We study the ac-conductivity in linear response theory in the general framework of ergodic magnetic Schrödinger operators. For the Anderson model, if the Fermi energy lies in the localization regime, we prove that the acconductivity is bounded from above by Cν 2 (log 1 ν ) d+2 at small frequencies ν. This is to be compared to Mott's formula, which predicts the leading term to be Cν 2 (log 1 ν ) d+1 . IntroductionThe occurrence of localized electronic states in disordered systems was first noted by Anderson in 1958 [An], who argued that for a simple Schrödinger operator in a disordered medium,"at sufficiently low densities transport does not take place; the exact wave functions are localized in a small region of space." This phenomenon was then studied by Mott, who wrote in 1968 [Mo1]: "The idea that one can have a continuous range of energy values, in which all the wave functions are localized, is surprising and does not seem to have gained universal acceptance." This led Mott to examine Anderson's result in terms of the Kubo-Greenwood formula for σ EF (ν), the electrical alternating current (ac) conductivity at Fermi energy E F and zero temperature, with ν being the frequency. Mott used its value at ν = 0 to reformulate localization: If a range of values of the Fermi energy E F exists in which σ EF (0) = 0, the states with these energies are said to be localized; if σ EF (0) = 0, the states are nonlocalized.Mott then argued that the direct current (dc) conductivity σ EF (0) indeed vanishes in the localized regime. In the context of Anderson's model, he studied the behavior of Re σ EF (ν) as ν → 0 at Fermi energies E F in the localization region (note Im σ EF (0) = 0). The result was the well-known Mott's formula for the ac-conductivity at zero temperature [Mo1], [Mo2], which we state as in

We present a rigorous study of the perfect Bose-gas in the presence of a homogeneous ergodic random potential. It is demonstrated that the Lifshitz tail behaviour of the one-particle spectrum reduces the critical dimensionality of the (generalized) Bose-Einstein Condensation (BEC) to d = 1. To tackle the Off-Diagonal Long-Range Order (ODLRO) we introduce the space average one-body reduced density matrix. For a one-dimensional Poisson-type random potential we prove that randomness enhances the exponential decay of this matrix in domain free of the BEC.

Fully epitaxial Fe(001)∕Fe3O4(001)∕MgO(001)∕Co micron-sized magnetic tunnel junctions have been elaborated on MgO(001) substrates. X-ray reflectivity and high-resolution transmission electron microscopy revealed a good quality and epitaxial growth of the stack with abrupt interfaces. The magnetotransport measurements exhibit a large negative tunneling magnetoresistance (TMR) value for magnetic tunnel junctions including an Fe3O4 layer and a MgO tunnel barrier (−8.5% at 300K and −22% at 80K). Moreover, the sign of the TMR changes with the applied bias. We discuss here the structural quality of the samples and the transport measurement results.

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