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

A quantitative study of spin noise spectroscopy in a classical gas of 41^{41}K atoms

We present a general derivation of the electron spin noise power spectrum in alkali gases as measured by optical Faraday rotation, which applies to both classical gases at high temperatures as well as ultracold quantum gases. We show that the spin-noise power spectrum is determined by an electron spin-spin correlation function, and we find that measurements of the spin-noise power spectra for a classical gas of $^{41}$K atoms are in good agreement with the predicted values. Experimental and theoretical spin noise spectra are directly and quantitatively compared in both longitudinal and transverse magnetic fields up to the high magnetic field regime (where Zeeman energies exceed the intrinsic hyperfine energy splitting of the $^{41}$K ground state)

Toeplitz Schur multipliers of Sp(L2(G))S_p (L^2 (G)) for p<1p<1

We study Toeplitz Schur miltipliers of Schatten-von Neumann class $S_p$ for $0 Mots-clé, Keywords : Schur multipliers, Schatten-von Neumann classes, commutative locally compact groups

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

Simulations of magnetic and magnetoelastic properties of Tb2Ti2O7 in paramagnetic phase

Magnetic and magnetoelastic properties of terbium titanate pyrochlore in paramagnetic phase are simulated. The magnetic field and temperature dependences of magnetization and forced magnetostriction in Tb2Ti2O7 single crystals and polycrystalline samples are calculated in the framework of exchange charge model of crystal field theory and a mean field approximation. The set of electron-deformation coupling constants has been determined. Variations of elastic constants with temperature and applied magnetic field are discussed. Additional strong softening of the crystal lattice at liquid helium temperatures in the magnetic field directed along the rhombic symmetry axis is predicted.Comment: 13 pages, 4 figures, 2 table