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

TL;DR

This paper constructs and analyzes eigenfunctions for a noncompact $SL(2, ext{C})$ spin chain, introducing a reflection operator that relates eigenfunctions of different chain lengths, with results expressed via hypergeometric functions.

Contribution

It introduces a method to construct eigenfunctions for $SL(2, ext{C})$ spin chains using a reflection operator, extending to chains with multiple sites.

Findings

Eigenfunctions expressed in hypergeometric functions of the complex field.

Orthogonality and completeness established for one-site eigenfunctions.

Derived equations for the reflection operator with $SL(2, ext{C})$-invariant $ ext{R}$-operator.

Abstract

We consider noncompact open $SL(2, \mathbb{C})$ spin chain and construct eigenfunctions of $B$-element of monodromy matrix for the simplest case of the chain with one site. The reflection operator appearing in this construction can be used to express eigenfunction for $n$ sites in terms of the eigenfunction for $n-1$ sites, this general result is briefly announced. We prove orthogonality and completeness of constructed eigenfunctions in the case of one site, express them in terms of the hypergeometric function of the complex field and derive the equation for the reflection operator with the general $SL(2,\mathbb{C})$-invariant $\mathbb{R}$-operator.