Uniqueness results for matrix-valued Schrödinger, Jacobi, and Dirac-type operators
Let g(z,x) denote the diagonal Green's matrix of a self-adjoint m × m matrix-valued Schrödinger operator H = -d2/dx2Im + Q in L2(ℝ)m, m ∈ ℕ. One of the principal results proven in this paper states that for a fixed x0 ∈ ℝ and z ∈ ℂ+, g(z,x0) and g′(z,x0) uniquely determine the matrix-valued m × m potential Q(x) for a.e. x ∈ ℝ. We also prove the following local version of this result. Let gj(z,x), j = 1, 2 be the diagonal Green's matrices of the self-adjoint Schrödinger operators Hj = -d2/dx2Im + Qj in L2(ℝ)m. Suppose that for fixed a > 0 and x0 ∈ ℝ, ∥g1(z,x0) - g2(z,x0)∥ℂm×m + ∥g′1(z,x0) - g′2(z,x0)∥ℂm×m = |z|→∞ O(e-2Im(z1/2)a) for z inside a cone along the imaginary axis with vertex zero and opening angle less than π/2, excluding the real axis. Then Q1(x) = Q2(x) for a.e. x ∈ [x0 - a,x0 + a]. Analogous results are proved for matrix-valued Jacobi and Dirac-type operators.
Gesztesy, F; Kiselev, A; Makarov, KA
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