We show persistence of both Anderson and dynamical localization in Schrödinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schrödinger operator with a non-positive ergodic random potential, and multiply the random potential by a decaying envelope function. If the envelope function decays slower than |$x$|^{-2}at infinity, we prove that the operator has infinitely many eigenvalues below zero. For envelopes decaying as |$x$|^{-α}at infinity, we determine the number of bound states below a given energy$E$<0, asymptotically as α↓0. To show that bound states located at the bottom of the spectrum are related to the phenomenon of Anderson localization in the corresponding ergodic model, we prove: (a) these states are exponentially localized with a localization length that is uniform in the decay exponent α; (b) dynamical localization holds uniformly in α.

With a given holomorphic section of a Hermitian vector bundle, one can associate a residue current by means of Cauchy–Fantappiè–Leray type formulas. In this paper we define products of such residue currents. We prove that, in the case of a complete intersection, the product of the residue currents of a tuple of sections coincides with the residue current of the direct sum of the sections.

In this paper we consider operators of the form$H$=λ(-$i$∇), with λ analytic in a strip and with some specific growth conditions at infinity, and prove Hardy type estimates in$L$^{2}(ℝ^{$n$}) with exponential weights. In fact we extend our previous results [19] from bounded analytic functions on a strip to analytic functions with polynomial growth in that strip.

The purpose of this note is to give an extension of the symbolic calculus of Melin for convolution operators on nilpotent Lie groups with dilations. Whereas the calculus of Melin is restricted to stratified nilpotent groups, the extension offered here is valid for general homogeneous groups. Another improvement concerns the$L$^{2}-boundedness theorem, where our assumptions on the symbol are relaxed. The zero-class conditions that we require are of the type $$|D^{\alpha}a(\xi)|\le C_{\alpha}\prod_{j=1}^R\rho_j(\xi)^{-|\alpha_j|},$$ where ρ_{$j$}are “partial homogeneous norms” depending on the variables ξ_{$k$}for$k$>$j$in the natural grading of the Lie algebra (and its dual) determined by dilations. Finally, the class of admissible weights for our calculus is substantially broader. Let us also emphasize the relative simplicity of our argument compared to that of Melin.

We give an elementary proof of the formula χ($K$^{$n$}A)=$n$^{3}σ($n$) for the Euler characteristic of the generalized Kummer variety$K$^{$n$}$A$, where σ($n$) denotes the sum of divisors function.