SciPost Phys. 10, 055 (2021) ·
published 4 March 2021
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.