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

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