Boundary Effects in Non-uniform Spin Chains

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.