  We give an explicit comparison of eigenvalues and eigenvectors of XY
Hamiltonians of an open linear spin-1/2 chain and a closed spin-1/2 ring with
periodic in space coefficients. It is shown that the Hamiltonian of a
k-periodic chain with nk-1 sites has (n-1)k multiplicity one eigenvalues which
are eigenvalues of multiplicity two for a Hamiltonian of a k-periodic ring with
2kn sites. For the corresponding eigenvectors in the case of a chain an
explicit expression in terms of eigenvectors of the ring Hamiltonian is given.
The remaining k-1 eigenvalues of the chain Hamiltonian and 2k eigenvalues of
the ring Hamiltonian, together with the corresponding eigenvectors, are
responsible for the difference between chain and ring models which displays in
the boundary effects at the ends of the chain and translation invariance of the
periodic ring.
