125 research outputs found
Nondegenerate three-dimensional complex Euclidean superintegrable systems and algebraic varieties
A classical (or quantum) second order superintegrable system is an integrable n-dimensional Hamiltonian system with potential that admits 2n−1 functionally independent second order constants of the motion polynomial in the momenta, the maximum possible. Such systems have remarkable properties: multi-integrability and multiseparability, an algebra of higher order symmetries whose representation theory yields spectral information about the Schrödinger operator, deep connections with special functions, and with quasiexactly solvable systems. Here, we announce a complete classification of nondegenerate (i.e., four-parameter) potentials for complex Euclidean 3-space. We characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints. The Euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation. There are exactly ten nondegenerate potentials. ©2007 American Institute of Physic
Harnack inequalities and B\^ocher-type theorems for conformally invariant fully nonlinear degenerate elliptic equations
We give a generalization of a theorem of B\^ocher for the Laplace equation to
a class of conformally invariant fully nonlinear degenerate elliptic equations.
We also prove a Harnack inequality for locally Lipschitz viscosity solutions
and a classification of continuous radially symmetric viscosity solutions.Comment: to appear in CPA
On the Singular Weyl-Titchmarsh Function of Perturbed Spherical Schroedinger Operators
We investigate the singular Weyl-Titchmarsh m-function of perturbed spherical
Schroedinger operators (also known as Bessel operators) under the assumption
that the perturbation satisfies . We show existence
plus detailed properties of a fundamental system of solutions which are entire
with respect to the energy parameter. Based on this we show that the singular
m-function belongs to the generalized Nevanlinna class and connect our results
with the theory of super singular perturbations.Comment: 35 page
Algebraic Integrability Conditions for Killing Tensors on Constant Sectional Curvature Manifolds
We use an isomorphism between the space of valence two Killing tensors on an
n-dimensional constant sectional curvature manifold and the irreducible
GL(n+1)-representation space of algebraic curvature tensors in order to
translate the Nijenhuis integrability conditions for a Killing tensor into
purely algebraic integrability conditions for the corresponding algebraic
curvature tensor, resulting in two simple algebraic equations of degree two and
three. As a first application of this we construct a new family of integrable
Killing tensors.Comment: 34 pages, no figure
Regular singular Sturm-Liouville operators and their zeta-determinants
We consider Sturm-Liouville operators on the line segment [0, 1] with general
regular singular potentials and separated boundary conditions. We establish
existence and a formula for the associated zeta-determinant in terms of the
Wronski- determinant of a fundamental system of solutions adapted to the
boundary conditions. This generalizes the earlier work of the first author,
treating general regular singular potentials but only the Dirichlet boundary
conditions at the singular end, and the recent results by Kirsten-Loya-Park for
general separated boundary conditions but only special regular singular
potentials.Comment: 38 pages, 2 figures; Completely revised according to the referees
comprehensive suggestions; v3: minor corrections, accepted for publication in
Journal of Functional Analysi
Stellar Population analysis from Broad-Band Colours
We have developed an analytical method to investigate the stellar populations
in a galaxy using the broad-band colours. The method enables us to determine
the relative contribution, spatial distribution and age for different stellar
populations and gives a hint about the dust distribution in a galaxy. We apply
this method to the irregular galaxy NGC 3077, using CCD images in U, B, V and R
filters.Comment: 8 pages, 11 ps. figs., use an_art.cls to appear in Astron. Nac
Multistep DBT and regular rational extensions of the isotonic oscillator
In some recent articles we developed a new systematic approach to generate
solvable rational extensions of primary translationally shape invariant
potentials. In this generalized SUSY QM partnership, the DBT are built on the
excited states Riccati-Schr\"odinger (RS) functions regularized via specific
discrete symmetries of the considered potential. In the present paper, we prove
that this scheme can be extended in a multistep formulation. Applying this
scheme to the isotonic oscillator, we obtain new towers of regular rational
extensions of this potential which are strictly isospectral to it. We give
explicit expressions for their eigenstates which are associated to the recently
discovered exceptional Laguerre polynomials and show explicitely that these
extensions inherit of the shape invariance properties of the original
potential
Invariant classification of the rotationally symmetric R-separable webs for the Laplace equation in Euclidean space
An invariant characterization of the rotationally symmetric R-separable webs
for the Laplace equation in Euclidean space is given in terms of invariants and
covariants of a real binary quartic canonically associated to the
characteristic conformal Killing tensor which defines the webs.Comment: 25 pages, recently submitted to the Journal of Mathematical Physic
Autocalibration with the Minimum Number of Cameras with Known Pixel Shape
In 3D reconstruction, the recovery of the calibration parameters of the
cameras is paramount since it provides metric information about the observed
scene, e.g., measures of angles and ratios of distances. Autocalibration
enables the estimation of the camera parameters without using a calibration
device, but by enforcing simple constraints on the camera parameters. In the
absence of information about the internal camera parameters such as the focal
length and the principal point, the knowledge of the camera pixel shape is
usually the only available constraint. Given a projective reconstruction of a
rigid scene, we address the problem of the autocalibration of a minimal set of
cameras with known pixel shape and otherwise arbitrarily varying intrinsic and
extrinsic parameters. We propose an algorithm that only requires 5 cameras (the
theoretical minimum), thus halving the number of cameras required by previous
algorithms based on the same constraint. To this purpose, we introduce as our
basic geometric tool the six-line conic variety (SLCV), consisting in the set
of planes intersecting six given lines of 3D space in points of a conic. We
show that the set of solutions of the Euclidean upgrading problem for three
cameras with known pixel shape can be parameterized in a computationally
efficient way. This parameterization is then used to solve autocalibration from
five or more cameras, reducing the three-dimensional search space to a
two-dimensional one. We provide experiments with real images showing the good
performance of the technique.Comment: 19 pages, 14 figures, 7 tables, J. Math. Imaging Vi
Non-dissipative electromagnetic medium with a double light cone
We study Maxwell's equations on a 4-manifold where the electromagnetic medium
is modelled by an antisymmetric (2, 2)-tensor kappa with real coefficients. In
this setting the Fresnel surface is a fourth order polynomial surface in each
cotangent space that acts as a generalisation of the light cone determined by a
Lorentz metric; the Fresnel surface parameterises electromagnetic wave-speeds
as a function of direction. The contribution of this paper is the complete
pointwise description of all electromagnetic medium tensors that satisfy the
following conditions:
(i) kappa is invertible,
(ii) kappa is skewon-free,
(iii) kappa is birefringent, that is, the Fresnel surface of kappa is the
union of two distinct light cones.
We show that there are only three classes of mediums with these properties.
Moreover, we give explicit expressions in local coordinates for each class
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