Application of Comparison Algebras in Construction of Commutative Composites of Symmetric Higher Order Differential Operators

In this paper, we have been interested in the commutativity and spectral properties of two higher order symmetric differential operators with unbounded coefficients. By constructing appropriate comparison algebras of the corresponding self-adjoint operator extensions and application of asymptotic integration, under suitable decay and growth conditions imposed on the coefficients, we have deduced that the operators commute if they are of the same order and the absolutely continuous spectrum of their self-adjoint extensions is the whole of real line. Using some examples, we have shown that some necessary conditions cannot be weakened further.