[Paper Review] Wave function for $GL(n,\mathbb{R})$ hyperbolic Sutherland model II. Dual Hamiltonians

S. Kharchev, S. Khoroshkin|arXiv (Cornell University)|Aug 11, 2021

Algebraic structures and combinatorial models8 references4 citations

TL;DR

This paper establishes the bispectral duality of the wave function for the $GL(n,\mathbb{R})$ hyperbolic Sutherland model by proving that the Mellin-Barnes integral representation of the wave function satisfies spectral equations for dual Ruijsenaars-Macdonald difference operators. The key result is that the wave function solves the spectral problem for these dual Hamiltonians with arbitrary coupling $g>1$, extending the known spectral properties of the original Sutherland Hamiltonian.

ABSTRACT

Recently we found Mellin-Barnes integrals, representing the wave function for $GL(n,\mathbb{R})$ hyperbolic Sutherland model. In present paper, we establish bispectral properties of this wave function with respect to dual Ruijesenaars-Macdonald operators.

Motivation & Objective

• To establish the bispectral properties of the wave function for the $GL(n,\mathbb{R})$ hyperbolic Sutherland model with arbitrary coupling $g>0$.
• To demonstrate that the Mellin-Barnes integral wave function satisfies spectral equations for dual Ruijsenaars-Macdonald difference operators.
• To extend the known spectral duality from the original Sutherland Hamiltonian to the dual Hamiltonian system defined by Ruijsenaars-Macdonald operators.
• To clarify the analytical structure of the wave function in both the $x$- and $\lambda$-variables, including analytic continuation and symmetry properties.

Proposed method

• The wave function is expressed as a product of a singular factor $\prod_{j<k}\operatorname{sh}^{g}|x_j - x_k|$ and a Mellin-Barnes integral $\Phi^{(g)}_{\lambda_1,\ldots,\lambda_n}(x_1,\ldots,x_n)$ over $\mathbb{R}^{n(n-1)/2}$, with gamma functions in the integrand.
• The dual Hamiltonians $\mathcal{H}_r^{(g)}(\lambda_1,\ldots,\lambda_n)$ are defined as Ruijsenaars-Macdonald difference operators with shifts in $\lambda_i$ by $+2$, involving products over subsets of indices and rational functions of $\lambda_i - \lambda_j$.
• The spectral equations are verified by showing that $\mathcal{H}_r^{(g)}\Psi^{(g)}_{\lambda_1,\ldots,\lambda_n} = e_r(e^{2x_1},\ldots,e^{2x_n})\Psi^{(g)}_{\lambda_1,\ldots,\lambda_n}$, where $e_r$ is the $r$-th elementary symmetric function.
• Analytic continuation is used to extend the wave function from imaginary $\lambda_i$ to real shifts in the difference operators, ensuring the spectral relation holds in a generalized sense.
• The proof relies on a recursive identity involving symmetric sums of rational functions, proven by induction and residue analysis, with vanishing residues at poles $u_i = v_a$.
• The method generalizes earlier results on bispectral problems in integrable systems, including those by Chalykh, Noumi, and Shiraishi, to the non-integer coupling case $g>1$.

Experimental results

Research questions

• RQ1 Does the Mellin-Barnes integral wave function for the $GL(n,\mathbb{R})$ hyperbolic Sutherland model satisfy spectral equations for dual Ruijsenaars-Macdonald difference operators?
• RQ2 Can the bispectral duality be extended from the original Sutherland Hamiltonian to the dual Hamiltonian system with arbitrary coupling $g>1$?
• RQ3 How do the spectral properties of the wave function behave under analytic continuation in the $\lambda$-parameters and shifts in the difference operators?
• RQ4 What is the precise relation between the integral representation of the wave function and the Ruijsenaars-Macdonald operators in the $\lambda$-space?
• RQ5 Do the dual Hamiltonians preserve the symmetry of the wave function under permutations of $x_k$?

Key findings

• The wave function $\Psi^{(g)}_{\lambda_1,\ldots,\lambda_n}(x_1,\ldots,x_n)$, defined via a Mellin-Barnes integral, satisfies the spectral equation $\mathcal{H}_r^{(g)}\Psi^{(g)}_{\lambda_1,\ldots,\lambda_n} = e_r(e^{2x_1},\ldots,e^{2x_n})\Psi^{(g)}_{\lambda_1,\ldots,\lambda_n}$ for all $r=1,\ldots,n$ and $g>1$.
• The wave function is analytic in a strip around $\mathbb{R}^n \subset \mathbb{C}^n$ and admits analytic continuation to real shifts in $\lambda_i$, enabling the spectral relation with difference operators.
• The dual Hamiltonians $\mathcal{H}_r^{(g)}$ are Ruijsenaars-Macdonald difference operators with shifts $T_{\lambda_i}f(\lambda_i) = f(\lambda_i + 2)$, and they act on the $\lambda$-parameters of the wave function.
• The wave function also satisfies the original Sutherland spectral equations: $H_1^{(g)}\Psi = (\sum_p \lambda_p)\Psi$ and $H_2^{(g)}\Psi = -\sum_p \lambda_p^2 \Psi$, confirming bispectral duality for both $H_r^{(g)}$ and $\mathcal{H}_r^{(g)}$.
• The proof of the key identity relies on residue analysis and symmetry, showing that the difference of symmetric rational functions vanishes identically, implying the identity holds for all $n$ by induction.
• The wave function’s invariance under permutations of $x_k$ is preserved, and the dual operators act on the $\lambda$-parameters, establishing a full bispectral duality between the $x$- and $\lambda$-spectral problems.