126 research outputs found
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.Comment: 13 page
New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies
Four new families of two-dimensional quantum superintegrable systems are
constructed from k-step extension of the harmonic oscillator and the radial
oscillator. Their wavefunctions are related with Hermite and Laguerre
exceptional orthogonal polynomials (EOP) of type III. We show that ladder
operators obtained from alternative construction based on combinations of
supercharges in the Krein-Adler and Darboux Crum ( or state deleting and
creating ) approaches can be used to generate a set of integrals of motion and
a corresponding polynomial algebra that provides an algebraic derivation of the
full spectrum and total number of degeneracies. Such derivation is based on
finite dimensional unitary representations (unirreps) and doesn't work for
integrals build from standard ladder operators in supersymmetric quantum
mechanics (SUSYQM) as they contain singlets isolated from excited states. In
this paper, we also rely on a novel approach to obtain the finite dimensional
unirreps based on the action of the integrals of motion on the wavefunctions
given in terms of these EOP. We compare the results with those obtained from
the Daskaloyannis approach and the realizations in terms of deformed oscillator
algebras for one of the new families in the case of 1-step extension. This
communication is a review of recent works.Comment: Contribution for the 30th International Colloquium on Group
Theoretical Methods in Physics (Group30) in Ghent (Belgium). Journal of
Physics: Conference Series (to appear
Supersymmetry as a method of obtaining new superintegrable systems with higher order integrals of motion
The main result of this article is that we show that from supersymmetry we
can generate new superintegrable Hamiltonians. We consider a particular case
with a third order integral and apply the Mielnik's construction in
supersymmetric quantum mechanics. We obtain a new superintegrable potential
separable in Cartesian coordinates with a quadratic and quintic integrals and
also one with a quadratic integral and an integral of order seven. We also
construct a superintegrable system written in terms of the fourth Painleve
transcendent with a quadratic integral and an integral of order seven.Comment: 16 page
New 1-step extension of the Swanson oscillator and superintegrability of its two-dimensional generalization
We derive a one-step extension of the well known Swanson oscillator that
describes a specific type of pseudo-Hermitian quadratic Hamiltonian connected
to an extended harmonic oscillator model. Our analysis is based on the use of
the techniques of supersymmetric quantum mechanics and address various
representations of the ladder operators starting from a seed solution of the
harmonic oscillator given in terms of a pseudo-Hermite polynomial. The role of
the resulting chain of Hamiltonians related via similarity transformation is
then exploited. In the second part we write down a two dimensional
generalization of the Swanson Hamiltonian and establish superintegrability of
such a system.Comment: accepted for publication in Physics Letters
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
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.Comment: 24 pages, 9 figures, 3 table
Deformed oscillator algebra approach of some quantum superintegrable Lissajous systems on the sphere and of their rational extensions
We extend the construction of 2D superintegrable Hamiltonians with separation
of variables in spherical coordinates using combinations of shift, ladder, and
supercharge operators to models involving rational extensions of the
two-parameter Lissajous systems on the sphere. These new families of
superintegrable systems with integrals of arbitrary order are connected with
Jacobi exceptional orthogonal polynomials (EOP) of type I (or II) and
supersymmetric quantum mechanics (SUSYQM). Moreover, we present an algebraic
derivation of the degenerate energy spectrum for the one- and two-parameter
Lissajous systems and the rationally extended models. These results are based
on finitely generated polynomial algebras, Casimir operators, realizations as
deformed oscillator algebras and finite-dimensional unitary representations.
Such results have only been established so far for 2D superintegrable systems
separable in Cartesian coordinates, which are related to a class of polynomial
algebras that display a simpler structure. We also point out how the structure
function of these deformed oscillator algebras is directly related with the
generalized Heisenberg algebras (GHA) spanned by the nonpolynomial integrals
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.Comment: 28 page
Exact solution of the p+ip Hamiltonian revisited: duality relations in the hole-pair picture
We study the exact Bethe Ansatz solution of the p+ip Hamiltonian in a form
whereby quantum numbers of states refer to hole-pairs, rather than
particle-pairs used in previous studies. We find an asymmetry between these
approaches. For the attractive system states in the strong pairing regime take
the form of a quasi-condensate involving two distinct hole-pair creation
operators. An analogous feature is not observed in the particle-pair picture.Comment: 19 pages, 2 figures, 2 table
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