# Uniqueness results for matrix-valued Schrödinger, Jacobi, and Dirac-type operators

Journal Article (Journal Article)

Let g(z,x) denote the diagonal Green's matrix of a self-adjoint m × m matrix-valued Schrödinger operator H = -d /dx I + Q in L (ℝ) , m ∈ ℕ. One of the principal results proven in this paper states that for a fixed x ∈ ℝ and z ∈ ℂ , g(z,x ) and g′(z,x ) 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 g (z,x), j = 1, 2 be the diagonal Green's matrices of the self-adjoint Schrödinger operators H = -d /dx I + Q in L (ℝ) . Suppose that for fixed a > 0 and x ∈ ℝ, ∥g (z,x ) - g (z,x )∥ℂ + ∥g′ (z,x ) - g′ (z,x )∥ℂ = |z|→∞ O(e ) for z inside a cone along the imaginary axis with vertex zero and opening angle less than π/2, excluding the real axis. Then Q (x) = Q (x) for a.e. x ∈ [x - a,x + a]. Analogous results are proved for matrix-valued Jacobi and Dirac-type operators. 2 2 2 m 2 2 2 m m×m m×m -2Im(z1/2)a m 0 + 0 0 j j m j 0 1 0 2 0 1 0 2 0 1 2 0 0

### Full Text

### Duke Authors

### Cited Authors

- Gesztesy, F; Kiselev, A; Makarov, KA

### Published Date

- August 23, 2002

### Published In

### Volume / Issue

- 239-240 /

### Start / End Page

- 103 - 145

### International Standard Serial Number (ISSN)

- 0025-584X

### Digital Object Identifier (DOI)

- 10.1002/1522-2616(200206)239:1<103::AID-MANA103>3.0.CO;2-F

### Citation Source

- Scopus