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

Large spin relaxation rates in trapped submerged-shell atoms

Spin relaxation due to atom-atom collisions is measured for magnetically trapped erbium and thulium atoms at a temperature near 500 mK. The rate constants for Er-Er and Tm-Tm collisions are 3.0 times 10^-10 cm^3 s^-1 and 1.1 times 10^-10 cm^3 s^-1, respectively, 2-3 orders of magnitude larger than those observed for highly magnetic S-state atoms. This is strong evidence for an additional, dominant, spin relaxation mechanism, electrostatic anisotropy, in collisions between these "submerged-shell" L > 0 atoms. These large spin relaxation rates imply that evaporative cooling of these atoms in a magnetic trap will be highly inefficient.Comment: 10 pages, 3 figure

Performance of the modified Becke-Johnson potential

Very recently, in the 2011 version of the Wien2K code, the long standing shortcome of the codes based on Density Functional Theory, namely, its impossibility to account for the experimental band gap value of semiconductors, was overcome. The novelty is the introduction of a new exchange and correlation potential, the modified Becke-Johnson potential (mBJLDA). In this paper, we report our detailed analysis of this recent work. We calculated using this code, the band structure of forty one semiconductors and found an important improvement in the overall agreement with experiment as Tran and Blaha [{\em Phys. Rev. Lett.} 102, 226401 (2009)] did before for a more reduced set of semiconductors. We find, nevertheless, within this enhanced set, that the deviation from the experimental gap value can reach even much more than 20%, in some cases. Furthermore, since there is no exchange and correlation energy term from which the mBJLDA potential can be deduced, a direct optimization procedure to get the lattice parameter in a consistent way is not possible as in the usual theory. These authors suggest that a LDA or a GGA optimization procedure is used previous to a band structure calculation and the resulting lattice parameter introduced into the 2011 code. This choice is important since small percentage differences in the lattice parameter can give rise to quite higher percentage deviations from experiment in the predicted band gap value.Comment: 10 pages, 2 figures, 5 Table

Regulation of CFTR ion channel gating by MgATP

AbstractSingle channel currents of wild-type CFTR reconstituted in lipid bilayers were recorded to study the temperature dependence of channel gating between +20°C and +40°C. The opening of the channel was highly temperature dependent and required an activation energy of about 100 kJ/mol. Closing of the channel was only weakly temperature dependent with an activation energy close to that of diffusion in water. We found no significant difference in the free energy between the open and closed states. Most of the excess energy needed to activate channel opening is used to diminish the entropy of the open state. This structural reorganization is initiated by ATP binding followed by interconversion to the open channel structure as the CFTR-ATP-Mg complex passes to the transition state for hydrolysis. The energy of the CFTR-ATP-Mg interaction in the transition state is responsible for the CFTR ion channel opening rather than the energy of ATP hydrolysis. Channel closing is a diffusion limited process and does not require additional ATP binding

Noise spectroscopy of optical microcavity

The intensity noise spectrum of the light passed through an optical microcavity is calculated with allowance for thermal fluctuations of its thickness. The spectrum thus obtained reveals a peak at the frequency of acoustic mode localized inside the microcavity and depends on the size of the illuminated area. The estimates of the noise magnitude show that it can be detected using the up-to-date noise spectroscopy technique.Comment: 10 pages, 1 figur

Characterizing normal crossing hypersurfaces

The objective of this article is to give an effective algebraic characterization of normal crossing hypersurfaces in complex manifolds. It is shown that a hypersurface has normal crossings if and only if it is a free divisor, has a radical Jacobian ideal and a smooth normalization. Using K. Saito's theory of free divisors, also a characterization in terms of logarithmic differential forms and vector fields is found and and finally another one in terms of the logarithmic residue using recent results of M. Granger and M. Schulze.Comment: v2: typos fixed, final version to appear in Math. Ann.; 24 pages, 2 figure

Approximating Nearest Neighbor Distances

Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the Euclidean cluster diameter is large. Therefore, it is preferred to assign smaller costs to the paths that stay close to the input points. In this paper, we consider the most natural metric with this property, which we call the nearest neighbor metric. Given a point set P and a path $\gamma$, our metric charges each point of $\gamma$ with its distance to P. The total charge along $\gamma$ determines its nearest neighbor length, which is formally defined as the integral of the distance to the input points along the curve. We describe a $(3+\varepsilon)$-approximation algorithm and a $(1+\varepsilon)$-approximation algorithm to compute the nearest neighbor metric. Both approximation algorithms work in near-linear time. The former uses shortest paths on a sparse graph using only the input points. The latter uses a sparse sample of the ambient space, to find good approximate geodesic paths.Comment: corrected author nam

Optical measurements of spin noise as a high resolution spectroscopic tool

The intrinsic fluctuations of electron spins in semiconductors and atomic vapors generate a small, randomly-varying "spin noise" that can be detected by sensitive optical methods such as Faraday rotation. Recent studies have demonstrated that the frequency, linewidth, and lineshape of this spin noise directly reveals dynamical spin properties such as dephasing times, relaxation mechanisms and g-factors without perturbing the spins away from equilibrium. Here we demonstrate that spin noise measurements using wavelength-tunable probe light forms the basis of a powerful and novel spectroscopic tool to provide unique information that is fundamentally inaccessible via conventional linear optics. In particular, the wavelength dependence of the detected spin noise power can reveal homogeneous linewidths buried within inhomogeneously-broadened optical spectra, and can resolve overlapping optical transitions belonging to different spin systems. These new possibilities are explored both theoretically and via experiments on spin systems in opposite limits of inhomogeneous broadening (alkali atom vapors and semiconductor quantum dots).Comment: 4 pages, 4 figure