# Quantum integrable systems

by Nenad Manojlović (Universidade do Algarve / GFMUL)

The study of exactly solvable quantum mechanical systems is at least as old as Quantum Mechanics. Over the last decades there has been a major development aimed at a unified treatment of many examples known previously and at a systematic constructions of new ones.

The study of quantum integrable models may be divided into two different steps. The first one is kinematics, it consists in the choice of appropriate algebras of observables and Hamiltonians. The second one is dynamical, it consists in the description of the spectra, the quantum integrals of motion, and their joint eigenvectors and various correlation functions. Drinfeld's theory of Quantum Groups has given an algebraic shape only to the kinematics of Quantum Inverse Scattering Method.

The Algebraic Bethe Ansatz is a powerfull algebraic tool which yields the spectrum and corresponding eigenstates in the applications of the Quantum Inverse Scattering Method to the models for which the highest weight type representations are relevant, like for example Gaudin models, quantum spin systems, etc.