Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators |
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Authors: | Steve Clark Fritz Gesztesy Walter Renger |
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Institution: | a Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, MO 65409, USA b Department of Mathematics, University of Missouri, Columbia, MO 65211, USA c Dr. Johannes Heidenhain GMBH, 83301 Traunreut, Germany |
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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 , 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. |
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Keywords: | primary 34E05 34B20 34L40 secondary 34A55 |
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