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

Abstract

In contrast to the setting considered by Adıgüzelov 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 \[ L_{o} (y(x)) := (-1)^{m} \big(A y(x)\big)^{(2m)} + A^{m}(y(x)), \] 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 $A y(x)$. 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.