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Intertwining Operators Associated to a Family of Differential-Reflection Operators

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Abstract

We introduce a family of differential-reflection operators \({\Lambda_{A, \varepsilon}}\) acting on smooth functions defined on \({{\mathbb R}}\). Here, A is a Sturm–Liouville function with additional hypotheses and \({-1 \leq \varepsilon \leq 1}\). For special pairs \({(A, \varepsilon),}\) we recover Dunkl’s, Heckman’s and Cherednik’s operators (in one dimension). The spectral problem for the operators \({\Lambda_{A, \varepsilon}}\) is studied. In particular, we obtain suitable growth estimates for the eigenfunctions of \({\Lambda_{A, \varepsilon}}\). As the operators \({\Lambda_{A, \varepsilon}}\), are a mixture of \({{\rm d}/{\rm d}x}\) and reflection operators, we prove the existence of an intertwining operator \({V_{A, \varepsilon}}\) between \({\Lambda_{A, \varepsilon}}\) and the usual derivative. The positivity of \({V_{A, \varepsilon}}\) is also established.

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Authors and Affiliations

  1. Institut Élie Cartan de Lorraine, Université de Lorraine-Nancy, B.P. 239, 54506, Vandoeuvres-lès-Nancy, France

    Salem Ben Said

  2. Laboratoire d’Analyse Mathématiques et Applications LR11ES11, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092, Tunis, Tunisia

    Asma Boussen & Mohamed Sifi

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  1. Salem Ben Said
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  2. Asma Boussen
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Correspondence to Salem Ben Said.

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Ben Said, S., Boussen, A. & Sifi, M. Intertwining Operators Associated to a Family of Differential-Reflection Operators. Mediterr. J. Math. 13, 4129–4151 (2016). https://doi.org/10.1007/s00009-016-0736-2

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  • DOI: https://doi.org/10.1007/s00009-016-0736-2

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