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 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].