  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 $H^1_{\rm per}([-\pi,\pi])$.
