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

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Title	Spectrum of a non-self-adjoint operator associated with the periodic heat equation
Authors/Editors*	Marina Chugunova and Dmitry Pelinovsky
Where published*	Journal of Mathematical Analysis and Applications
How published*	Journal
Year*	2007
Volume	-1
Number	-1
Pages
Publisher
Keywords
Link
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 $H^1_{\rm per}([-\pi,\pi])$.