Characteristic determinant and Manakov triple for the double elliptic integrable system

Abstract

Using the intertwining matrix of the IRF-Vertex correspondence we propose a determinant representation for the generating function of the commuting Hamiltonians of the double elliptic integrable system. More precisely, it is a ratio of the normally ordered determinants, which turns into a single determinant in the classical case. With its help we reproduce the recently suggested expression for the eigenvalues of the Hamiltonians for the dual to elliptic Ruijsenaars model. Next, we study the classical counterpart of our construction, which gives expression for the spectral curve and the corresponding L-matrix. This matrix is obtained explicitly as a weighted average of the Ruijsenaars and/or Sklyanin type Lax matrices with the weights as in the theta function series definition. By construction the L-matrix satisfies the Manakov triple representation instead of the Lax equation. Finally, we discuss the factorized structure of the L-matrix.

Duality in elliptic Ruijsenaars system and elliptic symmetric functions

“…A smarter approach is to use the generating function of the KS Hamiltonians in the determinant form [45], which can be written, in the ell-trig case, as…”

“…However, one has to find as many such Hamiltonians as there are x variables, and all these Hamiltonians have to commute. To put it differently, one has to construct a generating function of Hamiltonians that depends on an auxiliary spectral parameter u, and, for instance, in the case of KS Hamiltonians, this spectral parameter would be better to introduce in a tricky way [45]: H K S are made in a non-local way from simpler auxiliary operators O trig (u) depending already on the spectral parameter. As we demonstrate in this paper, the drop-out of u-dependence is a corollary of some θ -function identities.…”

On Cherednik and Nazarov-Sklyanin large N limit construction for integrable many-body systems with elliptic dependence on momenta

“…In our previous paper [34] different variants of determinant representations for (1.1)-(1.2) were proposed. Here we extend another set of algebraic constructions to the doubleelliptic case (1.1).…”

“…Notice that the matrix Pθ ω (uZ) appeared in our previous paper [34] as the one, whose determinant gives the generating function D N (u). Actually, Z = L RS is the Lax matrix of the quantum trigonometric Ruijsenaars-Schneider model [39].…”

“…Another feature of the Koroteev-Shakirov formulation is that it admits some algebraic constructions, which are widely known for the Calogero-Ruijsenaars family of integrable systems. In particular, it was shown in our previous paper [34] that the generating function of Hamiltonians has determinant representation, and the classical L-operator satisfies the Manakov equation instead of the standard Lax representation. For both formulations the commutativity of the Hamiltonians has not being proved yet, but verified numerically.…”