# On the Spectrality and Spectral Expansion of the Non-self-adjoint Mathieu-Hill Operator in All Real Line

## TL;DR

This paper analyzes the spectral properties of a non-self-adjoint Mathieu-Hill operator with complex potential, establishing conditions for spectral regularity and classifying its spectral decomposition based on potential characteristics.

## Contribution

It provides necessary and sufficient conditions for the operator to be asymptotically spectral and classifies spectral decompositions in terms of the potential.

## Key findings

- Operator has no spectral singularity at infinity under certain conditions
- Classification of spectral decomposition based on potential
- Conditions for the operator to be asymptotically spectral

## Abstract

In this paper we investigate the non-self-adjoint operator H generated in all real line by the Mathieu-Hill equation with a complex-valued potential. We find a necessary and sufficient conditions on the potential for which H has no spectral singularity at infinity and it is an asymptotically spectral operator. Moreover, we give a detailed classification, stated in term of the potential, for the form of the spectral decomposition of the operator H by investigating the essential spectral singularities.