Bispectral and (gl_N, gl_M) Dualities, Discrete Versus Differential

2006/05/07 by E. Mukhin, Mukhin, E., V. Tarasov +4 · 2 citations
Materials Science · Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Optical and Acousto-Optic Technologies #Quantum Algebra (math.QA) #Solid-state spectroscopy and crystallography #math.CA #math.QA

paper · pdf · doi:10.48550/arxiv.math/0605172

Latex, 48 pages

arxiv created 2006/05/07 · openalex publication_date 2006/05/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let V = < x^λ_ip_ij(x), i=1,...,n, j=1, ..., N_i > be a space of quasi-polynomials in x of dimension N=N₁+...+N_n. The regularized fundamental differential operator of V is the polynomial differential operator ∑_i=0^N A_N-i(x)(x \frac d dx)ⁱ annihilating V and such that its leading coefficient A₀ is a monic polynomial of the minimal possible degree. Let U = < z_a^u q_ab(u), a=1,...,m, b=1,..., M_a > be a space of quasi-exponentials in u of dimension M=M₁+...+M_m. The regularized fundamental difference operator of U is the polynomial difference operator ∑_i=0^M B_M-i(u)(τ_u)ⁱ annihilating U and such that its leading coefficient B₀ is a monic polynomial of the minimal possible degree. Here (τ_uf)(u)=f(u+1). Having a space V of quasi-polynomials with the regularized fundamental differential operator D, we construct a space of quasi-exponentials U = <z_a^uq_ab(u) > whose regularized fundamental difference operator is the difference operator ∑_i=0^N uⁱ A_N-i(τ_u). The space U is constructed from V by a suitable integral transform. Similarly, having U we can recover V by a suitable integral transform. Our integral transforms are analogs of the bispectral involution on the space of rational solutions to the KP hierarchy \citeW. As a corollary of the properties of the integral transforms we obtain a correspondence between solutions to the Bethe ansatz equations of two (gl_N, gl_M) dual quantum integrable models: one is the special trigonometric Gaudin model and the other is the special XXX model.

Cited by

• Large N limit of spectral duality in classical integrable systems
• Cactus flower spaces and monodromy of Bethe vectors

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