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We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as X(1)-Ja[...]texto impreso
Gómez-Ullate Otaiza, David ; Kamran, Niky ; Milson, Robert | Academic Press-Elsevier Science | 2010-05We prove an extension of Bochner's classical result that characterizes the classical polynomial families as eigenfunctions of a second-order differential operator with polynomial coefficients. The extended result involves considering differentia[...]texto impreso
García Ferrero, María Ángeles ; Gómez-Ullate Oteiza, David ; Milson, Robert | Elsevier Science | 2019-04-01It was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to [...]texto impreso
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 Hermi[...]texto impreso
Gómez-Ullate Otaiza, David ; Grandati, Yves ; Milson, Robert | American Institute of Physics | 2014-04Considering successive extensions of primary translationally shape invariant potentials, we enlarge the Krein-Adler theorem to mixed chains of state adding and state-deleting Darboux-Backlund transformations. It allows us to establish novel bi-l[...]texto impreso
We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the classical [...]texto impreso
We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvab[...]texto impreso
Our goal in this paper is to extend the theory of quasi-exactly solvable Schrodinger operators beyond the Lie-algebraic class. Let P-n be the space of nth degree polynomials in one variable. We first analyze exceptional polynomial subspaces M su[...]texto impreso
We prove that every rational extension of the quantum harmonic oscillator that is exactly solvable by polynomials is monodromy free, and therefore can be obtained by applying a finite number of state-deleting Darboux transformations on the harmo[...]texto impreso
Gómez-Ullate Otaiza, David ; Kasman, Alex ; Kuijlaars, Arno B. J. ; Milson, Robert | Academic Press-Elsevier Science | 2016-04The bispectral anti-isomorphism is applied to differential operators involving elements of the stabilizer ring to produce explicit formulas for all difference operators having any of the Hermite exceptional orthogonal polynomials as eigenfunctio[...]texto impreso
Gómez-Ullate Otaiza, David ; Kamran, Niky ; Milson, Robert | American Institute of Mathematical Sciences | 2007-05In this paper we derive structure theorems which characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful conc[...]texto impreso
We investigate the backward Darboux transformations (addition of the lowest bound state) of shape-invariant potentials on the line, and classify the subclass of algebraic deformations, those for which the potential and the bound states are simpl[...]texto impreso
It has been recently discovered that exceptional families of Sturm-Liouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of excepti[...]
