Non self-adjoint correct restrictions and extensions with real spectrum

TL;DR

This paper investigates conditions under which non self-adjoint operators, especially those related to Sturm-Liouville and Laplace operators, have real spectra and Riesz basis eigenvectors, extending the understanding of their spectral properties.

Contribution

It establishes criteria for the similarity of correct restrictions to self-adjoint operators and applies these results to specific differential operators.

Findings

Spectrum of non self-adjoint singularly perturbed operators is real

Eigenvectors form a Riesz basis in these cases

Results extend spectral theory for differential operators

Abstract

The work is devoted to the study of the similarity of a correct restriction to some self-adjoint operator in the case when the minimal operator is symmetric. The resulting theorem was applied to the Sturm-Liouville operator and the Laplace operator. It is shown that the spectrum of a non self-adjoint singularly perturbed operator is real and the corresponding system of eigenvectors forms a Riesz basis.