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Bispectral and (gl_N, gl_M) Dualities, Discrete Versus Differential | | E. Mukhin
; V. Tarasov
; A. Varchenko
; | | Date: |
7 May 2006 | | Subject: | Quantum Algebra; Classical Analysis and ODEs | | Abstract: | Let $V = < x^{lambda_i}p_{ij}(x), i=1,...,n, j=1, ..., N_i > $ be a space of quasi-polynomials in $x$ of dimension $N=N_1+...+N_n$. The regularized fundamental differential operator of $V$ is the polynomial differential operator $sum_{i=0}^N A_{N-i}(x)(x frac d {dx})^i$ annihilating $V$ and such that its leading coefficient $A_0$ 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 $M=M_1+...+M_m$. The regularized fundamental difference operator of $U$ is the polynomial difference operator $sum_{i=0}^M B_{M-i}(u)( au_u)^i$ annihilating $U$ and such that its leading coefficient $B_0$ is a monic polynomial of the minimal possible degree. Here $( au_uf)(u)=f(u+1)$.
Having a space $V$ of quasi-polynomials with the regularized fundamental differential operator $D$, we construct a space of quasi-exponentials $U = <z_a^{u}q_{ab}(u) >$ whose regularized fundamental difference operator is the difference operator $sum_{i=0}^N u^i A_{N-i}( au_u)$. 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 cite{W}.
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. | | Source: | arXiv, math/0605172 | | Services: | Forum | Review | PDF | Favorites | |
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