Provider: JSTOR http://www.jstor.org Database: JSTOR Content: text/plain; charset="us-ascii" TY - CHAP TI - Quasi-Periodic Localization on the Zd-lattice (d > 1) A2 - Bourgain, J. AB -

Consider quasi-periodicd-dimensional lattice Schrödinger operators${{H}_{\lambda }}(x)=\lambda \upsilon ({{x}_{1}}+{{n}_{1}}{{\omega }_{1}},\ldots ,{{x}_{d}}+{{n}_{d}}{{\omega }_{d}}){{\delta }_{n{n}'}}+\Delta \quad (n\in {{\bold Z}^{d}})\caption {(17.1)}$with Δ the lattice Laplacian on Zd, i.e.,$\begin{matrix} \Delta (n,{n}')=1\text{ if }\sum{\left| {{n}_{j}}-{{{{n}'}}_{j}} \right|=1} \\ =0\quad \text{otherwise} \\ \end{matrix}$

We assumeυa trigonometric polynomial or real analytic function on Td. More generally, one may consider operators of the form$H(x)=\lambda \upsilon ({{x}_{1}}+{{n}_{1}}{{\omega }_{1}},\ldots ,{{x}_{d}}+{{n}_{d}}{{\omega }_{d}}){{\delta }_{n{n}'}}+{{S}_{\phi }}\caption {(17.2)}$where${{S}_{\phi }}(n,{n}')=\hat{\phi }(n-{n}')$is a Toeplitz operator with real analytic symbol. We always assumeω∈ Tddiophantine.

On the Zd-lattice,d> 1, the transfer matrix approach to localization is not available, and in fact, all our results are perturbative. They will be obtained by adaptation of the method developed in Chapters 14 and 15 to thed-dimensional setting.

The main result is

EP - 132 PB - Princeton University Press PY - 2005 SN - 9780691120980 SP - 123 T2 - Green's Function Estimates for Lattice Schrodinger Operators and Applications. (AM-158) UR - http://www.jstor.org/stable/j.ctt7zvc64.20 Y2 - 2020/09/28/ ER -