We will consider infinite matrices indexed by Z (or Z^b) associated to a dynamical system in the sense that\[H={{(H{{(x)}_{m,n}})}_{m,n\in {\bold Z}}}\]satisfies

H(x)_m+1,n+1=H(Tx)_m,n

wherex∈ Ω, andTis an ergodicmeasure-preserving transformation of Ω. Typical settings considered here are\[\begin{array}{*{35}{l}} \Omega ={\bold T} & Tx=x+\omega \quad (1-\text{ frequency shift)} \\ \Omega ={{\bold T}^{d}} & Tx=x+\omega \quad (d-\text{ frequency shift)} \\ \Omega ={{\bold T}^{2}} & Tx=({{x}_{1}}+{{x}_{2}},{{x}_{2}}+\omega )\quad (\text{skewshift)} \\ \Omega ={{\bold T}^{2}} & Tx=Ax,\text{ where }A\in S{{L}_{2}}(\bold Z),\text{ hyperbolic} \\ \end{array}\]

Thus\[H{{(x)}_{m,n}}={{\phi }_{m-n}}({{T}^{m}}x)\caption {(1.0)}\]where theϕ_kare functions on Ω.

We will usually assume thatH(x) is self-adjoint, although many parts of our analysis are independent of this fact. Define

H_N=R_[1,N]HR_[1,N]

whereR_[1,N]= coordinate restriction to [1,N] ⊂ Z, and the associated Green’s functions are

G_N(E) = (H_N−E)⁻¹

(ifH_N−Eis an...

Consider 1D lattice Schrödinger operators of the form

H=υ(T^jx)δ_jj′+ Δ

Assume thatψ= (ψ_j)_j∈Zis a sequence satisfying

Hψ=Eψ

Then\[\left( \begin{matrix} {{\psi }_{n+1}} \\ {{\psi }_{n}} \\ \end{matrix} \right)={{M}_{n}}(E)\left( \begin{matrix} {{\psi }_{1}} \\ {{\psi }_{0}} \\ \end{matrix} \right)\]where\[{{M}_{n}}(E)={{M}_{n}}(E,x)=\prod\limits_{j=n}^{1}{\left( \begin{array}{*{35}{l}} \upsilon ({{T}^{j}}x)-E & -1 \\ 1 & 0 \\ \end{array} \right)}\caption {(2.1)}\]is the transfer (or fundamental) matrix.

Define further\[{{L}_{N}}(E)=\frac{1}{N}\int{\log \left\| {{M}_{N}}(x,E) \right\|dx}\caption {(2.2)}\]and\[\begin{align*} L(E)&=\underset{N\to \infty }{\mathop{\lim }}\,{{L}_{N}}(E) \\ &=\text{ Lyapounov exponent} \\ \end{align}\caption {(2.3)}\]

Observe that by submultiplicativity\[\left\| {{M}_{{{n}_{1}}+{{n}_{2}}}}(x,E) \right\|\le \left\| {{M}_{{{n}_{2}}}}({{T}^{{{n}_{1}}}}x,E \right\|. \left\| {{M}_{{{n}_{1}}}}(x,E) \right\|\]hence\[\begin{array}{*{35}{cl}} {{L}_{{{n}_{1}}+{{n}_{2}}}}(E) & \le \frac{{{n}_{1}}}{{{n}_{1}}+{{n}_{2}}}{{L}_{{{n}_{1}}}}(E)+\frac{{{n}_{2}}}{{{n}_{1}}+{{n}_{2}}}{{L}_{{{n}_{2}}}}(E) \\ L(E) & =\underset{n\to \infty }{\mathop{\lim }}\,{{L}_{n}}(E)\text{ exists} \\ \end{array}\]and by Kingman’s ergodic theorem (assumingTergodic) (see [K])\[L(E)=\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\log \left\| {{M}_{n}}(x,E) \right\|\quad x\ a.e.\]

There is the following relation betweenM_n(E) and determinants\[{{M}_{n}}(x,E)=\left[ \begin{array}{*{35}{cl}} \det ({{H}_{n}}(x)-E) & -\det\, ({{H}_{n-1}}(Tx)-E) \\ \det\, ({{H}_{n-1}}(x)-E) & -\det\, ({{H}_{n-2}}(Tx)-E) \\ \end{array} \right]\caption {(2.4)}\]

Define theintegrated density of statesas\[N(E)=\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{n}\#\left( ]-\infty ,E]\cap \text{Spec }{{H}_{n}}(x) \right)\quad x\ a.e.\]

The relation to the Lyapounov exponent is expressed by the Thouless formula (see [S], for instance)\[L(E)=\int{\log \left| E-{E}' \right|dN({E}')}\caption {(2.5)}\]

The convergence$\frac{1}{N}\log \left\| {{M}_{N}}(x,E) \right\|\to L(E)$can be made...

There is a particularly simple method to obtain lower bounds onL(E) in caseυ(x) is a trigonometric polynomial. The argument is based on Jensen’s inequality. We consider the exampleυ(x) = cosx.

Proposition 3.1.Consider

H(x) =λcos(nω+x)δ_nn′+ Δ

Then\[L(E)\ge \log \frac{\lambda }{2}\caption {(3.2)}\]

Proof. Write\[\begin{array}{*{35}{l}} {{L}_{N}}(E) & =\frac{1}{N}\int{\log \parallel \prod\limits_{N}^{1}{\left( \begin{array}{*{35}{cl}} \lambda \cos (\theta +j\omega )-E & -1 \\ 1 & 0 \\ \end{array} \right)}}\parallel d\theta \\ {} & =\frac{1}{N}\int_{\left| z \right|=1}{\log }\parallel \prod\limits_{N}^{1}{\left( \begin{array}{*{35}{cl}} \frac{\lambda }{2}{{e}^{-ij\omega }}+\frac{\lambda }{2}{{e}^{ij\omega }}{{z}^{2}}-Ez & -z \\ z & 0 \\ \end{array} \right)}\parallel \\ {} & \ge \frac{1}{N}\log \parallel {{\left( \begin{array}{*{35}{cl}} \frac{\lambda }{2}{{e}^{-ij\omega }} & 0 \\ 0 & 0 \\ \end{array} \right)}^{N}}\parallel \quad (\text{by Jensen)} \\ {} & =\log \frac{\lambda }{2}. \\ \end{array}\]

Remarks.

1. The argument clearly generalizes to trigonometric polynomials.

2. For$\upsilon (x)=\cos x,L(E)\ge \log \frac{\lambda }{2}$is optimal as energy-independent lower bound

It follows in particular thatL(E) >c> 0 for allEifλ> 2 (which is the regime of p.p. spectrum and Anderson localization). Ifυis given by a real analytic function on...

The material presented in this chapter appears in [B-G], [G-S] (see Chapter 1), and [B-G-S] with slightly different formulations and proofs.

Assumeu=u(x) 1-periodic with subharmonic extensionũ=ũ(z) to the strip |Imz| < 1 satisfying\[\left| u \right|\le 1,\quad \left| {\tilde{u}} \right|\le B\caption {(4.1)}\]

Apply Riesz representation onDas above.

Thus in particular for$\left| x \right|\le \frac{3}{4}$,\[u(x)=\int{\log \left| x-w \right|\mu (dw)+h(x)}\caption {(4.2)}\]where\[\frac{d\mu }{dw}=\Delta \tilde{u}\]

Henceμis a positive measure onD, andhis harmonic onD.

By (4.1),\[\left\| \mu \right\|\lesssim B\caption {(4.3)}\]\[\left| \partial _{x}^{(\alpha )}h \right|\lesssim B\text{ for }\left| x \right|<\frac{3}{4}\caption {(4.4)}\]

Denote\[\upsilon (x)=\int_{D}{\log \left| x-w \right|\mu (dw)}\]

Hence\[\begin{array}{*{35}{l}} {{\partial }_{x}}\upsilon & =\int{\frac{{x-}{{\operatorname{Re}}\,{w}}}{{{(x-\operatorname{Re}w)}^{2}}+{{(\operatorname{Im}w)}^{2}}}\mu (dw)} \\ {} & ={\mathcal H}[\nu ] \\ \end{array}\caption {(4.5)}\]

(${\mathcal H}$= Hilbert-transform) whereν≥ 0 is the measure on R given by\[\frac{d\nu }{dx}=\int{\frac{\left| \operatorname{Im}w \right|}{{{(x-\operatorname{Re}w)}^{2}}+{{(\operatorname{Im}w)}^{2}}}\mu (dw)}\]

Hence\[\left\| \nu \right\|\le \left\| \mu \right\|\lesssim B\caption {(4.6)}\]

Corollary 4.7.$\left| \hat{u}(k) \right|\lesssim \frac{B}{\left| k \right|}$....

This chapter is related primarily to [B-G] and [G-S] (see Chapter 1).

ForTgiven by a shiftTx=x+ω, withωsatisfying a diophantine condition, we establish inequalities of the form (2.6) for$\frac{1}{N}\log \left\| {{M}_{N}}(x,E) \right\|$. We will use the results from the previous chapter on subharmonic functions. We first treat the cased= 1 and thend= 2.

Theorem 5.1. Assume thatω∈ T satisfies aDC\[\left\| k\omega \right\|>c\frac{1}{\left| k \right|{{\left( \log (1+\left| k \right|) \right)}^{3}}}\, for\ k\in {\bold Z}\backslash \{0\}\caption {(5.2)}\]

Letυbe real analytic onTand\[{{M}_{N}}(x)=\prod\limits_{N}^{1}{\left( \begin{array}{ll} \upsilon (x+j\omega )-E & -1 \\ 1 & 0 \\ \end{array} \right)}\]

Then, forκ>N^−1/10,\[\text{mes}\left[ x\in {\bold T} \right|\left| \frac{1}{N}\log \parallel {{M}_{N}}(x)\parallel -{{L}_{N}}(E) \right|>\kappa ]

Remark. If (5.2) is weakened to a DC\[\left\| k\omega \right\|>c{{\left| k \right|}^{-A}}\text{ for }k\in {\bold Z}\backslash \{0\}\]we still get...

The main result of this chapter is Proposition 6.1 from [G-S] (see Chapter 1).

Proposition 6.1.Let${{A}_{1}},\cdots ,{{A}_{n}}$be a sequence in SL₂(R)satisfying the conditions\[\underset{1\le j\le n}{\mathop{\min }}\,\left\| {{A}_{j}} \right\|\ge \mu >n\caption {(6.2)}\]\[\underset{1\le j\le n}{\mathop{\max }}\,|\log \parallel {{A}_{j}}\parallel +\log \parallel {{A}_{j+1}}\parallel -\log \parallel {{A}_{j+1}}{{A}_{j}}\parallel |<\frac{1}{2}\log \mu \caption {(6.3)}\]

Then\[|\log \parallel {{A}_{n}}\cdots {{A}_{1}}\parallel +\sum\limits_{j=2}^{n-1}{\log \parallel {{A}_{j}}\parallel -}\sum\limits_{j=1}^{n-1}{\log \parallel {{A}_{j+1}}{{A}_{j}}\parallel |

Some notation used in the proof:

AssumeK∈SL₂(R). Denote$u_{K}^{\pm }$the normalized eigenvectors of$\sqrt{{K}^{\ast}{K}}$.

Thus\[Ku_{K}^{+}=\left\| K \right\|\upsilon _{K}^{+}\quad K\upsilon _{K}^{-}={{\left\| K \right\|}^{-1}}\upsilon _{K}^{-}\]where\[\left\| \upsilon _{K}^{+} \right\|=1=\left\| \upsilon _{K}^{-} \right\|\]

GivenK,M∈SL₂(R), denote further forε₁= ±1,ε₂= ±1\[{{b}^{{{\varepsilon }_{1}},{{\varepsilon }_{2}}}}(K,M)=\upsilon _{K}^{{{\varepsilon }_{1}}}\cdot u_{M}^{{{\varepsilon }_{2}}}\](only defined up to the sign).

Proof of Proposition 6.1. First, we observe that\[\begin{align*} \left\| MK \right\|\ge \left\| MKu_{K}^{+} \right\|&=\left\| K \right\|\left\| Mu_{K}^{+} \right\| \\ &\ge\left\| K \right\|({{b}^{+,+}}(K,M).\left\| M \right\|-{{\left\| M \right\|}^{-1}}) \\ \end{align}\]and also\[\left\| MK \right\|\le {{b}^{+,+}}(K,M)\left\| K \right\|.\left\| M \right\|+{{\left\| K \right\|}^{-1}}\left\| M \right\|+\left\| K \right\|{{\left\| M \right\|}^{-1}}\]

In particular\[\frac{\left\| {{A}_{j+1}}{{A}_{j}} \right\|}{\left\| {{A}_{j}} \right\|\left\| {{A}_{j+1}} \right\|}+\frac{1}{{{\left\| {{A}_{j+1}} \right\|}^{2}}}\ge {{b}^{+,+}}({{A}_{j}},{{A}_{j+1}})\ge \frac{\left\| {{A}_{j+1}}{{A}_{j}} \right\|}{\left\| {{A}_{j}} \right\|\left\| {{A}_{j+1}} \right\|}-\frac{1}{{{\left\| {{A}_{j}} \right\|}^{2}}}-\frac{1}{{{\left\| {{A}_{j+1}} \right\|}^{2}}}\caption {(6.5)}\]

Next, one gets for any vectoru\[{{A}_{n}}\cdots {{A}_{1}}u=\sum\limits_{{{\varepsilon }_{1}},\ldots ,{{\varepsilon }_{n}}=\pm 1}{{{\left\| {{A}_{n}} \right\|}^{{{\varepsilon }_{n}}}}\left[ \prod\limits_{j=1}^{n-1}{{{\left\| {{A}_{j}} \right\|}^{{{\varepsilon }_{j}}}}{{b}^{{{\varepsilon }_{j}},{{\varepsilon }_{j+1}}}}({{A}_{j}},{{A}_{j+1}})} \right]}\langle u_{{{A}_{1}}}^{{{\varepsilon }_{1}}},u\rangle \upsilon _{{{A}_{n}}}^{{{\varepsilon }_{n}}}\caption {(6.6)}\]

It follows from (6.5), (6.2),...

The results from this chapter appear in [B-G] and [G-S] (see Chapter 1).

The rotation vectorωis assumed to satisfy a diophantine condition (DC).

Results for generalωhave been obtained more recently in [B-J] and [B] but will not be presented here.

Again, let the transformationTbe given by theω-shift on T^d,d≥ 1, assumingωsatisfying a DC. The potentialυ=υ(x) is assumed real analytic on T^d.

Recall the LDT from Chapter 5 for\[{{M}_{N}}(E,x)=\prod\limits_{N}^{1}{\left( \begin{matrix} \upsilon (x+jw)-E & -1 \\ 1 & 0 \\ \end{matrix} \right)}\]

Thus, fixing a smallκ> 0 and takingNlarge enough, we have\[\text{mes }[x\in {{\bold T}^{d}}|\left| \frac{1}{N}\log \left\| {{M}_{N}}(E,x) \right\|-{{L}_{N}}(E) \right|>\kappa ]<{{e}^{-c{{N}^{\sigma }}}}\caption {(7.1)}\]for some constants...

The purpose of this chapter is to analyze in more detail the estimates of Chapter 7 for the small Lyapounov exponentL(E). We consider only the case of the 1-frequency shift model\[H(x)=\upsilon (x+{{n}_{\omega }}){{\delta }_{n{n}'}}+\Delta \quad \ (x,\omega \in {\bold T})\caption {(8.1)}\]withυ1-periodic and with bounded analytic extension onz=x+iy, |y| ≤ 1, say. Assume again the rotation numberωsatisfying (5.2)\[\left\| kw \right\|>c\frac{1}{\left| k \right|{{\left[ \log \left( 1+\left| k \right| \right) \right]}^{3}}}\text{ for }k\in \bold Z\backslash \{0\}\caption {(8.2)}\](this assumption may be replaced by weaker ones).

Proposition 8.3.Assume that the Lyapounov exponent L(·)of (8.1) satisfies

L(E) > 0for E∈ [E₁,E₂] ⊂ R

Then L(·)and the IDS N(·)are Holder continuous on[E₁,E₂]...

The purpose of this chapter is to summarize a number of results from the literature for later use. Some slightly weaker statements (which do suffice for our needs) also may be found in Bourgain and Goldstein [5] (in Related References) with proof.

Definition 9.1. A set${\mathcal S}\subset {{\bold R}^{n}}$is calledsemialgebraicif it is a finite union of sets defined by a finite number of polynomial equalities and inequalities. More precisely, let${\mathcal P}=\{{{P}_{1}},\ldots ,{{P}_{s}}\}\subset {\bold R}[{{X}_{1}},\ldots ,{{X}_{n}}]$be a family of real polynomials whose degrees are bounded byd. A (closed) semialgebraic set${\mathcal S}$is given by an expression\[{\mathcal S}=\bigcup\limits_{j}{\bigcap\limits_{\ell\in {{\mathcal L}_{j}}}{\{{{\bold R}^{n}}|{{P}_{\ell}}{{s}_{j\ell}}0\}},}\]where${{\mathcal L}_{j}}\subset \{1,\ldots ,s\}$ands_jℓ∈...

We will prove the following theorem.

Theorem 10.1.Consider the 1D lattice Schrödinger operator

H_ω(x) =υ(x+nω)δ_nn′+ Δ

where υ is an analytic potential onT^dand ω∈DC=DC_A,c⊂ T^drefers to frequency vectors satisfying a diophantine condition\[\left\| k.\omega \right\|>c{{\left| k \right|}^{-A}}for\ k\in {{\bold Z}^{d}}\backslash \{0\}\caption {(10.2)}\]

Assume that the Lyapounov exponent\[L(E)={{L}_{\omega }}(E)>{{c}_{0}}\caption {(10.3)}\]

for all ω∈DC and E∈R.

Fix x₀∈ T^d.Then, for almost all ω∈DC, H_ω(x₀)satisfies Anderson localization (i.e., H has p.p. spectrum with exponentially localized states).

Remarks.

1. The cased= 1, 2 was treated in [B-G] (see Chapter 1). The...

The preceding depends explicitly on the fundamental matrix formalism and hence requires nearest neighbor models (thus the off-diagonal is given by Δ). To be precise, the proof of Proposition 7.19 for the Green’s function is based on these techniques. Once this fact is established, our proof of localization would extend, for instance, more generally to Hamiltonians of the form\[H=\upsilon (x+n\omega ){{\delta }_{n{n}'}}+{{S}_{\phi }}\caption {(11.1)}\]with Δ replaced by a Toeplitz operatorS_ϕ\[{{S}_{\phi }}(n,{n}')=\hat{\phi }(n-{n}')\]withϕreal and decaying rapidly enough for |n| → ∞.

Our purpose here is to establish (nonperturbative) localization results in the generality of (11.1). We will first treat the case...

First, we recall some basic facts from spectral theory.

LetHbe a bounded self-adjoint operator on ℓ²(Z). Then, forz∈ C\SpecH, (H−z)⁻¹is analytic (hence in particular for Imz> 0), and we have forf∈ ℓ²\[\operatorname{Im}\langle {{(H-z)}^{-1}}f,f\rangle =\operatorname{Im}z.{{\left\| {{(H-z)}^{-1}}f \right\|}^{2}}\caption {(12.1)}\]

Thus

ϕ_f(z) = 〈(H−z)⁻¹f, f〉

is an analytic function on the upper half plane with Imϕ_f≥ 0 (ϕ_fis a so-called Herglotz function).

Therefore, one has a representation\[{{\phi }_{f}}(z)=\langle {{(H-z)}^{-1}}f,f\rangle =\int_{\bold R}{\frac{1}{\lambda -z}{{\mu }_{f}}(d\lambda )}\]whereμ_fis the spectral measure associated tof. Thus${{\mu }_{f}}\ge 0,\ \left\| {{\mu }_{f}} \right\|={{\left\| f \right\|}^{2}}$.

If forf, g∈ ℓ²we let\[{{\mu }_{f,g}}=\frac{1}{4}[{{\mu }_{f+g}}-{{\mu }_{f-g}}+i({{\mu }_{f+ig}}-{{\mu }_{f-ig}})]\]then\[\langle {{(H-z)}^{-1}}f,g\rangle =\int_{\bold R}{\frac{1}{\lambda -z}{{\mu }_{f,g}}(d\lambda )}\]...

Consider Schrödinger operators on Z\[{{H}_{\lambda }}(x)=\lambda \upsilon (x+n\omega ){{\delta }_{n{n}'}}+\Delta \caption {(13.1)}\]whereυis a nonconstant trigonometric polynomial on T^d. We proved that ifλ>λ₀(υ) then typically pure point spectrum with localization occurs. Ifd= 1, then forλ<λ₁(υ), one obtains purely (absolutely) continuous spectrum. These results are nonperturbative in the sense thatλ₀,λ₁do not depend onω(which is always assumed diophantine).

It turns out that ford≥ 2, we cannot expect such nonperturbative statements for the continuous spectrum at small disorder. Notice that the dual model of (13.1) is a lattice Schrödinger operator on the Z^d-lattice\[{{\tilde{H}}_{\lambda }}(x)=\cos (x+n.\omega )+\frac{\lambda }{2}{{S}_{\upsilon }}\caption {(13.2)}\]

Localization results...

The main result of this chapter is as follows:

Proposition 14.1.Let A(σ)be a self-adjoint N×N matrix function of a real parameter σ∈ [−δ,δ], satisfying the following conditions

(i)A(σ)is real analytic in σ, and there is a holomorphic extension to a strip\[\left| \operatorname{Re}z \right|<\delta ,\left| \operatorname{Im}z \right|<\gamma \caption {(14.2)}\]satisfying\[\left\| A(z) \right\|<{{B}_{1}}\caption {(14.3)}\]

(ii)For each σ∈ [−δ,δ], there is a subsetΛ ⊂ [1,N]s.t.\[\left| \Lambda \right|and\[\left\| {{({{R}_{[1,N]\backslash \Lambda }}A(\sigma ){{\mathcal R}_{[1,N]\backslash \Lambda }})}^{-1}} \right\|<{{B}_{2}}\caption {(14.5)}\]

(iii)\[\text{mes }[ \sigma \in [-\delta ,\delta ] \bigg\rvert \left\| A{{(\sigma )}^{-1}} \right\|>{{B}_{3}}]<{{10}^{-3}}\gamma {{(1+{{B}_{1}})}^{-1}}{{(1+{{B}_{2}})}^{-1}}\caption {(14.6)}\]Then, letting

κ< (1 +B₁+B₂)^−10M

we have\[\text{mes }\left[ \sigma \in \left[ -\frac{\delta }{2},\frac{\delta }{2} \right]\left| \left\| A{{(\sigma )}^{-1}} \right\|>\frac{1}{\kappa } \right. \right]<{{e}^{-\frac{c\log {{\kappa }^{-1}}}{M.\log (M+{{B}_{1}}+{{B}_{2}}+{{B}_{3}})}}}\caption {(14.7)}\]

Proof. Denote

δ₁= 10⁻²γ(1 +B₁)⁻¹(1 +B₂)⁻¹

Fix${{\sigma }_{0}}\in \left[ -\frac{\delta }{2},\frac{\delta }{2} \right]$. If$z\in {\bold C},\ \left| z-{{\sigma }_{0}} \right|<{{\delta }_{1}}$, it follows that...

We consider 1D lattice Schrödinger operatorsH(x),x∈ T^dassociated with a skew shift transformationT: T^d→ T^d, and thusH_m+1,n+1(x) =H_m,n(Tx). To simplify matters, letd= 2 and

T: T²→ T²: (x₁,x₂) ↦ (x₁+x₂,x₂+ω)

(the method applies equally well to higher-dimensional skew shift extensions).

We always assumeωsatisfying a DC. To avoid additional parameters, assume, say,\[\left\| k.\omega \right\|>c{{\left| k \right|}^{-2}}\,\text{for }k\in {\bold Z}\backslash \{0\}\caption {(15.0)}\]

H(xwill be given by\[H(x)=V({{T}^{n}}x){{\delta }_{n{n}'}}+\delta \Delta \caption {(15.1)}\]whereVis a real nonconstant trigonometric polynomial on T².

More generally, we will considerH(x) of the form\[{{H}_{nn}}(x)=V({{T}^{n}}x)\caption {(15.2)}\]and for...

We consider the time-dependent Schrödinger equation on T = R/Z\[i\frac{\partial \Psi (t,x)}{\partial t}=a\frac{{{\partial }^{2}}\Psi (t,x)}{\partial {{x}^{2}}}+ib\frac{\partial \Psi (t,x)}{\partial x}+V(t,x)\Psi (t,x)\caption {(16.1)}\]with potential\[V(t,x)=\kappa \cos 2\pi x\left( \sum\limits_{n\in \bold Z}{\delta (t-n)} \right)\caption {(16.2)}\]corresponding to a periodic sequence of kicks. The monodromy operatorWdefined by\[W\psi (t,x)=\Psi (t+1,x)\caption {(16.3)}\](= time-1 shift under the flow of (16.1)) is a unitary operator onL²(T). In the case of (16.1),Wis given by\[W={{U}_{a,b}}\cdot {{W}_{1,\kappa }}\caption {(16.4)}\]whereU_a,b\[{{U}_{a,b}}={{e}^{i\left( a\frac{{{d}^{2}}}{d{{x}^{2}}}+ib\frac{d}{dx} \right)}}\caption {(16.5)}\]and\[{{W}_{1,\kappa }}=\text{multiplication operator by}\ {{e}^{ik\cos 2\pi x}}\equiv \rho (x)\caption {(16.6)}\](see [Sin] and [Bel]).

After passing to Fourier transform,U_a,bbecomes a diagonal matrix\[{{U}_{ab}}={{e}^{-i(4{{\pi }^{2}}a{{n}^{2}}+2\pi bn)}}{{\delta }_{mn}}\caption {(16.7)}\]andW_1,κbecomes a Toeplitz matrix\[{{W}_{1,\kappa }}(m,n)=\hat{\rho }(m-n)\caption {(16.8)}\]where one easily verifies that\[\hat{\rho }(0)=1+0({{\kappa }^{2}})\ and\ |\hat{\rho }(k)|<\sqrt{\kappa }{{e}^{-c\left( \log \frac{1}{\kappa } \right)|k|}}\text{ for }{k}\in {\bold Z }\backslash \{ 0 \}\caption {(16.9)}\]

Hence, forκsmall,W_1,κis a perturbation of the identity with exponential off-diagonal decay....

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 Z^d, 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 T^d. 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∈ T^ddiophantine.

On the Z^d-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...

The problem is that of persistency ofb-dimensional tori in R^2b× R^2r-phase space for a real analytic HamiltonianHof the form\[\begin{array}{ll} H=H(I,\theta ,y)&=H({{I}_{1}},\ldots ,{{I}_{b}},{{\theta }_{1}},\ldots ,{{\theta }_{b}},{{y}_{1}},\ldots ,{{y}_{r}}) \\ &=\left\langle {{\lambda }_{0}},I \right\rangle +\sum\limits_{s=1\grave{\ }}^{r}{{{\mu }_{s}}|{{y}_{s}}{{|}^{2}}+|I{{|}^{2}}+\varepsilon {{H}_{1}}(I,\theta ,y)} \\ \end{array}\caption {(18.1)}\](the last term is perturbative), as considered in [E], [Kuk], [Pos], and [B1]. Here,I= (I₁, …,I_b),θ= (θ₁, …,θ_b) are action-angle variables for the “tangential” part of the phase space, andy= (y₁, …,y_r) are the “normal” coordinates. The vectorλ₀is diophantine and satisfying certain nonresonance conditions; in [E] and [Kuk], the conditions imposed are\[\left\langle {{\lambda }_{0}},k \right\rangle -{{\mu }_{s}}\ne 0 \quad (k\in {\bold Z}^{b},s=1,\ldots ,r)\caption {(18.2)}\]and also\[\left\langle {{\lambda }_{0}},k \right\rangle +{{\mu }_{s}}-{{\mu }_{{{s}'}}}\ne 0\quad (s\ne {s}')\caption {(18.3)}\]

In [B1], the persistency result (as stated in [E])...

We consider next the problem of persistency of finite-dimensional tori in infinite-dimensional phase space in the context of the nonlinear Schrödinger equation (NLS) with periodic boundary conditions. More specifically, consider a nonlinear perturbation of a linear equation of the form\[\frac{1}{i}{{q}_{t}}={\cal L}q+\varepsilon \frac{\partial {{H}_{1}}}{\partial \bar{q}}\caption {(19.1)}\]whereH₁=H₁(q, q̄) is a real polynomial expression inq, q̄with real coefficients.

Hereq= (q_n)_n∈Z^d. Select a finite set of modesn₁, …,n_b∈ Z^d. The operator${\mathcal L}$is given by a multiplier (μ_n)_n∈Z^d, where\[\left\{ \begin{array}{ll} {{\mu }_{{{n}_{j}}}}={{\lambda }_{j}}\quad (1\le j\le b) \\ {{\mu }_{n}}={{\left| n \right|}^{2}}\text{ for }n\in {{\bold Z}^{d}}\backslash \{{{n}_{1}},\ldots ,{{n}_{b}}\} \\ \end{array} \right.\caption {(19.2)}\](corresponding to the Δ-operator in the Schrödingermodel). Again,λ= (λ₁, …,λ_b) is a parameter taken...

Consider NLW with periodic boundary condition (x∈ T^d) of the form\[{{{y}_{tt}}}-{{\Delta}}y+\varepsilon {F}'(y)=0\caption {(20.1)}\]whereF(y) is a polynomial iny.

The construction of time-periodic solutions was achieved in arbitrary dimensiond(see [B1]). Quasi-periodic solutions were so far only produced ford= 1 (see [B2], [Kuk], and [Wa] with 1D Dirichlet bc). In this chapter we will indicate how the methods described in the preceding chapter may be used to treat this problem in general dimensiond. Rather than relying on amplitude-frequency modulation, extracting parameters from the nonlinearity, we discuss again nonlinear perturbations of a linear equation with parameters....