Quasi-exactly solvable potentials on the line and orthogonal polynomials

Journal of Mathematical Physics

Volumn 37, Issue 8, 1996, Pages 3954-3972

  Finkel, Federico ^a   González López, Artemio ^a   Rodríguez, Miguel A ^a  

^a UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030490980     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.531591     Document Type: Article

References (17)

1. C. M. Bender and G. V. Dunne, J. Math. Phys. 37, 6 (1996).
2. T. S. Chihara, An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978).
3. A. V. Turbiner, Commun. Math. Phys. 118, 467 (1988).
4. A. González-López, N. Kamran, and P. J. Olver, Commun. Math. Phys. 153, 117 (1993).
5. A. Krajewska, A. Ushveridze, and Z. Walczak, Preprint, hep-th 9601088, 1996.
6. A. González-López, N. Kamran, and P. J. Olver, Contemp. Math. 160, 113 (1994).
7. A. González-López, N. Kamran, and P. J. Olver, J. Phys. A 24, 3995 (1991).
8. N. Kamran and P. J. Olver, J. Math. Anal. Appl. 145, 342 (1990).
9. We assume that P is positive in some interval of the real line, so that the first integral in (12) is real.
10. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New Yoik, 1953), Vol. II.
11. Note, however, that a weakly orthogonal polynomial system need not satisfy a three-term recursion relation.
12. The moment problem in (-∞,∞) is never determined unless we impose the condition that ω is nondecreasing. Indeed, there are nonzero functions of bounded variation on (-∞,∞) whose moment functional (45) vanishes identically.
13. A classical theorem, Ref. 2, asserts that the moment problem of a positive-definite moment functional with bounded spectrum is determined; this theorem is not directly applicable in our case, however, since ℒ is not positive-definite.
14. k is positive for all k≥1, several classical criteria due to Carleman relate the growth rate of the moments with the determinacy of the moment problem. Again, these results are not relevant here because of (43).
15. J. E. L. Peck, SIAM Rev. 4, 135 (1962).
16. Y. Smirnov and A. V. Turbiner, Preprint, hep-th 9512002, 1995.
17. A. González-López, N. Kamran, and P. J. Olver, Philos. Trans. London Math. Soc. 354, 1165 (1996).