300 research outputs found
On a Generalized Oscillator System: Interbasis Expansions
This article deals with a nonrelativistic quantum mechanical study of a
dynamical system which generalizes the isotropic harmonic oscillator system in
three dimensions. The problem of interbasis expansions of the wavefunctions is
completely solved. A connection between the generalized oscillator system
(projected on the z-line) and the Morse system (in one dimension) is discussed.Comment: 23 pages, Latex File, to be published in International Journal of
Quantum Chemistr
On a Generalized Oscillator: Invariance Algebra and Interbasis Expansions
This article deals with a quantum-mechanical system which generalizes the
ordinary isotropic harmonic oscillator system. We give the coefficients
connecting the polar and Cartesian bases for D=2 and the coefficients
connecting the Cartesian and cylindrical bases as well as the cylindrical and
spherical bases for D=3. These interbasis expansion coefficients are found to
be analytic continuations to real values of their arguments of the
Clebsch-Gordan coefficients for the group SU(2). For D=2, the superintegrable
character for the generalized oscillator system is investigated from the points
of view of a quadratic invariance algebra.Comment: 13 pages, Latex file. Submitted for publication to Yadernaya Fizik
Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions
In this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems. We do this for the corresponding systems defined in Euclidean space and on the two-dimensional sphere. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup Cartesian and polar coordinate cases, and the details of the structure of the quadratic algebras. We also study the polynomial eigenfunctions in elliptic coordinates of the n-dimensional isotropic quantum oscillator
Coulomb-oscillator duality in spaces of constant curvature
In this paper we construct generalizations to spheres of the well known
Levi-Civita, Kustaanheimo-Steifel and Hurwitz regularizing transformations in
Euclidean spaces of dimensions 2, 3 and 5. The corresponding classical and
quantum mechanical analogues of the Kepler-Coulomb problem on these spheres are
discussed.Comment: 33 pages, LaTeX fil
Superintegrability on the two-dimensional hyperboloid
In this work we examine the basis functions for classical and quantum mechanical systems on the two-dimensional hyperboloid that admit separation of variables in at least two coordinate systems. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup spherical coordinate cases, and the details of the structure of the quadratic symmetry algebras
Superintegrability on the two dimensional hyperboloid II
This work is devoted to the investigation of the quantum mechanical systems
on the two dimensional hyperboloid which admit separation of variables in at
least two coordinate systems. Here we consider two potentials introduced in a
paper of C.P.Boyer, E.G.Kalnins and P.Winternitz, which haven't yet been
studied. We give an example of an interbasis expansion and work out the
structure of the quadratic algebra generated by the integrals of motion.Comment: 18 pages, LaTex; 1 figure (eps
Non Gaussian extrema counts for CMB maps
In the context of the geometrical analysis of weakly non Gaussian CMB maps,
the 2D differential extrema counts as functions of the excursion set threshold
is derived from the full moments expansion of the joint probability
distribution of an isotropic random field, its gradient and invariants of the
Hessian. Analytic expressions for these counts are given to second order in the
non Gaussian correction, while a Monte Carlo method to compute them to
arbitrary order is presented. Matching count statistics to these estimators is
illustrated on fiducial non-Gaussian "Planck" data.Comment: 4 pages, 1 figur
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