284 research outputs found
Spectra of observables in the q-oscillator and q-analogue of the Fourier transform
Spectra of the position and momentum operators of the Biedenharn-Macfarlane
q-oscillator (with the main relation aa^+-qa^+a=1) are studied when q>1. These
operators are symmetric but not self-adjoint. They have a one-parameter family
of self-adjoint extensions. These extensions are derived explicitly. Their
spectra and eigenfunctions are given. Spectra of different extensions do not
intersect. The results show that the creation and annihilation operators a^+
and a of the q-oscillator for q>1 cannot determine a physical system without
further more precise definition. In order to determine a physical system we
have to choose appropriate self-adjoint extensions of the position and momentum
operators.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Three variable exponential functions of the alternating group
New class of special functions of three real variables, based on the
alternating subgroup of the permutation group , is studied. These
functions are used for Fourier-like expansion of digital data given on lattice
of any density and general position. Such functions have only trivial analogs
in one and two variables; a connection to the functions of is shown.
Continuous interpolation of the three dimensional data is studied and
exemplified.Comment: 10 pages, 3 figure
Two-dimensional symmetric and antisymmetric generalizations of exponential and cosine functions
Properties of the four families of recently introduced special functions of
two real variables, denoted here by , and , are studied. The
superscripts and refer to the symmetric and antisymmetric functions
respectively. The functions are considered in all details required for their
exploitation in Fourier expansions of digital data, sampled on square grids of
any density and for general position of the grid in the real plane relative to
the lattice defined by the underlying group theory. Quality of continuous
interpolation, resulting from the discrete expansions, is studied, exemplified
and compared for some model functions.Comment: 22 pages, 10 figure
Eigenfunction Expansions of Functions Describing Systems with Symmetries
Physical systems with symmetries are described by functions containing
kinematical and dynamical parts. We consider the case when kinematical
symmetries are described by a noncompact semisimple real Lie group . Then
separation of kinematical parts in the functions is fulfilled by means of
harmonic analysis related to the group . This separation depends on choice
of a coordinate system on the space where a physical system exists. In the
paper we review how coordinate systems can be chosen and how the corresponding
harmonic analysis can be done. In the first part we consider in detail the case
when is the de Sitter group . In the second part we show how the
corresponding theory can be developed for any noncompact semisimple real Lie
group.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
(Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms
Four families of special functions, depending on n variables, are studied. We
call them symmetric and antisymmetric multivariate sine and cosine functions.
They are given as determinants or antideterminants of matrices, whose matrix
elements are sine or cosine functions of one variable each. These functions are
eigenfunctions of the Laplace operator, satisfying specific conditions at the
boundary of a certain domain F of the n-dimensional Euclidean space. Discrete
and continuous orthogonality on F of the functions within each family, allows
one to introduce symmetrized and antisymmetrized multivariate Fourier-like
transforms, involving the symmetric and antisymmetric multivariate sine and
cosine functions.Comment: 25 pages, no figures; LaTaX; corrected typo
E-Orbit Functions
We review and further develop the theory of -orbit functions. They are
functions on the Euclidean space obtained from the multivariate
exponential function by symmetrization by means of an even part of a
Weyl group , corresponding to a Coxeter-Dynkin diagram. Properties of such
functions are described. They are closely related to symmetric and
antisymmetric orbit functions which are received from exponential functions by
symmetrization and antisymmetrization procedure by means of a Weyl group .
The -orbit functions, determined by integral parameters, are invariant with
respect to even part of the affine Weyl group corresponding
to . The -orbit functions determine a symmetrized Fourier transform,
where these functions serve as a kernel of the transform. They also determine a
transform on a finite set of points of the fundamental domain of the
group (the discrete -orbit function transform).Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
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