Self-Adjoint Extensions of Symmetric Operators

Abstract

We consider self-adjoint extensions of composite symmetric differential operators through the Cayley transform. For fourth- and second-order operators, we compute deficiency indices and analyze spectral properties under different growth conditions. The results indicate when the composite operator’s deficiency indices equal the sum of its components, and when it lies in the limit point or limit circle cases. Applications of Levinson’s theorem provide explicit spectral characterizations, contributing to a rigorous functional analytic framework for operator extensions.