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Eigenvectors and scalar products for long range interacting spin chains II: the finite size effects

In this note, we study the eigenvectors and the scalar products the integrable long-range deformation of a XXX spin chain which is solved exactly by algebraic Bethe ansatz, and it coincides in the bulk with the Inozemtsev spin chain. At the closing point it contains a defect which effectively removes the wrapping interactions. Here we concentrate on determining the defect term for the first non-trivial order in perturbation in the deformation parameter and how it affects the Bethe ansatz equations. Our study is motivated by the relation with the dilatation operator of the N = 4 gauge theory in the su(2) sector.