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

C. M. Bender·S. Boettcher·V. M. Savage
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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.

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BibTeXRIS

C. M. Bender, S. Boettcher, V. M. Savage. 2000-05-10. Conjecture on the Interlacing of Zeros in Complex Sturm-Liouville Problems. https://doi.org/10.1063/1.1288247

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