We introduce a family of differential-reflection operators Λ A , 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 Λ A , ε is studied. In particular, we obtain suitable growth estimates for the eigenfunctions of Λ A , ε. As the operators Λ A , ε, are a mixture of d / d x and reflection operators, we prove the existence of an intertwining operator VA , ε between Λ A , ε and the usual derivative. The positivity of VA , ε is also established.
- Differential-reflection operators
- intertwining operators
- spectral problem
ASJC Scopus subject areas