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 −(ix)N potential, a quasiexactly-solvable −x4 potential, and an ix3 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.