Conjecture on the interlacing of zeros in complex Sturm-Liouville problems
- Department of Physics, Washington University, St. Louis, Missouri 63130 (United States)
- Department of Physics, Emory University, Atlanta, Georgia 30322 (United States)
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 Schroedinger 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){sup N} potential, a quasiexactly-solvable -x{sup 4} potential, and an ix{sup 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. (c) 2000 American Institute of Physics.
- OSTI ID:
- 20217446
- Journal Information:
- Journal of Mathematical Physics, Vol. 41, Issue 9; Other Information: PBD: Sep 2000; ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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