Generalized reflection coefficients

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.Comment: ref. [5] in the bibliography corrected (two coauthors were missing

The heat semigroup in the compact Heckman-Opdam setting and the Segal-Bargmann transform

In the first part of this paper, we study the heat equation and the heat kernel associated with the Heckman-Opdam Laplacian in the compact, Weyl-group invariant setting. In particular, this Laplacian gives rise to a Feller-Markov semigroup on a fundamental alcove of the affine Weyl group. The second part of the paper is devoted to the Segal-Bargmann transform in our context. A Hilbert space of holomorphic functions is defined such that the $L^2$-heat transform becomes a unitary isomorphism.Comment: 18 page