Complete spectrum of quantum integrable lattice models associated to $\mathcal{U}_{q} (\widehat{gl_{n}})$ by separation of variables

Abstract: In this paper we apply our new separation of variables approach to completely characterize the transfer matrix spectrum for quantum integrable lattice models associated to fundamental evaluation representations of $\mathcal{U}{q} (\widehat{gl{n}})$ with general quasi-periodic boundary conditions. We consider here the case of generic deformations associated to a parameter $q$ which is not a root of unity. The Separation of Variables (SoV) basis for the transfer matrix spectral problem is generated by using the action of the transfer matrix itself on a generic co-vector of the Hilbert space, following the general procedure described in our paper [1]. Such a SoV construction allows to prove that for general values of the parameters defining the model the transfer matrix is diagonalizable and with simple spectrum for any twist matrix which is also diagonalizable with simple spectrum. Then, using together the knowledge of such a SoV basis and of the fusion relations satisfied by the hierarchy of transfer matrices, we derive a complete characterization of the transfer matrix eigenvalues and eigenvectors as solutions of a system of polynomial equations of order $n+1$. Moreover, we show that such a SoV discrete spectrum characterization is equivalently reformulated in terms of a finite difference functional equation, the quantum spectral curve equation, under a proper choice of the set of its solutions. A construction of the associated Q-operator induced by our SoV approach is also presented.