This thesis is devoted to the study of self-adjoint extensions of certain symmetric differential operators. Such operators play a fundamental role in functional analysis and mathematical physics, as they describe observables and generate well-defined dynamics. We investigate both classical and new examples of symmetric, or formally symmetric, differential operators on the real line, half-lines, and punctured domains. In particular, we analyse first-order and second-order operators, and introduce new classes of third-order homogeneous and inhomogeneous differential operators, as well as systems of differential operators.
For each class of operators, we determine the deficiency indices and use them to characterize the existence of self-adjoint extensions. Whenever such extensions exist, they are described in terms of boundary conditions or parametrized via unitary maps between the corresponding deficiency subspaces.
A key aspect of this work is the establishment of an equivalence between different operator formulations. This allows us to transfer results from one problem to another, providing a unified framework in which properties that are difficult to analyse for a given operator can be deduced from a related one.
While classical methods based on boundary conditions are effective for lower-order operators, they turn out to be insufficient in the case of third-order differential operators. To overcome this difficulty, we adopt an alternative approach inspired by the theory of point interactions and singular perturbations. This requires the development of auxiliary results, some of which are established here for the first time.
Using this approach, we obtain a complete characterization of all self-adjoint extensions for several classes of third-order differential operators on punctured domains. These results constitute the main contribution of the thesis.
