Q-functions for lambda opers

doi:10.1007/s11005-025-01988-z

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Q-functions for lambda opers

2025-09-17

PRE

Davide Masoero *

E. Mukhin

Andrea Raimondo

DOI：10.1007/s11005-025-01988-z

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Abstract

En 中文

We consider the Schr & ouml;dinger operators which are constructed from the lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-opers corresponding to solutions of the sl<^>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{\mathfrak {sl}}_2$$\end{document} Gaudin Bethe Ansatz equations. We define and study the connection coefficients called the Q-functions. We conjecture that the Q-functions obtained from the lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-opers coincide with the Q-functions of the Bazhanov-Lukyanov-Zamolodchikov opers with the monster potential related to the quantum KdV flows. We give supporting evidence for this conjecture. In particular, we give a rigorous proof that the Q-functions of lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-opers satisfy the QQ and TQ relations.

Keywords:

ODE/IM correspondence

Affine Quantum Gaudin Models

Bethe Ansatz

Quantum KdV

Spectral Determinant

Complex WKB method