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

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

2010-12-17

arxiv.org/abs/1012.3954v1

Mathematics

Functional Analysis

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-639548