Bispectral Darboux Transformations: An Extension of the Krall Polynomials

International Mathematics Research Notices

Volumn , Issue 8, 1997, Pages

  Grünbaum, F Alberto ^a   Haine, Luc ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

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EID: 1842737033     PISSN: 10737928     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (36)

1. M. Adler and P. van Moerbeke, Birkhoff strata, Bäcklund transformations, and regularization of isospectral operators, Adv. Math. 108 (1994), 140-204.
2. _, Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials, Duke Math. J. 80 (1995), 863-911.
3. B. Bakalov, E. Horozov, and M. Yakimov, Bispectral algebras of commuting ordinary differential operators, Sofia preprint (1996), q-alg/9602011.
4. _, Bäcklund-Darboux transformations in Sato's Grassmannian, Sofia preprint (1996), q-alg/9602010.
5. 1+∞ algebra and the bispectral problem, to appear in Duke Math. J.
6. _, General methods for constructing bispectral operators, Phys. Lett. A 222 (1996), 59-66.
7. S. Bochner, Über Sturm-Liouvillesche Polynomsysteme, Math. Z. 29 (1929), 730-736.
8. J. J. Duistermaat and F. A. Grünbaum, Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177-240.
9. H. Flaschka and A. C. Newell, Monodromy- and spectrum-preserving deformations I, Comm. Math. Phys. 76 (1980), 65-116.
10. F. A. Grünbaum, "Differential equations in the spectral parameter: The higher order case" in Inverse Problems: An Interdisciplinary Study, Adv. Electron. Electron Phys. Suppl. 19, Academic Press, New York, 1987, 307-322.
11. _, The Kadomtsev-Petviashviliǐ equation: an alternative approach to the "rank two" solutions of Krichever and Novikov, Phys. Lett. A 139 (1989), 146-150.
12. _, "Darboux's method and some "rank two" explicit solutions of the KP equation", in Nonlinear Evolution Equations: Integrability and Spectral Methods, Proc. Nonlinear Sci., Manchester Univ. Press, Manchester, 1990, 271-277.
13. _, Time-band limiting and the bispectral problem, Comm. Pure Appl. Math. 47 (1994), 307-328.
14. _, Band-time-band limiting integral operators and commuting differential operators, Algebra i Analiz 8 (1996), 122-126.
15. F. A. Grünbaum and L. Haine, "Orthogonal polynomials satisfying differential equations: The role of the Darboux transformation" in Symmetries and Integrability of Difference Equations, D. Levi, L. Vinet, and P. Winternitz, eds., CRM Proc. Lecture Notes 9, Amer. Math. Soc., Providence, 1996, 143-154.
16. _, "A theorem of Bochner, revisited" in Algebraic Aspects of Integrable Systems: In memory of Irene Dorfman, A. S. Fokas and I. M. Gelfand, eds., Progr. Nonlinear Differential Equations 26, Birkhäuser, Boston, 1996, 143-172.
17. _, The q-version of a theorem of Bochner, J. Comput. Appl. Math. 68 (1996), 103-114.
18. _, Some functions that generalize the Askey-Wilson polynomials, to appear in Comm. Math. Phys.
19. E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944.
20. A. R. Its and V. Y. Novokshenov, The isomonodromic deformation method in the theory of Painlevé equations, Lecture Notes in Math. 1191, Springer-Verlag, Berlin, 1986.
21. A. Kasman and M. Rothstein, Bispectral Darboux transformations: the generalized Airy case, to appear in Phys. D.
22. A. M. Krall, Chebyshev sets of polynomials which satisfy an ordinary differential equation, SIAM Rev. 22 (1980), 436-441.
23. _, Orthogonal polynomials satisfying fourth order differential equations, Proc. Roy. Soc. Edinburgh Sect. A 87 (1981), 271-288.
24. H. L. Krall, Certain differential equations for Tchebycheff polynomials, Duke Math. J. 4 (1938), 705-718.
25. _, On orthogonal polynomials satisfying a certain fourth order differential equation, Pennsylvania State College Studies 1940:6 (1940), 1-24.
26. H. L. Krall and I. M. Sheffer, On pairs of related orthogonal polynomial sets, Math. Z. 86 (1965), 425-450.
27. L. L. Littlejohn, On the classification of differential equations having orthogonal polynomial solutions, Ann. Mat. Pura Appl. 138 (1984), 35-53.
28. _, The Krall polynomials: a new class of orthogonal polynomials, Quaestiones Math. 5 (1982), 255-265.
29. F. Magri and J. P. Zubelli, Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master symmetries for KdV, Comm. Math. Phys. 141 (1991), 329-351.
30. V. B. Matveev and M. A. Salle, Differential-difference evolution equations II, Darboux transformation for the Toda lattice, Lett. Math. Phys. 3 (1979), 425-429.
31. M. L. Mehta, Random Matrices, 2nd ed., Academic Press, Boston, MA, 1991.
32. D. Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev. 25 (1983), 379-393.
33. P. van Moerbeke, "Integrable foundations of string theory," in Lectures on Integrable Systems: In Memory of Jean-Louis Verdier, Proceedings of the CIMPA School, 10-28, June 1991 (Nice), O. Babelon, P. Cartier, and Y. Kosmann-Schwarzbach, eds., River Edge, NJ, 1994, 163-267.
34. G. Wilson, Bispectral commutative ordinary differential operators, J. Reine Angew. Math. 442 (1993), 177-204.
35. _, Collisions of Calogero-Moser particles and an adelic Grassmannian (with an appendix by I. G. Macdonald), 1996, preprint.
36. J. P. Zubelli, Differential equations in the spectral parameter for matrix differential operators, Phys. D 43 (1990), 269-287.