A-Type Open ${\rm SL}(2,\mathbb{C})$ Spin Chain

TL;DR

This paper constructs eigenfunctions for the noncompact open ${ m SL}(2, ext{C})$ spin chain using local Yang-Baxter operators, $Q$-operators, and Mellin-Barnes integrals, revealing their orthogonality and symmetry properties.

Contribution

It introduces a novel construction of eigenfunctions for the ${ m SL}(2, ext{C})$ spin chain, utilizing $Q$-operators and integral representations, advancing understanding of noncompact quantum integrable models.

Findings

Eigenfunctions constructed explicitly for the ${ m SL}(2, ext{C})$ spin chain.

Orthogonality and symmetry properties of eigenfunctions established.

Mellin-Barnes representation shown to be equivalent to coordinate representation.

Abstract

For the noncompact open ${\rm SL}(2,\mathbb{C})$ spin chain, the eigenfunctions of the special matrix element of monodromy matrix are constructed. The key ingredients of the whole construction are local Yang-Baxter $\mathcal{R}$-operators, $Q$-operator and raising operators obtained by reduction from the $Q$-operator. The calculation of various scalar products and the proof of orthogonality are based on the properties of $Q$-operator and demonstrate its hidden role. The symmetry of eigenfunctions with respect to reflection of the spin variable $s \to 1-s$ is established. The Mellin-Barnes representation for eigenfunctions is derived and equivalence with initial coordinate representation is proved. The transformation from one representation to another is grounded on the application of $A$-type Gustafson integral generalized to the complex field.