Functions of perturbed operators

We prove that if 0<\a<1 and $f$ is in the H\"older class \L_\a(\R), then for arbitrary self-adjoint operators $A$ and $B$ with bounded $A-B$, the operator $f(A)-f(B)$ is bounded and \|f(A)-f(B)\|\le\const\|A-B\|^\a. We prove a similar result for functions $f$ of the Zygmund class \L_1(\R): \|f(A+K)-2f(A)+f(A-K)\|\le\const\|K\|, where $A$ and $K$ 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 $f(A)-f(B)$ for f\in\L_\a(\R) and self-adjoint operators $A$ and $B$ such that $A-B$ 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 $L^2(\R_+)$ of the form (Q_\f f)(x)=\int_0^\be\f (\max\{x,y\})f(y)dy. We discuss necessary and sufficient conditions on $\phi$ to insure that $Q_{\phi}$ is bounded, compact, or in the Schatten-von Neumann class \bS_p, $1<p<\infty$. We also give necessary and sufficient conditions for $Q_{\phi}$ to be a finite rank operator. However, there is a kind of cut-off at $p=1$, and for membership in \bS_{p}, $0<p\leq1$, 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 $p=1$, we have a necessary condition and a sufficient condition, using $L^1$ and $L^2$ modulus of continuity, respectively, with a rather small gap in between. A second cut-off occurs at $p=1/2$: 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 $L^2(\R)$ and $\ell^2(\Z)$. We further study operations acting on bounded linear operators on $L^{2}(\R^{+})$ related to the class of operators Q_\f. In particular we study Schur multipliers given by functions of the form $\phi(\max\{x,y\})$ and we study properties of the averaging projection (Hilbert-Schmidt projection) onto the operators of the form Q_\f.Comment: 87 page