157 research outputs found
Localization and Toeplitz Operators on Polyanalytic Fock Spaces
The well know conjecture of {\it Coburn} [{\it L.A. Coburn, {On the
Berezin-Toeplitz calculus}, Proc. Amer. Math. Soc. 129 (2001) 3331-3338.}]
proved by {\it Lo} [{\it M-L. Lo, {The Bargmann Transform and Windowed Fourier
Transform}, Integr. equ. oper. theory, 27 (2007), 397-412.}] and {\it Englis}
[{\it M. Engli, Toeplitz Operators and Localization Operators,
Trans. Am. Math Society 361 (2009) 1039-1052.}] states that any {\it
Gabor-Daubechies} operator with window and symbol
quantized on the phase space by a {\it Berezin-Toeplitz} operator with window
and symbol coincides with a {\it Toeplitz} operator
with symbol for some polynomial differential operator .
Using the Berezin quantization approach, we will extend the proof for
polyanalytic Fock spaces. While the generation is almost mimetic for
two-windowed localization operators, the Gabor analysis framework for
vector-valued windows will provide a meaningful generalization of this
conjecture for {\it true polyanalytic} Fock spaces and moreover for
polyanalytic Fock spaces.
Further extensions of this conjecture to certain classes of Gel'fand-Shilov
spaces will also be considered {\it a-posteriori}.Comment: 23 page
Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle
With the aim of derive a quasi-monomiality formulation in the context of
discrete hypercomplex variables, one will amalgamate through a
Clifford-algebraic structure of signature the umbral calculus framework
with Lie-algebraic symmetries. The exponential generating function ({\bf EGF})
carrying the {\it continuum} Dirac operator D=\sum_{j=1}^n\e_j\partial_{x_j}
together with the Lie-algebraic representation of raising and lowering
operators acting on the lattice h\BZ^n is used to derive the corresponding
hypercomplex polynomials of discrete variable as Appell sets with membership on
the space Clifford-vector-valued polynomials. Some particular examples
concerning this construction such as the hypercomplex versions of falling
factorials and the Poisson-Charlier polynomials are introduced. Certain
applications from the view of interpolation theory and integral transforms are
also discussed.Comment: 24 pages. 1 figure. v2: a major revision, including numerous
improvements throughout the paper was don
Fischer Decomposition for Difference Dirac Operators
We establish the basis of a discrete function theory starting with a Fischer
decomposition for difference Dirac operators. Discrete versions of homogeneous
polynomials, Euler and Gamma operators are obtained. As a consequence we obtain
a Fischer decomposition for the discrete Laplacian
Fock Spaces, Landau Operators and the Regular Solutions of time-harmonic Maxwell equations
We investigate the representations of the solutions to Maxwell's equations
based on the combination of hypercomplex function-theoretical methods with
quantum mechanical methods. Our approach provides us with a characterization
for the solutions to the time-harmonic Maxwell system in terms of series
expansions involving spherical harmonics resp. spherical monogenics. Also, a
thorough investigation for the series representation of the solutions in terms
of eigenfunctions of Landau operators that encode dimensional spinless
electrons is given.
This new insight should lead to important investigations in the study of
regularity and hypo-ellipticity of the solutions to Schr\"odinger equations
with natural applications in relativistic quantum mechanics concerning massive
spinor fields.Comment: Exposition improved; Some typos corrected; Accepted for publication
in J.Phys.A (February 2011). http://www.mat.uc.pt/preprints/ps/p1047.pd
FISCHER DECOMPOSITION FOR DIFFERENCE DIRAC OPERATORS
We establish the basis of a discrete function theory starting with a Fischer decomposition for difference Dirac operators. Discrete versions of homogeneous polynomials, Euler and Gamma operators are obtained. As a consequence we obtain a Fischer decomposition for the discrete Laplacian. For the sake of simplicity we consider in the first part only Dirac operators which contain only forward or backward finite differences. Of course, these Dirac operators do not factorize the classic discrete Laplacian. Therefore, we will consider a different definition of a difference Dirac operator in the quaternionic case which do factorizes the discrete Laplacian
On fractional semidiscrete Dirac operators of L\'evy-Leblond type
In this paper we introduce a wide class of space-fractional and
time-fractional semidiscrete Dirac operators of L\'evy-Leblond type on the
semidiscrete space-time lattice (),
resembling to fractional semidiscrete counterparts of the so-called parabolic
Dirac operators. The methods adopted here are fairly operational, relying
mostly on the algebraic manipulations involving Clifford algebras, discrete
Fourier analysis techniques as well as standard properties of the analytic
fractional semidiscrete semigroup
, carrying
the parameter constraints and . The results obtained involve the study of Cauchy problems on
.Comment: 30 pages, 3 figures; several typos were corrected; Section 5 was
expande
- …