  Most studies of PT-symmetric quantum-mechanical Hamiltonians have considered
the Schroedinger eigenvalue problem on an infinite domain. This paper examines
the consequences of imposing the boundary conditions on a finite domain. As is
the case with regular Hermitian Sturm-Liouville problems, the eigenvalues of
the PT-symmetric Sturm-Liouville problem grow like $n^2$ for large $n$.
However, the novelty is that a PT eigenvalue problem on a finite domain
typically exhibits a sequence of critical points at which pairs of eigenvalues
cease to be real and become complex conjugates of one another. For the
potentials considered here this sequence of critical points is associated with
a turning point on the imaginary axis in the complex plane. WKB analysis is
used to calculate the asymptotic behaviors of the real eigenvalues and the
locations of the critical points. The method turns out to be surprisingly
accurate even at low energies.
