Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. III. Localization properties

Consider the Anderson Hamiltonian H V = κ∆ V + ξ(·) on the multidimensional lattice torus V increasing to the whole of lattice, where ξ(·) is an i.i.d. potential with distribution function F . For K = 1, 2, . . ., let ψ(·; λ K, V ) be the eigenfunction of H V associated with the Kth largest eigenvalue λ K, V , and let z K, V ∈ V be the coordinate of the Kth larger value ξ K, V of ξ(·) in V . It is well-known that if F satisfies the condition log − log(1 − F (t)) = o(t) and some additional conditions on regular variation and continuity at infinity, then ψ(·; λ K, V ) is (asymptotically) completely localized at the site z τ (K), V , as a localization centre for the eigenfunction for some (random) τ (K) = τ V (K) 1. In this paper, we study the asymptotic behavior in probability of the indices τ V (K) as V increases and K 1 is fixed. In particular, we show that if F satisfies the condition − log(1 − F (t)) = O(t 3 ) (resp., −t −3 log(1 − F (t)) → ∞) and additional regularity conditions at infinity, then τ V (K) = O(1) (resp., τ V (K) → ∞) with high probability. For Weibull's and double exponential types distributions, we obtain the first order expansion formulas for log τ V (K)

Sub-surface modifications in silicon with ultra-short pulsed lasers above 2 µm

Nonlinear optical phenomena in silicon such as self-focusing and multi-photon absorption are strongly dependent on the wavelength, energy, and duration of the exciting pulse, especially for wavelengths >2µm. We investigate the sub-surface modification of silicon using ultra-short pulsed lasers at wavelengths in the range of 1950–2400 nm, at a pulse duration between 2 and 10 ps and pulse energy varying from 1 µJ to 1 mJ. We perform numerical simulations and experiments using fiber-based lasers built in-house that operate in this wavelength range for the surface and sub-surface processing of Si-wafers. The results are compared to the literature data at 1550 nm. Due to a dip in the nonlinear absorption spectrum and a peak in the spectrum of the third-order nonlinearity, the wavelengths between 2000 and 2200 nm prove to be more favorable for creating sub-surface modifications in silicon. This is the case even though those wavelengths do not allow as tight focusing as those at 1550 nm. This is compensated for by an increased self-focusing due to the nonlinear Kerr-effect around 2100 nm at high light intensities, characteristic for ultra-short pulses

Labour Market and Social Policy in Italy: Challenges and Changes. Bertelsmann Policy Brief #2016/02

vEight years after the outbreak of the financial crisis, Italy has still to cope with and overcome a plethora of economic and social challenges. On top of this, it faces an unfavourable demographic structure and severe disparities between its northern and southern regions. Some promising reforms have recently been enacted, specifically targeting poverty and social exclusion. However, much more remains to be done on the way towards greater economic stability and widely shared prosperity