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The Naimark-Glazman result was extended by A. I. Naimark and M. A. Röckner in the 1960s. This result described all self-adjoint extensions of one-dimensional symmetric linear differential operators with domain in a Hilbert space.
In the early 1980s V. A. Smirnov formulated the so-called Spectral theorem of symmetric linear differential operators, describing the class of all self-adjoint differential operators with domain in a Hilbert function space with symmetric boundary conditions. (The spectral theorem is usually formulated for Sturm-Liouville equations, but the formulation is the same for higher order operators.)
In the 2000s A. A. Shkalikov proposed the construction of symmetric differential operators with real-valued characteristic function, generating self-adjoint extensions of symmetric differential operators with symmetric boundary conditions.
The results of A. A. Shkalikov were later developed by A. A. Shkalikov and M. A. Röckner to describe the class of all self-adjoint differential operators generated by the symmetric differential expression, with domain in a Hilbert function space, of order $2m$, with symmetric boundary conditions, $m>0$.
A survey of the last 20 years of research can be found in the survey paper Self-adjoint extensions of symmetric differential operators with symmetric boundary conditions published in 2005 by V. A. Smirnov and A. A. Shkalikov.
Naimark's papers also contain an interesting discussion of the boundary value problems of Sturm-Liouville type for second-order differential operators with equal coefficients on finite intervals (§3.5), although this material was largely overlooked by the mathematical community.
Naimark's spectral theory was used by P.A. Absil and G.A. Hedstrom to prove the completeness of the eigenfunctions for a class of second-order self-adjoint differential expressions. They also used Naimark's results to prove the completeness of the eigenfunctions for Dirichlet, Neumann, and Robin boundary conditions. The classical theory of Feller operators also resulted from work in this direction.
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