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

## TL;DR

This paper studies the spectral properties of a new class of Sturm-Liouville problems with transmission conditions, establishing eigenvalue asymptotics and the Abel basis property of root functions, with novel results even in the continuous case.

## Contribution

It introduces a new approach to analyze the spectrum of nonselfadjoint Sturm-Liouville problems with transmission conditions, deriving eigenvalue asymptotics and basis properties.

## Key findings

- Eigenvalues have specific asymptotic formulas.
- Root functions form an Abel basis.
- Results are new even without transmission conditions.

## Abstract

This work investigates spectrum and root functions (that is, eigen- and associated functions) of a Sturm-Liouville problem involving an abstract linear operator (nonselfadjoint in general) in the equation together with supplementary transmission conditions at the one interior singular point. So, the problem under consideration is not pure differential pro lem. At first we establish isomorphism and coerciveness with respect to the spectral parameter for the corresponding nonhomogeneous problem. Then by suggesting an own our method we prove that the spectrum of the considered problem is discrete and derive an asymptotic approximation formulas for the eigenvalues. We must note that Asymptotics of the eigenvalues of such type problems is investigated at first in literature in the present work and the obtained results are new even in the continuous case (i.e. without transmission conditions). Finally it is shown that the system of root functions form an Abel basis of the corresponding Hilbert space.