It is a general property of elliptic differential operators with periodic coefficients, that their spectra are formed by union of closed intervals called spectral bands (see [12], [14]) possibly separated by gaps. One of the challenging questions of the spectral theory of periodic operators is to find out whether or not the number of gaps in the spectrum of a given operator is finite. The state...

In this introductory review we discuss recent dynamical tests of the AdS 5 × S 5 string/N = 4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the limit of large angular momenta. The energies of the folded and circular spinning string solutions rotating on a S 3 within the S 5 a...

We generalise the fusion procedure for the A (1) N−1 open spin chain (N > 2) and we show that the transfer matrix satisfies a crossing property. We use these results to solve the A (1) N−1 open spin chain with Uq (SU(N )) symmetry by means of the analytical Bethe ansatz method. Our results coincide with the known ones obtained via the nested Bethe ansatz.

A Bethe Ansatz study of a self dual ZN spin model is undertaken for even spin system. One has to solve a coupled system of Bethe Ansatz Equations (BAE) involving zeroes of two families of transfer matrices. A numerical study on finite size lattices is done for identification of elementary excitations over the Ferromagnetic and Antiferromagnetic ground states. The free energies for both Ferromag...

N = 4 supersymmetric Yang-Mills operators carrying large charges are dual to semiclassical strings in AdS5 × S. The spectrum of anomalous dimensions of very large operators has been calculated solving the Bethe ansatz equations in the thermodynamic limit, and matched to energies of string solitons. We have considered finite size corrections to the Bethe equations, that should correspond to quan...

Establishing the completeness of a Bethe Ansatz solution for an exactly solved model is a perennial challenge, which is typically approached on a case by case basis. For the rational, spin-1/2 Richardson– Gaudin system it will be shown that completeness can be deduced through a direct approach. This method does not depend on knowledge of the distribution of Bethe roots, such as a string hypothe...

It is shown that the numbers of off-diagonal solutions to the Uq(X (r) N ) Bethe equation at q = 0 coincide with the coefficients in the recently introduced canonical power series solution of the Q-system. Conjecturally the canonical solutions are characters of the KR (Kirillov-Reshetikhin) modules. This implies that the numbers of off-diagonal solutions agree with the weight multiplicities, wh...

Exact solutions for eigenvalues and eigenfunctions of some nuclear many-body systems are found by using an infinite-dimensional, Lie-algebraic approach based on the corresponding Bethe ansatz. Applications of the theory, including solutions of some nuclear pairing problems and U(5) W SO(6) transitional Hamiltonians of the interacting boson model, are given. The relationship between this new met...

The general expression for the local matrix of a quantum chain L(θ) with the site space in any representation of su(3) is obtained. This is made by generalizing L(θ) from the fundamental representation and imposing the fulfilment of the Yang-Baxter equation. With these operators and using a generalization of the nested Bethe ansatz, the Bethe equations for a multistate quantum chain combining t...

We investigate different ways of generating approximate solutions to the pairwise Markov random field (MRF) selection problem. We focus mainly on the inverse Ising problem, but discuss also the somewhat related inverse Gaussian problem because both types of MRF are suitable for inference tasks with the belief propagation algorithm (BP) under certain conditions. Our approach consists in to take ...