Abstract Gustafson’s integrals are multidimensional generalizations of the classical Mellin–Barnes integrals. We show that some of these integrals arise from relations between matrix elements in Sklyanin’s representation of separated variables in spin chain models. We also present several new integrals.
We propose a non-standard separation of variables for the classical integrable XXX and XXZ spin chains with degenerate twist matrix.We show that for the case of such twist matrices one can interchange the role of classical separating functions A(u) and B(u) and construct a new full set of separated variables, satisfying simpler equation of separation and simpler Abel equations in comparison with the standard separated variables of Sklyanin.We show that for certain cases of the twist matrices the constructed separated variables can be directly identified with action-angle coordinates.
Using the Sklyanin-Kharchev-Lebedev method of Separation of Variables adapted to the cyclic Baxter--Bazhanov--Stroganov or τ(2)-model, we derive factorized formulae for general finite-size Ising model spin matrix elements, proving a recent conjecture by Bugrij and Lisovyy.
Abstract Integrable sl sl (N) spin chains, which we consider in this paper, are not only the prototypical example of quantum integrable systems but also systems with a wide range of applications. For these models we use the Functional Separation of Variables (FSoV) technique with a new tool called Character Projection to compute all matrix elements of a complete set of operators, which we call principal operators, in the basis diagonalising the tower of conserved charges as determinants in Q-functions. Building up on these results we then derive similar determinant forms for the form-factors of combinations of multiple principal operators between arbitrary factorizable states, which include, in particular, off-shell Bethe vectors and Bethe vectors with arbitrary twists. We prove that the set of principal operators generates the complete spin chain Yangian. Furthermore, we derive the representation of these operators in the SoV bases allowing one to compute correlation functions with an arbitrary number of principal operators. Finally, we show that the available combinations of multiple insertions includes Sklyanin’s SoV B operator. As a result, we are able to derive the B operator for sl sl (N) spin chains using a minimal set of ingredients, namely the FSoV method and the structure of the SoV basis.