I'm looking for a simple example of differential (possibly generalized) eigenvalue problem with complex and discrete spectrum. The function space should be complex functions of real variable and if possible I would like to have homogenous boundary conditions.

I know that the most basic example of differential eigenvalue problem is $$ -u''(t)=\lambda u(t), \qquad t\in[0,1], $$ $$ u(0) = 0, \qquad u(1) = 0, $$ where eigenvalues are $\lambda_n = n^2 \pi^2$, $n\in\mathbb{N}$ and corresponding eigenfunctions are $u_n(t) = \sin(n\pi t)$. The spectrum is discrete but real.

I have come up with this example of generalized eigenvalue problem with complex eigenvalues: $$ \begin{bmatrix} u(t) \\ v(t) \end{bmatrix}' = \lambda\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} u(t) \\ v(t) \end{bmatrix}, \qquad t\in[0,1], $$ $$ u(0) = 0, \qquad v'(0) = 0. $$ The eigenvalues of this problem are $\lambda\in\mathbb{C}$ and corresponding eigenfunctions are $[\sin(\lambda t), \cos(\lambda t)]^\mathrm{T}$. The spectrum is now complex but continuous.

Is there a simple example of (generalized) eigenvalue problem with complex and discrete spectrum?

Note: By generalized eigenvalue problem I mean a problem of the form $Ax=\lambda Bx$, where $A$ and $B$ are some differential operators.


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