Steve Clark, Fritz Gesztesy, and Walter Renger
Trace Formulas and Borg-Type Theorems for Matrix-Valued Jacobi and Dirac
Finite Difference Operators
(94K, LaTeX)
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^\pm 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}] \cup [E_-^{1/2},E_+^{1/2}], 0 \leq E_- < E_+.