27 research outputs found
Multiple algebraisations of an elliptic Calogero-Sutherland model
Recently, Gomez-Ullate et al. (1) have studied a particular N-particle
quantum problem with an elliptic function potential supplemented by an external
field. They have shown that the Hamiltonian operator preserves a finite
dimensional space of functions and as such is quasi exactly solvable (QES). In
this paper we show that other types of invariant function spaces exist, which
are in close relation to the algebraic properties of the elliptic functions.
Accordingly, series of new algebraic eigenfunctions can be constructed.Comment: 9 Revtex pages, 3 PS-figures; Summary, abstract and conclusions
extende
Families of superintegrable Hamiltonians constructed from exceptional polynomials
We introduce a family of exactly-solvable two-dimensional Hamiltonians whose
wave functions are given in terms of Laguerre and exceptional Jacobi
polynomials. The Hamiltonians contain purely quantum terms which vanish in the
classical limit leaving only a previously known family of superintegrable
systems. Additional, higher-order integrals of motion are constructed from
ladder operators for the considered orthogonal polynomials proving the quantum
system to be superintegrable
Polynomial algebras and exact solutions of general quantum non-linear optical models I: Two-mode boson systems
We introduce higher order polynomial deformations of Lie algebra. We
construct their unitary representations and the corresponding single-variable
differential operator realizations. We then use the results to obtain exact
(Bethe ansatz) solutions to a class of 2-mode boson systems, including the
Boson-Einstein Condensate models as special cases. Up to an overall factor, the
eigenfunctions of the 2-mode boson systems are given by polynomials whose roots
are solutions of the associated Bethe ansatz equations. The corresponding
eigenvalues are expressed in terms of these roots. We also establish the
spectral equivalence between the BEC models and certain quasi-exactly solvable
Sch\"ordinger potentials.Comment: 20 pages, final version to appear in J. Phys. A: Math. Theor
Higher Order Quantum Superintegrability: a new "Painlev\'e 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 . 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 , 4
and 5 these equations have the Painlev\'e property. We conjecture that this is
true for all . The two integrals X and Y commute with the Hamiltonian,
but not with each other. Together they generate a polynomial algebra (for any
) of integrals of motion. We show how this algebra can be used to calculate
the energy spectrum and the wave functions.Comment: 23 pages, submitted as a contribution to the monographic volume
"Integrability, Supersymmetry and Coherent States", a volume in honour of
Professor V\'eronique Hussin. arXiv admin note: text overlap with
arXiv:1703.0975
Extended Krein-Adler theorem for the translationally shape invariant potentials
Considering successive extensions of primary translationally shape invariant potentials, we enlarge the Krein-Adler theorem to mixed chains of state adding and state-deleting Darboux-Backlund transformations. It allows us to establish novel bi-linear Wronskian and determinantal identities for classical orthogonal polynomials