Abstract. We consider the XXX spin-1 2 Heisenberg chain on the circle with an arbitrary twist. We characterize its spectral problem using the modified algebraic Bethe anstaz and study the scalar product between the Bethe vector and its dual. We obtain modified Slavnov and Gaudin-Korepin formulas for the model. Thus we provide a first example of such formulas for quantum integrable models without U(1) symmetry characterized by an inhomogenous Baxter T-Q equation. The study of quantum integrable models with U(1) symmetry by the Bethe ansatz (BA) methods In the case of models without U(1) symmetry, the usual BA techniques in general fail to provide a complete description of the spectrum 1 . Thus alternative methods have been developed, for instance, the separation of variables (SoV) 1 In some cases, some gauge transformation can allow to apply the ABA, see for example the XYZ spin chai
