Abstract
Consider the Wronskians of the classical Hermite polynomials
where \({k_i=\lambda_i+n-i, \,\, i=1,\ldots, n}\) and \({\lambda=(\lambda_1, \ldots, \lambda_n)}\) is a partition. Gómez-Ullate et al. showed that for a special class of partitions the corresponding polynomials are orthogonal and dense among all polynomials with respect to a certain inner product, but in contrast to the usual case have some degrees missing (so-called exceptional orthogonal polynomials). We generalise their results to all partitions by considering complex contours of integration and non-positive Hermitian products. The corresponding polynomials are orthogonal and dense in a finite-codimensional subspace of \({\mathbb C[x]}\) satisfying certain quasi-invariance conditions. A Laurent version of exceptional orthogonal polynomials, related to monodromy-free trigonometric Schrödinger operators, is also presented.
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References
Adler V.E.: A modification of Crum’s method. Theor. Math. Phys. 101, 1381–1386 (1995)
Berest Y.Y., Lutsenko I.M.: Huygens’ principle in Minkowski spaces and soliton solutions of the Korteweg-de Vries equation. Commun. Math. Phys. 190, 113–132 (1997)
Bochner S.: Über Sturm-Liouvillesche Polynomsysteme. Math. Z. 29, 730–736 (1929)
Chalykh O.A., Feigin M.V., Veselov A.P.: Multidimensional Baker-Akhiezer functions and Huygens’ principle. Commun. Math. Phys. 206, 533–566 (1999)
Crum M.M.: Associated Sturm-Liouville systems. Q. J. Math. Oxford Ser. 6, 121–127 (1955)
Cruz-Barroso R., Diaz Mendoza C., Orive R.: Orthogonal Laurent polynomials. A new algebraic approach. J. Math. Anal. Appl. 408, 40–54 (2013)
Duistermaat J.J., Grünbaum F.A.: Differential equations in the spectral parameter. Commun. Math. Phys. 103, 177–240 (1986)
Felder G., Hemery A.D., Veselov A.P.: Zeros of Wronskians of Hermite polynomials and Young diagrams. Phys. D 241, 2131–2137 (2012)
Felder G., Willwacher T.: Jointly orthogonal polynomials. J. Lond. Math. Soc. 91(3), 750–768 (2015)
Gibbons J., Veselov A.P.: On the rational monodromy-free potentials with sextic growth. J. Math. Phys. 50(1), 013513 (2009)
Gómez-Ullate D., Kamran N., Milson R.: An extension of Bochner’s problem: exceptional invariant subspaces. J. Approx. Theory 162, 987–1006 (2010)
Gómez-Ullate D., Grandati Y., Milson R.: Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials. J. Phys. A 47, 015203 (2014)
García-Ferrero M.Á., Gómez-Ullate D.: Oscillation theorems for the Wronskian of an arbitrary sequence of eigenfunctions of Schrödinger’s equation. Lett. Math. Phys. 105, 551–573 (2015)
Grinevich P.G., Novikov S.P.: Singular soliton operators and indefinite metrics. Bull. Braz. Math. Soc. N. Ser. 44(4), 809–840 (2013)
Oblomkov A.A.: Monodromy-free Schrödinger operators with quadratically increasing potential. Theor. Math. Phys. 121, 1574–1584 (1999)
Olshanetsky M.A., Perelomov A.M.: Quantum integrable systems related to Lie algebras. Phys. Rep. 94, 313–404 (1983)
Reed M., Simon B.: Methods of modern mathematical physics. I. Functional analysis 2nd edn. Academic Press, New York (1980)
Szegö, G.: Orthogonal polynomials. AMS Colloquium Publ., AMS (1939)
Veselov A.P.: On Stieltjes relations, Painlevé-IV hierarchy and complex monodromy. J. Phys. A 34, 3511–3519 (2001)
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Haese-Hill, W.A., Hallnäs, M.A. & Veselov, A.P. Complex Exceptional Orthogonal Polynomials and Quasi-invariance. Lett Math Phys 106, 583–606 (2016). https://doi.org/10.1007/s11005-016-0828-8
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DOI: https://doi.org/10.1007/s11005-016-0828-8