| Summary: | We construct quantum Separation of Variables (SoV) bases for both the
fundamental inhomogeneous $gl_{\mathcal{M}|\mathcal{N}}$ supersymmetric
integrable models and for the inhomogeneous Hubbard model both defined with
quasi-periodic twisted boundary conditions given by twist matrices having
simple spectrum. The SoV bases are obtained by using the integrable structure
of these quantum models,i.e. the associated commuting transfer matrices,
following the general scheme introduced in [1]; namely, they are given by set
of states generated by the multiple action of the transfer matrices on a
generic co-vector. The existence of such SoV bases implies that the
corresponding transfer matrices have non degenerate spectrum and that they are
diagonalizable with simple spectrum if the twist matrices defining the
quasi-periodic boundary conditions have that property. Moreover, in these SoV
bases the resolution of the transfer matrix eigenvalue problem leads to the
resolution of the full spectral problem, i.e. both eigenvalues and
eigenvectors. Indeed, to any eigenvalue is associated the unique (up to a
trivial overall normalization) eigenvector whose wave-function in the SoV bases
is factorized into products of the corresponding transfer matrix eigenvalue
computed on the spectrum of the separate variables. As an application, we
characterize completely the transfer matrix spectrum in our SoV framework for
the fundamental $gl_{1|2}$ supersymmetric integrable model associated to a
special class of twist matrices. From these results we also prove the
completeness of the Bethe Ansatz for that case. The complete solution of the
spectral problem for fundamental inhomogeneous $gl_{\mathcal{M}|\mathcal{N}}$
supersymmetric integrable models and for the inhomogeneous Hubbard model under
the general twisted boundary conditions will be addressed in a future
publication.
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