# Intertwining operators associated to a family of differential-reflection operators

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1 Analyse IECL - Institut Élie Cartan de Lorraine 2 Analyse Mathématique et Applications

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 $\varepsilon\in \mathbb R.$ 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 mixture of $d-dx$and reflection operators, we prove the existence of an intertwiningoperator $V {A,\varepsilon}$ between $\Lambda {A, \varepsilon}$ and theusual derivative. The positivity of $V {A,\varepsilon}$ is also established.

Keywords : Differential-reflection operators spectral problem intertwining operators

Autor: Salem Ben Said - Asma Boussen - Mohamed Sifi -

Fuente: https://hal.archives-ouvertes.fr/

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