arXiv subjects

Ian Marquette

Publications and source records attributed to Ian Marquette.

At least 73 records · Page 4Linked to original sources

arXiv2021

Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures

The recent interest in the study of higher-rank polynomial algebras related to $n$-dimensional classical and quantum superintegrable systems with coalgebra symmetry and their connection with the generalised Racah algebra $R(n)$, a higher-rank generalisation of the rank one Racah algebra $R(3)$, raises the problem of understanding the role played by the $n - 2$ quadratic subalgebras generated by the left and right Casimir invariants (sometimes referred as universal quadratic substructures) from this new perspective. Such subalgebra structures play a signficant role in the algebraic derivation of spectrum of quantum superintegrable systems. In this work, we tackle this problem and show that the above quadratic subalgebra structures can be understood, at a fixed $n > 3$, as the images of $n - 2$ injective morphisms of $R(3)$ into $R(n)$. We show that each of the $n-2$ quadratic subalgebras is isomorphic to the rank one Racah algebra $R(3)$. As a byproduct, we also obtain an equivalent presentation for the universal quadratic subtructures generated by the partial Casimir invariants of the coalgebra. The construction, which relies on explicit (symplectic or differential) realisations of the generators, is performed in both the classical and the quantum cases.

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arXiv2021

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.

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arXiv2020

Embedding of the Racah Algebra R($\boldsymbol{n}$) and Superintegrability

The rank-$1$ Racah algebra $R(3)$ plays a pivotal role in the theory of superintegrable systems. It appears as the symmetry algebra of the $3$-parameter system on the $2$-sphere from which all second-order conformally flat superintegrable models in $2$D can be obtained by means of suitable limits and contractions. A higher rank generalization of $R(3)$, the so-called rank $n-2$ Racah algebra $R(n)$, has been considered recently and showed to be the symmetry algebra of the general superintegrable model on the $(n-1)$-sphere. In the present work, we show that such an algebraic structure naturally arises as embedded inside a larger quadratic algebra characterizing $n$D superintegrable models with non-central terms. This is shown both in classical and quantum mechanics through suitable (symplectic or differential) realisations of the Racah and additional generators. Among the main results, we present an explicit construction of the complete symmetry algebras for two families of $n$-dimensional maximally superintegrable models, the Smorodinsky-Winternitz system and the generalized Kepler-Coulomb system. For both families, the underlying symmetry algebras are higher-rank quadratic algebras containing the Racah algebra $R(n)$ as subalgebra. High-order algebraic relations among the generators of the full quadratic algebras are also obtained both in the classical and quantum frameworks. These results should shed new light to the further understanding of the structures of quadratic algebras in the context of superintegrable systems.

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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

Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems

The notion of hidden symmetry algebra used in the context of exactly solvable systems is re-examined from the purely algebraic way, analyzing subspaces of commuting polynomials that generate finite-dimensional quadratic algebras. By construction, these algebras do not depend on the choice of realizations by vector fields of the underlying Lie algebra, allowing to propose a procedure to analyze polynomial algebras as those subspaces in an enveloping algebra that commute with a given algebraic Hamiltonian.

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arXiv2020

Polynomial algebras from $su(3)$ and the generic model on the two sphere

Construction of superintegrable systems based on Lie algebras have been introduced over the years. However, these approaches depend on explicit realisations, for instance as a differential operators, of the underlying Lie algebra. This is also the case for the construction of their related symmetry algebra which take usually the form of a finitely generated quadratic algebra. These algebras often display structure constants which depend on the central elements and in particular on the Hamiltonian. In this paper, we develop a new approach reexamining the case of the generic superintegrable systems on the 2-sphere for which a symmetry algebra is known to be the Racah algebra $R(3)$. Such a model is related to the 59 $2D$ superintegrable systems on conformally flat spaces and their 12 equivalence classes. We demonstrate that using further polynomials of degree 2,3 and 4 in the enveloping algebra of $su(3)$ one can generate an algebra based only on abstract commutation relations of $su(3)$ Lie algebra without explicit constraints on the representations or realisations. This construction relies on the maximal Abelian subalgebra, also called MASA, which are the Cartan generators and their commutant. We obtain a new 6-dimensional cubic algebra where the structure constant are integer numbers which reduce from a quartic algebra for which the structure constant depend on the Cartan generator and the Casimir invariant. We also present other form of the symmetry algebra using the quadratic and cubic Casimir invariants of $su(3)$. It reduces as the known quadratic Racah algebra $R(3)$ only when using an explicit realization. This algebraic structure describe the symmetry of the generic superintegrable systems on the 2 sphere. We also present a contraction to another 6-dimensional cubic algebra which would corresponding to the symmetry algebra of a Smorodinsky-Winternitz model.

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arXiv2020

A new way to classify 2D higher order quantum superintegrable systems

We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation $HΨ\equiv (Δ_2 +V)Ψ=EΨ$ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. We apply the method, as an example, to revisit the Tremblay and Winternitz (2010) derivation of the Painlevé VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and, as new results, we give a listing of the possible potentials on the 2-sphere that separate in spherical coordinates and 2-hyperbolic (two-sheet) potentials separating in horocyclic coordinates. In particular, we show that the Painlevé VI potential also appears for a 3rd order superintegrable system on the 2-sphere that separates in spherical coordinates, as well as a 3rd order superintegrable system on the 2-hyperboloid that separates in spherical coordinates and one that separates in horocyclic coordinates. Our aim is to develop tools for analysis and classification of higher order superintegrable systems on any 2D Riemannian space, not just Euclidean space.

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arXiv2020

A new approach to analysis of 2D higher order quantum superintegrable systems

We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation $HΨ\equiv (Δ_2 +V)Ψ=EΨ$ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. Most of this paper is devoted to describing the method. Details will be provided elsewhere. As examples we revisit the Tremblay and Winternitz derivation of the Painlevé VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and we show that the Painlevé VI potential also appears for a 3rd order superintegrable system on the 2-sphere that separates in spherical coordinates, as well as a 3rd order superintegrable system on the 2-hyperboloid that separates in spherical coordinates and one that separates in horocyclic coordinates. The purpose of this project is to develop tools for analysis and classification of higher order superintegrable systems on any 2D Riemannian space, not just Euclidean space.

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arXiv2020

Fourth Painlevé and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians

In this work, we introduce a new realization of exactly-solvable time-dependent Hamiltonians based on the solutions of the fourth Painlevé and the Ermakov equations. The latter is achieved by introducing a shape-invariant condition between an unknown quantum invariant and a set of third-order intertwining operators with time-dependent coefficients. The new quantum invariant is constructed by adding a deformation term to the well-known parametric oscillator invariant. Such a deformation depends explicitly on time through the solutions of the Ermakov equation, which ensures the regularity of the new time-dependent potential of the Hamiltonian at each time. On the other hand, with the aid of the proper reparametrization, the fourth Painlevé equation appears, the parameters of which dictate the spectral behavior of the quantum invariant. In particular, the eigenfunctions of the third-order ladder operators lead to several sequences of solutions to the Schrödinger equation, determined in terms of the solutions of a Riccati equation, Okamoto polynomials, or nonlinear bound states of the derivative nonlinear Schrödinger equation. Remarkably, it is noticed that the solutions in terms of the nonlinear bound states lead to a quantum invariant with equidistant eigenvalues, which contains both an (N+1)-dimensional and an infinite sequence of eigenfunctions. The resulting family of time-dependent Hamiltonians is such that, to the authors' knowledge, have been unnoticed in the literature of stationary and nonstationary systems.

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arXiv2019

Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras

We present a new method for constructing $D$-dimensional minimally superintegrable systems based on block coordinate separation of variables. We give two new families of superintegrable systems with $N$ ($N\leq D$) singular terms of the partitioned coordinates and involving arbitrary functions. These Hamiltonians generalize the singular oscillator and Kepler systems. We derive their exact energy spectra via separation of variables. We also obtain the quadratic algebras satisfied by the integrals of motion of these models. We show how the quadratic symmetry algebras can be constructed by novel application of the gauge transformations from those of the non-partitioned cases. We demonstrate that these quadratic algebraic structures display an universal nature to the extent that their forms are independent of the functions in the singular potentials.

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arXiv2019

Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras

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.

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arXiv2019

Generalized conformal pseudo-Galilean algebras and their Casimir operators

A generalization $\mathfrak{Gal}_{\ell}(p,q)$ of the conformal Galilei algebra $\mathfrak{g}_{\ell}(d)$ with Levi subalgebra isomorphic to $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{so}(p,q)$ is introduced and a virtual copy of the latter in the enveloping algebra of the extension is constructed. Explicit expressions for the Casimir operators are obtained from the determinant of polynomial matrices. For the central factor $\overline{\mathfrak{Gal}}_{\ell}(p,q)$, an exact formula giving the number of invariants is obtained and a procedure to compute invariants functions that do not depend on variables of the Levi subalgebra is developed. It is further shown that such solutions determine complete sets of invariants provided that the relation $d\leq 2\ell+2$ is satisfied.

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arXiv2019

Higher Order Quantum Superintegrability: a new "Painlevé conjecture"

We review recent results on superintegrable quantum systems in a two-dimensional Euclidean space with the following properties. They are integrable because they allow the separation of variables in Cartesian coordinates and hence allow a specific integral of motion that is a second order polynomial in the momenta. Moreover, they are superintegrable because they allow an additional integral of order $N>2$. Two types of such superintegrable potentials exist. The first type consists of "standard potentials" that satisfy linear differential equations. The second type consists of "exotic potentials" that satisfy nonlinear equations. For $N= 3$, 4 and 5 these equations have the Painlevé property. We conjecture that this is true for all $N\geq3$. The two integrals X and Y commute with the Hamiltonian, but not with each other. Together they generate a polynomial algebra (for any $N$) of integrals of motion. We show how this algebra can be used to calculate the energy spectrum and the wave functions.

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arXiv2018

Ladder operators and coherent states for multi-step supersymmetric rational extensions of the truncated oscillator

We construct ladder operators, $\tilde{C}$ and $\tilde{C^\dagger}$, for a multi-step rational extension of the harmonic oscillator on the half plane, $x\ge0$. 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, we construct the coherent states, eigenvectors of $\tilde{C}$ with generally complex eigenvalues, $z$, as superpositions of a subset of the energy eigenvectors. Then we calculate the properties of these coherent states, looking for classical or non-classical behaviour. We calculate the energy expectation as a function 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 function for a particular choice of $z$. For these coherent states on one arm of a beamsplitter, we calculate the two excitation number distribution and the linear entropy of the output state. We plot the standard deviations in $x$ and $p$ and find no squeezing in the regime considered. By plotting the Mandel $Q$ parameter for the coherent states as a function of $|z|$, we find that the number statistics is sub-Poissonian.

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arXiv2018

Two-dimensional superintegrable systems from operator algebras in one dimension

We develop new constructions of 2D classical and quantum superintegrable Hamiltonians allowing separation of variables in Cartesian coordinates. In classical mechanics we start from two functions on a one-dimensional phase space, a natural Hamiltonian $H$ and a polynomial of order $N$ in the momentum $p.$ We assume that their Poisson commutator $\{H,K\}$ vanishes, is a constant, a constant times $H$, or a constant times $K$. In the quantum case $H$ and $K$ are operators and their Lie commutator has one of the above properties. We use two copies of such $(H,K)$ pairs to generate two-dimensional superintegrable systems in the Euclidean space $E_2$, allowing the separation of variables in Cartesian coordinates. All known separable superintegrable systems in $E_2$ can be obtained in this manner and we obtain new ones for $N=4.$

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arXiv2018

Extended Laplace-Runge-Lentz vectors, new family of superintegrable systems and quadratic algebras

We present a useful proposition for discovering extended Laplace-Runge-Lentz vectors of certain quantum mechanical systems. We propose a new family of superintegrable systems and construct their integrals of motion. We solve these systems via separation of variables in spherical coordinates and obtain their exact energy eigenvalues and the corresponding eigenfunctions. We give the quadratic algebra relations satisfied by the integrals of motion. Remarkably these algebra relations involve the Casimir operators of certain higher rank Lie algebras in the structure constants.

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arXiv2018

On Casimir Operators of Conformal Galilei Algebras

In previous work, we introduced an algorithm that utilises differential operator realisations to find polynomial Casimir operators of Lie algebras. In this article we build on this work by applying the algorithm to several classes of finite dimensional conformal Galilei algebras with central extension. In these cases we highlight the utility of an algebra anti-automorphism, and give relevant details through key examples.

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arXiv2018

Quantum superintegrable system with a novel chain structure of quadratic algebras

We analyse the $n$-dimensional superintegrable Kepler-Coulomb system with non-central terms. We find a novel underlying chain structure of quadratic algebras formed by the integrals of motion. We identify the elements for each sub-structure and obtain the algebra relations satisfied by them and the corresponding Casimir operators. These quadratic sub-algebras are realized in terms of a chain of deformed oscillators with factorized structure functions. We construct the finite-dimensional unitary representations of the deformed oscillators, and give an algebraic derivation of the energy spectrum of the superintegrable system.

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