  We present a new approach to construct the separate variables basis leading
to the full characterization of the transfer matrix spectrum of quantum
integrable lattice models. The basis is generated by the repeated action of the
transfer matrix itself on a generically chosen state of the Hilbert space. The
fusion relations for the transfer matrix, stemming from the Yang-Baxter algebra
properties, provide the necessary closure relations to define the action of the
transfer matrix on such a basis in terms of elementary local shifts, leading to
a separate transfer matrix spectral problem. Hence our scheme extends to the
quantum case a key feature of Liouville-Arnold classical integrability
framework where the complete set of conserved charges defines both the level
manifold and the flows on it leading to the construction of action-angle
variables. We work in the framework of the quantum inverse scattering method.
As a first example of our approach, we give the construction of such a basis
for models associated to Y(gln) and argue how it extends to their trigonometric
and elliptic versions. Then we show how our general scheme applies concretely
to fundamental models associated to the Y(gl2) and Y(gl3) R-matrices leading to
the full characterization of their spectrum. For Y(gl2) and its trigonometric
deformation a particular case of our method reproduces Sklyanin's construction
of separate variables. For Y(gl3) it gives new results, in particular through
the proper identification of the shifts acting on the separate basis. We stress
that our method also leads to the full characterization of the spectrum of
other known quantum integrable lattice models, including in particular
trigonometric and elliptic spin chains, open chains with general integrable
boundaries, and further higher rank cases that we will describe in forthcoming
publications.
