Abstract
The spectrum of short-closed chains up to N=12 are studied by exact diagonalization to obtain the spin-wave spectrum of the Hamiltonian H=2J Sigma i=1Nsi.si+1+2J alpha Sigma i=1Nsi.si+2, -1.0<or= alpha <or=1.0 with Jor=0.3 and alpha <or=-0.5. A few remarks are made concerning the low-lying excitations.

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