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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
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
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Quasi-exactly solvable Schrodinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods. Such operators lie in the enveloping algebra of a finite-dimensional Lie algebra of first order differe[...]texto impreso
Finkel Morgenstern, Federico ; González López, Artemio ; Kamran, Niky ; Rodríguez González, Miguel Ángel | American Institute of Physics | 1999-07In this paper we point out a close connection between the Darboux transformation and the group of point transformations which preserve the form of the time-dependent Schroumldinger equation (TDSE). In our main result, we prove that any pair of t[...]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 first establish some general results connecting real and complex Lie algebras ofirst-order diferential operators. These are applied to completely classify all finite-dimensional real Lie algebras of first-order diferential operators in R^2 .[...]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
A generalization of the classical one-dimensional Darboux transformation to arbitrary n- dimensional oriented Riemannian manifolds is constructed using an intrinsic formulation based on the properties of twisted Hodge Laplacians. The classical t[...]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[...]
