15,740 research outputs found
Functions of perturbed operators
We prove that if 0<\a<1 and is in the H\"older class \L_\a(\R), then
for arbitrary self-adjoint operators and with bounded , the
operator is bounded and \|f(A)-f(B)\|\le\const\|A-B\|^\a. We
prove a similar result for functions of the Zygmund class \L_1(\R):
\|f(A+K)-2f(A)+f(A-K)\|\le\const\|K\|, where and are self-adjoint
operators. Similar results also hold for all H\"older-Zygmund classes
\L_\a(\R), \a>0. We also study properties of the operators for
f\in\L_\a(\R) and self-adjoint operators and such that belongs
to the Schatten--von Neumann class \bS_p. We consider the same problem for
higher order differences. Similar results also hold for unitary operators and
for contractions.Comment: 6 page
Chasing 'Slow Light'
A critical review of experimental studies of the so-called 'slow light'
arising due to anomalously high steepness of the refractive index dispersion
under conditions of electromagnetically induced transparency or coherent
population oscillations is presented. It is shown that a considerable amount of
experimental evidence for observation of the 'slow light' is not related to the
low group velocity of light and can be easily interpreted in terms of a
standard model of interaction of light with a saturable absorber.Comment: 17 pages, 8 figures, to be published in June issue of Phisics:
Uspekhi under the title "Notes on Slow Light
Comment on ``A quantum-classical bracket that satisfies the Jacobi identity'' [J. Chem. Phys. 124, 201104 (2006)]
It shown that the quantum-classical dynamical bracket recently proposed in J.
Chem. Phys. 124, 201104 (2006) fails to satisfy the Jacobi identity.Comment: 2 pages, no figure
An Interesting Class of Operators with unusual Schatten-von Neumann behavior
We consider the class of integral operators Q_\f on of the form
(Q_\f f)(x)=\int_0^\be\f (\max\{x,y\})f(y)dy. We discuss necessary and
sufficient conditions on to insure that is bounded, compact,
or in the Schatten-von Neumann class \bS_p, . We also give
necessary and sufficient conditions for to be a finite rank
operator. However, there is a kind of cut-off at , and for membership in
\bS_{p}, , the situation is more complicated. Although we give
various necessary conditions and sufficient conditions relating to
Q_{\phi}\in\bS_{p} in that range, we do not have necessary and sufficient
conditions. In the most important case , we have a necessary condition and
a sufficient condition, using and modulus of continuity,
respectively, with a rather small gap in between. A second cut-off occurs at
: if \f is sufficiently smooth and decays reasonably fast, then \qf
belongs to the weak Schatten-von Neumann class \wS{1/2}, but never to
\bS_{1/2} unless \f=0.
We also obtain results for related families of operators acting on
and .
We further study operations acting on bounded linear operators on
related to the class of operators Q_\f. In particular we
study Schur multipliers given by functions of the form and
we study properties of the averaging projection (Hilbert-Schmidt projection)
onto the operators of the form Q_\f.Comment: 87 page
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