BC-type open $SL(2,\mathbb{C})$ spin chain

TL;DR

This paper diagonalizes the boundary operator of a noncompact open $SL(2, ext{C})$ spin chain, constructing eigenfunctions with symmetry properties and proving their orthogonality and completeness using Mellin-Barnes integrals.

Contribution

It introduces a novel method to construct eigenfunctions for the open $SL(2, ext{C})$ spin chain with boundary interactions, utilizing $ ext{K}$- and $ ext{R}$-operators and establishing their symmetry and completeness.

Findings

Eigenfunctions are explicitly constructed and shown to be orthogonal.

Symmetries of eigenfunctions under permutations, reflections, and parameter transformations are established.

Completeness of the eigenfunctions is proven using Mellin-Barnes integral representation.

Abstract

We diagonalize the $B$-element of monodromy matrix for noncompact open $SL(2,\mathbb{C})$ spin chain with boundary interaction. The monodromy matrix is defined in terms of $SL(2,\mathbb{C})$ $L$-operator and boundary $K$-matrix. The eigenfunctions of $B$-operator are constructed iteratively using raising $\Lambda$-operators. The key role in the calculations plays the Baxter $Q$-operator commuting with the $B$-operator. The main building blocks for $\Lambda$- and $Q$-operators are $\mathcal{K}$-operator -- the general solution of reflection equation and $\mathcal{R}$-operator -- the reduction of the general solution of the Yang-Baxter equation. Two types of the symmetry of eigenfunctions are established. The first kind is the invariance under permutations and reflections of spectral variables, or in other words, under the action of Weyl group of B and C root systems. The second kind is the symmetry with respect to transformation $(s,g) \to (1-s,1-g)$, where $s$ is the spin variable and $g$ is the parameter of $K$-matrix. We prove that obtained system of eigenfunctions is orthogonal and complete. The calculation of the scalar product of eigenfunction is given in initial coordinate representation. We derive the Mellin-Barnes integral representation for eigenfunctions and use it to prove the comleteness.