Conjecture on the Interlacing of Zeros in Complex Sturm-Liouville Problems

TL;DR

This paper investigates the interlacing of zeros in eigenfunctions of complex Sturm-Liouville problems, suggesting a universal pattern that may influence understanding of eigenfunction completeness in non-Hermitian quantum systems.

Contribution

It provides the first numerical evidence of interlacing zeros in complex Sturm-Liouville problems and conjectures universality across different potentials.

Findings

Zeros exhibit interlacing pattern in all studied cases

Pattern appears consistent across different complex potentials

Conjecture of universality of interlacing pattern

Abstract

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.