We prove the commutativity of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. We also show thair linear independence for the numbers of spins larger than 4. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments.
Local Identities Involving Jacobi Elliptic Functions
本文导出了一些涉及Jacobi椭圆函数的任意秩的局部恒等式，并利用它们得到了一些新的结果。首先，我们提出了一个替代的，更简单的推导我们最近发现的循环恒等式，随着扩展到几个新的循环恒等式的任意秩。其次，我们得到了循环恒等式的一个推广，其中连续项具有乘法相位因子exp（2 i π/s），其中s是任意整数。第三，我们通过推导四个类似于我们先前讨论的循环和的主恒等式的局部“主恒等式”来系统化局部恒等式。第四，我们指出，许多地方的身份可以被认为是精确离散的标准非线性微分方程所满足的雅可比椭圆函数。最后，我们得到了一些明确的答案，一些明确的积分和更简单的形式，不定积分涉及雅可比椭圆函数。
We propose and prove a set of identities for ${\rm GL}_M$ elliptic $R$-matrix (in the fundamental representation). In the scalar case ($M=1$) these are elliptic function identities derived by S.N.M. Ruijsenaars as necessary and sufficient conditions for his kernel identity underlying construction of integral solutions to quantum spinless Ruijsenaars-Schneider model. In this respect the result of the present paper can be considered as the first step towards constructing solutions of quantum eigenvalue problem for the anisotropic spin Ruijsenaars model.
We establish a bilinear framework for elliptic soliton solutions which are composed by the Lamé-type plane wave factors. $τ$ functions in Hirota's form are derived and vertex operators that generate such $τ$ functions are presented. Bilinear identities are constructed and an algorithm to calculate residues and bilinear equations is formulated. These are investigated in detail for the KdV equation and sketched for the KP hierarchy. Degenerations by the periods of elliptic functions are investigated, giving rise to the bilinear framework associated with trigonometric/hyperbolic and rational func…
We present four infinite families of mutually commuting difference operators which include the deformed elliptic Ruijsenaars operators. The trigonometric limit of this kind of operators was previously introduced by Feigin and Silantyev. They provide a quantum mechanical description of two kinds of relativistic quantum mechanical particles which can be identified with particles and anti-particles in an underlying quantum field theory. We give direct proofs of the commutativity of our operators and of some other fundamental properties such as kernel function identities. In particular, we give a …
We describe integrable elliptic q-deformed anisotropic long-range spin chain. The derivation is based on our recent construction for commuting anisotropic elliptic spin Ruijsenaars-Macdonald operators. We prove that the Polychronakos freezing trick can be applied to these operators, thus providing the commuting set of Hamiltonians for long-range spin chain constructed by means of the elliptic Baxter-Belavin ${\rm GL}_M$ $R$-matrix. Namely, we show that the freezing trick is reduced to a set of elliptic function identities, which are then proved. These identities can be treated as conditions fo…
We derive and study Yangian Ward identities for the infinite class of four-point ladder integrals and their Basso-Dixon generalisations. These symmetry equations follow from interpreting the respective Feynman integrals as correlation functions in the bi-scalar fishnet theory. Alternatively, the presented identities can be understood as anomaly equations for a momentum space conformal symmetry. The Ward identities take the form of inhomogeneous extensions of the partial differential equations defining the Appell hypergeometric functions. We employ a manifestly conformal tensor reduction in ord…
The aim of this note is to provide a full space quadratic external field extension of a classical result of Marcel Riesz for the equilibrium measure on a ball with respect to Riesz s-kernels. We address the case s=d-3 for arbitrary dimension d, in particular the logarithmic kernel in dimension 3. The equilibrium measure for this full space external field problem turns out to be a radial arcsine distribution supported on a ball with a special radius. As a corollary, we obtain new integral identities involving special functions such as elliptic integrals and more generally hypergeometric functio…
