Bispectral and (gl_N, gl_M) Dualities, Discrete Versus Differential
| dc.creator | Mukhin, E. | |
| dc.creator | Tarasov, V. | |
| dc.creator | Varchenko, A. | |
| dc.date | 2006-05-07 | |
| dc.date.accessioned | 2026-07-07T07:13:58Z | |
| dc.date.available | 2026-07-07T07:13:58Z | |
| dc.description | Let $V = < x^{λ_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 $\sum_{i=0}^M B_{M-i}(u)(τ_u)^i$ annihilating $U$ and such that its leading coefficient $B_0$ 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 $U = <z_a^{u}q_{ab}(u) >$ whose regularized fundamental difference operator is the difference operator $\sum_{i=0}^N u^i A_{N-i}(τ_u)$. 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 \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. | |
| dc.description | Latex, 48 pages | |
| dc.identifier | https://arxiv.org/abs/math/0605172 | |
| dc.identifier | http://arxiv.org/abs/math/0605172 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/112686 | |
| dc.subject | Quantum Algebra | |
| dc.subject | Classical Analysis and ODEs | |
| dc.title | Bispectral and (gl_N, gl_M) Dualities, Discrete Versus Differential | |
| dc.type | text |