Conjecture on the interlacing of zeros in complex Sturm–Liouville problems

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 Schrödinger 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 2(ix) potential, a quasiexactly-solvable 2x potential, and an ix 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. © 2000 American Institute of Physics. @S0022-2488~00!04309-7#