On extensions and spectral problems for fourth order differential operator equation

TL;DR

This paper investigates the spectral properties and boundary conditions of a fourth-order differential operator, providing criteria for spectrum types, asymptotic spectrum behavior, and a new method for evaluating regularized traces.

Contribution

It introduces a comprehensive analysis of selfadjoint extensions, spectrum characterization, and a novel approach for regularized trace calculations for such operators.

Findings

Criteria for purely discrete or continuous spectrum

Asymptotic behavior of the spectrum analyzed

New method for regularized trace evaluation

Abstract

The aim of the paper is firstly to study domains of definitions in terms of boundary conditions of minimal and maximal operators, as well as selfadjoint extensions of a minimal operator associated with the fourth-order differential operator equation. Further, we give necessary and sufficient conditions for that operators to have a purely discrete or continuous spectrum, to exist extension with resolvent from $\sigma_p,$ study asymptotics of spectrum in case pure discrete spectrum. Finally, give the new and more general method for evaluations of regularized traces of operators with discrete spectrum associated with one class boundary value problems.