Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators

TL;DR

This paper investigates symmetric operators with maximal defect subspaces, showing that under certain conditions their self-adjoint extensions lack continuous spectrum in an interval, with applications to differential operators and spectral theory.

Contribution

It establishes conditions under which self-adjoint extensions have no continuous spectrum and extends known results to differential operators with singular endpoints and arbitrary equal deficiency indices.

Findings

Self-adjoint extensions have no continuous spectrum in certain intervals.

The point spectrum of extensions is nowhere dense in those intervals.

Counterexample to a conjecture by Hartman and Wintner on Sturm-Liouville spectra.

Abstract

Let $\gH$ be a Hilbert space and let $A$ be a simple symmetric operator in $\gH$ with equal deficiency indices $d:=n_\pm(A)<\infty$. We show that if, for all $\l$ in an open interval $I\subset\bR$, the dimension of defect subspaces $\gN_\l(A)(=\Ker (A^*-\l))$ coincides with $d$, then every self-adjoint extension $\wt A\supset A$ has no continuous spectrum in $I$ and the point spectrum of $\wt A$ is nowhere dense in $I$. Application of this statement to differential operators makes it possible to generalize the known results by Weidmann to the case of an ordinary differential expression with both singular endpoints and arbitrary equal deficiency indices of the minimal operator. Moreover, we show in the paper, that an old conjecture by Hartman and Wintner on the spectrum of a self-adjoint Sturm - Liouville operator is not valid.