Conjecture on the Interlacing of Zeros in Complex Sturm-Liouville Problems
Bender, C. M. ; Boettcher, S. ; Savage, V. M.
arXiv, 0005012 / Harvested from arXiv
Text on arXiv
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.
Publié le : 2000-05-10
Classification:  Mathematical Physics,  Condensed Matter,  High Energy Physics - Lattice,  High Energy Physics - Theory 
@article{0005012,
     author = {Bender, C. M. and Boettcher, S. and Savage, V. M.},
     title = {Conjecture on the Interlacing of Zeros in Complex Sturm-Liouville
  Problems},
     journal = {arXiv},
     volume = {2000},
     number = {0},
     year = {2000},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0005012}
}

Bender, C. M.; Boettcher, S.; Savage, V. M. Conjecture on the Interlacing of Zeros in Complex Sturm-Liouville
  Problems. arXiv, Tome 2000 (2000) no. 0, . http://gdmltest.u-ga.fr/item/0005012/