6 research outputs found
An infinite family of superintegrable Hamiltonians with reflection in the plane
We introduce a new infinite class of superintegrable quantum systems in the
plane. Their Hamiltonians involve reflection operators. The associated
Schr\"odinger equations admit separation of variables in polar coordinates and
are exactly solvable. The angular part of the wave function is expressed in
terms of little -1 Jacobi polynomials. The spectra exhibit "accidental"
degeneracies. The superintegrability of the model is proved using the
recurrence relation approach. The (higher-order) constants of motion are
constructed and the structure equations of the symmetry algebra obtained.Comment: 19 page
Supersymmetric Quantum Mechanics with Reflections
We consider a realization of supersymmetric quantum mechanics where
supercharges are differential-difference operators with reflections. A
supersymmetric system with an extended Scarf I potential is presented and
analyzed. Its eigenfunctions are given in terms of little -1 Jacobi polynomials
which obey an eigenvalue equation of Dunkl type and arise as a q-> -1 limit of
the little q-Jacobi polynomials. Intertwining operators connecting the wave
functions of extended Scarf I potentials with different parameters are
presented.Comment: 17 page
Infinite families of superintegrable systems separable in subgroup coordinates
A method is presented that makes it possible to embed a subgroup separable
superintegrable system into an infinite family of systems that are integrable
and exactly-solvable. It is shown that in two dimensional Euclidean or
pseudo-Euclidean spaces the method also preserves superintegrability. Two
infinite families of classical and quantum superintegrable systems are obtained
in two-dimensional pseudo-Euclidean space whose classical trajectories and
quantum eigenfunctions are investigated. In particular, the wave-functions are
expressed in terms of Laguerre and generalized Bessel polynomials.Comment: 19 pages, 6 figure
Jordan algebras and orthogonal polynomials
We illustrate how Jordan algebras can provide a framework for the
interpretation of certain classes of orthogonal polynomials. The big -1 Jacobi
polynomials are eigenfunctions of a first order operator of Dunkl type. We
consider an algebra that has this operator (up to constants) as one of its
three generators and whose defining relations are given in terms of
anticommutators. It is a special case of the Askey-Wilson algebra AW(3). We
show how the structure and recurrence relations of the big -1 Jacobi
polynomials are obtained from the representations of this algebra. We also
present ladder operators for these polynomials and point out that the big -1
Jacobi polynomials satisfy the Hahn property with respect to a generalized
Dunkl operator.Comment: 11 pages, 30 reference
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