Quasi-exactly solvable systems and orthogonal polynomials

Journal of Mathematical Physics

Volumn 37, Issue 1, 1996, Pages 6-11

  Bender, Carl M ^a   Dunne, Gerald V ^b  

^a WASHINGTON UNIVERSITY   (United States)

^b UNIVERSITY OF CONNECTICUT   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030553424     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.531373     Document Type: Article

References (16)

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2. A. Turbiner, Commun. Math. Phys. 118, 467 (1988).
3. A. Morozov, A. Perelomov, A. Roslyi, M. Shifman, and A. Turbiner, Int. J. Mod. Phys. A5, 803 (1990).
4. A. Turbiner, Contemp. Math. 160, 263 (1994).
5. A. G. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mechanics (Institute of Physics, Bristol, 1993).
6. F. Iachello, Nucl. Phys. A 560, 23 (1993); Contemp. Math. 160, 151 (1994).
7. F. Iachello, Nucl. Phys. A 560, 23 (1993); Contemp. Math. 160, 151 (1994).
8. Here, monic means that the coefficient of the highest power of E is one.
9. Higher Transcendental Functions, edited by A. Erdélyi (McGraw-Hill, New York, 1953), Vol. II, Chap. 10.
10. Note that the actual domain of integration over E in (12) is from - ∞ to + ∞, but depending on the weight function this domain may be further truncated. Knowing the domain corresponds to knowing the explicit weight function, which is in general a difficult problem. However, knowledge of the domain is unnecessary for our algebraic approach.
11. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (National Bureau of Standards, Washington, 1970), Chap. 23.
12. C. M. Bender and K. A. Milton, J. Math. Phys. 35, 364 (1994).
13. C. M. Bender, L. R. Mead, and S. S. Pinsky, J. Math. Phys. 28, 509 (1987).
14. n(E) discussed in this paper. Indeed, if the domain of the weight function for a set of orthogonal polynomials is the whole real line then the polynomials must be the Hermite polynomials (see Ref. 14).
15. C. M. Bender and J. Feinberg, J. Math. Phys. 36, 3106 (1995).
16. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), Sec. 8.6 and Prob. 8.56.