8,546 research outputs found
Generalized uncertainty relations and coherent and squeezed states
Characteristic uncertainty relations and their related squeezed states are
briefly reviewed and compared in accordance with the generalizations of three
equivalent definitions of the canonical coherent states. The standard SU(1,1)
coherent states are shown to be the unique states that minimize the
Schroedinger uncertainty relation for every pair of the three generators and
the Robertson relation for the three generators. The characteristic uncertainty
inequalities are naturally extended to the case of several states. It is shown
that these inequalities can be written in the equivalent complementary form.Comment: 14 pages, two columns revtex, no figure
Equality cases for the uncertainty principle in finite Abelian groups
We consider the families of finite Abelian groups \ZZ/p\ZZ\times \ZZ/p\ZZ,
\ZZ/p^2\ZZ and \ZZ/p\ZZ\times \ZZ/q\ZZ for two distinct prime
numbers. For the two first families we give a simple characterization of all
functions whose support has cardinality while the size of the spectrum
satisfies a minimality condition. We do it for a large number of values of
in the third case. Such equality cases were previously known when divides
the cardinality of the group, or for groups \ZZ/p\ZZ.Comment: Mistakes have been corrected. This paper has been accepted for
publication in Acta Sci. Math. (Szeged
Hypoelliptic functional inequalities
In this paper we derive a variety of functional inequalities for general
homogeneous invariant hypoelliptic differential operators on nilpotent Lie
groups. The obtained inequalities include Hardy, Rellich,
Hardy-Littllewood-Sobolev, Galiardo-Nirenberg, Caffarelli-Kohn-Nirenberg and
Trudinger-Moser inequalities. Some of these estimates have been known in the
case of the sub-Laplacians, however, for more general hypoelliptic operators
almost all of them appear to be new as no approaches for obtaining such
estimates have been available. Moreover, we obtain several versions of local
and global weighted Trudinger-Moser inequalities with remainder terms, critical
Hardy and weighted Gagliardo-Nirenberg inequalities, which appear to be new
also in the case of the sub-Laplacian. Curiously, we also show the equivalence
of many of these critical inequalities as well as asymptotic relations between
their best constants. The approach developed in this paper relies on
establishing integral versions of Hardy inequalities on homogeneous groups, for
which we also find necessary and sufficient conditions for the weights for such
inequalities to be true. Consequently, we link such integral Hardy inequalities
to different hypoelliptic inequalities by using the Riesz and Bessel kernels
associated to the described hypoelliptic operators.Comment: 58 page
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