2002/01/03 by E. D. Belokolos, F. Gesztesy, Fritz Gesztesy +8
Computer Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Matrix Theory and Algorithms #Quantum and electron transport phenomena #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.SP
paper · pdf · doi:10.48550/arxiv.math/0201019
LaTeX, 28 pages
arxiv created 2002/01/03 · openalex publication_date 2002/01/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a generalization of the well-known theorems by Borg and Hochstadt for periodic self-adjoint Schrödinger operators without a spectral gap, respectively, one gap in their spectrum, in the matrix-valued context. Our extension of the theorems of Borg and Hochstadt replaces the periodicity condition of the potential by the more general property of being reflectionless (the resulting potentials then automatically turn out to be periodic and we recover Després' matrix version of Borg's result). In addition, we assume the spectra to have uniform maximum multiplicity (a condition automatically fulfilled in the scalar context considered by Borg and Hochstadt). Moreover, the connection with the stationary matrix KdV hierarchy is established. The methods employed in this paper rely on matrix-valued Herglotz functions, Weyl--Titchmarsh theory, pencils of matrices, and basic inverse spectral theory for matrix-valued Schrödinger operators.