Singular boundary conditions for Sturm–Liouville operators via perturbation theory

Abstract

We show that all self-adjoint extensions of semi-bounded Sturm-Liouville operators with limit-circle endpoint(s) can be obtained via an additive singular form bounded self-adjoint perturbation of rank equal to the deficiency indices, say d ∈ {1, 2}. This characterization generalizes the well-known analog for semi-bounded Sturm-Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written aswhere A 0 is a distinguished self-adjoint extension and Θ a self-adjoint linear relation in C d . The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form bounded with respect to A 0 , i.e. it belongs to H −1 (A 0 ), with possible 'infinite coupling.' A boundary triple and compatible boundary pair for the symmetric operator are constructed to ensure the perturbation is well-defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations Θ. The merging of boundary triples with perturbation theory provides a more holistic view of the operator's matrix-valued spectral measures: identifying not just the location of the spectrum, but also certain directional information. As an example, self-adjoint extensions of the classical Jacobi differential equation (which has two limit-circle endpoints) are obtained and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.

The spectrum of self-adjoint extensions associated with exceptional Laguerre differential expressions

“…Differences stem from two important processes: shifting Maya diagrams into canonical/conjugate canonical position, and evaluating Wronskians with these shifted diagrams at x D 0. Other spectral information, such as the strength of the point masses in the spectral measure, can also be obtained from the m-function, see e.g., [7].…”

“…For completeness, some basics from the theory of boundary triples are included here. The content of this subsection is also found in [3,7], which the interested reader may consult for more details. For the purposes of this paper, we restrict ourselves to the simplified case of Sturm-Liouville differential operators rather than more general linear relations.…”

“…/ can be obtained without exploiting (2.13) by simply switching the definitions of the maps 0 and 1 . A similar construction that the reader may find useful for more general Sturm-Liouville operators is available in [7,12]. Now, fix a fundamental system .u 1 .…”

“…Remark 7.10. Perturbation theory says that the spectrum of L 0 will smoothly shift to the spectrum of L 1 as the parameter increases and eigenvalues of all self-adjoint extensions will be simple, see, e.g., [7,38,39].…”

“…The Weyl m-function associated with an operator provides all of the relevant spectral data. Here we are primarily focused on only the location of eigenvalues, but is possible to extract the spectral measure and even eigenfunctions [7]. The challenge of applying the theory in this context is that solutions to the general XOP must be normalized.…”