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91.
In this paper we show that for a.e. x∈[ 0,2 π) the operators defined on as
and with Dirichlet condition ψ− 1= 0, have pure point spectrum in with exponentially decaying eigenfunctions where δ > 0 and are small. As it is a simple consequence of known techniques that for small λ one has [− 2 +δ, 2−δ]⊂ spectrum (H(x)) for a.e.x∈[ 0, 2 π), we thus established Anderson localization on the spectrum up to the edges and the center. More general potentials
than cosine can be treated, but only those energies with nonzero spectral density are allowed. Finally, we prove the same
result for operators on the whole line ℤ with potential , where A:?2→?2 is a hyperbolic toral automorphism, F∈C
1(?2), ∫F= 0, and λ small. The basis for our analysis is an asymptotic formula for the Lyapunov exponent for λ→ 0 by Figotin–Pastur,
and generalized by Chulaevski–Spencer. We combine this asymptotic expansion with certain martingale large deviation estimates
in order to apply the methods developed by Bourgain and Goldstein in the quasi-periodic case.
Received: 28 January 2000 / Accepted: 14 June 2000 相似文献
92.
In this paper we establish relaxation of an arbitrary 1-equivariant wave map from ${\mathbb{R}^{1+3}_{t,x}{\setminus} (\mathbb{R}\times B(0,1))\to S^3}$ of finite energy and with a Dirichlet condition at r = 1, to the unique stationary harmonic map in its degree class. This settles a recent conjecture of Bizoń, Chmaj, Maliborski who observed this asymptotic behavior numerically. 相似文献
93.
We consider the radial free wave equation in all dimensions and derive asymptotic formulas for the space partition of the energy, as time goes to infinity. We show that the exterior energy estimate, which Duyckaerts et al. obtained in odd dimensions (Duyckaerts et al., J Eur Math Soc 13:533–599, 2011; J Eur Math Soc, 2013) fails in even dimensions. Positive results for restricted classes of data are obtained. 相似文献