Spectral analysis of a higher-order self-adjoint Differential operator with unbounded operator coefficients

Abstract

In contrast to the setting considered by Ad & imath;guzelov and Sezer [4], where the differential operator involves classical scalar derivatives followed by multiplication with a self-adjoint unbounded operator, this study investigates a structurally distinct operator-differential model. The dual appearance of the unbounded operator both inside the highest-order derivatives and as an independent power term has not been systematically investigated in the literature. This structural feature induces a fundamentally different functional-analytic framework, leading to novel spectral properties and domain regularity requirements. Specifically, we examine expressions of the form L-o(y(x)) := (-1)(m)(Ay(x))((2m)) + A(m)(y(x)), where the operator A appears both inside the highest-order derivatives and as a power term. This formulation modifies the spectral characteristics and imposes distinct regularity conditions on the domain. Although the analytical techniques employed are analogous to those in [4], the operator structure considered here falls into a different class, requiring boundary conditions directly on Ay(x). The paper establishes the fundamental spectral framework for this setting, including explicit eigenvalue-eigenfunction formulas, symmetry, self-adjointness, and lower semi-boundedness of the associated operator.