Spectral theory of commutative higher order difference operators with unbounded coefficients

Abstract

We have established the necessary and sufficient conditions for any two even higher order symmetric difference maps to generate commuting minimal difference operators. We have done this through construction of appropriate comparison algebras of the self-adjoint operator extensions of the minimal operators generated and application of asymptotic summation. The results show that if the first difference on the coefficients tends to zero whenever the coefficients are allowed to be unbounded and that the difference maps considered have the same order, then they generate minimal operators that commute and the corresponding self-adjoint operators commute too. We have further shown that the self-adjoint operator extensions of the respective minimal operators can be expressed as the composite of the independent self-adjoint operator extensions if the generated minimal difference operators have closed ranges. Finally, we have shown that the spectra of these self-adjoint operator extensions are the whole of the real line if the coefficients are unbounded. These results therefore, extend the existing results in the continuous case to discrete setting.