53 research outputs found
Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold
In this paper we prove that the two dimensional superintegrable systems with
quadratic integrals of motion on a manifold can be classified by using the
Poisson algebra of the integrals of motion. There are six general fundamental
classes of superintegrable systems. Analytic formulas for the involved
integrals are calculated in all the cases. All the known superintegrable
systems are classified as special cases of these six general classes.Comment: LaTeX, 72 pages. Extended version of the published version in JM
Quasi-exactly solvable problems and the dual (q-)Hahn polynomials
A second-order differential (q-difference) eigenvalue equation is constructed
whose solutions are generating functions of the dual (q-)Hahn polynomials. The
fact is noticed that these generating functions are reduced to the (little
q-)Jacobi polynomials, and implications of this for quasi-exactly solvable
problems are studied. A connection with the Azbel-Hofstadter problem is
indicated.Comment: 15 pages, LaTex; final version, presentation improved, title changed,
to appear in J.Math.Phy
A q-Deformed Schr\"odinger Equation
We found hermitian realizations of the position vector , the angular
momentum and the linear momentum , all behaving like
vectors under the algebra, generated by and . They are
used to introduce a -deformed Schr\" odinger equation. Its solutions for the
particular cases of the Coulomb and the harmonic oscillator potentials are
given and briefly discussed.Comment: 14 pages, latex, no figure
Electron shielding of the nuclear magnetic moment in hydrogen-like atom
The correction to the wave function of the ground state in a hydrogen-like
atom due to an external homogenous magnetic field is found exactly in the
parameter . The projection of the correction to the wave
function of the state due to the external homogeneous magnetic field
is found for arbitrary . The projection of the correction to the
wave function of the state due to the nuclear magnetic moment is
also found for arbitrary . Using these results, we have calculated the
shielding of the nuclear magnetic moment by the electron.Comment: 15 page
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
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
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
On some nonlinear extensions of the angular momentum algebra
Deformations of the Lie algebras so(4), so(3,1), and e(3) that leave their
so(3) subalgebra undeformed and preserve their coset structure are considered.
It is shown that such deformed algebras are associative for any choice of the
deformation parameters. Their Casimir operators are obtained and some of their
unitary irreducible representations are constructed. For vanishing deformation,
the latter go over into those of the corresponding Lie algebras that contain
each of the so(3) unitary irreducible representations at most once. It is also
proved that similar deformations of the Lie algebras su(3), sl(3,R), and of the
semidirect sum of an abelian algebra t(5) and so(3) do not lead to associative
algebras.Comment: 22 pages, plain TeX + preprint.sty, no figures, to appear in J.Phys.
Superintegrable Systems with a Third Order Integrals of Motion
Two-dimensional superintegrable systems with one third order and one lower
order integral of motion are reviewed. The fact that Hamiltonian systems with
higher order integrals of motion are not the same in classical and quantum
mechanics is stressed. New results on the use of classical and quantum third
order integrals are presented in Section 5 and 6.Comment: To appear in J. Phys A: Mathematical and Theoretical (SPE QTS5
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