Deformed oscillator algebras for two dimensional quantum superintegrable systems

Quantum superintegrable systems in two dimensions are obtained from their classical counterparts, the quantum integrals of motion being obtained from the corresponding classical integrals by a symmetrization procedure. For each quantum superintegrable systema deformed oscillator algebra, characterized by a structure function specific for each system, is constructed, the generators of the algebra being functions of the quantum integrals of motion. The energy eigenvalues corresponding to a state with finite dimensional degeneracy can then be obtained in an economical way from solving a system of two equations satisfied by the structure function, the results being in agreement to the ones obtained from the solution of the relevant Schrodinger equation. The method shows how quantum algebraic techniques can simplify the study of quantum superintegrable systems, especially in two dimensions.Comment: 22 pages, THES-TP 10/93, hep-the/yymmnn

Three dimensional quadratic algebras: Some realizations and representations

Four classes of three dimensional quadratic algebras of the type \lsb Q_0 , Q_\pm \rsb $=$ $\pm Q_\pm$, \lsb Q_+ , Q_- \rsb $=$ $aQ_0^2 + bQ_0 + c$, where $(a,b,c)$ are constants or central elements of the algebra, are constructed using a generalization of the well known two-mode bosonic realizations of $su(2)$ and $su(1,1)$. The resulting matrix representations and single variable differential operator realizations are obtained. Some remarks on the mathematical and physical relevance of such algebras are given.Comment: LaTeX2e, 23 pages, to appear in J. Phys. A: Math. Ge

Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb problem

We show that there exist some intimate connections between three unconventional Schr\"odinger equations based on the use of deformed canonical commutation relations, of a position-dependent effective mass or of a curved space, respectively. This occurs whenever a specific relation between the deforming function, the position-dependent mass and the (diagonal) metric tensor holds true. We illustrate these three equivalent approaches by considering a new Coulomb problem and solving it by means of supersymmetric quantum mechanical and shape invariance techniques. We show that in contrast with the conventional Coulomb problem, the new one gives rise to only a finite number of bound states.Comment: 22 pages, no figure. Archive version is already official. Published by JPA at http://stacks.iop.org/0305-4470/37/426

Casimir Effect as a Test for Thermal Corrections and Hypothetical Long-Range Interactions

We have performed a precise experimental determination of the Casimir pressure between two gold-coated parallel plates by means of a micromachined oscillator. In contrast to all previous experiments on the Casimir effect, where a small relative error (varying from 1% to 15%) was achieved only at the shortest separation, our smallest experimental error ($\sim 0.5$%) is achieved over a wide separation range from 170 nm to 300 nm at 95% confidence. We have formulated a rigorous metrological procedure for the comparison of experiment and theory without resorting to the previously used root-mean-square deviation, which has been criticized in the literature. This enables us to discriminate among different competing theories of the thermal Casimir force, and to resolve a thermodynamic puzzle arising from the application of Lifshitz theory to real metals. Our results lead to a more rigorous approach for obtaining constraints on hypothetical long-range interactions predicted by extra-dimensional physics and other extensions of the Standard Model. In particular, the constraints on non-Newtonian gravity are strengthened by up to a factor of 20 in a wide interaction range at 95% confidence.Comment: 17 pages, 7 figures, Sixth Alexander Friedmann International Seminar on Gravitation and Cosmolog

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.Comment: 8 pages, presented at ICGTMP 28, accepted for j.conf.serie

Connection Between Type A and E Factorizations and Construction of Satellite Algebras

Recently, we introduced a new class of symmetry algebras, called satellite algebras, which connect with one another wavefunctions belonging to different potentials of a given family, and corresponding to different energy eigenvalues. Here the role of the factorization method in the construction of such algebras is investigated. A general procedure for determining an so(2,2) or so(2,1) satellite algebra for all the Hamiltonians that admit a type E factorization is proposed. Such a procedure is based on the known relationship between type A and E factorizations, combined with an algebraization similar to that used in the construction of potential algebras. It is illustrated with the examples of the generalized Morse potential, the Rosen-Morse potential, the Kepler problem in a space of constant negative curvature, and, in each case, the conserved quantity is identified. It should be stressed that the method proposed is fairly general since the other factorization types may be considered as limiting cases of type A or E factorizations.Comment: 20 pages, LaTeX, no figure, to be published in J. Phys.

An SU(2) Analog of the Azbel--Hofstadter Hamiltonian

Motivated by recent findings due to Wiegmann and Zabrodin, Faddeev and Kashaev concerning the appearence of the quantum U_q(sl(2)) symmetry in the problem of a Bloch electron on a two-dimensional magnetic lattice, we introduce a modification of the tight binding Azbel--Hofstadter Hamiltonian that is a specific spin-S Euler top and can be considered as its ``classical'' analog. The eigenvalue problem for the proposed model, in the coherent state representation, is described by the S-gap Lam\'e equation and, thus, is completely solvable. We observe a striking similarity between the shapes of the spectra of the two models for various values of the spin S.Comment: 19 pages, LaTeX, 4 PostScript figures. Relation between Cartan and Cartesian deformation of SU(2) and numerical results added. Final version as will appear in J. Phys. A: Math. Ge

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

From Quantum Affine Symmetry to Boundary Askey-Wilson Algebra and Reflection Equation

Within the quantum affine algebra representation theory we construct linear covariant operators that generate the Askey-Wilson algebra. It has the property of a coideal subalgebra, which can be interpreted as the boundary symmetry algebra of a model with quantum affine symmetry in the bulk. The generators of the Askey-Wilson algebra are implemented to construct an operator valued $K$- matrix, a solution of a spectral dependent reflection equation. We consider the open driven diffusive system where the Askey-Wilson algebra arises as a boundary symmetry and can be used for an exact solution of the model in the stationary state. We discuss the possibility of a solution beyond the stationary state on the basis of the proposed relation of the Askey-Wilson algebra to the reflection equation

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 $N>2$. 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 Painlev\'e 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.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