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.
Highlights
The double elliptic model [17,18,19,20,21,22,23,24,25,26] is an integrable system with an elliptic dependence on both – positions of particles and their momenta
A geometrical approach was used based on the studies of spectral curves and Seiberg-Witten differentials [37]. In this way the Dell Hamiltonians where proposed in terms of higher genus theta-functions with a dynamical period matrices
In this paper, using the Hamiltonians (1.1), we construct a generalization of the Macdonald determinant operator for the Dell system and study its applications
Summary
The double elliptic (or Dell) model [17,18,19,20,21,22,23,24,25,26] is an integrable system with an elliptic dependence on both – positions of particles and their momenta. A geometrical approach was used based on the studies of spectral curves and Seiberg-Witten differentials [37] In this way the Dell Hamiltonians where proposed in terms of higher genus theta-functions with a dynamical period matrices. On the contrary does not appeal to the explicit form of the wavefunctions and is mostly focused on the generating function itself It is based on the usage of the intertwining matrix Ξ(z) of the IRF-Vertex correspondence (see (9) for their explicit form) and the Hasegawa’s factorization formula [39,40,41]. IRF-Vertex correspondence provides relation between dynamical and non-dynamical quantum (or classical) R-matrices as a special twisted gauge transformation with the matrix g(z), relating the Lax operator (1.4) with the one of the Sklyanin type [65, 66]
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