  We study both analytically and numerically the spectrum of inhomogeneous
strings with $\mathcal{PT}$-symmetric density. We discuss an exactly solvable
model of $\mathcal{PT}$-symmetric string which is isospectral to the uniform
string; for more general strings, we calculate exactly the sum rules $Z(p)
\equiv \sum_{n=1}^\infty 1/E_n^p$, with $p=1,2,\dots$ and find explicit
expressions which can be used to obtain bounds on the lowest eigenvalue. A
detailed numerical calculation is carried out for two non-solvable models
depending on a parameter, obtaining precise estimates of the critical values
where pair of real eigenvalues become complex.
