arXiv subjects

Ian Marquette

Publications and source records attributed to Ian Marquette.

At least 109 records · Page 6Linked to original sources

arXiv2014

Quadratic algebra structure and spectrum of a new superintegrable system in N-dimension

We introduce a new superintegrable Kepler-Coulomb system with non-central terms in $N$-dimensional Euclidean space. We show this system is multiseparable and allows separation of variables in hyperspherical and hyperparabolic coordinates. We present the wave function in terms of special functions. We give a algebraic derivation of spectrum of the superintegrable system. We show how the $so(N+1)$ symmetry algebra of the $N$-dimensional Kepler-Coulomb system is deformed to a quadratic algebra with only 3 generators and structure constants involving a Casimir operator of $so(N-1)$ Lie algebra. We construct the quadratic algebra and the Casimir operator. We show this algebra can be realized in terms of deformed oscillator and obtain the structure function which yields the energy spectrum.

math-ph↗

arXiv2014

Ground-state Bethe root densities and quantum phase transitions

Exactly solvable models provide a unique method, via qualitative changes in the distribution of the ground-state roots of the Bethe Ansatz equations, to identify quantum phase transitions. Here we expand on this approach, in a quantitative manner, for two models of Bose--Einstein condensates. The first model deals with the interconversion of bosonic atoms and molecules. The second is the two-site Bose--Hubbard model, widely used to describe tunneling phenomena in Bose--Einstein condensates. For these systems we calculate the ground-state root density. This facilitates the determination of analytic forms for the ground-state energy, and associated correlation functions through the Hellmann--Feynman theorem. These calculations provide a clear identification of the quantum phase transition in each model. For the first model we obtain an expression for the molecular fraction expectation value. For the two-site Bose--Hubbard model we find that there is a simple characterisation of condensate fragmentation.

nlin.SI↗

arXiv2014

Combined state-adding and state-deleting approaches to type III multi-step rationally-extended potentials: applications to ladder operators and superintegrability

Type III multi-step rationally-extended harmonic oscillator and radial harmonic oscillator potentials, characterized by a set of $k$ integers $m_1$, $m_2$, \ldots, $m_k$, such that $m_1 < m_2 < \cdots < m_k$ with $m_i$ even (resp.\ odd) for $i$ odd (resp.\ even), are considered. The state-adding and state-deleting approaches to these potentials in a supersymmetric quantum mechanical framework are combined to construct new ladder operators. The eigenstates of the Hamiltonians are shown to separate into $m_k+1$ infinite-dimensional unitary irreducible representations of the corresponding polynomial Heisenberg algebras. These ladder operators are then used to build a higher-order integral of motion for seven new infinite families of superintegrable two-dimensional systems separable in cartesian coordinates. The finite-dimensional unitary irreducible representations of the polynomial algebras of such systems are directly determined from the ladder operator action on the constituent one-dimensional Hamiltonian eigenstates and provide an algebraic derivation of the superintegrable systems whole spectrum including the level total degeneracies.

math-ph↗

arXiv2014

On realizations of polynomial algebras with three generators via deformed oscillator algebras

We present the most general polynomial Lie algebra generated by a second order integral of motion and one of order M, construct the Casimir operator, and show how the Jacobi identity provides the existence of a realization in terms of deformed oscillator algebra. We also present the classical analog of this construction for the most general Polynomial Poisson algebra. Two specific classes of such polynomial algebras are discussed that include the symmetry algebras observed for various 2D superintegrable systems.

math-ph↗

arXiv2014

New quasi-exactly solvable class of generalized isotonic oscillators

We introduce a new family of quasi-exactly solvable generalized isotonic oscillators which are based on the pseudo-Hermite exceptional orthogonal polynomials. We obtain exact closed-form expressions for the energies and wavefunctions as well as the allowed potential parameters for the first two members of the family using the Bethe ansatz method. Numerical calculations of the energies reveal that member potentials have multiple quasi-exactly solvable eigenstates and the number of states for higher members are parameter dependent.

math-ph↗

arXiv2013

Quartic Poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras

We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog and we extend Daskaloyannis' construction in obtained in context of quadratic algebras and we obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite dimensional unitary irreductible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptionnal orthogonal polynomials introduced recently.

math-ph↗

arXiv2012

New families of superintegrable systems from Hermite and Laguerre exceptional orthogonal polynomials

In recent years, many exceptional orthogonal polynomials (EOP) were introduced and used to construct new families of 1D exactly solvable quantum potentials, some of which are shape invariant. In this paper, we construct from Hermite and Laguerre EOP and their related quantum systems new 2D superintegrable Hamiltonians with higher-order integrals of motion and the polynomial algebras generated by their integrals of motion. We obtain the finite-dimensional unitary representations of the polynomial algebras and the corresponding energy spectrum. We also point out a new type of degeneracies of the energy levels of these systems that is associated with holes in sequences of EOP.

math-ph↗

arXiv2012

Singular Isotonic Oscillator, Supersymmetry and Superintegrability

In the case of a one-dimensional nonsingular Hamiltonian $H$ and a singular supersymmetric partner $H_a$, the Darboux and factorization relations of supersymmetric quantum mechanics can be only formal relations. It was shown how we can construct an adequate partner by using infinite barriers placed where are located the singularities on the real axis and recover isospectrality. This method was applied to superpartners of the harmonic oscillator with one singularity. In this paper, we apply this method to the singular isotonic oscillator with two singularities on the real axis. We also applied these results to four 2D superintegrable systems with second and third-order integrals of motion obtained by Gravel for which polynomial algebras approach does not allow to obtain the energy spectrum of square integrable wavefunctions. We obtain solutions involving parabolic cylinder functions.

math-ph↗

arXiv2012

Generalised Heine-Stieltjes and Van Vleck polynomials associated with degenerate, integrable BCS models

We study the Bethe Ansatz/Ordinary Differential Equation (BA/ODE) correspondence for Bethe Ansatz equations that belong to a certain class of coupled, nonlinear, algebraic equations. Through this approach we numerically obtain the generalised Heine-Stieltjes and Van Vleck polynomials in the degenerate, two-level limit for four cases of exactly solvable Bardeen-Cooper-Schrieffer (BCS) pairing models. These are the s-wave pairing model, the p+ip-wave pairing model, the p+ip pairing model coupled to a bosonic molecular pair degree of freedom, and a newly introduced extended d+id-wave pairing model with additional interactions. The zeros of the generalised Heine-Stieltjes polynomials provide solutions of the corresponding Bethe Ansatz equations. We compare the roots of the ground states with curves obtained from the solution of a singular integral equation approximation, which allows for a characterisation of ground-state phases in these systems. Our techniques also permit for the computation of the roots of the excited states. These results illustrate how the BA/ODE correspondence can be used to provide new numerical methods to study a variety of integrable systems.

nlin.SI↗

arXiv2012

Integrability of an extended d+id-wave pairing Hamiltonian

We introduce an integrable Hamiltonian which is an extended d+id-wave pairing model. The integrability is deduced from a duality relation with the Richardson-Gaudin (s-wave) pairing model, and associated to this there exists an exact Bethe ansatz solution. We study this system using the continuum limit approach and solve the corresponding singular integral equation obtained from the Bethe ansatz solution. We also conduct a mean-field analysis and show that results from these two approaches coincide for the ground state in the continuum limit. We identify instances of the integrable system where the excitation spectrum is gapless, and discuss connections to non-integrable models with d+id-wave pairing interactions through the mean-field analysis.

cond-mat.supr-con↗

arXiv2012

Generalized five-dimensional Kepler system, Yang-Coulomb monopole and Hurwitz transformation

The 5D Kepler system possesses many interesting properties. This system is superintegrable and also with a $su(2)$ nonAbelian monopole interaction (Yang-Coulomb monopole). This system is also related to a 8D isotropic harmonic oscillator by a Hurwitz transformation. We introduce a new superintegrable Hamiltonian that consists in a 5D Kepler system with new terms of Smorodinsky-Winternitz type. We obtain the integrals of motion of this systems. They generate a quadratic algebra with structure constants involving the Casimir operator of a $so(4)$ Lie algebra. We also show that this system remains superintegrable with a $su(2)$ nonAbelian monopole (generalized Yang-Coulomb monopole). We study this system using parabolic coordinates and obtain from Hurwitz transformation its dual that is a 8D singular oscillator. This 8D singular oscillator is also a new superintegrable system and multiseparable. We obtained its quadratic algebra that involves two Casimir operators of $so(4)$ Lie algebras. This correspondence is used to obtain algebraically the energy spectrum of the generalized Yang Coulomb monopole.

math-ph↗

arXiv2012

Classical ladder operators, polynomial Poisson algebras and classification of superintegrable systems

We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems respectively with a third and a fourth order ladder operators satisfying polynomial Heisenberg algebras. These systems are written in terms of the solutions of quartic and quintic equations. We use these results to present two new families of superintegrable systems and examples of trajectories that are deformed Lissajous's figures.

math-ph↗

arXiv2011

Generalized Heisenberg Algebras, SUSYQM and Degeneracies: Infinite Well and Morse Potential

We consider classical and quantum one and two-dimensional systems with ladder operators that satisfy generalized Heisenberg algebras. In the classical case, this construction is related to the existence of closed trajectories. In particular, we apply these results to the infinite well and Morse potentials. We discuss how the degeneracies of the permutation symmetry of quantum two-dimensional systems can be explained using products of ladder operators. These products satisfy interesting commutation relations. The two-dimensional Morse quantum system is also related to a generalized two-dimensional Morse supersymmetric model. Arithmetical or accidental degeneracies of such system are shown to be associated to additional supersymmetry.

math-ph↗

arXiv2011

Generalized Kaluza-Klein monopole, quadratic algebras and ladder operators

We present a generalized Kaluza-Klein monopole system. We solve this quantum superintegrable systems on a Euclidean Taub Nut manifold using the separation of variables of the corresponding Schroedinger equation in spherical and parabolic coordinates. We present the integrals of motion of this system, the quadratic algebra generated by these integrals, the realization in term of a deformed oscillator algebra using the Daskaloyannis construction and the energy spectrum. The structure constants and the Casimir operator are functions not only of the Hamiltonian but also of other two integrals commuting with all generators of the quadratic algebra and forming an Abelian subalgebra. We present an other algebraic derivation of the energy spectrum of this system using the factorization method and ladder operators.

math-ph↗

arXiv2010

Quadratic Algebra Approach to Relativistic Quantum Smorodinsky-Winternitz Systems

There exist a relation between the Klein-Gordon and the Dirac equations with scalar and vector potentials of equal magnitude (SVPEM) and the Schrodinger equation. We obtain the relativistic energy spectrum for the four Smorodinsky-Winternitz systems from the quasi-Hamiltonian and the quadratic algebras obtained by Daskaloyannis in the non-relativistic context. We point out how results obtained in context of quantum superintegrable systems and their polynomial algebras may be applied to the quantum relativistic case. We also present the symmetry algebra of the Dirac equation for these four systems and show that the quadratic algebra obtained is equivalent to the one obtained from the quasi-Hamiltonian.

math-ph↗

arXiv2010

An infinite family of superintegrable systems from higher order ladder operators and supersymmetry

We will discuss how we can obtain new quantum superintegrable Hamiltonians allowing the separation of variables in Cartesian coordinates with higher order integrals of motion from ladder operators. We will discuss also how higher order supersymmetric quantum mechanics can be used to obtain systems with higher order ladder operators and their polynomial Heisenberg algebra. We will present a new family of superintegrable systems involving the fifth Painleve transcendent which possess fourth order ladder operators constructed from second order supersymmetric quantum mechanics. We present the polynomial algebra of this family of superintegrable systems.

math-ph↗

arXiv2010

Generalized MICZ-Kepler system, duality, polynomial and deformed oscillator algebras

We present the quadratic algebra of the generalized MICZ-Kepler system in three-dimensional Euclidean space $E_{3}$ and its dual the four dimensional singular oscillator in four-dimensional Euclidean space $E_{4}$. We present their realization in terms of a deformed oscillator algebra using the Daskaloyannis construction. The structure constants are in these cases function not only of the Hamiltonian but also of other integrals commuting with all generators of the quadratic algebra. We also present a new algebraic derivation of the energy spectrum of the MICZ-Kepler system on the three sphere $S^{3}$ using a quadratic algebra. These results point out also that results and explicit formula for structure functions obtained for quadratic, cubic and higher order polynomial algebras in context of two-dimensional superintegrable systems may be applied to superintegrable systems in higher dimensions with and without monopoles.

math-ph↗

arXiv2010

Construction of classical superintegrable systems with higher order integrals of motion from ladder operators

We construct integrals of motion for multidimensional classical systems from ladder operators of one-dimensional systems. This method can be used to obtain new systems with higher order integrals. We show how these integrals generate a polynomial Poisson algebra. We consider a one-dimensional system with third order ladders operators and found a family of superintegrable systems with higher order integrals of motion. We obtain also the polynomial algebra generated by these integrals. We calculate numerically the trajectories and show that all bounded trajectories are closed.

math-ph↗