arXiv subjects

Sarah Post

Publications and source records attributed to Sarah Post.

At least 19 recordsLinked to original sources

arXiv2026

Painlevé VI superintegrability and Heun-Racah algebras

We study the two fourth-order quantum superintegrable systems in the Euclidean plane that separate in polar coordinates and whose angular potentials are determined by the general four-parameter sixth Painlevé transcendent. One system has an oscillator radial term and the other an extended-Coulomb radial term. For both systems we give the fourth-order quantum integral explicitly, derive the cubic symmetry algebra and its Casimir, and construct a deformed-oscillator realization. At fixed energy the two algebras are specializations of the Heun--Racah algebra. Their structure functions contain the same quartic angular factor, determined by the Painlevé~VI parameters, while different radial factors distinguish oscillator and Coulomb dynamics. The Coulomb integral is obtained by transforming the Cartesian Hermitian symmetry at the operator level. The lower-order terms generated by the polar coordinate change are essential for commutation with the Hamiltonian. The two quantum systems are also related by the Levi-Civita transformation and coupling-constant metamorphosis, which gives a direct correspondence between their Painlevé parameters, algebras, Casimirs and representations. Earlier exceptional-Jacobi models arise at special rational parameter values, whereas the results here apply to the full four-parameter Painlevé~VI family. We also compare the quantum results with the corresponding classical Poisson algebras.

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arXiv2026

Quasi-exactly solvable deformations of quantum systems associated with exceptional orthogonal polynomials

Quasi-exactly solvable (QES) deformations of harmonic and singular oscillators have been widely studied via a range of approaches. In this paper, we obtain families of new QES deformations of solvable quantum systems associated with exceptional orthogonal polynomials (EOPs). The construction builds upon the theory of Darboux-Crum transformations and the classification of exactly solvable quantum systems associated with EOPs of Hermite type. It is shown that the deformations break the exact solvability of the undeformed systems and introduce model parameters into the deformed systems that permit the existence of a finite number of polynomial solutions whose roots satisfy systems of algebraic equations. The new families presented and studied consist of deformations of quantum systems related to Hermite EOPs of type III with arbitrary codimensions. We present polynomial and rational deformations and analyze, in each case, the conditions for quasi-exact solvability in terms of Bethe ansatz equations and parameter constraints. In general, the structures of the underlying polynomial solutions are no longer associated with well-known classical orthogonal polynomials. So for the general cases, we mainly focus on presenting the new approach as well as the Bethe ansatz equations and the constraints for model parameters. As applications, we construct particular families of QES deformations related to solvable models allowing one, two, and up to four gaps, and obtain the closed form expressions for their wavefunctions and spectra. We analyze the existence of QES solutions in the spaces of model parameters, providing information on the number of solutions for given parameters.

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arXiv2026

Realization embeddings of the rank two Racah algebra into the rank two Jacobi algebra

We construct an explicit embedding of the rank two Racah algebra $\mathfrak{R}_2$ into the rank two Jacobi algebra $\mathfrak{J}_2$. Working in the differential model of $\mathfrak{J}_2$ on the triangle, we recall that the subalgebra structure of $\mathfrak{J}_2$ is organized by a pentagon whose four upper edges carry rank one Jacobi algebras and whose base carries a rank one Racah algebra. Promoting the two multiplication operators to Racah type generators by tridiagonalization turns every edge of the pentagon into a rank one Racah algebra and realizes $\mathfrak{R}_2$ inside $\mathfrak{J}_2$. We determine the common eigenfunctions of different commuting pairs explicitly as products of Gauss hypergeometric functions and Jacobi polynomials. Different overlap coefficients are computed involving univariate Wilson polynomials as well as, for one pair of bases, bivariate Racah polynomials of Tratnik type. We also recall how $\mathfrak{J}_2$ arises as a contraction of the rank two Racah algebra $\mathfrak{R}_2$.

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arXiv2024

The Racah Algebra of Rank 2: Properties, Symmetries and Representation

The goals of this paper are threefold. First, we provide a new ''universal'' definition for the Racah algebra of rank 2 as an extension of the rank-1 Racah algebra where the generators are indexed by subsets and any three disjoint indexing sets define a subalgebra isomorphic to the rank-1 case. With this definition, we explore some of the properties of the algebra including verifying that these natural assumptions are equivalent to other defining relations in the literature. Second, we look at the symmetries of the generators of the rank-2 Racah algebra. Those symmetries allows us to partially make abstraction of the choice of the generators and write relations and properties in a different format. Last, we provide a novel representation of the Racah algebra. This new representation requires only one generator to be diagonal and is based on an expansion of the split basis representation from the rank-1 Racah algebra.

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arXiv2023

Special function models of indecomposable $sl(2)$ representations: The Laguerre Case

In this paper, we point out connections between certain types of indecomposable representations of $sl(2)$ and generalizations of well-known orthogonal polynomials. Those representations take the form of infinite dimensional chains of weight or generalised weight spaces, for which the Cartan generator acts in a diagonal way or via Jordan blocks. The other generators of the Lie algebras $sl(2)$ act as raising and lowering operators but are now allowed to relate the different chains as well. In addition, we construct generating functions, we calculate the action of the Casimir invariant and present relations to systems of non-homogeneous second-order coupled differential equations. We present different properties as higher-order linear differential equations for building blocks taking the form of one variable polynomials. We also present insight into the zeroes and recurrence relations.

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arXiv2020

A fourth-order superintegrable system with a rational potential related to Painleve VI

In this paper, we investigate in detail a superintegrable extension of the singular harmonic oscillator whose wave functions can be expressed in terms of exceptional Jacobi polynomials. We show that this Hamiltonian admits a fourth-order integral of motion and use the classification of such systems to show that the potential gives a rational solution associated with the sixth Painlevé equation. Additionally, we show that the integrals of the motion close to form a cubic algebra and describe briefly deformed oscillator representations of this algebra.

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arXiv2020

An Infinite-Dimensional $\square_q$-Module Obtained from the $q$-Shuffle Algebra for Affine $\mathfrak{sl}_2$

Let $\mathbb F$ denote a field, and pick a nonzero $q \in \mathbb F$ that is not a root of unity. Let $\mathbb Z_4=\mathbb Z/4 \mathbb Z$ denote the cyclic group of order 4. Define a unital associative ${\mathbb F}$-algebra $\square_q$ by generators $\lbrace x_i \rbrace_{i \in \mathbb Z_4}$ and relations $$\frac{q x_i x_{i+1}-q^{-1}x_{i+1}x_i}{q-q^{-1}} = 1,\qquad x^3_i x_{i+2} - \lbrack 3 \rbrack_q x^2_i x_{i+2} x_i + \lbrack 3 \rbrack_q x_i x_{i+2} x^2_i -x_{i+2} x^3_i = 0,$$ where $\lbrack 3 \rbrack_q = \big(q^3-q^{-3}\big)/\big(q-q^{-1}\big)$. Let $V$ denote a $\square_q$-module. A vector $ξ\in V$ is called NIL whenever $x_1 ξ= 0 $ and $x_3 ξ=0$ and $ξ\not=0$. The $\square_q$-module $V$ is called NIL whenever $V$ is generated by a NIL vector. We show that up to isomorphism there exists a unique NIL $\square_q$-module, and it is irreducible and infinite-dimensional. We describe this module from sixteen points of view. In this description an important role is played by the $q$-shuffle algebra for affine $\mathfrak{sl}_2$.

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arXiv2017

A higher rank extension of the Askey-Wilson Algebra

A novel generalization of the Askey-Wilson algebra is presented and shown to be associated with coproducts in the quantum algebra $U_q(su(1,1))$. This algebra has 15 non-commuting generators given by $Q^{(A)}$, with $A\subset \{1,2,3,4\}$ and their 5 linearly independent inversions generated by the algebra automorphism $q\rightarrow q^{-1}$, $E\leftrightarrow F.$ The set of generators can be split into operators fixed under inversion, and those with an orientation under this inversion. We then show that the generators will either commute or satisfy q-commutator relations linear in the generators, with the restriction that parity operators commute with generators of opposite parity and the q-commutation relations are between those with the same parity. Finally, we give a novel algebra expression satisfied by the generators involving only the natural generators, i.e. those arising from the coupling scheme.

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arXiv2017

Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials

We introduce an extended Kepler-Coulomb quantum model in spherical coordinates. The Schrödinger equation of this Hamiltonian is solved in these coordinates and it is shown that the wave functions of the system can be expressed in terms of Laguerre, Legendre and exceptional Jacobi polynomials (of hypergeometric type). We construct ladder and shift operators based on the corresponding wave functions and obtain their recurrence formulas. These recurrence relations are used to construct higher-order, algebraically independent integrals of motion to prove superintegrability of the Hamiltonian. The integrals form a higher rank polynomial algebra. By constructing the structure functions of the associated deformed oscillator algebras we derive the degeneracy of energy spectrum of the superintegrable system.

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arXiv2015

An algebraic interpretation of the multivariate $q$-Krawtchouk polynomials

The multivariate quantum $q$-Krawtchouk polynomials are shown to arise as matrix elements of "$q$-rotations" acting on the state vectors of many $q$-oscillators. The focus is put on the two-variable case. The algebraic interpretation is used to derive the main properties of the polynomials: orthogonality, duality, structure relations, difference equations and recurrence relations. The extension to an arbitrary number of variables is presented

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arXiv2015

Racah Polynomials and Recoupling Schemes of $\mathfrak{su}(1,1)$

The connection between the recoupling scheme of four copies of $\mathfrak{su}(1,1)$, the generic superintegrable system on the 3 sphere, and bivariate Racah polynomials is identified. The Racah polynomials are presented as connection coefficients between eigenfunctions separated in different spherical coordinate systems and equivalently as different irreducible decompositions of the tensor product representations. As a consequence of the model, an extension of the quadratic algebra ${\rm QR}(3)$ is given. It is shown that this algebra closes only with the inclusion of an additional shift operator, beyond the eigenvalue operators for the bivariate Racah polynomials, whose polynomial eigenfunctions are determined. The duality between the variables and the degrees, and hence the bispectrality of the polynomials, is interpreted in terms of expansion coefficients of the separated solutions.

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arXiv2015

Invariant Classification and Limits of Maximally Superintegrable Systems in 3D

The invariant classification of superintegrable systems is reviewed and utilized to construct singular limits between the systems. It is shown, by construction, that all superintegrable systems on conformally flat, 3D complex Riemannian manifolds can be obtained from singular limits of a generic system on the sphere. By using the invariant classification, the limits are geometrically motivated in terms of transformations of roots of the classifying polynomials.

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arXiv2014

$q$-Rotations and Krawtchouk polynomials

An algebraic interpretation of the one-variable quantum $q$-Krawtchouk polynomials is provided in the framework of the Schwinger realization of $\mathcal{U}_{q}(sl_{2})$ involving two independent $q$-oscillators. The polynomials are shown to arise as matrix elements of unitary "$q$-rotation" operators expressed as $q$-exponentials in the $\mathcal{U}_{q}(sl_{2})$ generators. The properties of the polynomials (orthogonality relation, generating function, structure relations, recurrence relation, difference equation) are derived by exploiting the algebraic setting. The results are extended to another family of polynomials, the affine $q$-Krawtchouk polynomials, through a duality relation.

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arXiv2014

Quantum Integrals from Coalgebra Structure

Quantum versions of the hydrogen atom and the harmonic oscillator are studied on non Euclidean spaces of dimension N. 2N-1 integrals, of arbitrary order, are constructed via a multi-dimensional version of the factorization method, thus confirming the conjecture of D Riglioni 2013 J. Phys. A: Math. Theor. 46 265207. The systems are extended via coalgebra extension of sl(2) representations, although not all integrals are expressible in these generators. As an example, dimensional reduction is applied to 4D systems to obtain extension and new proofs of the superintegrability of known families of Hamiltonians.

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arXiv2013

Contractions of 2D 2nd Order Quantum Superintegrable Systems and the Askey Scheme for Hypergeometric Orthogonal Polynomials

We show explicitly that all 2nd order superintegrable systems in 2 dimensions are limiting cases of a single system: the generic 3-parameter potential on the 2-sphere, S9 in our listing. We extend the Wigner-Inönü method of Lie algebra contractions to contractions of quadratic algebras and show that all of the quadratic symmetry algebras of these systems are contractions of that of S9. Amazingly, all of the relevant contractions of these superintegrable systems on flat space and the sphere are uniquely induced by the well known Lie algebra contractions of e(2) and so(3). By contracting function space realizations of irreducible representations of the S9 algebra (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems, and using Wigner's idea of "saving" a representation, we obtain the full Askey scheme of hypergeometric orthogonal polynomials. This relationship directly ties the polynomials and their structure equations to physical phenomena. It is more general because it applies to all special functions that arise from these systems via separation of variables, not just those of hypergeometric type, and it extends to higher dimensions.

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arXiv2013

Classical and Quantum Superintegrability with Applications

A superintegrable system is, roughly speaking, a system that allows more integrals of motion than degrees of freedom. This review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta. We present a classification of second-order superintegrable systems in two-dimensional Riemannian and pseudo-Riemannian spaces. It is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis. The determining equations for the existence of integrals of motion of arbitrary order in real Euclidean space $E_2$ are presented and partially solved for the case of third-order integrals. A systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order. The algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally. The relation between superintegrability and the classification of orthogonal polynomials is analyzed.

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arXiv2012

A superintegrable finite oscillator in two dimensions with SU(2) symmetry

A superintegrable finite model of the quantum isotropic oscillator in two dimensions is introduced. It is defined on a uniform lattice of triangular shape. The constants of the motion for the model form an SU(2) symmetry algebra. It is found that the dynamical difference eigenvalue equation can be written in terms of creation and annihilation operators. The wavefunctions of the Hamiltonian are expressed in terms of two known families of bivariate Krawtchouk polynomials; those of Rahman and those of Tratnik. These polynomials form bases for SU(2) irreducible representations. It is further shown that the pair of eigenvalue equations for each of these families are related to each other by an SU(2) automorphism. A finite model of the anisotropic oscillator that has wavefunctions expressed in terms of the same Rahman polynomials is also introduced. In the continuum limit, when the number of grid points goes to infinity, standard two-dimensional harmonic oscillators are obtained. The analysis provides the $N\rightarrow \infty$ limit of the bivariate Krawtchouk polynomials as a product of one-variable Hermite polynomials.

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arXiv2012

Infinite families of superintegrable systems separable in subgroup coordinates

A method is presented that makes it possible to embed a subgroup separable superintegrable system into an infinite family of systems that are integrable and exactly-solvable. It is shown that in two dimensional Euclidean or pseudo-Euclidean spaces the method also preserves superintegrability. Two infinite families of classical and quantum superintegrable systems are obtained in two-dimensional pseudo-Euclidean space whose classical trajectories and quantum eigenfunctions are investigated. In particular, the wave-functions are expressed in terms of Laguerre and generalized Bessel polynomials.

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