Reflection operator and hypergeometry I: $SL(2, \mathbb{R})$ spin chain

TL;DR

This paper studies the open $SL(2, \mathbb{R})$ spin chain, focusing on the one-particle case, and introduces reflection operators linked to hypergeometric functions, establishing their properties and connections to integral transforms.

Contribution

It provides new representations of the reflection operator, proves orthogonality and completeness of eigenfunctions, and connects them to hypergeometric transforms in the $SL(2, \mathbb{R})$ spin chain context.

Findings

Derived multiple representations of the reflection operator

Proved orthogonality and completeness of eigenfunctions

Connected eigenfunctions to the index hypergeometric transform

Abstract

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.