  In the present article we study the form factors of quantum integrable
lattice models solvable by the separation of variables (SoV) method. It was
recently shown that these models admit universal determinant representations
for the scalar products of the so-called separate states (a class which
includes in particular all the eigenstates of the transfer matrix). These
results permit to obtain simple expressions for the matrix elements of local
operators (form factors). However, these representations have been obtained up
to now only for the completely inhomogeneous versions of the lattice models
considered. In this article we give a simple algebraic procedure to rewrite the
scalar products (and hence the form factors) for the SoV related models as
Izergin or Slavnov type determinants. This new form leads to simple expressions
for the form factors in the homogeneous and thermodynamic limits. To make the
presentation of our method clear, we have chosen to explain it first for the
simple case of the $XXX$ Heisenberg chain with anti-periodic boundary
conditions. We would nevertheless like to stress that the approach presented in
this article applies as well to a wide range of models solved in the SoV
framework.
