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We prove some local estimates on the trace of spectral projectors for random Schr\"odinger operators restricted to cubes $\Lambda \subset R^d$. We also present a new proof of the spectral averaging result based on analytic perturbation theory. Together, these provide another proof of the Wegner estimate with an explicit form of the constant and an alternate proof of the Birman-Solomyak formula. We also use these results to prove the Lipschitz continuity of the local density of states function for a restricted family of random Schr\"odinger operators on cubes $\Lambda \subset R^d$, for $d \geq 1$. The result holds for low energies without a localization assumption but is not strong enough to extend to the infinite-volume limit., Comment: To appear in a volume dedicated to the memory of Erik Balslev

We introduce quantum hypercube states, a class of continuous-variable quantum states that are generated as orthographic projections of hypercubes onto the quadrature phase-space of a bosonic mode. In addition to their interesting geometry, hypercube states display phase-space features much smaller than Planck's constant, and a large volume of Wigner-negativity. We theoretically show that these features make hypercube states sensitive to displacements at extremely small scales in a way that is surprisingly robust to initial thermal occupation and to small separation of the superposed state-components. In a high-temperature proof-of-principle optomechanics experiment we observe, and match to theory, the signature outer-edge vertex structure of hypercube states., Comment: Main consists of 5 pages and 5 figures. Supplementary material consists of 5 pages and 6 figures

We present some results on the monotonicity of some traces involving functions of self-adjoint operators with respect to the natural ordering of their associated quadratic forms. We also apply these results to complete a proof of the Wegner estimate for continuum models of random Schr\"odinger operators as given in \cite{co-hi1}.

We review various formulations of conductivity for one-particle Hamiltonians and relate them to the current-current correlation measure. We prove that the current-current correlation measure for random Schr\"odinger operators has a density at coincident energies provided the energy lies in a localization regime. The density vanishes at such energies and an upper bound on the rate of vanishing is computed. We also relate the current-current correlation measure to the localization length.

We study the contribution of different types of scattering sources to the extraordinary Hall effect. Scattering by magnetic nano-particles embedded in normal-metal matrix, insulating impurities in magnetic matrix, surface scattering and temperature dependent scattering are experimentally tested. Our new data, as well as previously published results on a variety of materials, are fairly interpreted by a simple modification of the skew scattering model.

A low concentration of cobalt clusters with a fcc structure and containing almost one thousand atoms are embedded in two different metallic matrices: platinum and niobium. Samples have been prepared using a co-deposition technique. Cobalt clusters preformed in the gas phase and matrix atoms are simultaneously deposited on a silicon substrate under Ultra High Vacuum conditions. This original technique allows to prepare nanostructured systems from miscible elements such as Co/Pt and Co/Nb in which clusters keep a pure cobalt core surrounded with an alloyed interface. Magnetic measurements performed using a Vibrating Sample Magnetometer (VSM) reveal large differences in the magnetic properties of cobalt clusters in Pt and Nb pointing out the key role of cluster/matrix interfaces., Comment: 7 pages (LaTeX), 12 PostScript figures, 1 PostScript table

We prove the existence of localized states at the edges of the bands for the two-dimensional Landau Hamiltonian with a random potential, of arbitrary disorder, provided that the magnetic field is sufficiently large. The corresponding eigenfunctions decay exponentially with the magnetic field and distance. We also prove that the integrated density of states is Lipschitz continuous away from the Landau energies. The proof relies on a Wegner estimate for the finite-area magnetic Hamiltonians with random potentials and exponential decay estimates for the finite-area Green's functions. The proof of the decay estimates for the Green's functions uses fundamental results from two-dimensional bond percolation theory., Comment: 34 pages,CPT-94/P.3061,LaTex