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6 can be also generalized to the continuous case in a similar way. Acknowledgements. I thank F. Germinet for stimulating discussions on the subject of the paper. 56 S. Tcheremchantsev References 1. : Localization at weal disorder: Some elementary bounds. Rev. Math. Phys. 6, 1163–1182 (1994) 2. : Finite-volume fractional-moment criteria for Anderson localization. To appear in Commun. Math. Phys. 3. Damanik, D. : Multi-scale analysis implies strong dynamical localization. Preprint (1999) 4. De Bièvre, S.
4. 1. For a given vector ψ ∈ HM consider the function R(n) = supk |γk gk (n)|, where |gk (n)|2 = Kn (1) |ek (x)|2 dx ≤ 1, n ∈ Zd and γk = ψ, ek . If the function R(n) is fast decaying then DL(ψ) hold. Proof. As the function R(n) is fast decaying, for any r > 0, (|n| + 1)r |γk gk (n)|2 ≤ |γk |2 wk (r) = (|n| + 1)r R 2 (n) ≤ C(r) < +∞. n n On the other hand, wk (r) ≥ Dk r/d after reordering. Therefore, |γk | ≤ C(m)k −m for any m > 0. 17) Next, as ∞ |ψ(t, x)| ≤ |γk ek (x)|, k=1 for any n ∈ Zd the bound holds: ∞ |ψ(t, x)| dx ≤ Kn (1) 2 |γk gk (n)| 2 ≡ S 2 (n).
This work was supported in part by the NATO collaborative research grant CRG. 960188. Work supported in part by the National Science Foundation under grant number PHY-9802510. Permanent address: Physics Department, Penn State University, University Park, PA 16802, USA. 58 M. Günaydin, K. Koepsell, H. Nicolai In this paper, we will present a novel construction involving the maximally extended Lie group E8(8) . This construction of E8(8) together with the corresponding construction of E8(−24) contain all previous examples of generalized space-times based on exceptional Lie groups, and at the same time goes beyond the framework of Jordan algebras.