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In this review we demonstrate how the algebraic Bethe ansatz is used for thecalculation of the energy spectra and form factors (operator matrix elements inthe basis of Hamiltonian eigenstates) in exactly solvable quantum systems. Asexamples we apply the theory to several models of current interest in the studyof Bose-Einstein condensates, which have been successfully created usingultracold dilute atomic gases. The first model we introduce describes Josephsontunneling between two coupled Bose-Einstein condensates. It can be used notonly for the study of tunneling between condensates of atomic gases, but forsolid state Josephson junctions and coupled Cooper pair boxes. The theory isalso applicable to models of atomic-molecular Bose-Einstein condensates, withtwo examples given and analysed. Additionally, these same two models arerelevant to studies in quantum optics. Finally, we discuss the model ofBardeen, Cooper and Schrieffer in this framework, which is appropriate forsystems of ultracold fermionic atomic gases, as well as being applicable forthe description of superconducting correlations in metallic grains withnanoscale dimensions. In applying all of the above models to physicalsituations, the need for an exact analysis of small scale systems isestablished due to large quantum fluctuations which render mean-fieldapproaches inaccurate.

journal of physics. a, mathematical and general/journal of physics. a, mathematical and general

Algebraic Bethe ansatz method for the exact calculation of energy spectra and form factors: applications to models of Bose Einstein condensates and metallic nanograins