A conjecture on exceptional orthogonal polynomials
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2013
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[1] V. E. Adler, A modification of Crum’s method, Theor. Math. Phys. 101 (1994) 1381–1386.
[2] S. Bochner, Uber Sturm-Liouvillesche Polynomsysteme, ¨ Math. Z. 29 (1929), 730-736.
[3] JF Cariñeena, A. M. Perelomov, M. F. Rañaada and M. Santander, A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator, J. Phys. A 41 (2008) 085301.
[4] S. Y. Dubov, V. M. Eleonskii, and N. E. Kulagin, Equidistant spectra of anharmonic oscillators, Sov. Phys. JETP 75 (1992) 446–451 Chaos 4 (1994) 47– 53.
[5] D. Dutta and P. Roy, Conditionally exactly solvable potentials and exceptional orthogonal polynomials, J. Math. Phys. 51 (2010) 042101.
[6] D. Dutta and P. Roy, Information entropy of conditionally exactly solvable potentials, J. Math. Phys 52 (2011) 032104.
[7] J. M. Fellows and R. A. Smith, Factorization solution of a family of quantum nonlinear oscillators, J. Phys. A 42 (2009) 335303.
[8] D. Gómez-Ullate, N. Kamran and R. Milson, Quasi-exact solvability and the direct approach to invariant subspaces. J. Phys. A 38(9) (2005) 2005– 2019.
[9] D. Gómez-Ullate, N. Kamran and R. Milson, The Darboux transformation and algebraic deformations of shape-invariant potentials, J. Phys. A 37 (2004), 1789–1804.
[10] D. Gómez-Ullate, N. Kamran and R. Milson, Supersymmetry and algebraic Darboux transformations, J. Phys. A 37 (2004), 10065–10078.
[11] D. Gómez-Ullate, N. Kamran and R. Milson, Quasi-exact solvability in a general polynomial setting, Inverse Problems, 23 (2007) 1915–1942.
[12] D. Gómez-Ullate, N. Kamran, and R. Milson, An extension of Bochner’s problem: exceptional invariant subspaces J. Approx. Theory 162 (2010) 987–1006.
[13] D. Gómez-Ullate, N. Kamran, and R. Milson, An extended class of orthogonal polynomials defined by a Sturm-Liouville problem , J. Math. Anal. Appl. 359 (2009) 352–367.
[14] D. Gómez-Ullate, N. Kamran, and R. Milson, Exceptional orthogonal polynomials and the Darboux transformation, J. Phys. A 43 (2010) 434016.
[15] D. Gómez-Ullate, N. Kamran, and R. Milson, Two-step Darboux transformations and exceptional Laguerre polynomials, J. Math. Anal. Appl., 387 (2012), 410–418.
[16] D. Gómez-Ullate, N. Kamran, and R. Milson, On orthogonal polynomials spanning a non-standard flag, in Algebraic aspects of Darboux transformations, quantum integrable systems and supersymmetric quantum mechanics. P. AcostaHumánez et al. Eds. Contemp. Math. 563 (2012) 51–72.
[17] A. González-López, N. Kamran and P.J. Olver, Normalizability of one-dimensional quasi-exactly solvable Schrdinger operators, Comm. Math. Phys. 153 (1993), no. 1, 117146.
[18] Y. Grandati, Multistep DBT and regular rational extensions of the isotonic oscillator, arXiv:1108.4503
[19] Y. Grandati, Solvable rational extensions of the isotonic oscillator , Ann Phys. 326 (2011) 2074–2090.
[20] F.A. Grünbaum and L. Haine, Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation, Symmetries an Integrability of Differential Equations, CRM Proc. Lecture Notes, 9, Amer. Math. Soc. Providence, RI, 1996, 143–154.
[21] C.-L Ho, Dirac(-Pauli), Fokker-Planck equations and exceptional Laguerre polynomials, Annals of Physics, 326,( 2011), 797-807.
[22] N. Kamran, and P.J. Olver, Lie algebras of differential operators and Lie-algebraic potentials, J. Math. Anal. Appl. 145 (1990), no. 2, 342356
[23] M.G. Krein, Doklady acad. Nauk. CCCP, 113 (1957) 970-973.
[24] P. Lesky, Die Charakterisierung der klassischen orthogonalen Polynome durch Sturm-Liouvillesche Differentialgleichungen, Arch. Rat. Mech. Anal. 10 (1962), 341-352.
[25] C. Quesne, Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry, J. Phys. A 41 (2008) 392001–392007
[26] C. Quesne, Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics, SIGMA 5 (2009), 084, 24 pages.
[27] C. Quesne, Higher-order SUSY, exactly solvable potentials, and exceptional orthogonal polynomials, Mod. Phys. Lett. A 26 (2011) 1843-1852.
[28] S. Odake and R. Sasaki, Infinitely many shape invariant potentials and new orthogonal polynomials, Phys. Lett. B 679 (2009) 414–417.
[29] S. Odake and R. Sasaki, Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials, Phys. Lett. B 682 (2009), 130–136.
[30] S. Odake and R. Sasaki, Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials, J. Math. Phys. 51 (2010), 053513.
[31] S. Odake and R. Sasaki, Another set of infinitely many exceptional (Xm) Laguerre polynomials, Phys. Lett. B 684 (2010), 173–176.
[32] S. Odake and R. Sasaki, Exactly solvable quantum mechanics and infinite families of multi- indexed orthogonal polynomials, Phys. Lett. B 702 (2011), 164–170.
[33] R. Sasaki, S. Tsujimoto, and A. Zhedanov, Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux- Crum transformations, J. Phys. A 43 (2010), 315204.
[34] A. Turbiner, Quasi-exactly-solvable problems and sl(2) algebra, Comm. Math. Phys. 118 (1988), no. 3, 467-474
Abstract
Exceptional orthogonal polynomial systems (X-OPSs) arise as eigenfunctions of Sturm-Liouville problems, but without the assumption that an eigenpolynomial of every degree is present. In this sense, they generalize the classical families of Hermite, Laguerre, and Jacobi, and include as a special case the family of CPRS orthogonal polynomials introduced by Cariena et al. (J. Phys. A 41:085301, 2008). We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux-Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X-2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X-2-OPSs. The classification includes all cases known to date plus some new examples of X-2-Laguerre and X-2-Jacobi polynomials.
Description
© Springer. The research of DGU was supported in part by MICINN-FEDER grant MTM2009- 06973 and CUR-DIUE grant 2009SGR859. The research of NK was supported in part by NSERC grant RGPIN 105490-2011. The research of RM was supported in part by NSERC grant RGPIN-228057-2009.