Stability estimates for determination of potential from the impedance boundary map

We study the impedance boundary map (or Robin-to-Robin map) for the Schrödinger equation in open bounded demain at fixed energy in multidimensions. At least, in dimension d ≥ 3, we give global stability estimates for determining potential from these boundary data and, as corollary, from the Cauchy data set. Our results include also, in particular, an extension of the Alessandrini identity to the case of the impedance boundary map. We consider the Schrödinger equation −∆ψ + v(x)ψ = Eψ, x ∈ D, E ∈ R, (0.1) where D is an open bounded domain in R, d ≥ 2, with ∂D ∈ C, (0.2) v ∈ L∞(D). (0.3) We consider the impedance boundary map M̂α = M̂α,v(E) defined by M̂α[ψ]α = [ψ]α−π/2 (0.4) for all sufficiently regular solutions ψ of equation (0.1) in D̄ = D ∪ ∂D, where [ψ]α = [ψ(x)]α = cosαψ(x)− sinα ∂ψ ∂ν |∂D(x), x ∈ ∂D, α ∈ R (0.5) and ν is the outward normal to ∂D. One can show(see Lemma 2.2) that there is not more than a countable number of α ∈ R such that E is an eigenvalue for the operator −∆ + v in D with the boundary condition cosαψ|∂D − sinα ∂ψ ∂ν |∂D = 0. (0.6) Therefore, for any energy level E we can assume that for some fixed α ∈ R E is not an eigenvalue for the operator −∆ + v in D with boundary condition (0.6) (0.7) and, as a corollary, M̂α can be defined correctly. Note that the impedance boundary map M̂α is reduced to the Dirichlet-toNeumann(DtN) map if α = 0 and is reduced to the Neumann-to-Dirichlet(NtD) map if α = π/2. The map M̂α can be called also as the Robin-to-Robin map. General Robin-to-Robin map was considered, in particular, in [9].