We construct quantum Separation of Variables (SoV) bases for both the
fundamental inhomogeneous %
gl_{\mathcal{M}|\mathcal{N}}glℳ|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 actions 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 separated variables.
As an application, we characterize completely the transfer matrix
spectrum in our SoV framework for the fundamental
gl_{1|2}gl1|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}}glℳ|𝒩
supersymmetric integrable models and for the inhomogeneous Hubbard model
under the general twisted boundary conditions will be addressed in a
future publication.
TRACE MAPS, INVARIANTS, AND SOME OF THEIR APPLICATIONS