  The zeros of the eigenfunctions of self-adjoint Sturm-Liouville eigenvalue
problems interlace. For these problems interlacing is crucial for completeness.
For the complex Sturm-Liouville problem associated with the Schrodinger
equation for a non-Hermitian PT-symmetric Hamiltonian, completeness and
interlacing of zeros have never been examined. This paper reports a numerical
study of the Sturm-Liouville problems for three complex potentials, the large-N
limit of a -(ix)^N potential, a quasi-exactly-solvable -x^4 potential, and an
ix^3 potential. In all cases the complex zeros of the eigenfunctions exhibit a
similar pattern of interlacing and it is conjectured that this pattern is
universal. Understanding this pattern could provide insight into whether the
eigenfunctions of complex Sturm-Liouville problems form a complete set.
