemannian manifold (d etermined by the coefficients of the principal part), where the principal p art is the Laplacian. In our second step, we employ explicit Carleman estim ates at the differential level to take care of the variable first-order (en ergy level) terms. In our third step, we employ micro-local analysis yieldi ng sharp trace estimate to remove artificial geometrical conditions on the controlled part of the boundary in the Neumann case.