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

Abstract

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A, B in the self-adjoint Jacobi operator H =AS + +A − S − +B (with S ± the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E − , E + ], E − < E + , we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric selfadjoint Dirac difference operators D with spectrum given by −E 145 Borg-type result for periodic scalar Jacobi operators to the class of reflectionless matrix-valued Jacobi operators.

A Borg-Type Theorem Associated With Orthogonal Polynomials on the Unit Circle

“…The extension of Theorem 1.2 to matrix-valued reflectionless Jacobi operators (and a corresponding result for Dirac-type difference operators) has recently been obtained in [11].…”

Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday

“…For further information on these circle of ideas see [16,17,37,44,45,46,47,50,51,52,53,55,59,69,107,111,112,113,124,132,145]. In particular, we mention also the review by Fritz [38].…”

Matrix Sturm-Liouville operators with boundary conditions dependent on the spectral parameter

“…The all above mentioned papers related with the differential and difference equations are of scalar coefficients. Spectral analysis of the selfadjoint differential and difference equations with matrix coefficients are studied in [10,11,14]. The spectral analysis of the non-selfadjoint operator, generated in L 2 (R + ) by (1.1) and the boundary condition…”