Uniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators

Abstract

Let g(z, x) denote the diagonal Green's matrix of a self‐adjoint m × m matrix‐valued Schrödinger operator One of the principal results proven in this paper states that for a fixed x0 ∈ ℝ and all 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 Suppose that for fixed a > 0 and x0 ∈ ℝ, 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.

Uniqueness of the potential function for the vectorial Sturm-Liouville equation on a finite interval

“…For the case m ≥ 2, some interesting results had been obtained (see [10][11][12][13][14][15][16][17][18][19][20]). In particular, for m = 2 and Q(x) is a two-by-two real symmetric matrix-valued smooth functions defined in the interval [0, π] Shen [18] showed that five spectral data can determine Q(x) uniquely.…”

Construction of the Solution of the Inverse Spectral Problem for a System Depending Rationally on the Spectral Parameter, Borg–Marchenko-Type Theorem and Sine-Gordon Equation

“…An important series of papers by Gesztesy, Simon and coauthors on the high energy asymptotics of the Weyl functions and local Borg-Marchenkotype uniqueness results has initiated a growing interest in this important domain (see [6,[8][9][10]16,17,19,26,38,46,47] and references therein). The Weyl-Titchmarsh theory for a non-self-adjoint case (the skew-self-adjoint Dirac type system) has been studied in [7,20,34,35] and the Borg-Marchenko-type results for this system have been published in [39].…”

Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators

“…As discussed in detail in [23], while various aspects of inverse spectral theory for scalar Schrödinger, Jacobi, and Dirac-type operators, and more generally, for 2 × 2 Hamiltonian systems, are well understood by now, the corresponding theory for such operators and Hamiltonian systems with m × m, m ∈ N, matrix-valued coefficients is still largely a wide open field. A particular inverse spectral theory aspect we have in mind is that of determining isospectral sets (manifolds) of such systems.…”

“…Inverse spectral and scattering theory for matrix-valued finite difference systems and its intimate connection to matrixvalued orthogonal polynomials and the moment problem are treated in [1,2], [4,Section VII.2], [16][17][18]21,39,40,[45][46][47], [49,Chapters 8], [50]. A number of uniqueness theorems for matrix-valued Jacobi operators were proved in [23] (cf. also [29]).…”