  In this work we consider open $SL(2, \mathbb{R})$ spin chain, mainly the
simplest case of one particle. Eigenfunctions of the model can be constructed
using the so-called reflection operator. We obtain several representations of
this operator and show its relation to the hypergeometric function. Besides, we
prove orthogonality and completeness of one-particle eigenfunctions and connect
them to the index hypergeometric transform. Finally, we briefly state the
formula for the eigenfunctions in many-particle case.
