What is an example of a symmetric operator with non-real spectrum? I've been wracking my brains but all the examples I can think of with differential operators end up having either real eigen-values and/or intractable (possibly) complex eigen-values.
For example, take
\begin{align}
D(T)&=\{u\in L^2(0,\pi)~|~u''\in L^2(0,\pi),iu(0)=u'(0),u(\pi)=0\}, \\
Tu&=-u''.
\end{align}
Then integrating by parts shows it's symmetric, and the function $\sin(\kappa x)-i\kappa\cos(\kappa x)$ is an eigen-function for the eigen-value $\kappa^2$ which must satisfy $\tan(\kappa\pi)=i\kappa$. From here I must show that valid $\kappa$'s must be non-real (we exclude $0$ since $T$ has kernel $\{0\}$). I am not sure how to proceed.
Is there a better way? Is it too much of me to ask to be able to compute explicitly a complex eigen-value, as opposed to simply establishing the existence of some other piece of the spectrum, e.g., continuous, essential, etc.?

$i\frac{d}{dx}$ defined on functions of compact support in $(0,\infty)$ (i.e. with no boundary condition at $0$).
The intuition is that $i\frac{d}{dx}$ is the infinitesimal generator of translations on $(-\infty,\infty)$ but this can't work on $(0,\infty)$ .
