Invariant Subspaces of the Shrödinger Operator with a Finite Support Potential
Tóm tắt
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
$$\lambda$$
.
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