Exceptional Orthogonal Polynomials Research Articles

Connecting exceptional orthogonal polynomials of different kind

The known asymptotic relations interconnecting Jacobi, Laguerre, and Hermite classical orthogonal polynomials are generalized to the corresponding exceptional orthogonal polynomials of codimension m. It is proved that Xm-Laguerre exceptional orthogonal polynomials of type I, II, or III can be obtained as limits of Xm-Jacobi exceptional orthogonal polynomials of the same type. Similarly, Xm-Hermite exceptional orthogonal polynomials of type III can be derived from Xm-Jacobi or Xm-Laguerre ones. The quadratic transformations expressing Hermite classical orthogonal polynomials in terms of Laguerre ones is also extended to even X2m-Hermite exceptional orthogonal polynomials.

Representations of quadratic Heisenberg-Weyl algebras and polynomials in the fourth Painlevé transcendent

<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>

One continuous parameter family of Dirac–Lorentz scalar potentials associated with exceptional orthogonal polynomials

We extend our recent works [Int. J. Mod. Phys. A 38, 2350069 (2023)] and obtain one-parameter [Formula: see text] family of rationally extended Dirac–Lorentz scalar potentials with their explicit solutions in terms of [Formula: see text] exceptional orthogonal polynomials. We further show that as the parameter [Formula: see text] or [Formula: see text], we get the corresponding rationally extended Pursey and the rationally extended Abraham–Moses-type of scalar potentials, respectively, which have one bound state less than the starting scalar potentials.

Rational extensions of an oscillator-shaped quantum well potential in a position-dependent mass background

We show that a recently proposed oscillator-shaped quantum well model associated with a position-dependent mass can be solved by applying a point canonical transformation to the constant-mass Schrödinger equation for the Scarf I potential. On using the known rational extension of the latter connected with X 1-Jacobi exceptional orthogonal polynomials, we build a rationally-extended position-dependent mass model with the same spectrum as the starting one. Some more involved position-dependent mass models associated with X 2-Jacobi exceptional orthogonal polynomials are also considered.

Time and band limiting for exceptional polynomials

The “time-and-band limiting” commutative property was found and exploited by D. Slepian, H. Landau and H. Pollak at Bell Labs in the 1960's, and independently by M. Mehta and later by C. Tracy and H. Widom in Random matrix theory. The property in question is the existence of local operators with simple spectrum that commute with naturally appearing global ones.Here we give a general result that insures the existence of a commuting differential operator for a given family of exceptional orthogonal polynomials satisfying the “bispectral property”. As a main tool we go beyond bispectrality and make use of the notion of Fourier Algebras associated to the given sequence of exceptional polynomials. We illustrate this result with two examples, of Hermite and Laguerre type, exhibiting also a nice Perline's form for the commuting differential operator.

DEK-type orthogonal polynomials and a modification of the Christoffel formula

In this note we revisit one of the first known examples of exceptional orthogonal polynomials that was introduced by Dubov, Eleonskii, and Kulagin in relation to nonharmonic oscillators with equidistant spectra. We dissect the DEK polynomials using the discrete Darboux transformations and unravel a characterization bypassing the differential equation that defines the DEK polynomials. This characterization also leads to a family of general orthogonal polynomials with missing degrees and this approach manifests its relation to biorthogonal polynomials introduced by Iserles and Nørsett, which are applicable to a whole range of problems in computational and applied analysis. We also obtain a modification of the Christoffel formula for this family since its classical form cannot be applied in this case.

Rational extensions of the Dunkl oscillator in the plane and exceptional orthogonal polynomials

It is shown that rational extensions of the isotropic Dunkl oscillator in the plane can be obtained by adding some terms either to the radial equation or to the angular one obtained in the polar coordinates approach. In the former case, the isotropic harmonic oscillator is replaced by an isotropic anharmonic one, whose wave functions are expressed in terms of [Formula: see text]-Laguerre exceptional orthogonal polynomials. In the latter, it becomes an anisotropic potential, whose explicit form has been found in the simplest case associated with [Formula: see text]-Jacobi exceptional orthogonal polynomials.

Rationally-extended Dunkl oscillator on the line

It is shown that the extensions of exactly-solvable quantum mechanical problems connected with the replacement of ordinary derivatives by Dunkl ones and with that of classical orthogonal polynomials by exceptional orthogonal ones can be easily combined. For such a purpose, the example of the Dunkl oscillator on the line is considered and three different types of rationally-extended Dunkl oscillators are constructed. The corresponding wavefunctions are expressed in terms of exceptional orthogonal generalized Hermite polynomials, defined in terms of the three different types of X m -Laguerre exceptional orthogonal polynomials. Furthermore, the extended Dunkl oscillator Hamiltonians are shown to be expressible in terms of some extended Dunkl derivatives and some anharmonic oscillator potentials.

Solutions of one-dimensional Dirac equation associated with exceptional orthogonal polynomials and the parametric symmetry

We consider one-dimensional Dirac equation with rationally extended scalar potentials corresponding to the radial oscillator, the trigonometric Scarf and the hyperbolic Pöschl–Teller potentials and obtain their solution in terms of exceptional orthogonal polynomials. Further, in the case of the trigonometric Scarf and the hyperbolic Pöschl–Teller cases, a new family of Dirac scalar potentials is generated using the idea of parametric symmetry and their solutions are obtained in terms of conventional as well as exceptional orthogonal polynomials.

Convergence Rates of Exceptional Zeros of Exceptional Orthogonal Polynomials

We consider the zeros of exceptional orthogonal polynomials (XOP). Exceptional orthogonal polynomials were originally discovered as eigenfunctions of second order differential operators that exist outside the classical Bochner–Brenke classification due to the fact that XOP sequences omit polynomials of certain degrees. This omission causes several properties of the classical orthogonal polynomial sequences to not extend to the XOP sequences. One such property is the restriction of the zeros to the convex hull of the support of the measure of orthogonality. In the XOP case, the zeros that exist outside the classical intervals are called exceptional zeros and they often converge toward easily identifiable limit points as the degree becomes large. We deduce the exact rate of convergence and verify that certain estimates that previously appeared in the literature are sharp.

Supersymmetry and shape invariance of exceptional orthogonal polynomials

We discuss the exceptional Laguerre and the exceptional Jacobi orthogonal polynomials in the framework of the supersymmetric quantum mechanics (SUSYQM). We express the differential equations for the Jacobi and the Laguerre exceptional orthogonal polynomials (EOP) as the eigenvalue equations and make an analogy with the time independent Schrödinger equation to define “Hamiltonians” enables us to study the EOPs in the framework of the SUSYQM and to realize the underlying shape invariance associated with such systems. We show that the underlying shape invariance symmetry is responsible for the solubility of the differential equations associated with these polynomials.

Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators

We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining higher order ladder operators. One feature of these algebras is that they preserve by construction some aspects of the structure of the gl(n) Lie algebra. Among the classes of Hamiltonians arising in this framework are various deformations of harmonic oscillator and singular oscillator related to exceptional orthogonal polynomials and even Painlevé and higher order Painlevé analogs. As an explicit example, we investigate a new three-dimensional superintegrable system related to Hermite exceptional orthogonal polynomials of type III. Among the main results is the determination of the degeneracies of the model in terms of the finite-dimensional irreducible representations of the polynomial algebra.

Exceptional Gegenbauer polynomials via isospectral deformation

AbstractIn this paper, we show how to construct exceptional orthogonal polynomials (XOP) using isospectral deformations of classical orthogonal polynomials. The construction is based on confluent Darboux transformations, where repeated factorizations at the same eigenvalue are allowed. These factorizations allow us to construct Sturm–Liouville problems with polynomial eigenfunctions that have an arbitrary number of real‐valued parameters. We illustrate this new construction by exhibiting the class of deformed Gegenbauer polynomials, which are XOP families that are isospectral deformations of classical Gegenbauer polynomials.

Rational extension of many particle systems

In this talk, we briefly review the rational extension of many particle systems, and is based on a couple of our recent works. In the first model, the rational extension of the truncated Calogero-Sutherland (TCS) model is discussed analytically. The spectrum is isospectral to the original system and the eigenfunctions are completely expressed in terms of exceptional orthogonal polynomials (EOPs). In the second model, we discuss the rational extension of a quasi exactly solvable (QES) N-particle Calogero model with harmonic confining interaction. New long-range interaction to the rational Calogero model is included to construct this QES many particle system using the technique of supersymmetric quantum mechanics (SUSYQM). Under a specific condition, infinite number of bound states are obtained for this system, and corresponding bound state wave functions are written in terms of EOPs.

Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials

We extend and generalize the construction of Sturm–Liouville problems for a family of Hamiltonians constrained to fulfill a third-order shape-invariance condition and focusing on the ‘−2x/3’ hierarchy of solutions to the fourth Painlevé transcendent. Such a construction has been previously addressed in the literature for some particular cases but we realize it here in the most general case. The corresponding potential in the Hamiltonian operator is a rationally extended oscillator defined in terms of the conventional Okamoto polynomials, from which we identify three different zero-modes constructed in terms of the generalized Okamoto polynomials. The third-order ladder operators of the system reveal that the complete set of eigenfunctions is decomposed as a union of three disjoint sequences of solutions, generated from a set of three-term recurrence relations. We also identify a link between the eigenfunctions of the Hamiltonian operator and a special family of exceptional Hermite polynomial.

A New Property of Exceptional Orthogonal Polynomials

A New Property of Exceptional Orthogonal Polynomials

Supersymmetry and Shape Invariance of Exceptional Orthogonal Polynomials

Supersymmetry and Shape Invariance of Exceptional Orthogonal Polynomials

One parameter family of rationally extended isospectral potentials

We start from a given one dimensional rationally extended shape invariant potential associated with Xm exceptional orthogonal polynomials and using the idea of supersymmetry in quantum mechanics, we obtain one continuous parameter (λ) family of rationally extended strictly isospectral potentials. We illustrate this construction by considering three well known rationally extended potentials, two with pure discrete spectrum (the extended radial oscillator and the extended Scarf-I) and one with both the discrete and the continuous spectrum (the extended generalized Pöschl–Teller) and explicitly construct the corresponding one continuous parameter family of rationally extended strictly isospectral potentials. Further, in the special case of λ=0 and −1, we obtain two new exactly solvable rationally extended potentials, namely the rationally extended Pursey and the rationally extended Abraham–Moses potentials respectively. We illustrate the whole procedure by discussing in detail the particular case of the X1 rationally extended one parameter family of potentials including the corresponding Pursey and the Abraham Moses potentials.

Spin and pseudospin symmetries in radial Dirac equation and exceptional hermite polynomials

We have generalized the solutions of the radial Dirac equation with a tensor potential under spin and pseudospin symmetry limits to exceptional orthogonal Hermite polynomials family. We have obtained new general rational potential models which are the generalization of the nonlinear isotonic potential families and also energy dependent.

Exceptional Legendre Polynomials and Confluent Darboux Transformations

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of ''exceptional'' degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question. The main novelty of the paper is the novel construction that allows for exceptional polynomial families with an arbitrary number of real parameters.