Intertwining Operators Associated to a Family of Differential-Reflection Operators

Abstract

We introduce a family of differential-reflection operators ΛA,ε acting on smooth functions defined on R. Here, A is a Sturm-Liouville function with additional hypotheses and −1 ≤ ε ≤ 1. For special pairs (A, ε), we recover Dunkl's, Heckman's and Cherednik's operators (in one dimension). The spectral problem for the operators ΛA,ε is studied. In particular, we obtain suitable growth estimates for the eigenfunctions of ΛA,ε. As the operators ΛA,ε are a mixture of d/dx and reflection operators, we prove the existence of an intertwining operator VA,ε between ΛA,ε and the usual derivative. The positivity of VA,ε is also established.Mathematics Subject Classification. 34K99, 34B25, 33E30.