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

Pavel V. Antonenko^ab, Sergey È. Derkachov^b, Pavel A. Valinevich<sup>b

a</sup> Leonhard Euler International Mathematical Institute, Pesochnaya nab. 10, 197022 St. Petersburg, Russia
^b Saint-Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia

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.

Keywords: open spin chain, principal series representations, Mellin–Barnes integrals.

MSC: 81R12, 17B80, 33C70

Received: August 13, 2025; in final form December 8, 2025; Published online December 21, 2025

Language: English

DOI: 10.3842/SIGMA.2025.107

ArXiv: 2507.09568