Self-adjoint domains, symplectic geometry, and limit-circle solutions

Everitt and Markus characterized the domains of self-adjoint operator realizations of very general even and odd order symmetric ordinary differential equations in terms of Lagrangian subspaces of symplectic spaces. Recently, for the even order case with real coefficients, Wang, Sun and Zettl constructed limit-circle (LC) solutions and Hao, Wang, Sun and Zettl characterized the self-adjoint domains in terms of LC solutions. These LC solutions are higher order analogues of the celebrated Titchmarsh–Weyl limit-circle solutions in the second-order case. This LC characterization has been used to obtain information about the discrete, continuous, and essential spectra of these operators. In this paper we investigate the connection between these two very different kinds of characterizations and thus add the methods of symplectic geometry to the techniques available for the investigation of the spectrum of self-adjoint operators.