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arXiv · 1406.1481

Generalized reflection coefficients

Christian Remling
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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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Christian Remling. 2014-06-09. Generalized reflection coefficients. https://doi.org/10.1007/s00220-015-2341-9

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