Construction of symmetric third order difference operators on a Hilbert space

Abstract

We have constructed a symmetric third order difference equation and studied the absolutely continuous spectrum of the self adjoint subspace extension generated using the subspace and decomposition theories. In particular, we have shown that under some decay and growth conditions, the self-adjoint subspace extension exists with discrete spectrum on l2(N) while the absolutely continuous spectrum contained on half line of multiplicity one exists on l2(Z) if we use decomposition technique. We have given a number of examples to reinforce our results.