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 self-adjoint Dirac difference operators D with spectrum given by - E + 1 / 2 ,- E - 1 / 2 ∪ E - 1 / 2 , E + 1 / 2 , 0 ⩽ E - < E + . Our approach is based on trace formulas and matrix-valued (exponential) Herglotz representation theorems. As a by-product of our techniques we obtain the extension of Flaschka's Borg-type result for periodic scalar Jacobi operators to the class of reflectionless matrix-valued Jacobi operators.