Abstract
I consider general reflection coefficients for arbitrary one-dimensional whole line differential or difference operators of order 2. These reflection coefficients are semicontinuous functions of the operator: their absolute value can only go down when limits are taken. This implies a corresponding semicontinuity result for the absolutely continuous spectrum, which applies to a very large class of maps. In particular, we can consider shift maps (thus recovering and generalizing a result of Last–Simon) and flows of the Toda and KdV hierarchies (this is new). Finally, I evaluate an attempt at finding a similar general setup that gives the much stronger conclusion of reflectionless limit operators in more specialized situations.
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Communicated by P. Deift
CR’s work has been supported by NSF Grant DMS 1200553.
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Remling, C. Generalized Reflection Coefficients. Commun. Math. Phys. 337, 1011–1026 (2015). https://doi.org/10.1007/s00220-015-2341-9
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DOI: https://doi.org/10.1007/s00220-015-2341-9