4,119 research outputs found
Discrete coherent states for higher Landau levels
We consider the quantum dynamics of a charged particle evolving under the
action of a constant homogeneous magnetic field, with emphasis on the discrete
subgroups of the Heisenberg group (in the Euclidean case) and of the SL(2, R)
group (in the Hyperbolic case). We investigate completeness properties of
discrete coherent states associated with higher order Euclidean and hyperbolic
Landau levels, partially extending classic results of Perelomov and of
Bargmann, Butera, Girardello and Klauder. In the Euclidean case, our results
follow from identifying the completeness problem with known results from the
theory of Gabor frames. The results for the hyperbolic setting follow by using
a combination of methods from coherent states, time-scale analysis and the
theory of Fuchsian groups and their associated automorphic forms.Comment: Revised for Annals of Physic
Restrictions and extensions of semibounded operators
We study restriction and extension theory for semibounded Hermitian operators
in the Hardy space of analytic functions on the disk D. Starting with the
operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D)
of measure zero, there is a densely defined Hermitian restriction of zd/dz
corresponding to boundary functions vanishing on F. For every such restriction
operator, we classify all its selfadjoint extension, and for each we present a
complete spectral picture.
We prove that different sets F with the same cardinality can lead to quite
different boundary-value problems, inequivalent selfadjoint extension
operators, and quite different spectral configurations. As a tool in our
analysis, we prove that the von Neumann deficiency spaces, for a fixed set F,
have a natural presentation as reproducing kernel Hilbert spaces, with a
Hurwitz zeta-function, restricted to FxF, as reproducing kernel.Comment: 63 pages, 11 figure
Wavelet frames, Bergman spaces and Fourier transforms of Laguerre functions
The Fourier transforms of Laguerre functions play the same canonical role in
wavelet analysis as do the Hermite functions in Gabor analysis. We will use
them as analyzing wavelets in a similar way the Hermite functions were recently
by K. Groechenig and Y. Lyubarskii in "Gabor frames with Hermite functions, C.
R. Acad. Sci. Paris, Ser. I 344 157-162 (2007)". Building on the work of K.
Seip, "Beurling type density theorems in the unit disc, Invent. Math., 113,
21-39 (1993)", concerning sampling sequences on weighted Bergman spaces, we
find a sufficient density condition for constructing frames by translations and
dilations of the Fourier transform of the nth Laguerre function. As in
Groechenig-Lyubarskii theorem, the density increases with n, and in the special
case of the hyperbolic lattice in the upper half plane it is given by b\log
a<\frac{4\pi}{2n+\alpha}, where alpha is the parameter of the Laguerre
function.Comment: 15 page
Hardy's paradox tested in the spin-orbit Hilbert space of single photons
We test experimentally the quantum ``paradox'' proposed by Lucien Hardy in
1993 [Phys. Rev. Lett. 71, 1665 (1993)] by using single photons instead of
photon pairs. This is achieved by addressing two compatible degrees of freedom
of the same particle, namely its spin angular momentum, determined by the
photon polarization, and its orbital angular momentum, a property related to
the optical transverse mode. Because our experiment involves a single particle,
we cannot use locality to logically enforce non-contextuality, which must
therefore be assumed based only on the observables' compatibility. On the other
hand, our single-particle experiment can be implemented more simply and allows
larger detection efficiencies than typical two-particle ones, with a potential
future advantage in terms of closing the detection loopholes.Comment: 7 pages, 5 figures and 1 tabl
Polynomial Ensembles and Recurrence Coefficients
Polynomial ensembles are determinantal point processes associated with (non
necessarily orthogonal) projections onto polynomial subspaces. The aim of this
survey article is to put forward the use of recurrence coefficients to obtain
the global asymptotic behavior of such ensembles in a rather simple way. We
provide a unified approach to recover well-known convergence results for real
OP ensembles. We study the mutual convergence of the polynomial ensemble and
the zeros of its average characteristic polynomial; we discuss in particular
the complex setting. We also control the variance of linear statistics of
polynomial ensembles and derive comparison results, as well as asymptotic
formulas for real OP ensembles. Finally, we reinterpret the classical algorithm
to sample determinantal point processes so as to cover the setting of
non-orthogonal projection kernels. A few open problems are also suggested.Comment: 23 page
The Spectrum of Volterra-type integration operators on generalized Fock spaces
We describe the spectrum of certain integration operators acting on general-
ized Fock spaces
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