Completeness of SoV Representation for $\mathrm{SL}(2,\mathbb R)$ Spin Chains

TL;DR

This paper introduces a new method leveraging Gustafson's integrals to prove the unitarity of the separation of variables transform for infinite-dimensional $ ext{SL}(2, ext{R})$ spin chains, enhancing understanding of their spectral properties.

Contribution

It develops a novel approach based on integral evaluations to establish the completeness of the SoV representation for $ ext{SL}(2, ext{R})$ spin chains.

Findings

Proves unitarity of the SoV transform for these models.

Establishes completeness of the SoV representation.

Provides a new analytical framework for infinite-dimensional integrable models.

Abstract

This work develops a new method, based on the use of Gustafson's integrals and on the evaluation of singular integrals, allowing one to establish the unitarity of the separation of variables transform for infinite-dimensional representations of rank one quantum integrable models. We examine in detail the case of the $\mathrm{SL}(2,\mathbb R)$ spin chains.