Invariant Subspaces of the Shrödinger Operator with a Finite Support Potential

We consider a Hamiltonian of a system of two fermions on a three-dimensional lattice $$\mathbb{Z}^{3}$$ with a finite spherically symmetric potential. It is proven that the corresponding Shrödinger operator $$H(\mathbf{k}),$$ where $$\mathbf{k}\in(-\pi,\pi]^{3}$$ is the total quasimomentum of the system, has four invariant subspaces $$L_{123}^{-},\,\,L_{1}^{-},\,\,L_{2}^{-},\,\,L_{3}^{-}$$ and it has no eigenfunctions in $$L_{123}^{-}$$ . We also show that the operator $$H(\mathbf{\Lambda}),\,\,\mathbf{\Lambda}=(\pi-2\lambda,\pi-2\lambda,\pi-2\lambda)$$ has four different threefold eigenvalues for small $$\lambda$$ .