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In the present paper, we consider the inverse problem of a Schrödinger equation defined on an open, bounded, connected set Ω of a complete,  n -dimensional Riemannian manifold  M  with metric  g . It is subject to a non-homogenous Dirichlet boundary term. We aim at determining the potential coefficient by means of a Neumann boundary measurement on a portion Γ 1  of the boundary Γ of Ω. Under sharp conditions on the complementary part Γ 0  = Γ∖Γ 1 , and under weak regularity requirements on the data, we establish two canonical results of the inverse problem: (i) global uniqueness and (ii) global Lipschitz stability. The lower bound inequality corresponding to the upper bound inequality contained in (ii) is also given. Our proofs rely on four main ingredients: (a) sharp Carleman estimate at the  H  1 -level for Schrödinger equations on Riemannian manifolds [Control Methods in PDE-Dynamical Systems (Snowbird 2005), Contemp. Math. 426, American Mathematical Society, Providence (2007), 339–404], (b) related continuous observability inequality at the  H  1 -level [Control Methods in PDE-Dynamical Systems (Snowbird 2005), Contemp. Math. 426, American Mathematical Society, Providence (2007), 339–404], (c) a continuous observability inequality at the  L  2 -level [J. Inverse Ill-Posed Probl. 12 (2004), 43–123], [Functional Analysis and Evolution Equations, Birkhäuser, Basel (2008), 613–636], (d) optimal regularity theory for Schrödinger equations with Dirichlet boundary data ([Differential Integral Equations 5 (1992), 521–535], [Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Encyclopedia Math. Appl. 2, Cambridge University Press, Cambridge, 2000] as well as the new Theorem 3.6).

journal of inverse and ill-posed problems

Global uniqueness and stability in determining the electric potential coefficient of an inverse problem for Schrödinger equations on Riemannian manifolds