Spectral Analysis of a High-Order Self-Adjoint Differential Operator with Unbounded Operator Coefficients
Hilbert space, self adjoint operator, closed operator, spectrum.
Abstract
In contrast to the setting considered by Adıg¨uzelov and Sezer [4],
where the differential operator involves classical scalar derivatives followed by
multiplication with a self-adjoint unbounded operator, the present study extends
the analysis to a more general operator-differential structure. Specifically, we
examine expressions of the form Loy(x) := (−1)m(Ay(x))2m + Amy(x), where
the operator appears inside the highest-order derivatives and also as a power
term. This formulation leads to modified spectral characteristics and imposes
stronger regularity conditions on the domain. While the structural properties
of the operator differ from those in earlier studies, the spectral decomposition
methods and core analytical techniques remain parallel, allowing the results to
encompass a broader class of operator-differential problems in Hilbert space.
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