The zeros of the eigenfunctions of self-adjoint Sturm-Liouville eigenvalue
problems interlace. For these problems interlacing is crucial for completen
ess. For the complex Sturm-Liouville problem associated with the Schrodinge
r equation for a non-Hermitian PT-symmetric Hamiltonian, completeness and i
nterlacing of zeros have never been examined. This paper reports a numerica
l study of the Sturm-Liouville problems for three complex potentials, the l
arge-N limit of a -(ix)(N) potential, a quasiexactly-solvable -x(4) potenti
al, and an ix(3) potential. In all cases the complex zeros of the eigenfunc
tions exhibit a similar pattern of interlacing and it is conjectured that t
his pattern is universal. Understanding this pattern could provide insight
into whether the eigenfunctions of complex Sturm-Liouville problems form a
complete set. (C) 2000 American Institute of Physics. [S0022-2488(00)04309-
7].