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Reflectionless operators in one dimension are particularly amenable to inverse scattering and are intimately related to integrable systems like KdV and Toda. Recent work has indicated a strong (but not equivalent) relationship between reflectionless operators and almost periodic potentials with absolutely continuous spectrum. This makes the realm of reflectionless operators a natural place to begin addressing Deift’s conjecture on integrable flows with almost periodic initial conditions and Simon’s conjecture on gems of spectral theory establishing correspondences between certain coefficient and spectral properties.
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David Damanik, Fritz Gesztesy, Peter Yuditskii, Mini-Workshop: Reflectionless Operators: The Deift and Simon Conjectures. Oberwolfach Rep. 14 (2018), no. 4, pp. 2943–2985