arXiv subjects

David Gomez-Ullate

Publications and source records attributed to David Gomez-Ullate.

At least 19 recordsLinked to original sources

arXiv2025

Classification of exceptional Jacobi polynomials

We provide a full classification scheme for exceptional Jacobi operators and polynomials. The classification contains six degeneracy classes according to whether $α,β$ or $α\pmβ$ assume integer values. Exceptional Jacobi operators are in one-to-one correspondence with spectral diagrams, a combinatorial object that describes the number and asymptotic behaviour at the endpoints of $(-1,1)$ of all quasi-rational eigenfunctions of the operator. With a convenient indexing scheme for spectral diagrams, explicit Wronskian and integral construction formulas are given to build the exceptional operators and polynomials from the information encoded in the spectral diagram. In the fully degenerate class $α,β\in\mathbb N_0$ there exist exceptional Jacobi operators with an arbitrary number of continuous parameters. The classification result is achieved by a careful description of all possible rational Darboux transformations that can be performed on exceptional Jacobi operators.

math.CA↗

arXiv2020

Spectral Theory of Exceptional Hermite Polynomials

In this paper we revisit exceptional Hermite polynomials from the point of view of spectral theory, following the work initiated by Lance Littlejohn. Adapting a result of Deift, we provide an alternative proof of the completeness of these polynomial families. In addition, using equivalence of Hermite Wronskians we characterize the possible gap sets for the class of exceptional Hermite polynomials.

math.CA↗

arXiv2020

Rational solutions of Painleve systems

Although the solutions of Painlevé equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions, there do exist rational solutions for specialized values of the equation parameters. A very successful approach in the study of rational solutions to Painlevé equations involves the reformulation of these scalar equations into a symmetric system of coupled, Riccati-like equations known as dressing chains. Periodic dressing chains are known to be equivalent to the $A_N$-Painlevé system, first described by Noumi and Yamada. The Noumi-Yamada system, in turn, can be linearized as using bilinear equations and $τ$-functions; the corresponding rational solutions can then be given as specializations of rational solutions of the KP hierarchy. The classification of rational solutions to Painlevé equations and systems may now be reduced to an analysis of combinatorial objects known as Maya diagrams. The upshot of this analysis is a an explicit determinental representation for rational solutions in terms of classical orthogonal polynomials. In this paper we illustrate this approach by describing Hermite-type rational solutions of Painlevé of the Noumi-Yamada system in terms of cyclic Maya diagrams. By way of example we explicitly construct Hermite-type solutions for the PIV, PV equations and the $A_4$ Painlevé system.

math-ph↗

arXiv2019

Corrigendum on the proof of completeness for exceptional Hermite polynomials

Exceptional orthogonal polynomials are complete families of orthogonal polynomials that arise as eigenfunctions of a Sturm-Liouville problem. Antonio Durán discovered a gap in the original proof of completeness for exceptional Hermite polynomials, that has propagated to analogous results for other exceptional families. In this paper we provide an alternative proof that follows essentially the same arguments, but provides a direct proof of the key lemma on which the completeness proof is based. This direct proof makes use of the theory of trivial monodromy potentials developed by Duistermaat and Grünbaum and Oblomkov.

math.CA↗

arXiv2019

Ladder operators and rational extensions

This note presents the classification of ladder operators corresponding to the class of rational extensions of the harmonic oscillator. We show that it is natural to endow the class of rational extensions and the corresponding intertwining operators with the structure of a category REXT. The combinatorial data for this interpretation is realized as a functor MD $\to$ REXT, where MD refers to the set of Maya diagrams appropriately endowed with categorical structure. Our formalism allows us to easily reproduce and extend earlier results on ladder operators.

math-ph↗

arXiv2018

Shape invariance and equivalence relations for pseudowronskians of Laguerre and Jacobi polynomials

In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be interpreted as the general transcription of shape invariance and specific discrete symmetries acting on the parameters of the isotonic oscillator and Darboux-Poschl-Teller potential.

math.CA↗

arXiv2017

Durfee rectangles and pseudo-Wronskian equivalences for Hermite polynomials

We study an equivalence class of iterated rational Darboux transformations applied on the harmonic oscillator, showing that many choices of state adding and state deleting transformations lead to the same transformed potential. As a by-product, we derive new identities between determinants whose entries are Hermite polynomials. These identities have a combinatorial interpretation in terms of Maya diagrams, partitions and Durfee rectangles, and serve to characterize the equivalence class of rational Darboux transformations. Since the determinants have different orders, we analyze the problem of finding the minimal order determinant in each equivalence class, or equivalently, the minimum number of Darboux transformations. The solution to this problem has an elegan graphical interpretation. The results are applied to provide alternative and more efficient representations for exceptional Hermite polynomials and rational solutions of the Painlevé IV equation.

math.CA↗

arXiv2013

Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials

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 harmonic oscillator. Equivalently, every exceptional orthogonal polynomial system of Hermite type can be obtained by applying a Darboux-Crum transformation to the classical Hermite polynomials. Exceptional Hermite polynomial systems only exist for even codimension 2m, and they are indexed by the partitions λof m. We provide explicit expressions for their corresponding orthogonality weights and differential operators and a separate proof of their completeness. Exceptional Hermite polynomials satisfy a 2l+3 recurrence relation where l is the length of the partition λ. Explicit expressions for such recurrence relations are given.

math-ph↗

arXiv2012

On orthogonal polynomials spanning a non-standard flag

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.

math-ph↗

arXiv2012

A conjecture on Exceptional Orthogonal Polynomials

Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials. We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux-Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPS. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials.

math-ph↗

arXiv2011

Two-step Darboux transformations and exceptional Laguerre polynomials

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 exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformations. The construction is illustrated with an example of a 2-step Darboux transformation of the classical Laguerre polynomials, which gives rise to a new orthogonal polynomial system indexed by two integer parameters. For particular values of these parameters, the classical Laguerre and the type II $X_\ell$-Laguerre polynomials are recovered.

math-ph↗

arXiv2011

Understanding complex dynamics by means of an associated Riemann surface

We provide an example of how the complex dynamics of a recently introduced model can be understood via a detailed analysis of its associated Riemann surface. Thanks to this geometric description an explicit formula for the period of the orbits can be derived, which is shown to depend on the initial data and the continued fraction expansion of a simple ratio of the coupling constants of the problem. For rational values of this ratio and generic values of the initial data, all orbits are periodic and the system is isochronous. For irrational values of the ratio, there exist periodic and quasi-periodic orbits for different initial data. Moreover, the dependence of the period on the initial data shows a rich behavior and initial data can always be found such the period is arbitrarily high.

nlin.CD↗

arXiv2010

Exceptional orthogonal polynomials and the Darboux transformation

We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described $X_m$ Laguerre polynomials in terms of an isospectral Darboux transformation. We also show that the shape-invariance of these new polynomial families is a direct consequence of the permutability property of the Darboux-Crum transformation.

math-ph↗

arXiv2009

An Extension of Bochner's Problem: Exceptional Invariant Subspaces

A classical result due to Bochner characterizes the classical orthogonal polynomial systems as solutions of a second-order eigenvalue equation. We extend Bochner's result by dropping the assumption that the first element of the orthogonal polynomial sequence be a constant. This approach gives rise to new families of complete orthogonal polynomial systems that arise as solutions of second-order eigenvalue equations with rational coefficients. The results are based on a classification of exceptional polynomial subspaces of codimension one under projective transformations.

math-ph↗

arXiv2009

An extended class of orthogonal polynomials defined by a Sturm-Liouville problem

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 $X_1$-Jacobi and $X_1$-Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the the compact interval $[-1,1]$ or the half-line $[0,\infty)$, respectively, and they are a basis of the corresponding $L^2$ Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second order operator has a complete set of polynomial eigenfunctions $\{p_i\}_{i=1}^\infty$, then it must be either the $X_1$-Jacobi or the $X_1$-Laguerre Sturm-Liouville problem. A Rodrigues-type formula can be derived for both of the $X_1$ polynomial sequences.

math-ph↗

arXiv2007

Asymptotically isochronous systems

Mechanisms are elucidated underlying the existence of dynamical systems whose generic solutions approach asymptotically (at large time) isochronous evolutions: all their dependent variables tend asymptotically to functions periodic with the same fixed period. We focus on two such mechanisms, emphasizing their generality and illustrating each of them via a representative example. The first example belongs to a recently discovered class of integrable indeed solvable many-body problems. The second example consists of a broad class of (generally nonintegrable) models obtained by deforming appropriately the well-known (integrable and isochronous) many-body problem with inverse-cube two-body forces and a one-body linear ("harmonic oscillator") force.

nlin.SI↗

arXiv2007

Two novel classes of solvable many-body problems of goldfish type with constraints

Two novel classes of many-body models with nonlinear interactions "of goldfish type" are introduced. They are solvable provided the initial data satisfy a single constraint (in one case; in the other, two constraints): i. e., for such initial data the solution of their initial-value problem can be achieved via algebraic operations, such as finding the eigenvalues of given matrices or equivalently the zeros of known polynomials. Entirely isochronous versions of some of these models are also exhibited: i.e., versions of these models whose nonsingular solutions are all completely periodic with the same period.

nlin.SI↗

arXiv2006

Quasi-exact solvability in a general polynomial setting

Our goal in this paper is to extend the theory of quasi-exactly solvable Schrodinger operators beyond the Lie-algebraic class. Let $\cP_n$ be the space of n-th degree polynomials in one variable. We first analyze "exceptional polynomial subspaces" which are those proper subspaces of $\cP_n$ invariant under second order differential operators which do not preserve $\cP_n$. We characterize the only possible exceptional subspaces of codimension one and we describe the space of second order differential operators that leave these subspaces invariant. We then use equivalence under changes of variable and gauge transformations to achieve a complete classification of these new, non-Lie algebraic Schrodinger operators. As an example, we discuss a finite gap elliptic potential which does not belong to the Treibich-Verdier class.

nlin.SI↗