Asymptotics of the eigenvalues and Abel basis property of the root functions of new type Sturm-Liouville problems

Abstract

The main goal of this study is to investigate the main spectral properties of a new type Sturm–Liouville problems(SLP’s). The problems studied here differs from the classical SLP’s in that, the equation contain an abstract linear operator which can be non-selfadjoint and unbounded in the Hilbert space of square-integrable functions, and the boundary conditions contain an additional transmission conditions at an internal singular point. So, SLP’s under consideration are not purely differential. We emphasize that this type of non-classical SLP’s which includes an abstract linear operator in differential equation, was studied by the authors of this work for the first time in the literature. Naturally, the study of such type non-classical SLP’s are much more complicated than the classical purely differential SLP’s, because it is not clear how to apply the known methods of the Sturm–Liouville theory to problems of this type. The main difficulties lie in the derivation of such important spectral properties as the discreteness of the spectrum and the completeness of the corresponding eigenfunctions. First, we establish isomorphism and coerciveness with respect to the spectral parameter for the corresponding nonhomogeneous problem. Based on these results and using our own approaches we prove that the spectrum of the main problem is discrete. Then we derive some asymptotic formulas for the eigenvalues. Finally it is shown that the system of root functions (i.e. eigen and associated functions) form an Abel basis of order α for all α>1. The obtained results are new even in the case when the problem under consideration does not contain an additional transmission conditions. © 2022, Adv. Theor. Math. Phys. All rights reserved.