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.

中文翻译：

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$ gl _ {\ mathcal {M} | \ mathcal {N}} $和Hubbard模型的变量库分离

我们为基本非均匀$ gl _ {\ mathcal {M} | \ mathcal {N}} $超对称可积模型和非均质Hubbard模型（均使用准周期扭曲边界条件定义）构建变量的量子分离（SoV）基通过具有简单光谱的扭曲矩阵 SoV基是通过使用这些量子模型的可积结构获得的，即相关联的通勤转移矩阵，遵循[1]中介绍的一般方案。也就是说，它们是由传递矩阵对通用协矢量的多重作用所生成的一组状态给出的。这样的SoV基的存在意味着相应的传递矩阵具有简并谱，并且如果定义准周期边界条件的扭曲矩阵具有该特性，则它们可以用简单谱对角化。此外，在这些SoV基数中，传递矩阵特征值问题的解决方案导致了整个频谱问题的解决方案，即特征值和特征向量。实际上，任何特征值都与唯一的特征向量相关（最大程度地归一化），其SoV基中的波函数被分解为在单独变量的频谱上计算出的相应传递矩阵特征值的乘积。作为应用，我们在SoV框架中完全描述了与特殊类型的扭曲矩阵相关的基本$ gl_ {1 | 2} $超对称可积模型的传递矩阵谱。从这些结果中，我们还证明了Bethe Ansatz在该情况下的完整性。基本的非均匀$ gl _ {\ mathcal {M} | \ mathcal {N}} $超对称可积模型和一般扭曲边界条件下的非均匀Hubbard模型的光谱问题的完整解决方案将在以后的出版物中讨论。