Connecting exceptional orthogonal polynomials of different kind

A simple derivation is presented of the four families of infinitely many shape-invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. The Darboux–Crum transformations are applied to connect the well-known shape-invariant Hamiltonians of the radial oscillator and the Darboux–Pöschl–Teller potential to the shape-invariant potentials of Odake–Sasaki. Dutta and Roy derived the two lowest members of the exceptional Laguerre polynomials by this method. The method is expanded to its full generality and many other ramifications, including the aspects of the generalized Bochner problem and the bispectral property of the exceptional orthogonal polynomials, are discussed.

In recent years, one of the most interesting developments in quantum mechanics has been the construction of new exactly solvable potentials connected with the appearance of families of exceptional orthogonal polynomials (EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre and Hermite) classical orthogonal polynomials, which start with a constant, the EOP families begin with some polynomial of degree greater than or equal to one, but still form complete, orthogonal sets with respect to some positive-definite measure. We show how they may appear in the bound-state wavefunctions of some rational extensions of well-known exactly solvable quantum potentials. Such rational extensions are most easily constructed in the framework of supersymmetric quantum mechanics (SUSYQM), where they give rise to a new class of translationally shape invariant potentials. We review the most recent results in this field, which use higher-order SUSYQM. We also comment on some recent re-examinations of the shape invariance condition, which are independent of the EOP construction problem.

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 families of Jacobi, Laguerre, and Hermite polynomials. Unlike the classical families, these new examples, called exceptional orthogonal polynomials, feature non-standard polynomial flags; the lowest degree polynomial has degree m > 0. In this paper we review the classification of codimension m = 1 exceptional polynomials, and give a novel, compact proof of the fundamental classification theorem for codimension 1 polynomial flags. As well, we describe the mechanism or rational factorizations of 2nd order operators as the analogue of the Darboux transformation in this context. We finish with the example of higher codimension generalization of Jacobi polynomials and perform the complete analysis of parameter values for which these families have non-singular weights.

We show that the exceptional orthogonal polynomials can be viewed as confluent limits of the generalized Schur polynomials introduced by Sergeev and Veselov.

Exceptional Hahn and Jacobi orthogonal polynomials

We construct exact solutions and rational extensions to quantum systems of generalized nonlinear oscillators. Our method is based on a connection between nonlinear oscillator systems and Schrödinger models for certain hyperbolic potentials. The rationally extended models admit discrete spectrums and corresponding closed form solutions are expressed through Jacobi type Xm exceptional orthogonal polynomials.

We examine time-dependent Schrodinger equation with oscillating boundary condition. More specifically, we use separation of variable technique to construct time-dependent rationally extended Poschl–Teller potential (whose solutions are given by in terms of $$X_1$$ Jacobi exceptional orthogonal polynomials) and its supersymmetric partner, namely the Poschl–Teller potential. We have obtained exact solutions of the Schrodinger equation with the above-mentioned potentials subjected to some boundary conditions of the oscillating type. A number of physical quantities like the average energy, probability density, expectation values, etc., have also been computed for both the systems and compared with each other.

Wronskian Appell polynomials and symmetric functions

We construct ladder operators, C̃ and C̃†, for a multistep rational extension of the harmonic oscillator on the half plane, x ≥ 0. These ladder operators connect all states of the spectrum in only infinite-dimensional representations of their polynomial Heisenberg algebra. For comparison, we also construct two different classes of ladder operator acting on this system that form finite-dimensional as well as infinite-dimensional representations of their respective polynomial Heisenberg algebras. For the rational extension, we construct the position wavefunctions in terms of exceptional orthogonal polynomials. For a particular choice of parameters and for the three lowest weights μ = −5, −3, and 5, we construct the coherent states, eigenvectors of C̃ with generally complex eigenvalues, z, as superposition of subsets of the energy eigenvectors. Then, we calculate the properties of these coherent states, looking for classical or nonclassical behavior. We calculate the energy expectations as functions of |z|. We plot position probability densities for the coherent states and for the even and odd cat states formed from these coherent states. We plot the Wigner functions for a particular choice of z. For these coherent states on one arm of a beamsplitter, we calculate the two excitation number distributions and the linear entropies of the output states. We plot the standard deviations in x and find squeezing in the regime considered in one of the cases. By plotting the Mandel Q parameters for the coherent states as functions of |z|, we find that the number statistics is sub-Poissonian in all cases.

Rationally extended shape invariant potentials in arbitrary D-dimensions are obtained by using point canonical transformation (PCT) method. The bound-state solutions of these exactly solvable potentials can be written in terms of <em>X<sub>m</sub></em> Laguerre or <em>X<sub>m</sub></em> Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property.

<p>We provide new insights into the solvability property of a Hamiltonian involving the fourth Painlevé transcendent and its derivatives. This Hamiltonian is third-order shape invariant and can also be interpreted within the context of second supersymmetric quantum mechanics. In addition, this Hamiltonian admits third-order lowering and raising operators. We have considered the case when this Hamiltonian is irreducible, i.e., when no special solutions exist for given parameters $ \alpha $ and $ \beta $ of the fourth Painlevé transcendent $ P_{IV}(x, \alpha, \beta) $. This means that the Hamiltonian does not admit a potential in terms of rational functions (or the hypergeometric type of special functions) for those parameters. In such irreducible cases, the ladder operators are as well involving the fourth Painlevé transcendent and its derivative. An important case for which this occurs is when the second parameter (i.e., $ \beta $) of the fourth Painlevé transcendent $ P_{IV}(x, \alpha, \beta) $ is strictly positive, $ \beta > 0 $. This Hamiltonian was studied for all hierarchies of rational solutions that come in three families connected to the generalized Hermite and Okamoto polynomials. The explicit form of ladder, the associated wavefunctions involving exceptional orthogonal polynomials, and recurrence relations were also completed described. Much less is known for the irreducible case, in particular the excited states. Here, we developed a description of the induced representations based on various commutator identities for the highest and lowest weight type representations. We also provided for such representations a new formula concerning the explicit form of the related excited states from the point of view of the Schrödinger equation as two-variables polynomials that involve the fourth Painlevé transcendent and its derivative.</p>

Résumé. In recent years, progress toward the classification of superintegrable systems with higher order integrals of motion has been made. In particular, a complete classification of all exotic potentials with a third or a fourth order integrals, and allowing separation of variables in Cartesian coordinates. All doubly exotic potentials with a fifth order integral have also been completely classified. It has been demonstrated how the Chazy class of third order differential equations plays an important role in solving determining equations. Moreover, taking advantage of various operator algebras defined as Abelian, Heisenberg, Conformal and Ladder case of operator algebras, we re-derived these models. These new techniques also provided further examples of superintegrable Hamiltonian with integrals of arbitrary order. It has been conjectured that all quantum superintegrable potentials that do not satisfy any linear equation satisfy nonlinear equations having the Painlevé property. In addition, it has been discovered that their integrals naturally generate finitely generated polynomial algebras and the representations can be exploited to calculate the energy spectrum. For certain very interesting cases associated with exceptional orthogonal polynomials, these algebraic structures do not allow to calculate the full spectrum and degeneracies. It has been demonstrated that alternative sets of integrals which can be build and used to provide a complete solution. This this allow to make another conjecture i.e. that higher order superintegrable systems can be solved algebraically, they require alternative set of integrals than the one provided by a direct approach.

Several orthogonal polynomials have limit forms in which Hermite polynomials show up. Examples are limits with respect to certain parameters of the Jacobi and Laguerre polynomials. In this paper we are interested in more details of these limits and we give asymptotic representations of several orthogonal polynomials in terms of Hermite polynomials. In fact we give finite exact representations that have an asymptotic character. From these representations the well-known limits can be derived easily. Approximations of the zeros of the Gegenbauer polynomials $C_n^{gamma(x)$ and Laguerre polynomials $L_n^{alpha(x)$ are derived (for large values of $gamma$ and $alpha$, respectively) in terms of zeros of the Hermite polynomials and compared with numerical values. We also consider the Jacobi polynomials and the so-called Tricomi-Carlitz polynomials.

A previous construction of regular rational extensions of the trigonometric Darboux-Pöschl-Teller potential, obtained by one-step Darboux transformations using seed functions associated with the para-Jacobi polynomials of Calogero and Yi, is generalized by considering m-step Darboux transformations. As a result, some novel families of exceptional orthogonal polynomials depending on m discrete parameters, as well as m continuous real ones λ1, λ2, …, λm, are obtained. The restrictions imposed on these parameters by the rational extensions regularity conditions are studied in detail.

Constrained orthogonal polynomials have been recently introduced in the study of the Hohenberg–Kohn functional to provide basis functions satisfying particle number conservation for an expansion of the particle density. More generally, we define block orthogonal (BO) polynomials which are orthogonal, with respect to a first Euclidean inner product, to a given i-dimensional subspace of polynomials associated with the constraints. In addition, they are mutually orthogonal with respect to a second Euclidean inner product. We recast the determination of these polynomials into a general problem of finding particular orthogonal bases in an Euclidean vector space endowed with distinct inner products. An explicit two step Gram–Schmidt orthogonalization (G-SO) process to determine these bases is given. By definition, the standard block orthogonal (SBO) polynomials are associated with a choice of equal to the subspace of polynomials of degree less than i. We investigate their properties, emphasizing similarities to and differences from the standard orthogonal polynomials. Applications to classical orthogonal polynomials will be given in forthcoming papers.