Spectrum of a non-self-adjoint operator associated with the periodic heat equation

TL;DR

This paper analyzes the spectrum of a non-self-adjoint operator related to the periodic heat equation, revealing eigenvalue properties and the limitations of eigenfunction bases in certain function spaces.

Contribution

It provides a rigorous spectral analysis of a specific non-self-adjoint operator and demonstrates the eigenfunctions' completeness but not forming a basis in the relevant function space.

Findings

Eigenvalues are purely imaginary and simple.

Eigenfunctions become nearly linearly dependent for large eigenvalues.

Eigenfunctions do not form a basis in the function space.

Abstract

We study the spectrum of the linear operator $L = - \partial_{\theta} - \epsilon \partial_{\theta} (\sin \theta \partial_{\theta})$ subject to the periodic boundary conditions on $\theta \in [-\pi,\pi]$. We prove that the operator is closed in $L^2([-\pi,\pi])$ with the domain in $H^1_{\rm per}([-\pi,\pi])$ for $|\epsilon| < 2$, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in $H^1_{\rm per}([-\pi,\pi])$.