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M. Chugunova and D. Pelinovsky

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

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Journal of Mathematical Analysis and its Applications 342, 970-988 (2008)

**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]).

**Keywords**:

non-self-adjoint operators, advection-diffusion equation,
closure and spectrum, numerical approximations of eigenvalues,
stability and ill-posedness