Spectral extrema and Lifshitz tails for non monotonous alloy type models
نویسندگان
چکیده
In the present note, we determine the ground state energy and study the existence of Lifshitz tails near this energy for some non monotonous alloy type models. Here, non monotonous means that the single site potential coming into the alloy random potential changes sign. In particular, the random operator is not a monotonous function of the random variables. Résumé. Cet article est consacré à la détermination de l’énergie de l’état fondamental et à l’étude de possibles asymptotiques de Lifshitz au voisinage de cette énergie pour certains modèles d’Anderson continus non monotones. Ici, non monotone signifie que le potentiel de simple site entrant dans la composition du potentiel aléatoire change de signe. En particulier, l’opérateur aléatoire n’est pas une fonction monotone des variables aléatoires. 0 Introduction and results In this paper, we consider the continuous alloy type (or Anderson) random Schrödinger operator: Hω = −∆+ Vω where Vω(x) = ∑ γ∈Zd ωγV (x− γ) (0.1) on Rd, d ≥ 1, where V is the site potential, and (ωγ)γ∈Zd are the random coupling constants. Throughout this paper, we assume (H1) (1) V : Rd → R is Lp (where p = 2 if d ≤ 3 and p > d/2 if d > 3), non identically vanishing and supported in (−1/2, 1/2)d ; LAGA, U.M.R. 7539 C.N.R.S, Institut Galilée, Université de Paris-Nord, 99 Avenue J.-B. Clément, F-93430 Villetaneuse, France et Institut Universitaire de France. Email: [email protected] Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, Japan 153-8914. Email: [email protected]
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تاریخ انتشار 2008