# Green's Function Estimates for Lattice Schrodinger Operators and Applications. (AM-158)

J. Bourgain
Pages: 200
https://www.jstor.org/stable/j.ctt7zvc64

1. Front Matter
(pp. i-ii)
(pp. iii-iv)
3. Acknowledgment
(pp. v-viii)
4. Chapter One Introduction
(pp. 1-10)

We will consider infinite matrices indexed by Z (or Zb) 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 theϕkare functions on Ω.

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

HN=R[1,N]HR[1,N]

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

GN(E) = (HNE)−1

(ifHNEis an...

5. Chapter Two Transfer Matrix and Lyapounov Exponent
(pp. 11-14)

Consider 1D lattice Schrödinger operators of the form

H=υ(Tjx)δjj+ Δ

Assume thatψ= (ψj)j∈Zis a sequence satisfying

=

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 betweenMn(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...

6. Chapter Three Herman’s Subharmonicity Method
(pp. 15-18)

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(+x)δnn+ 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...

7. Chapter Four Estimates on Subharmonic Functions
(pp. 19-24)

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

8. Chapter Five LDT for Shift Model
(pp. 25-28)

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...

9. Chapter Six Avalanche Principle in SL2(R)
(pp. 29-30)

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: AssumeKSL2(R). Denoteu_{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,MSL2(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),...

10. Chapter Seven Consequences for Lyapounov Exponent, IDS, and Green’s Function
(pp. 31-38)

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 Td,d≥ 1, assumingωsatisfying a DC. The potentialυ=υ(x) is assumed real analytic on Td.

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...

11. Chapter Eight Refinements
(pp. 39-48)

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]...

12. Chapter Nine Some Facts about Semialgebraic Sets
(pp. 49-54)

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${{\mathcal L}_{j}}\subset \{1,\ldots ,s\}$andsjℓ∈...

13. Chapter Ten Localization
(pp. 55-64)

We will prove the following theorem.

Theorem 10.1.Consider the 1D lattice Schrödinger operator

Hω(x) =υ(x+)δnn+ Δ

where υ is an analytic potential onTdand ωDC=DCA,c⊂ Tdrefers 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 ωDC and ER.

Fix x0∈ Td.Then, for almost all ωDC, Hω(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...

14. Chapter Eleven Generalization to Certain Long-Range Models
(pp. 65-74)

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 operatorSϕ${{S}_{\phi }}(n,{n}')=\hat{\phi }(n-{n}')$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...

15. Chapter Twelve Lyapounov Exponent and Spectrum
(pp. 75-86)

First, we recall some basic facts from spectral theory.

LetHbe a bounded self-adjoint operator on ℓ2(Z). Then, forz∈ C\SpecH, (Hz)−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

ϕf(z) = 〈(Hz)−1f, f

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

Therefore, one has a representation${{\phi }_{f}}(z)=\langle {{(H-z)}^{-1}}f,f\rangle =\int_{\bold R}{\frac{1}{\lambda -z}{{\mu }_{f}}(d\lambda )}$whereμfis 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$\langle {{(H-z)}^{-1}}f,g\rangle =\int_{\bold R}{\frac{1}{\lambda -z}{{\mu }_{f,g}}(d\lambda )}$...

16. Chapter Thirteen Point Spectrum in Multifrequency Models at Small Disorder
(pp. 87-96)

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 Td. 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 Zd-lattice${{\tilde{H}}_{\lambda }}(x)=\cos (x+n.\omega )+\frac{\lambda }{2}{{S}_{\upsilon }}\caption {(13.2)}$

Localization results...

17. Chapter Fourteen A Matrix-Valued Cartan-Type Theorem
(pp. 97-104)

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...

18. Chapter Fifteen Application to Jacobi Matrices Associated with Skew Shifts
(pp. 105-116)

We consider 1D lattice Schrödinger operatorsH(x),x∈ Tdassociated with a skew shift transformationT: Td→ Td, and thusHm+1,n+1(x) =Hm,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...

19. Chapter Sixteen Application to the Kicked Rotor Problem
(pp. 117-122)

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$W={{U}_{a,b}}\cdot {{W}_{1,\kappa }}\caption {(16.4)}$whereUa,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,Ua,bbecomes a diagonal matrix${{U}_{ab}}={{e}^{-i(4{{\pi }^{2}}a{{n}^{2}}+2\pi bn)}}{{\delta }_{mn}}\caption {(16.7)}$andW1,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,W1,κis a perturbation of the identity with exponential off-diagonal decay....

20. Chapter Seventeen Quasi-Periodic Localization on the Zd-lattice (d > 1)
(pp. 123-132)

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...

21. Chapter Eighteen An Approach to Melnikov’s Theorem on Persistency of Nonresonant Lower Dimension Tori
(pp. 133-142)

The problem is that of persistency ofb-dimensional tori in R2b× R2r-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, …,Ib),θ= (θ1, …,θb) are action-angle variables for the “tangential” part of the phase space, andy= (y1, …,yr) 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])...

22. Chapter Nineteen Application to the Construction of Quasi-Periodic Solutions of Nonlinear Schrödinger Equations
(pp. 143-158)

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= (qn)n∈Zd. Select a finite set of modesn1, …,nb∈ Zd. The operator${\mathcal L}$is given by a multiplier (μn)n∈Zd, 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, …,λb) is a parameter taken...

23. Chapter Twenty Construction of Quasi-Periodic Solutions of Nonlinear Wave Equations
(pp. 159-168)

Consider NLW with periodic boundary condition (x∈ Td) 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....

24. Appendix
(pp. 169-173)