DIFFUSION IN THE MEAN FOR MARKOV SCHR ¨ODINGER EQUATIONS

By

Franklin Zakary Tilocco

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics – Doctor of Philosophy

2020

ABSTRACT

DIFFUSION IN THE MEAN FOR MARKOV SCHR ¨ODINGER EQUATIONS

By

Franklin Zakary Tilocco

We consider the evolution of a quantum particle hopping on a cubic lattice in any dimension

and subject to a potential consisting of a periodic part and a random part that fluctuates

stochastically in time.

If the random potential evolves according to a stationary Markov

process, we obtain diffusive scaling for moments of the position displacement, with a diffusion

constant that grows as the inverse square of the disorder strength at weak coupling. More

generally, we show that a central limit theorem holds such that the square amplitude of the

wave packet converges, after diffusive rescaling, to a solution of a heat equation. We also

consider how the addition of a random, stochastically evolving, potential leads to diffusive

propagation in the random dimer and trimmed Anderson models.

Copyright by
FRANKLIN ZAKARY TILOCCO
2020

ACKNOWLEDGMENTS

The results presented here are based on two papers. The first is a joint work with Jeffrey

Schenker and Shiwen Zhang, [34]. The second, which is a joint work with Jeffrey Schenker,

is currently being written.

I would like to thank Jeff for his support and guidance.

iv

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 2 MODELS AND STATEMENT OF MAIN RESULTS . . . . . . . . .
2.0.1 Periodic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.0.2 Anomalous Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

CHAPTER 3 HISTORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Periodic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Anomalous Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .

3.2.1 Dimer Model
3.2.2 Trimmed Anderson Model

CHAPTER 4 RIGOROUS ANALYSIS OF DIFFUSION FOR PERIODIC PO-

4.1 General assumptions and statement of the main result

TENTIALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
4.1.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 General result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 Augmented space analysis

4.2.1 The Markov semigroup on augmented spaces and the Pillet-Feynman-

Kac formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Vector valued Fourier Analysis . . . . . . . . . . . . . . . . . . . . . .
4.3 Spectral analysis on the augmented space . . . . . . . . . . . . . . . . . . . .

Spectral analysis of (cid:98)K0 . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Block decomposition of (cid:98)L0 . . . . . . . . . . . . . . . . . . . . . . . .

4.3.1

4.3.3

Spectral gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Proof of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Diffusive scaling and reality of the diffusion matrix . . . . . . . . . .
4.4.3 Limiting behavior of D(λ) for small λ . . . . . . . . . . . . . . . . . .

1

5
5
8

12
12
15
15
15

16
16
16
21
22

22
25
32
32
35
37
46
46
48
53

CHAPTER 5 NUMERICAL ANALYSIS OF DIFFUSION IN MARKOV SCHR ¨ODINGER
57
57
59
59
59

EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .

5.2.1 Dimer Model
5.2.2 Trimmed Anderson Model

CHAPTER 6 CONCLUSIONS AND CONJECTURES . . . . . . . . . . . . . . . .

62

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

v

APPENDIX A DECOMPOSITION OF THE SECOND MOMENTS AND THE

PROOF OF LEMMA 4.4.2 . . . . . . . . . . . . . . . . . . . .

65

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

vi

LIST OF FIGURES

Figure 1.1: Graphical representation of different types of transport. . . . . . . . . . .

Figure 2.1: Graphical representation of the dimer model potential . . . . . . . . . . .

Figure 2.2: The different transport regimes of the random dimer model. The system
exhibits ballistic motion when ∆ε = 0 (black), superdiffusive motion
when 0 < ∆ε < 2 (blue), diffusive motion when ∆ε = 2 (purple), and
localization when ∆ε > 2 (red).

. . . . . . . . . . . . . . . . . . . . . . .

Figure 2.3: Subdiffusive and localized transport in the trimmed Anderson model.
The purple line, which corresponds to subdiffusive propagation, is ob-
tained by taking Γ = 2Z. The blue line, which corresponds to local-
j∈Z Bj with Bj = {8j, 1 + 8j, 2 +
8j, 3 + 8j}.

ization, is obtained by letting Γ =(cid:83)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.4: Graphical representation of the trimmed Anderson model for Γ = 2Z.

Figure 4.1: Spectral gap of (cid:98)L0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

3

9

9

10

11

38

Figure 5.1: The diffusion constant as a function of time and disorder strength. . . . .

59

Figure 5.2: The average diffusion constant as a function of the disorder strength for

the dimer model.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Figure 5.3: The average diffusion constant as a function of the disorder strength for

the trimmed Anderson model. . . . . . . . . . . . . . . . . . . . . . . . .

60

vii

CHAPTER 1

INTRODUCTION

Diffusive propagation is expected, and observed, to emerge from wave motion in a ran-

dom medium in a variety of situations. The general intuition behind this expectation is

that repeated scattering due to the random medium leads to a loss of coherence, which in

a multi-scattering expansion or path integral formulation suggests a relation with random

walks and diffusion. This intuition is notoriously difficult to make precise in the context of a

static random environment. Indeed, proving the emergence of diffusion for the Schr¨odinger
wave equation with a weakly disordered potential, in dimension d ≥ 3, is one of the key

outstanding open problems of mathematical physics. For a random environment that fluc-

tuates stochastically in time, the analysis is simpler and diffusive propagation has proved

amenable to rigorous methods. Heuristically, this simplification is to be expected because

time fluctuations suppress recurrence effects in path expansions.

The present work is a continuation of a project initiated by Schenker and collaborators

[23, 18, 27, 33, 17] in which diffusive propagation has been shown to occur for solutions to a

tight binding Schr¨odinger equation with a random potential evolving stochastically in time.

In [23, 27], the following stochastic Schr¨odinger equation on

ψ : Zd → C | (cid:88)

x∈Zd

 ,

|ψ(x)|2 < ∞

(1.0.1)

(cid:96)2(Zd) :=

was considered:

i∂tψt(x) = H0ψt(x) + λV (x, t)ψt(x),

(1.0.2)

with H0 a (non-random) translation invariant Schr¨odinger operator, V (x, t) a zero-mean
random potential with time dependent stochastic fluctuations, and λ ≥ 0 a coupling constant.

These models had been considered previously by Tcheremchantsev [36, 37], who obtained

1

diffusive bounds for the p-th position moment,

(cid:11) :=

t

(cid:88)

x∈Zd

up to logarithmic corrections:

(cid:10)Xp
(ln t)ν− (cid:46) (cid:88)

p
2

t

x∈Zd

|x|p E(|ψt(x)|2),

(1.0.3)

|x|p E(|ψt(x)|2) (cid:46) t

p
2 (ln t)ν+,

t → ∞.

(1.0.4)

Here ν+, ν− > 0 are positive constants and E(·) denotes averaging with respect to the random
potential V .

Given suitable hypotheses on H0 and V , it was proven in [23] that diffusive scaling
(without logarithmic corrections, i.e., ν− = ν+ = 0) holds for p = 2, while in [27] diffusive

scaling was shown to hold for all moments. Furthermore, it was observed that at weak
disorder, λ → 0, the corresponding diffusion constant D has the asymptotic form

D ∼ C
λ2 ,

(1.0.5)

for some constant C. The divergence of D as λ → 0 seen in Equation (1.0.5) is to be expected,

since the translation invariant Schr¨odinger operator H0 on its own leads to ballistic transport.

In [33], a more subtle situation is considered where the environment is a superposition of

two parts:

i∂tψt(x) = H0ψt(x) + U (x)ψt(x) + λV (x, t)ψt(x),

(1.0.6)

where U is a static random potential that, at λ = 0, gives rise to Anderson localization

(absence of transport). In this particular case, it was observed that diffusion scaling holds

and that the diffusion constant has asymptotic form

D ∼ Cλ2.

(1.0.7)

Taken together, the results in [23, 27, 33] suggest that solutions to (1.0.6) with a general

potential U should satisfy diffusion with a diffusion constant whose asymptotic behavior in

the small λ limit is governed by the dynamics of the static Schr¨odinger operator H0+U . Here

2

Figure 1.1: Graphical representation of different types of transport.

we study this idea in the context of models in the form of Equation (1.0.6). In particular, we
shall rigorously study the case where U is periodic, which in the λ → 0 limit, leads to ballistic

motion. In addition, we shall numerically study two special cases of the Anderson model:

the random dimer model and the trimmed Anderson model. In the absence of disorder, the

former leads to superdiffusive motion, i.e.,

(cid:68)
(cid:68)

X2
t

X2
t

(cid:69) ∼ tρ,
(cid:69) ∼ tρ,

ρ > 1,

ρ < 1.

(1.0.8)

(1.0.9)

while the latter leads to subdiffusive motion, i.e.,

Figure 1.1 displays the relationship between ballistic, superdiffusive, diffusive, and subdiffu-

sive motion. In these three cases we show diffusive propagation for the evolution whenever
λ (cid:54)= 0. Furthermore, we obtain an asymptotic relationship between the diffusion constant

and the coupling constant.

Remark 1.0.1. In the absence of disorder, all three cases that we consider exhibit anomalous

diffusion, i.e.,

(1.0.10)

(cid:68)

X2
t

(cid:69) ∼ tρ

3

for ρ (cid:54)= 1. However, whenever discussing anomalous diffusion below we exclude the ballistic
(cid:52)
(U periodic) case from consideration.1

The remainder of this document is organized as follows: In Chapter 2 we describe the

types of systems that we are interested in analyzing and state our main results. Chapter 3

discusses related works and their relation to the problems considered here. Equation (1.0.6)

with periodic U is the subject of Chapter 4 and Equation (1.0.6) for the anomalous cases

is the subject of Chapter 5. Finally in Chapter 6 we conjecture about diffusive propagation

for solutions to Equation (1.0.6) with generic U .

1Here, and throughout, the (cid:52) symbol is used to denote the end of a remark.

4

CHAPTER 2

MODELS AND STATEMENT OF MAIN RESULTS

2.0.1 Periodic Potentials

First we consider solutions to Equation (1.0.6) with {U (x)}
potential. Recall that given p = (pj)d

x∈Zd a real valued p-periodic
>0, a function U : Zd (cid:55)→ R is called p-periodic

j=1 ∈ Zd

if

U (x + pjej) = U (x)

(2.0.1)

for all 1 ≤ j ≤ d and x ∈ Zd, where ej denotes the standard basis of Zd. Without loss of
generality, we assume that pj ≥ 2 for some j. Otherwise, U is constant and the problem
reduces to that studied in [23]. Throughout this paper, we denote by U the multiplication
operator, (U ψ)(x) = U (x)ψ(x) for ψ(x) ∈ (cid:96)2(Zd).

The analysis below will be applicable to a broad class of operators H0 and V (x, t). To

avoid technicalities in this chapter, let us state the main results in terms of hopping H0 given
by the standard discrete Laplacian on Zd,

(−∆ψ)(x) =

ψ(y),

(2.0.2)

(cid:88)

y:(cid:107)x−y(cid:107)=1

and potential V (x, t) given by the following so-called Markovian “flip process,” which is a

non-trivial, and somewhat typical, example of a potential satisfying the general requirements.

In general, the random potential is given by V (x, t) = vx(ω(t)), where ω(t) is an evolving

point in an auxiliary state space Ω. For the flip process, we take the state space Ω =
{−1, 1}Zd
; and vx(ω) = ωx, the xth coordinate of ω. Thus the potential V (x, t) = vx(ω(t))
takes only the values ±1. The process ω(t) is obtained by putting independent, identical

Poisson processes at each site x, and allowing each coordinate ωx to flip sign at the arrival

5

times t1(x) ≤ t2(x) ≤ ··· of the Poisson process. Now the general equation (1.0.6) becomes:

i∂tψt(x) =

ψt(y) + U (x)ψt(x) + λvx(ω(t))ψt(x).

(2.0.3)

(cid:88)

y:(cid:107)y−x(cid:107)=1

Remark 2.0.1. The general assumptions we require of the potential are set out in Section

4.1. They allow for a process vx(ω(t)) which is correlated from site to site and need not take

discrete values. A somewhat typical example satisfying our general assumptions is given by
(x, t) (cid:55)→ vx(ω(t)) such that

1. at any fixed time t, the field x (cid:55)→ vx(ω(t)) is distributed according to the Gibbs state
of a translation-invariant, finite-range lattice Hamiltonian h at a temperature T for

which there is a unique Gibbs state (the high temperature regime); and

2. the evolution t (cid:55)→ {vx(ω(t))|x ∈ Zd} is given by a continuous-time Glauber-type

dynamics for h, preserving the Gibbs state at temperature T .

As long as the lattice Hamiltonian h includes terms coupling the field at different sites, the
(cid:52)

resulting dynamics are correlated from site to site.

A sign of diffusive propagation is the existence of a diffusion constant for solutions to

Equation (2.0.3),

|x|2E(|ψt(x)|2),

x

D := lim
t→∞
characterized by the relationship x ∼ √
t in the mean amplitude of evolving wave packets.
Here, and throughout this introduction, E(·) denotes averaging with respect to the Poisson
flipping times t1(x) ≤ t2(x) ≤ ··· and the initial values {ωx(0)}
and uniform in {−1, 1}.

x∈Zd, taken independent

(2.0.4)

We will show below that the limit in Equation (2.0.4) exists for any p-periodic potential U

and any disorder strength λ > 0, furthermore we show that D > 0. To give an unambiguous

definition, one may take the initial value ψ0(x) = δ0(x). However, as we will show, the limit

remains the same for any other choice of (normalized) ψ0 with(cid:80)

x |x|2|ψ0(x)|2 < ∞.

6

(cid:88)

1
t

We refer to the existence of a finite, positive diffusion constant as in Equation (2.0.4) as

diffusive scaling. More generally, we have the following

Theorem 2.0.2 (Central limit theorem). For any periodic potential U and λ > 0, there is
a positive definite d × d matrix D = D(λ, u) such that for any bounded continuous function
f : Rd → R and any normalized ψ0 ∈ (cid:96)2(Zd) we have

(cid:88)

(cid:19)

(cid:18) x√

E(cid:16)|ψt(x)|2(cid:17)

f

(cid:90)

(cid:18) 1

(cid:19) d

2

(cid:68)
(cid:69)
r, D−1r

− 1
2
e

lim
t→∞

t

x∈Zd

where ψt(x) is the solution to Equation (2.0.3) with initial value ψ0. If furthermore(cid:80)

x(1 +
|x|2)|ψ0(x)|2 < ∞, then diffusive scaling, Equation (2.0.4), holds with the diffusion constant

Rd

2π

dr,

(2.0.5)

=

f (r)

D(λ) = lim
t→∞

1
t

|x|2E(cid:16)|ψt(x)|2(cid:17)

(cid:88)

x∈Zd

= tr D(λ).

(2.0.6)

Moreover, Equation (2.0.5) extends to quadratically bounded continuous f with supx(1 +
|x|2)−1|f (x)| < ∞.

It is well known that if λ = 0 in (2.0.3), then the free periodic Schr¨odinger equation has

Bloch-wave solutions and exhibits ballistic motion by Floquet theory, see [2, 8]:

|x|2(cid:12)(cid:12)(cid:12)(cid:68)

(cid:88)

x∈Zd

lim
t→∞

1
t2

δx, e−it(∆+U )δ0

(cid:69)(cid:12)(cid:12)(cid:12)2 ∈ (0,∞).

(2.0.7)

Indeed, strong ballistic motion was obtained for ∆ + U in [8]. That is, if X is the position
operator and X(t) = eit(∆+U )Xe−it(∆+U ) its Heisenberg evolution, then there exists a
bounded, self-adjoint operator Q, with ker(Q) = {0}, such that for any ψ in the domain of

X,

lim
t→∞

1
t

X(t)ψ = Qψ.

If we extend the definition of D(λ) in (2.0.6) to λ = 0, then D(0) = ∞. We are primarily
interested here in the regime λ ∼ 0, although we will demonstrate diffusion for all λ > 0.

However, for small λ the diffusion constant will be large and has the following asymptotic
behavior as λ → 0:

7

Theorem 2.0.3. Under the hypotheses of Theorem 2.0.2, there is a positive definite d × d
matrix D0 such that

D(λ) =

1
λ2

D0 + o(1)

and D(λ) = tr D(λ) =

1
λ2

tr D0 + o(1)

as λ → 0. (2.0.8)

(cid:16)

(cid:17)

(cid:16)

(cid:17)

The conclusions of Theorems 2.0.2 and 2.0.3 are true for Equation (1.0.6) under much

more general assumptions on the hopping H0 and the time dependent stochastic potential

V (x, t). We will state the general assumptions and results in Section 4.1.

2.0.2 Anomalous Diffusion

Recall that the Anderson model on (cid:96)2(Z) is given by a Hamiltonian of the form

where α =(cid:0)α(x)(cid:1)

x∈Z ⊂ [−1, 1]
Z

Hα = −∆ + gUα,

(2.0.9)

is a collection of independent and identically distributed

(i.i.d.) random variables, Uα(x) = α(x), and g > 0. It is known that for almost every choice

of U and any nonzero value of g the eigenfuctions of (2.0.9) are localized; in particular,

|x|2|(cid:104)δx, e−itHαδ0(cid:105)|2 < ∞.

(2.0.10)

(cid:88)

x∈Z

sup
t≥0

It follows immediately from (2.0.10) that, over large time scales, any solution to (2.0.9)

will have a diffusion constant equal to zero. As mentioned in the introduction, we will be

interested in two special cases of (2.0.9): the random dimer model and the trimmed Anderson

model. Both of these cases, which occur with probability zero, exhibit transport properties

different than the almost sure case.

variables(cid:0)α(x)(cid:1)

The random dimer model is obtained from (2.0.9) by selecting a realization of the random

x∈Z from the set {εa, εb}Z ⊂ [−1, 1]
Z

with the additional requirement that

α(2x) = α(2x + 1),

(2.0.11)

for every integer x, see figure 2.1. To simplify notation we will absorb g into the site energies

εa, εb and thus allow them to take any real value. Assuming ψ0 = δ0(x), the magnitude of
|εa − εb| determines the transport properties of the random dimer model (see figure 2.2):

8

Figure 2.1: Graphical representation of the dimer model potential

1. When |εa − εb| = 0, the transport is ballistic.
2. When 0 < |εa − εb| < 2, the transport is superdiffusive. Specifically,

(cid:69) ∼ t3/2.

(cid:68)

X2
t

(2.0.12)

3. When |εa − εb| = 2, the transport is diffusive.
4. When |εa − εb| > 2, the transport is localized.

Figure 2.2: The different transport regimes of the random dimer model. The system
exhibits ballistic motion when ∆ε = 0 (black), superdiffusive motion when 0 < ∆ε < 2
(blue), diffusive motion when ∆ε = 2 (purple), and localization when ∆ε > 2 (red).

The trimmed Anderson model can be characterized as the Anderson model without a

covering condition. Specifically, this means that the potential is only defined on a subset

9

10-210-110010110210310410510-410-21001021041061081010Γ (cid:40) Z,

Uα(n) =

α(n), n ∈ Γ

n ∈ Γc

0,

.

(2.0.13)

The transport properties of this model depend on Γ as well as the strength of the coupling

(cid:91)

j∈J

g. For example, if

Γc =

Bj,

(2.0.14)

where each Bj is connected, has finite length, and dist(Bi, Bj) ≥ 3 for i (cid:54)= j, then no
non-trivial solution can be supported on Γc and localization occurs, see [13] and figure 2.3.
Presently, we will be interested in the case Γ = 2Z, see figure 2.4. Using the numerical

methods outlined in Chapter 5, we find that the solutions to (2.0.9) exhibit subdiffusive

propagation, see figure 2.3.

The blue line, which corresponds to localization, is obtained by letting Γ =(cid:83)

Figure 2.3: Subdiffusive and localized transport in the trimmed Anderson model. The
purple line, which corresponds to subdiffusive propagation, is obtained by taking Γ = 2Z.
j∈Z Bj with
Bj = {8j, 1 + 8j, 2 + 8j, 3 + 8j}.

10

0    5000 1000015000200002500050010001500200025003000Figure 2.4: Graphical representation of the trimmed Anderson model for Γ = 2Z.

In Chapter 5, we consider

i∂tψt(x) = Hαψt(x) + λvx(ω(t))ψt(x)

(2.0.15)

where Hα is given by either the random dimer model or the trimmed Anderson model, and

vx(ω(t)) is the “flip process”. We will show numerically that for λ > 0 these models exhibit

diffusive scaling. In particular, we will see that in the small λ limit the diffusion constant

scales as

for the random dimer model, and as

D ∼ λ−1,

D ∼ λ1.186,

for the trimmed Anderson model.

(2.0.16)

(2.0.17)

11

CHAPTER 3

HISTORY

3.1 Periodic

A brief history of related studies is as follows: In 1974, Ovchinnikov and Erikman obtained

diffusion for a Gaussian Markov (“white noise”) potential [29]. This result was generalized

by Madhukar and Post [26] to include models with site diagonal and nearest-neighbor off-

diagonal disorder. In the 80s, Pillet obtained results on transience of the wave in related

models and derived a Feynman-Kac representation [30] which we employ here. Using Pillet’s

Feynman-Kac formula, Tchermentchansev [36, 37] showed that position moments exhibit

diffusive scaling, up to logarithmic corrections for any bounded potential U (x) in (1.0.6):

s
2 (ln t)ν+,

t → ∞.

(3.1.1)

(ln t)ν− (cid:46) (cid:88)

1

s
2

t

|x|sE(cid:16)|ψt(x)|2(cid:17) (cid:46) t

x

The case U (x) ≡ 0 (or equivalently, p = (1,··· , 1)) was considered by Schenker in [23],
where (3.1.1) was shown to hold for s = 2 with ν− = ν+ = 0. Moreover, the central limit

theorem (2.0.5) and the asymptotic behavior (2.0.8) were also obtained in [23]. The proof

in [23] was revisited by Musselman and Schenker in [27] to obtain diffusive scaling for all

position moments of the mean wave amplitude. The models studied in [23] are special cases

of those considered here.

For a certain class of random potentials U (x), including the case of an i.i.d. potential,

diffusive scaling and the central limit theorem were proven by Schenker in [33]. Moreover, if

H0 + U exhibits Anderson localization, then O(λ2) asymptotics (1.0.7) were proved for the

diffusion constant. The arguments in [33] do not require strict independence of the static

potential at different sites. However, the Equivalence of Twisted Shifts assumption taken

in [33] excludes p-periodic background potentials, as well as almost-periodic background

potentials. The periodic case falls in an intermediate regime between the period-free case

12

and the i.i.d. case. This is a motivation to revisit the proofs in [23] and [33] and develop the

current approach to the p-periodic case, for both diffusive scaling and limiting behavior.

In [17], Fr¨ohlich and Schenker used the techniques of [33] to study diffusion for a lattice

particle governed by a Lindblad equation describing jumps in momentum driven by interac-

tion with a heat bath. In some sense, this is the quantum analogue of the classical dynamics

of a disordered oscillator system perturbed by noise in the form of a momentum jump pro-

cess, considered in [3, 4] and reviewed in [5]. A key feature of the noise in [3, 4] is that energy

is conserved in the system with noise; this is necessary so that one can speak about heat

flux. By contrast, in the present work, and in [23, 27, 33, 17], energy conservation is broken

by the noise. Indeed the only conserved quantity for the evolution we consider is quantum

probability; and it is this quantity which is subject to diffusive transport. (In comparing the

present work with results on Markovian limit master equations as in [3, 4, 17], it is useful to

note that in the formal derivation of quantum or classical master equations one obtains the

square of the coupling to the heat bath multiplying the Lindbladian or stochastic term. Thus

it is the square of the coupling λ2 which should be compared with the coupling constants in
[3, 4, 17]. The scaling D(λ) ∼ D0λ−2 seen here is consistent with the inverse linear scaling
seen in those works.)

That diffusive transport emerges from (1.0.6) depends on the fact that it is a lattice,

or tight-binding, equation. A time-dependent potential coupled with the unbounded kinetic

energy present in continuum models can lead to stochastic acceleration resulting in super-

diffusive, or even super-ballistic, transport. Stochastic acceleration has been well studied in

the context of classical systems, see for example [1, 32, 35]. For quantum systems in the

continuum, transport has been studied in the context of Gaussian white-noise potentials
[15, 16, 20, 19], for which the super-ballistic transport (cid:104)x2(cid:105) ∼ t3 has been proved.

There are also parallel works on diffusion for the continuum Schr¨odinger equation with

Markovian forcing and periodic boundary conditions in space, e.g., [14]. One physical inter-

pretation of this continuous model is as a rigid rotator coupled to a classical heat bath. In

13

[14], the Hs norm of the wave function is shown to behave as ts/4. It is interesting to point

out that, as in the present work, the existence of a spectral gap for the Markov generator

is essential both for their analysis and the results. In many models with Markovian forcing,

the potential V (x, t) is quite rough. However, Bourgain studied the case where V (x, t) is

analytic/smooth in x and quasi-periodic/smooth in t. In [6], he showed that energy may

grow logarithmically. The reader is refereed to [10, 28, 38], for more work on Sobolev norm

growth and controllability of Schr¨odinger equations with time-dependent potentials.

The proof presented here is a generalization of that in [23]. Some of the arguments

are essentially standard fare and parallel the work of [23] closely. However, there are three

places in the proof where some substantially new arguments were needed. First, the Fourier

analysis (see Section 4.2.2) in our work is more subtle and requires careful consideration

due to the periodic potential. The extension developed here is of independent interest and

may benefit the future study of the limit-periodic and quasi-periodic cases. Secondly, the

proof of a spectral gap (Lemma 4.3.8) and the proof of the main results in Section 4.4 are

technically more involved in the current work. The interaction between the periodic part and

the hopping terms complicates the block decomposition on the augmented space. Finally, in

the present proof, the analysis of the asymptotic behavior of the diffusion constant is quite a

bit more involved. In [23], (1.0.5) essentially follows from a formula derived for the diffusion

constant in the midst of the proof of diffusion. Unfortunately, Theorem 2.0.3 in the p-period

case does not have such a simple proof and is obtained by a new approach. The proof is

based on an interesting observation linking the ballistic motion of the unperturbed part to

the diffusive scaling. This observation is part of the motivation behind the conjecture given

in Chapter 6 on the more general situations, linking the transport exponent to the limiting

behavior of the diffusion constant.

14

3.2 Anomalous Diffusion

3.2.1 Dimer Model

The random dimer model was first considered by Dunlap, Phillips, and Wu in [9]. They

argue that whenever the two site energies εa, εb, satisfy the inequality

−1 < εa − εb < 1,

(3.2.1)

√

there will be

N eigenstates with localization length of order N . These extended states

contribute to transport and have diffusion constant which scales like t1/2. This, in turn,

implies superdiffusive scaling with (cid:10)X2

(cid:11) ∼ t3/2. Numerical confirmation of this result is

t

provided in the same paper. Bovier [7] confirms this result with the use of invariant measures

and perturbation theory. Rigorous confirmation of superdiffusive scaling is given by a lower

bound on transport proven by Jitomirskaya, Schulz-Baldes, and Stolz [22], and an upper

bound proven by Jitomirskaya and Schulz-Baldes [21].

3.2.2 Trimmed Anderson Model

Three particularly relevant articles on the trimmed Anderson model are [12, 13, 31].

In

[31], Rojas-Molina provides Wegner estimates for the trimmed Anderson model and proves

dynamical localization at the bottom of the spectrum for strong disorder. Elgart and Klein

prove similar results in [12] for the trimmed Anderson model plus an arbitrary bounded

background potential. Finally, Elgart and Sodin [13] examine how the strength of the disor-

der g and the density of the sub-lattice Γ influence transport away from the bottom of the
spectrum. Furthermore, [13] explores the possibility of delocalization, in dimension d ≥ 3,

for strong disorder.

15

RIGOROUS ANALYSIS OF DIFFUSION FOR PERIODIC POTENTIALS

CHAPTER 4

This chapter considered the case when U is periodic. In Section 4.1, a more general class

of operators is introduced and the main result, Theorem 4.1.11, which generalizes Theorems

2.0.2 and 2.0.3, is formulated. In Section 4.2 the basic analytic tools of “augmented space

analysis,” developed previously in [23, 33], are reviewed. In Section 4.3, we present the heart

of our argument: a block decomposition to study the spectral gap of the induced operator on

the augmented space. Section 4.4 is devoted to a proof of the main result. Certain technical

results used below are collected in Appendix A.

4.1 General assumptions and statement of the main result

We study a more general class of equations with hopping terms other than nearest neigh-

bor and a perturbing potential V that is not necessarily the “flip process.” More precisely,

we shall consider equation (1.0.6) in the form

i∂tψt(x) = H0ψt(x) + U (x)ψt(x) + λVx(ω(t))ψt(x)

(4.1.1)

Here U is the real-valued, p-periodic potential as in (2.0.1) for some p ∈ Zd

>0; H0 is a self-

adjoint, short-ranged, translation invariant hopping operator with non-zero hopping along a
set of vectors that generate Zd; Vx(ω(t)) is a time-dependent random potential that fluctuates
according to a stationary Markov process ω(t); and λ ≥ 0 is a coupling constant used to

set the strength of the disorder. These assumptions will be made precise in the section that

follows.

4.1.1 Assumptions

Assumption 4.1.1 (Probability space). Throughout, let (Ω, µ) be a probability space, on
which the additive group Zd acts through a collection of µ-measure preserving maps. That

16

is, for each x ∈ Zd there is a µ-measure preserving map, τx : Ω → Ω, where τ0 is the identity
map and τx ◦ τy = τx+y for each x, y ∈ Zd. We refer to the maps τx, x ∈ Zd as “disorder
translations.”

Assumption 4.1.2 (Markov dynamics). The space Ω is a compact Hausdorff space, µ is
a Borel measure and for each α ∈ Ω there is a probability measure Pα on the σ-algebra
generated by Borel-cylinder subsets of the path space P(Ω) = Ω[0,∞). Furthermore, the

collection of these measures has the following properties

1. Right continuity of paths: For each α ∈ Ω, with Pα probability one, every path t (cid:55)→ ω(t)

is right continuous and has initial value ω(0) = α.

2. Shift invariance in distribution: For each α ∈ Ω and x ∈ Zd, Pτxα = Pα ◦ S−1

x , where

Sx({ω(t)}t≥0) = {τxω(t)}t≥0 is the shift τx lifted to path space P(Ω).

3. Stationary Markov property: There is a filtration {Ft}t≥0 on the Borel σ-algebra of

P(Ω) such that ω(t) is Ft measurable and

(cid:0){ω(t + s)}t≥0 ∈ E(cid:12)(cid:12)Fs

(cid:1) = P

ω(s)(E)

Pα

for any measurable E ⊂ P(Ω) and any s > 0.

4. Invariance of µ: For any Borel measurable E ⊂ Ω and each t > 0,

(cid:90)

Ω

Pα(ω(t) ∈ E) µ(dα) = µ(E).

average(cid:82)

We use Eα(·) to denote averaging with respect to Pα and E (·) to denote the combined
Eα(·) µ(dα) over the Markov paths and the initial value of the process. Invariance
of µ under the dynamics is equivalent to the identity E (f (ω(t))) = E (f (ω(0))) for f ∈
L1(Ω). An important tool for studying Markov processes is conditioning on the value of a

Ω

process at a given time. The proper definition can be found in, e.g. [33]. Conditioning on
the value of the processes at t = 0 determines the initial value: E (·|ω(0) = α) = Eα(·). To

17

the process {ω(t)}t≥0, there is associated a Markov semigroup, obtained by averaging over
the initial value conditioned on the value of the process at later times:

Stf (α) := E (f (ω(0))|ω(t) = α) .

As is well known, St is a strongly continuous contraction semi-group on Lp(Ω) for 1 ≤ p < ∞.
The semigroup St has a generator

Bf := lim
t↓0

(f − Stf ) ,

1
t

(4.1.2)

defined on the domain D(B) where the right hand side exists in the L2-norm.1 By the Lumer-

Phillips theorem, B is a maximally accretive operator. Note that St1 = 1 by definition, where
1(α) = 1 for all α ∈ Ω. The invariance of µ under the process {ω(t)}t≥0 implies further that
†
t 1 = 1. It follows that
S

(cid:26)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)

Ω

L2

0(Ω) :=

f ∈ L2(Ω)

f (α)µ(dα) = 0

(cid:27)

is invariant under the semi-group St and its adjoint S

†
t . We assume that B is sectorial and

strictly dissipative on L2

0(Ω).

Assumption 4.1.3 (Sectoriality of B). There are b, γ ≥ 0 such that

for all f ∈ D(B). Here (cid:104)f, g(cid:105) =(cid:82) f gdµ denotes the inner product on L2(Ω).

|Im(cid:104)f, Bf(cid:105)| ≤ γ Re(cid:104)f, Bf(cid:105) + b(cid:107)f(cid:107)2

Assumption 4.1.4 (Gap condition for B). There is T > 0 such that

Re(cid:104)f, Bf(cid:105) ≥ 1
T

for all f ∈ D(B).

(cid:90)

(cid:13)(cid:13)(cid:13)(cid:13)f −

(cid:13)(cid:13)(cid:13)(cid:13)2

f dµ

Ω

L2(Ω)

(4.1.3)

(4.1.4)

1 Note Equation (4.1.2) defines a generator with positive real part; while, it is common

in probability theory to define a generator with negative real part.

18

(B − z)−1 = (cid:82) ∞

Remark 4.1.5. 1) Given the generator B we formally write the semigroup St as e−tB. 2)
The resolvent of the semigroup e−tB is the operator valued analytic function R(z)
:=

0 etze−tBdt, which is defined and satisfies (cid:107)R(z)(cid:107) ≤ 1|Re z| when Re z < 0.
Sectoriality is equivalent to the existence of a analytic continuation of R(z) to z ∈ C \ Kb,γ
with the bound (cid:107)R(z)(cid:107) ≤ dist−1(z, Kb,γ) where Kb,γ is the sector {Re z ≥ 0} ∩ {|Im z| ≤
b + γ|Re z|} (see [24, Theorem V.3.2]). In particular Assumption 4.1.3 holds (with b = 0

and γ = 0) if the Markov dynamics is reversible, in which case B is self-adjoint. 3) The

Assumption 4.1.6 (Translation covariance, boundedness and non-degeneracy of the po-

tential). The potentials Vx(ω) appearing in the Schr¨odinger equation (4.1.1) are given by

L2
0(Ω)

− t
T .

(cid:13)(cid:13)(cid:13)(cid:13) ≤ e

gap assumption implies that the restriction of B to L2

(cid:13)(cid:13)(cid:13)(cid:13)St|
Vx(ω) = v(τxω) where v ∈ L∞(Ω). We assume that (cid:107)v(cid:107)∞ = 1,(cid:82)

is non-degenerate in the sense that there is χ > 0 such that

(cid:13)(cid:13)(cid:13)B−1(v(τx·) − v(τy·))

(cid:13)(cid:13)(cid:13)L2(Ω)

for all x, y ∈ Zd, x (cid:54)= y.

0(Ω) is strictly accretive, and thus that
(cid:52)

Ω v(ω)µ(dω) = 0, and v

≥ χ

(4.1.5)

Remark 4.1.7. Since the Markov process is translation invariant, B commutes with the

translations Txf (α) = f (τxα) of L2(Ω). Thus (4.1.5) is equivalent to

(cid:13)(cid:13)(cid:13)B−1(v(τx·) − v(·))

(cid:13)(cid:13)(cid:13)L2(Ω)

≥ χ.

(4.1.6)
for all x ∈ Zd, x (cid:54)= 0. The non-degeneracy essentially amounts to requiring that B−1(vτx)
are uniformly non-parallel to B−1(v) for x (cid:54)= 0.
In particular, the condition is trivially
satisfied if for example if the processes v(τxω(t)) and v(ω(t)) are independent for x (cid:54)= 0, as
(cid:52)
in the “flip process”.

Assumption 4.1.8 (Translation invariance and non-degeneracy of the hopping terms). The
hopping operator, H0, on (cid:96)2(Zd) is defined by

H0ψ(x) =

h(x − ξ)ψ(ξ).

(4.1.7)

(cid:88)

ξ(cid:54)=x

19

(4.1.8)

(4.1.9)

(cid:110)

spanZ (supph) = Zd,

(cid:111)

ξ ∈ Zd : h(ξ) (cid:54)= 0

.

where supph =

Remark 4.1.9. 1) It follows from (1) and (2) that (cid:98)h(k) = (cid:80)

x e−ik·xh(x) is a real-valued
C2 function on the torus [0, 2π)d. In particular, H0 is a bounded self-adjoint operator with
(cid:107)H0(cid:107)

(cid:96)2(Zd)→(cid:96)2(Zd)

= maxk |(cid:98)h(k)| and
(cid:107)(cid:98)h(cid:107)∞,(cid:107)(cid:98)h(cid:48)(cid:107)∞,(cid:107)(cid:98)h(cid:48)(cid:48)(cid:107)∞ ≤ (cid:88)

ξ∈Zd\{0}

Additionally, the hopping kernel h : Zd \ {0} → C is

1. Self-adjoint:

2. Short range:

3. Non-degenerate:

h(−ξ) = h(ξ);

(cid:88)

ξ∈Zd\{0}

|ξ|2|h(ξ)| < ∞;

(1 + |ξ|2)|h(ξ)| < ∞.

(4.1.10)

2) It is natural to assume that supp h can generate the entire Zd lattice, otherwise the system
(cid:52)

can always be reduced a direct sum of systems over several sub-lattices.

Below we will need the following simple consequence of the non-degeneracy of h:

Proposition 4.1.10. For each non-zero k ∈ Rd,

(cid:88)
Proof. Suppose on the contrary that (cid:80)

ξ∈Zd

|k · ξ|2|h(ξ)|2 > 0.

that k · ξ = 0 for all ξ ∈ supp h, violating the non-degeneracy of h.

(4.1.11)

ξ∈Zd |k · ξ|2|h(ξ)|2 = 0 for some k (cid:54)= 0. It follows

20

4.1.2 General result

The main result is the following

Theorem 4.1.11 (Central limit theorem). For any periodic potential U and λ > 0, there is
a positive definite d × d matrix D = D(λ, U ) such that for any bounded continuous function
f : Rd → R and any normalized ψ0 ∈ (cid:96)2(Zd) we have

(cid:88)

(cid:68)
r, D−1r
where ψt(x) is the solution to Equation (4.1.1). If furthermore(cid:80)

E(cid:16)|ψt(x)|2(cid:17)

lim
t→∞

x∈Zd

− 1
2

e

f (r)

(cid:18) 1

(cid:19) d

2

x(1 + |x|2)|ψ0(x)|2 < ∞,

(cid:19)

(cid:18) x√

t

dr,

(4.1.12)

(cid:90)

=

Rd

2π

(cid:69)

f

then diffusive scaling Equation (2.0.4) holds with the diffusion constant

D(λ) = lim
t→∞

1
t

|x|2E(cid:16)|ψt(x)|2(cid:17)

(cid:88)

x∈Zd

= tr D(λ).

(4.1.13)

Moreover, Equation (4.1.12) extends to quadratically bounded continuous f with supx(1 +
|x|2)−1|f (x)| < ∞.

(cid:12)(cid:12)(cid:12)(cid:68)

x2
j

δx, e−it(H0+U )δ0

(cid:69)(cid:12)(cid:12)(cid:12)2

dt > 0,

j = 1··· , d,

(4.1.14)

Assume further that

lim
T→∞

2
T 3

(cid:90) ∞

0

(cid:16)

− 2t

e

T (cid:88)
(cid:17)

x∈Zd

then there is a positive definite d × d matrix D0 such that

D(λ) =

1
λ2

D0 + o(1)

and D(λ) = tr D(λ) =

1
λ2

(cid:16)

(cid:17)

tr D0 + o(1)

as λ → 0.

(4.1.15)

Remark 4.1.12. 1) In the case with the short range hopping H0 and periodic U , the strong
limit of all the j-th velocity operators limt t−1 Xj(ψt) always exist, which implies the exis-
tence of the limit in (4.1.14). We say H0 + U has ballistic motion if the limit in (4.1.14)

is positive. 2) δ0 in (4.1.14) can be replaced by any ψ0 with compact support. 3) There
always exists a semi-positive definite d × d matrix D0 such that (4.1.15) holds regardless of
(4.1.14). If (4.1.14) is true for j ∈ S with S ⊂ {1, 2,··· , d}, then the restriction of D0 on
S × S is positive definite, and we still have D(λ) ∼ λ−2 since tr D0 > 0.
(cid:52)

21

4.2 Augmented space analysis

4.2.1 The Markov semigroup on augmented spaces and the Pillet-Feynman-Kac

formula

As in the works [23, 33], our analysis of the Schr¨odinger equation, Equation (4.1.1), is based
on a formula of Pillet [30] for E(ρt), where ρt(x, y) = ψt(x)ψt(y) is the density matrix
corresponding to a solution ψt to Equation (4.1.1). Pillet’s formula relates E(ρt) to matrix

elements of a contraction semi-group on the “augmented space”

H := L2(Ω;HS(Zd)),

(4.2.1)

where HS(Zd) denotes the Hilbert-Schmidt ideal in the bounded operators on (cid:96)2(Zd).

The term “augmented space” refers to a space of functions obtained by “augmenting”
functions defined on X = Zd or X = Zd×Zd by allowing dependence on the disorder ω ∈ Ω.

More specifically, it refers to spaces of the form

Definition 4.2.1 (Definition 3.1 of [33]). Let (B(X),(cid:107)·(cid:107)B(X)) be a Banach space of functions
on X whose norm satisfies

1. If g ∈ B(X) and 0 ≤ |f (x)| ≤ |g(x)| for every x ∈ X, then f ∈ B(X) and (cid:107)f(cid:107)B(X) ≤

(cid:107)g(cid:107)B(X).

2. For every x ∈ X, the evaluation x (cid:55)→ f (x) is a continuous linear functional on B(X).

For p ≥ 1, the augmented space Bp(X × Ω) is the set of maps F : X × Ω → C such that
x → (cid:107)F (x,·)(cid:107)Lp(Ω) ∈ B(X).

A general theory of such spaces is developed in [33]. In particular, it is shown there that

Bp(X × Ω) is a Banach space under the norm

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)
(cid:18)(cid:90)

Ω

(cid:107)F(cid:107)Bp(X×Ω) :=

(cid:19) 1

p

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)B(X)

,

|F (x, ω)|p µ(dω)

22

with (cid:107)F(cid:107)Bp(X×Ω) ≤ (cid:0)(cid:82)

Ω (cid:107)F (·, ω)(cid:107)pµ(dx)(cid:1) 1

It follows that Lp(Ω;B) ⊂
Bp(X × Ω), although in general equality may not hold. For B(X) = (cid:96)p(X) and 1 ≤ q ≤ ∞,
we denote Bq(X) by (cid:96)p;q(X). Then, for 1 ≤ p < ∞,

p [33, Prop. 3.1].

(cid:96)p;p(X × Ω) = Lp(Ω; (cid:96)p(X)) = Lp(X × Ω),

where we take the product measure Counting Measure × µ on X × Ω [33, Prop 3.2].
particular, (cid:96)2;2(X × Ω) is a Hilbert space with inner product

In

(cid:104)F, G(cid:105) =

F (x, ω)G(x, ω)µ(dω).

Another space that will play an important role below is (cid:96)∞;1(X × Ω) which is the space of

(cid:90)

(cid:88)

x∈X

Ω

(cid:90)

maps with

(cid:107)F(cid:107)(cid:96)∞;1 := sup
x∈X

Ω

|F (x, ω)|µ(dω) < ∞.

Returning now to H = L2(Ω;HS(Zd)), we note that we may think of an element F ∈ H

as a C-valued map on

via the identification

M := Zd × Zd × Ω,

F (x, y, ω) := (cid:104)δx, F (ω)δy(cid:105).

(4.2.2)

(4.2.3)

It follows from [33, Prop. 3.2] that

provided M is given the product measure m =

H = (cid:96)2;2(Zd × Zd × Ω) = L2(M ),

(cid:16)

counting measure on Zd × Zd(cid:17) × µ.

We define operators K, U and V that lift the commutators with H0, U and Vω to H:

KF (ω) := [H0, F (ω)] , UF (ω) := [U, F (ω)] ,

and VF (ω) := [Vω, F (ω)] .

(4.2.4)

The following proposition follows immediately from Equation (4.2.4).

23

Proposition 4.2.2. The operators K, U and V are self-adjoint, bounded and are given by

the following explicit expressions

KF (x, y, ω) =

(cid:88)

ξ(cid:54)=0

h(ξ) [F (x − ξ, y, ω) − F (x, y − ξ, ω)] ,

UF (x, y, ω) = [U (x) − U (y)] F (x, y, ω)

VF (x, y, ω) = (cid:2)v(τxω) − v(τyω)(cid:3) F (x, y, ω),

and

for any F ∈ L2(M ).

(4.2.5)

(4.2.6)

(4.2.7)

The final ingredient for Pillet’s formula is the lift of the Markov generator B to L2(M ).
Throughout, we will use e−tB to denote the Markov semigroup lifted to the augmented space
Bp(X × Ω), with B the corresponding generator. This semigroup is defined by

e−tBF (x, α) := EΩ (F (x, ω(0))| ω(t) = α) .

(4.2.8)

In particular, given φ ∈ B(X) and f ∈ Lp(Ω) we have

e−tB(φ ⊗ f ) = φ ⊗ e−tBf,

where φ ⊗ f denotes the function

(φ ⊗ f )(x, ω) := φ(x)f (ω).

Proposition 4.2.3 (Prop. 3.3 of [33]). The semigroup e−tB is contractive and positivity
preserving on Bp(X × Ω) and B is sectorial on L2(X × Ω), with the same constants b and

γ as appear in Assumption 4.1.3.

Pillet’s formula expresses the average of the time dependent dynamics (2.0.3) in terms

of the semi-group on L2(M ) generated by L = iK + iU + iλV + B.

Lemma 4.2.4 (Pillet’s formula [30]). Let

L := iK + iU + iλV + B

(4.2.9)

24

on the domain D(B) ⊂ L2(M ). Then L is maximally accretive and sectorial and if ρt =
ψt (cid:104)ψt, ·(cid:105) is the density matrix corresponding to a solution ψt to Equation (4.1.1) with ψ0 ∈
(cid:96)2(Zd), then

E (ρt|ω(t) = α) = [e−tL (ρ0 ⊗ 1)](α),

(4.2.10)

where 1(α) = 1 for all α. Consequently, we have

(cid:90)

Ω

E (ρt) =

(cid:104)
e−tL (ρ0 × 1)

(cid:105)

(ω)µ(dω).

(4.2.11)

Furthermore, for a solution ψt to Equation (4.1.1), we have

E(cid:16)

(cid:17)

ψt(x)ψt(y)

=

(cid:68)

δx ⊗ δy ⊗ 1, e−tL(cid:0)ψ0 ⊗ ψ0 ⊗ 1(cid:1)(cid:69)
(cid:12)(cid:12)(cid:12)δx ⊗ δx ⊗ 1, e−tLρ0 ⊗ 1
(cid:69)

L2(M )

.

δx ⊗ δx ⊗ 1, e−tLρ0 ⊗ 1

In particular, we have

E (ρt(x, x)) =

(cid:68)

(4.2.12)

H .

(4.2.13)

Remark 4.2.5. Here and below we will use tensor product notation for elements of (cid:96)2(Zd×Zd),

[φ ⊗ ψ](x, y) = φ(x)ψ(y).

Thus a rank one operator ψ (cid:104)φ, ·(cid:105) ∈ HS(Zd) corresponds to ψ ⊗ φ.

(cid:52)

For the derivation of this result, we refer the reader to [33, Lemmas. 3.5 and 3.6]. In
[33], the term U is different, stemming as it does there from the background static random

potential. However, an essentially identical proof works in the present context.

4.2.2 Vector valued Fourier Analysis

For each ξ ∈ Zd, we define the (simultaneous position and disorder) shift operator

SξΨ(x, y, ω) := Ψ(x − ξ, y − ξ, τξω)

(4.2.14)

for any function Ψ defined on Zd × Zd × Ω.

25

Proposition 4.2.6. The map ξ (cid:55)→ Sξ is a unitary representation of the additive group Zd
on the Hilbert space H, and for every ξ ∈ Zd

(cid:2)Sξ,K(cid:3) = (cid:2)Sξ,V(cid:3) = (cid:2)Sξ, B(cid:3) = 0.

The potential term U only commutes with a subgroup of translations Sξ, corresponding

to translation over a period of the potential. For ξ ∈ Zd let
p ◦ ξ := (p1ξ1, . . . , pdξd)

and

pZd = {p ◦ ξ : ξ ∈ Zd}.

Then

Proposition 4.2.7. For every ξ ∈ Zd,(cid:2)Sp◦ξ,U(cid:3) = 0.

(4.2.15)

(4.2.16)

Because of Props. 4.2.6, 4.2.7, a suitable Floquet transform will give a fibre decomposition

of the various operators K, U, V and B. Let Td = [0, 2π)d denote the torus,

(cid:99)M := Zd × Ω,

and let Zp = Zp1 × ··· × Zpd denote the fundamental cell of the periodicity group on Zd.
Note that (cid:96)2(Zp) ∼= C⊗p := Cp1 ⊗ ··· ⊗ Cpd. Using this identification, let πσ : C⊗p → C
be the coordinate evaluation map associated to a point σ = (σ1,··· , σd) ∈ Zp. For f, g ∈
L2((cid:99)M ; C⊗p), we use the natural inner product on L2((cid:99)M ; C⊗p)

(cid:88)

σ∈Zp

=

(cid:104)f, g(cid:105)

(cid:104)πσf, πσg(cid:105)

L2((cid:99)M ;C)

L2((cid:99)M ;C⊗p)
be a map (cid:98)Ψk : (cid:99)M → C⊗p as follows:
(cid:88)
πσ(cid:98)Ψk(x, ω) :=
e−i k·(p◦ξ+σ) Sp◦ξ+σΨ(x, 0, ω)
(cid:88)

ξ∈Zd

e−i k·n Ψ(x − n,−n, τnω),

=

n∈pZd+σ

26

Given Ψ ∈ L2(M ) and k ∈ Td, the Floquet transform of Ψ ∈ L2(M ) at k is defined to

.

(4.2.17)

(4.2.18)

for each σ ∈ Zp. Initially we define this Floquet transform on the augmented space

W1(M ) :=

F : M → C

|F (x + y, y, ω)|µ(dω) < ∞

.

(4.2.19)

(cid:40)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup

x

(cid:90)

(cid:88)

y

(cid:41)

The basic results of Fourier analysis are naturally extended to this Floquet transform. In

particular, if F ∈ W1(M ), then (cid:98)Fk ∈ (cid:96)∞;1((cid:99)M ) for each k and k (cid:55)→ (cid:98)Fk is continuous.

Furthermore, Plancherel’s Theorem,

(cid:90)

(cid:107)(cid:98)Fk(cid:107)2

L2((cid:99)M )

holds for F ∈ W1(M )(cid:84) L2(M ), where ν denotes normalized Lebesgue measure on the torus

L2(M )

ν(dk),

Td

=

(cid:107)F(cid:107)2

Td. Thus, the Floquet transform extends naturally to L2(M ). Throughout the rest of the

paper, we assume that the Floquet transform is properly defined on L2(M ). For more details

of this extension in a similar context, we refer readers to Section 3 in [33].

One may easily compute

(cid:88)

(cid:104)

(cid:105)
πσ(cid:98)Ψk(x − ξ, ω) − e−ik·ξπσ−ξ(cid:98)Ψk(x − ξ, τξω)

;

h(ξ)

ξ(cid:54)=0

(cid:91)(KΨ)k(x, ω) =
(cid:91)(UΨ)k(x, ω) = (u(x − σ) − u(−σ)) πσ(cid:98)Ψk(x, ω);
(cid:91)(VΨ)k(x, ω) = (v(τxω) − v(ω)) πσ(cid:98)Ψk(x, ω);
(cid:91)(BΨ)k(x, ω) =B πσ(cid:98)Ψk(x, ω),

πσ

πσ

πσ

πσ

where on the right hand side, B acts on πσ(cid:98)Ψk as in Equation (4.2.8). With the above
computations in mind, let (cid:98)Kk, (cid:98)U, and (cid:98)V denote the following operators on functions φ :
(cid:99)M → C⊗p:

πσ((cid:98)Kk φ) (x, ω) =

(cid:104)

h(ξ)

πσφ(x − ξ, ω) − e−ik·ξπσ−ξφ(x − ξ, τξω)

(cid:88)
πσ ((cid:98)U φ)(x, ω) = (u(x − σ) − u(−σ)) πσφ(x, ω);

ξ(cid:54)=0

(cid:105)

;

(4.2.20)

(4.2.21)

(4.2.22)

and

((cid:98)Vφ)(x, ω) = (v(τxω) − v(ω)) φ(x, ω).

27

We now present three lemmas (Lemmas 4.2.8-4.2.12), which describe the basic properties

Lemmas 3.13-3.15 of [33], with the main difference being that here we consider the vector

of the operators (cid:98)Kk, (cid:98)U, and (cid:98)V. These results are the adaptation to the present context of
valued space L2((cid:99)M ; C⊗p) instead of L2((cid:99)M ; C). We omit the details of the proofs here.
Lemma 4.2.8. Let (cid:99)M = Zd × Ω,(cid:98)Kk, (cid:98)U and (cid:98)V be given as above, then
1. (cid:98)Kk, (cid:98)U and (cid:98)V are bounded on (cid:96)∞;1((cid:99)M ; C⊗p).
2. (cid:98)Kk, (cid:98)U and (cid:98)V are bounded and self-adjoint on L2((cid:99)M ; C⊗p) with the following bounds:
(cid:13)(cid:13)(cid:13)(cid:98)U(cid:13)(cid:13)(cid:13)L2((cid:99)M ;C⊗p)
((cid:100)UΨ)k = (cid:98)U(cid:98)Ψk

3. If Ψ ∈ L2(M ; C) and let (cid:98)Ψk be given as in (4.2.18), then

(cid:13)(cid:13)(cid:13)(cid:98)V(cid:13)(cid:13)(cid:13)L2((cid:99)M ;C⊗p)
((cid:100)VΨ)k = (cid:98)V(cid:98)Ψk

≤ 2(cid:107)(cid:98)h(cid:107)∞,

(cid:13)(cid:13)(cid:13)L2((cid:99)M ;C⊗p)
(cid:91)(KΨ)k = (cid:98)Kk(cid:98)Ψk,

≤ 2(cid:107)u(cid:107)∞,

≤ 2

(cid:13)(cid:13)(cid:13)(cid:98)Kk

and

for ν-almost every k ∈ Td.

Because the Markov process has a distribution invariant under the shifts, the Markov

semigroup commutes with Floquet transform:
Lemma 4.2.9 (Lemma 3.14,[33]). Let the Markov semigroup e−tB be defined as in Equation

(4.2.8). Then,

(cid:92)(cid:2)e−tBΨ(cid:3)

k = e−tB(cid:98)Ψk

for Ψ ∈ L2(M ) and ν-almost every k ∈ Td.

Lemma 4.2.10. Let (cid:98)Kk be given as in (4.2.20) with h that satisfies (4.1.8). Then the map
k (cid:55)→ (cid:98)Kk is C2 on Td, considered either as a map into the bounded operators on (cid:96)∞;1((cid:99)M ; C⊗p)
or as a map into the bounded operators on L2((cid:99)M ; C⊗p).
Moreover, we have the explicit expression for the derivatives for any φ(x, ω) ∈ L2((cid:99)M ; C⊗p)

, k ∈ Td and 1 ≤ i, j ≤ d:

πσ∂kj(cid:98)Kkφ(x, ω) = i

(cid:88)

ξ(cid:54)=0

ξj h(ξ) e−ik·ξπσ−ξφ(x − ξ, τξω),

(4.2.23)

28

πσ∂ki

(cid:88)
(cid:13)(cid:13)(cid:13)∂ki
(cid:13)(cid:13)(cid:13) ≤ (cid:107)(cid:98)h(cid:48)(cid:107)∞,
where (cid:107)(cid:98)h(cid:48)(cid:107)∞,(cid:107)(cid:98)h(cid:48)(cid:48)(cid:107)∞ are bounded in (4.1.10).

∂kj(cid:98)Kkφ(x, ω) =
(cid:13)(cid:13)(cid:13)∂kj(cid:98)Kk

with bounds

ξ(cid:54)=0

In particular, let

ξi ξj h(ξ) e−ik·ξπσ−ξφ(x − ξ, τξω).

∂kj(cid:98)Kk

(cid:13)(cid:13)(cid:13) ≤ (cid:107)(cid:98)h(cid:48)(cid:48)(cid:107)∞,

−→
−→
1 ∈ C⊗p be the vector with πσ
1 = 1 for all σ ∈ Zp. Then
(cid:88)
ξj h(ξ) δξ ⊗ −→
(cid:88)
ξ(cid:54)=0
ξi ξj h(ξ) δξ ⊗ −→

∂kj(cid:98)K0 δ0 ⊗ −→
∂kj(cid:98)K0 δ0 ⊗ −→

1 ⊗ 1 = i

1 ⊗ 1 =

1 ⊗ 1.

1 ⊗ 1,

∂ki

ξ(cid:54)=0

(4.2.24)

(4.2.25)

(4.2.26)

(4.2.27)

−→
1 q ∈
Remark 4.2.11. Throughout the rest of the paper, we will frequently use the notation
Cq for any q ∈ Z>0 to indicate the constant vector in Cq with all entries 1 and write
−→
(cid:52)
1 =

−→
1 ⊗p for simplicity.

Putting these results together we obtain

Lemma 4.2.12. For each k ∈ Td, let

(cid:98)Lk := i(cid:98)Kk + i(cid:98)U + iλ(cid:98)V + B

(4.2.28)

Furthermore

on the domain D(B) ⊂ L2((cid:99)M ; C⊗p). Then (cid:98)Lk is maximally accretive on L2((cid:99)M ; C⊗p).
1. For t > 0, k (cid:55)→ e−t(cid:98)Lk is

a) a C2 map from Td into the contractions on L2((cid:99)M ; C⊗p); and
b) a C2 map from Td into the bounded operators on (cid:96)∞;1((cid:99)M ; C⊗p).
2. The operators {(cid:98)Lk}
f ∈ L2((cid:99)M ; C⊗p)
where γ, b are given as in (4.1.3) and b(cid:48) = 2b + 2(cid:107)(cid:98)h(cid:107)∞ + 2(cid:107)u(cid:107)∞ + 2λ.

k∈Td are uniformly sectorial; that is for every k ∈ Td and every
(cid:68)

(cid:69)(cid:12)(cid:12)(cid:12) ≤ γ Re

f, (cid:98)Lkf

f, (cid:98)Lkf

+ b(cid:48)(cid:107)f(cid:107)2
L2

(cid:12)(cid:12)(cid:12)Im

(4.2.29)

(cid:69)

(cid:68)

29

3. If Ψ ∈ W1(M ), then

e−t(cid:98)Lk(cid:98)Ψk :=

(cid:92)(cid:2)e−tLΨ(cid:3)

k

(4.2.30)

for every k ∈ Td. For Ψ ∈ L2(M ), Equation (4.2.30) holds for ν-almost every k.

Combining (4.2.30) with Pillet’s formula (Lemma 4.2.4), we obtain the following Floquet

transformed Pillet formula in vector form:

Lemma 4.2.13 (Floquet transformed Pillet formula). Let ψ0 ∈ (cid:96)2(Zd) and define(cid:98)ρ0;k(x) ∈

C⊗p for x ∈ Zd, k ∈ Td as

πσ(cid:98)ρ0;k(x) :=

(cid:88)

n∈pZd+σ

e−ik·nψ0(x − n)ψ0(−n), σ ∈ Zp.

(4.2.31)

e−ik·yE(cid:16)

(cid:17)
ψt(x − y)ψt(−y)

Then(cid:88)

y∈Zd

(cid:68)

=

δx ⊗ −→

L2((cid:99)M ;C⊗p)

In particular, for every k ∈ Td,

1 ⊗ 1, e−t(cid:98)Lk(cid:0)(cid:98)ρ0;k ⊗ 1(cid:1)(cid:69)
where ψt is the solution to Equation (4.1.1) with initial condition ψ0. Here e−t(cid:98)Lk((cid:98)ρ0;k⊗ 1) ∈
(cid:96)∞;1((cid:99)M ; C⊗p) for each k and is in L2((cid:99)M ; C⊗p) for ν-almost every k.
eik·xE(cid:16)|ψt(x)|2(cid:17)
1 ⊗ 1, e−t(cid:98)Lk(cid:0)(cid:98)ρ0;k ⊗ 1(cid:1)(cid:69)
δ0 ⊗ −→
(cid:16)
(cid:17)
e−tL(ρ0 ⊗ 1)
(cid:17)
(cid:16)
e−tL(ρ0 ⊗ 1)
(cid:69)
δx ⊗ δy ⊗ 1, e−tL(ρ0 ⊗ 1)

L2((cid:99)M ;C⊗p)
(cid:68)
(cid:69)
δx ⊗ δy, e−tL(ρ0 ⊗ 1(ω))

Pillet’s formula (4.2.12) can be rewritten as

E (Ψ(x, y,·)) =

= E(cid:16)

Proof. Let Ψ(x, y, ω) =

L2(Zd×Zd)

.

(cid:88)

x∈Zd

(cid:90)
(cid:68)

Ω

(cid:68)

=

.

(4.2.33)

(x, y, ω) µ(dω)

(x, y, ω) =

=

L2(Zd×Zd×Ω)

ψt(x)ψt(y)

.

,

(4.2.32)

(cid:17)

30

We note that ρ0 ⊗ 1 ∈ W1(M ) and that e−tL is a bounded operator on W1(M ) (see [33,
Lemma 3.9]). Thus Ψ ∈ W1(M ) and its Floquet transform

πσ(cid:98)Ψk(x, ω) =

(cid:88)

n∈pZd+σ

e−i k·n Ψ(x − n,−n, τnω).

is continuous in k. Direct computation shows that

(cid:90)

Td

eik·y πσ(cid:98)Ψk(x, ω) ν(dk) = Ψ(x − y,−y, τyω)δpZ+σ(y).
(cid:88)

e−ik·yΨ(x − y,−y, τyω) = πσ(cid:98)Ψk(x, ω),

y∈pZd+σ

Thus, by the Fourier-inversion formula,

and (cid:88)

y∈pZd+σ

e−ik·yE (Ψ(x − y,−y,·)) = πσE(cid:16)(cid:98)Ψk(x,·)
(cid:17)

=

(cid:68)

δx ⊗ 1, πσ(cid:98)Ψk

(cid:69)

L2((cid:99)M ;C)

for every k ∈ Td.

On the other hand, by (4.2.30), for Φ = ρ0 ⊗ 1, we have

(cid:98)Ψk =

(e−tLΦ)k = e−t(cid:98)Lk(cid:98)Φk,

(cid:92)

(cid:88)

where

πσ(cid:98)Φk = πσ

(cid:92)(ρ0 ⊗ 1)k(x, ω) =

e−i k·n ψ0(x − n)ψ0(−n) ⊗ 1.

Clearly, (cid:98)Φk = (cid:98)ρ0;k ⊗ 1, by the definition (4.2.31) of (cid:98)ρ0;k. Putting everything together, we
have (cid:88)

n∈pZd+σ

(cid:68)

(cid:17)

e−ik·yE(cid:16)
e−ik·yE(cid:16)

ψt(x − y)ψt(−y)
(cid:17)

ψt(x − y)ψt(−y)

=

y∈pZd+σ

(cid:88)

y∈Zd

(cid:69)
δx ⊗ 1, πσ e−t(cid:98)Lk(cid:98)ρ0;k ⊗ 1
(cid:69)
1 ⊗ 1, e−t(cid:98)Lk(cid:98)ρ0;k ⊗ 1

δx ⊗ −→

L2((cid:99)M ;C)

.

L2((cid:99)M ;C⊗p)

.

Finally, summing over σ in the periodicity cell Zp, we find that

=

(cid:68)

31

4.3 Spectral analysis on the augmented space

4.3.1 Spectral analysis of (cid:98)K0
The spectral analysis of (cid:98)Lk plays an important role in studying the diffusive scaling of this
model. We begin by showing that 0 is an eigenvalue of (cid:98)K0. This observation allows us to
write down a block decomposition and to find a spectral gap for (cid:98)L0 in the two sections that
The key observation regarding (cid:98)K0 is the following:
Lemma 4.3.1. Let x ∈ Zd and −→w ∈ C⊗p. Then

follow.

(cid:88)

(cid:98)K0 δx ⊗ −→w ⊗ 1 =
j=1(Apj )ξj with Ap the p × p right shift matrix,

h(ξ) δx−ξ ⊗ (I − A−ξ

ξ(cid:54)=0

p )−→w ⊗ 1,

p =(cid:78)d

where Aξ

(4.3.1)

(4.3.2)



0 1

0 0
...
...

0 0

1 0

0

1
. . .

0

0

 .

··· 0
··· 0
...
. . .
··· 1
··· 0

Ap :=

ξ(cid:54)=0

(cid:88)
(cid:88)
(cid:88)

ξ(cid:54)=0

ξ(cid:54)=0

=

=

Proof. This follows from direct computation:

πσ(cid:98)K0(δx ⊗ −→w ⊗ 1) =

h(ξ)[πσδx−ξ ⊗ −→w ⊗ 1 − πσ−ξδx−ξ ⊗ −→w ⊗ 1]
h(ξ)δx−ξ ⊗ [πσ − πσ−ξ]−→w ⊗ 1
p )−→w ⊗ 1.
h(ξ) δx−ξ ⊗ πσ(I − A−ξ

To proceed we need to consider the matrices Aξ

p. We begin with Ap, the p× p right shift.

32

Lemma 4.3.2. Let m ∈ Z, p ∈ Z>0. The matrix Am
eigenvalues,

p = (Ap)m has

p

gcd(m,p) distinct

2πi (cid:96)m
p ,

e

(cid:96) = 0, 1,··· ,

p

gcd(m, p)

− 1,

(4.3.3)

each of multiplicity gcd(m, p).

Proof. Since Ap

p = 1, it suffices to restrict our attention to 0 < m < p. The eigenvalues of

Ap are all p-th roots of unity

2πi (cid:96)

p , (cid:96) = 0, 1,··· , p − 1,

λ(cid:96) = e

and each eigenvalue has multiplicity one. The corresponding eigenvectors are the elements of

the discrete Fourier basis. For 1 < m < p, it follows from the spectral mapping theorem that
(cid:96) = λm
(cid:96)(cid:48)
gcd(m,p) for some integer n. Finally, since |(cid:96) − (cid:96)(cid:48)| < p, it follows that

for (cid:96) = 0, 1,··· , p − 1. From here, it is easy to verify that λm

p has eigenvalues λm
Am
(cid:96)
whenever |(cid:96) − (cid:96)(cid:48)| =

np

there are

p

gcd(m,p) distinct eigenvalues each of multiplicity gcd(m, p).

This result has an immediate extension to Ap, the tensor product of right shift operators.

(cid:78)d
Corollary 4.3.3. If p = (p1,··· , pd) ∈ Zd

mj
pj has eigenvalues

j=1 A

>0 and m = (m1,··· md) ∈ Zd, then Am

p :=

d(cid:89)

j=1

(cid:96)j mj
pj

;

2πi

e

(cid:96)j = 0, 1,··· ,

pj

gcd(mj, pj)

− 1.

j=1 is the standard basis on Zd, then
Ker(I − Aej

p ) = Cp1 ⊗ ··· ⊗ {−→
d(cid:92)

Ker(I − Aej

p ) = span{−→
1 }.

1 pj} ⊗ ··· ⊗ Cpd.

(4.3.4)

(4.3.5)

In particular, if (ej)d

Note that, by Equation (4.3.5),

j=1

The following lemma extends this result to a collection Amj
m1, . . . , mk generate Zd.

p , j = 1, . . . , k, where the vectors

33

Lemma 4.3.4. Let m1,··· , mk ∈ Zd, n1,··· , nk ∈ Z, and M = n1m1 + ··· + nkmk for
some k ≥ 1. Then, we have

k(cid:92)

j=1

Ker(I − Amj

p ) ⊂ Ker(I − AM
p ).

Ker(I − Aej

p ) = span{−→
1 }.

In particular, if m1,··· , mk generate Zd, then
d(cid:92)

Ker(I − Amj

p ) =

k(cid:92)
Proof. Suppose w ∈(cid:84)k

j=1

j=1

w = Amj

p w =

Repeated application of (4.3.8) yields

j=1 Ker(I − Amj

p ), then for each j = 1, 2,··· , k,

(cid:16)Amj

p

(cid:17)nj

w = Anj mj

p

w.

(4.3.6)

(4.3.7)

(4.3.8)

(4.3.9)

w = An1m1

p

= Ankmk

p

w = AM

p w.

Thus, w ∈ Ker(I − AM
p ).

If m1,··· , mk generate Zd, then (4.3.6) implies the first equality in (4.3.7). The second

equality follows from Corollary 4.3.3 since

d(cid:92)

Ker(I − Aej

p ) =

d(cid:92)

(cid:16)Cp1 ⊗ ··· ⊗ −→

1 pj ⊗ ··· ⊗ Cpd

(cid:17)

= span{−→
1 }.

j=1

j=1

We return now to consideration of (cid:98)K0. The non-degenerate support condition (4.1.9)
guarantees that the hopping kernel, h, is non-zero on a spanning set, {ξj}j∈J , of Zd. Com-
p )−→w = 0 for all ξ with h(ξ) (cid:54)= 0
bining this fact with Lemma 4.3.4, we can see that (I − A−ξ
if and only if −→w (cid:107) −→
Corollary 4.3.5. Let x ∈ Zd and −→w ∈ C⊗p. Then (cid:98)K0(δx ⊗ −→w ⊗ 1) = 0 if and only if
−→w (cid:107) −→

1 . Moreover, there is c0 > 0 such that for −→w ⊥ −→

1 . In particular, Lemma 4.3.1 leads to the following

(cid:13)(cid:13)(cid:13)(cid:98)K0(δx ⊗ −→w ⊗ 1)

1 ,

(cid:13)(cid:13)(cid:13)2 ≥ c0(cid:107)−→w(cid:107)2.

34

Proof. By Lemma 4.3.1, we have

(cid:13)(cid:13)(cid:13)(cid:98)K0(δx ⊗ −→w ⊗ 1)

(cid:13)(cid:13)(cid:13)2

=

(cid:88)

ξ

|h(ξ)|2(cid:13)(cid:13)(cid:13)(I − A

p )−→w
−ξ

(cid:13)(cid:13)(cid:13)2

.

The right hand side is a quadratic form Q(w) on the finite dimensional space C⊗p. Further-
more, by Lemma 4.3.1, Q(w) vanishes only if w (cid:107) −→
fact, by Lemma 4.3.2 the smallest eigenvalue of Q(w) on {−→
d(cid:88)

1 . The lower bound (4.3.9) follows. In

(cid:88)

1 }⊥ is

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)1 − exp

−2πi

(cid:96)jξj
pj

j=1

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

.

c0 = min
(cid:96)∈Zp\0

ξ

|h(ξ)|2

Thus c0 (cid:54)= 0 and Equation (4.3.9) holds.
4.3.2 Block decomposition of (cid:98)L0
In the previous section, we showed that δ0 ⊗ −→
for (cid:98)U and (cid:98)V. Finally, the Markov generator satisfies B1 = B†1 = 0. Therefore,

1 ⊗ 1 is an eigenvector of (cid:98)K0 corresponding

to the eigenvalue 0. Using (4.2.21) and (4.2.22), it is easy to check that this claim also holds

(cid:98)L0 δ0 ⊗ −→

1 ⊗ 1 = (cid:98)L†

0 δ0 ⊗ −→

1 ⊗ 1 = 0.

To further analyze the spectrum of (cid:98)Lk we will use a block decomposition associated to
the following direct sum decomposition of L2((cid:99)M ; C⊗p) ∼= (cid:96)2(Zd) ⊗ C⊗p ⊗ L2(Ω):

(4.3.10)

(cid:96)2(Zd) ⊗ C⊗p ⊗ L2(Ω) = (cid:98)H0 ⊕ (cid:98)H1 ⊕ (cid:98)H2 ⊕ (cid:98)H3,

(4.3.11)

where

(cid:98)H0 := span{δ0 ⊗ −→
(cid:98)H1 := δ0 ⊗ {−→

1 }⊥ ⊗ 1,

1 ⊗ 1},

(cid:98)H2 := {δ0}⊥ ⊗ C⊗p ⊗ 1 = (cid:96)2(Zd\{0}) ⊗ C⊗p ⊗ 1,

35

and

(cid:98)H3 :=

(cid:16)(cid:98)H0 ⊕ (cid:98)H1 ⊕ (cid:98)H2

(cid:17)⊥

(cid:26)

(cid:90)

=

Ψ(x, ω) :

Ψ(x, ω)dµ(ω) = 0

Note that dim (cid:98)H0 = 1, dim (cid:98)H1 = p1 ··· pd − 1, and dim (cid:98)H2 = dim (cid:98)H3 = ∞.
We will write operators on L2((cid:99)M ; C⊗p) as 4 × 4 matrices of operators acting between
the various spaces (cid:98)Hj, j = 0, 1, 2, 3. Throughout we will use the notation:
1. Pj = the orthogonal projection onto (cid:98)Hj,

Ω

.

(cid:27)

2. P⊥

j = 1 − Pj.
In particular, P = P⊥

3 = P0 + P1 + P2 is the orthogonal projection of L2((cid:99)M ; C⊗p) onto the

space (cid:98)H0 ⊕ (cid:98)H1 ⊕ (cid:98)H2 = (cid:96)2(Zd) ⊗ C⊗p ⊗ 1 of “non-random” functions:

P Ψ(x) =

Ψ(x, ω)dµ(ω).

Then P3 = P⊥ = 1 − P is the projection onto the space of mean zero functions (cid:98)H3.
Lemma 4.3.6. On (cid:98)H0 ⊕ (cid:98)H1 ⊕ (cid:98)H2 ⊕ (cid:98)H3 the operators (cid:98)K0,(cid:98)U,(cid:98)V, and B have following block

Ω

decomposition




0

(cid:98)K0 =

(cid:98)V =

0

0

0

P1(cid:98)K0P2
0 P2(cid:98)K0P1 P2(cid:98)K0P2

0

0

0

0

0

0

0

P3(cid:98)K0P3

 ,

0 0

0

0

0

0

0 0

P2(cid:98)VP3
0 0 P3(cid:98)VP2 P3(cid:98)VP3

0 0

0

Proof. The eigenvalue equation (4.3.10) gives

P0 T = T P0 = 0

36

(cid:90)

 ,

 ,

0

0

0

P3(cid:98)UP3



(cid:98)U =

0 0

0

0

0 0

0 0 P2(cid:98)UP2



0 0

0 0 0

0 0 0

0 0 0

0

0

0

0

 .

0 0 0 P3BP3

and B =

for T = (cid:98)K0,(cid:98)U,(cid:98)V, B,(cid:98)L0. From the definition (4.2.20) of (cid:98)K0 we see that this operator is
x and any F, G ∈ L2(Ω; Cp). Thus P1(cid:98)K0P1 = 0. The definitions (4.2.21), (4.2.22) of (cid:98)U, (cid:98)V
imply that they vanish on δ0 ⊗ F , so

(cid:69)
δx ⊗ F, (cid:98)K0δx ⊗ G

“off-diagonal” with respect to position, in the sense that

= 0 for any

(cid:68)

P1(cid:98)U = P1(cid:98)V = 0, (cid:98)UP1 = (cid:98)VP1 = 0.

Since (cid:98)K0,(cid:98)U are “non-random”, we have for j = 0, 1, 2,

Pj(cid:98)K0P3 = 0, P3(cid:98)K0Pj = 0, Pj(cid:98)UP3 = 0, P3(cid:98)UPj = 0.

Since (cid:98)V is mean zero on L2(Ω) and B1 = B†1 = 0, we have
3 B = BP⊥

3 (cid:98)VP⊥

3 = 0, P⊥

P⊥

3 = 0.

Corollary 4.3.7. On (cid:98)H the operator (cid:98)L0 = i(cid:98)K0 + i(cid:98)U + iλ(cid:98)V + B has block decomposition



(cid:98)L0 =

0

0

0

0

iP1(cid:98)K0P2
0 iP2(cid:98)K0P1 P2(i(cid:98)K0 + i(cid:98)U)P2
iλP3(cid:98)VP2

0

0

0

0

0

iλP2(cid:98)VP3
P3(cid:98)L0P3

 .

(4.3.12)

4.3.3 Spectral gap

With the block decomposition (4.3.12), we are now in a position to prove that (cid:98)L0 has a
Lemma 4.3.8. If λ > 0, then 0 is a non-degenerate eigenvalue of (cid:98)L0 and there is g > 0

spectral gap.

such that

σ((cid:98)L0) = {0} ∪ Σ+

with Σ+ ⊂ {z : Re z > g}. For λ small, there is c = c(p,(cid:107)(cid:98)h(cid:107)∞,(cid:107)u(cid:107)∞, γ, T, b) > 0 such

that g ≥ cλ2.

37

Im z

σ((cid:98)L0) (cid:51) 0

g

σ((cid:98)L0)\{0} ⊂ {z : Re z > g } ∩ N+

Re z

Figure 4.1: Spectral gap of (cid:98)L0

by the sectoriality of B,

Before proceeding to the proof of the lemma, we note that the sectoriality of B places

further restrictions on Σ+. Indeed, Re(cid:98)L0 = Re B ≥ 0 in the sense of quadratic forms. Thus,
(cid:110)(cid:104)Φ,(cid:98)L0Φ(cid:105)(cid:111)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13)(cid:98)K0 + (cid:98)U + λ(cid:98)V(cid:13)(cid:13)(cid:13) + |Im{(cid:104)Φ, BΦ(cid:105)}|
(cid:12)(cid:12)(cid:12)Im
if (cid:107)Φ(cid:107) = 1. It follows that the numerical range Num((cid:98)L0) =

≤ 2(cid:107)(cid:98)h(cid:107)∞ + 2(cid:107)u(cid:107)∞ + 2λ + γ Re(cid:104)Φ,(cid:98)L0Φ(cid:105),
(cid:69)(cid:12)(cid:12)(cid:12) (cid:107)Φ(cid:107) = 1
(cid:111)

contained in

(cid:110)(cid:68)
Φ, (cid:98)L0Φ
N+ := {z : Re z ≥ 0 and | Im z| ≤ 2(cid:107)(cid:98)h(cid:107)∞ + 2(cid:107)u(cid:107)∞ + 2λ + γ Re z}.
Since σ((cid:98)L0) ⊂ Num((cid:98)L0), we find that Σ+ ⊂ {Re z > g} ∩ N+ —see Figure 4.1.
To prove Lemma 4.3.8, it suffices to show that the restriction of (cid:98)L0 to (cid:98)H⊥

0 = (cid:98)H1⊕(cid:98)H2⊕(cid:98)H3,

(4.3.13)

is

(4.3.14)



0

iP1(cid:98)K0P2
iP2(cid:98)K0P1 P2(i(cid:98)K0 + i(cid:98)U)P2
iλP3(cid:98)VP2

0

 ,

0

iλP2(cid:98)VP3
P3(cid:98)L0P3

J =

has spectrum contained in {Re z > g}.

38

Lemma 4.3.9. There is g > 0, such that whenever Re z < g,

1. Γ3 − z is boundedly invertible on (cid:98)H3, where Γ3 = P3(cid:98)L0P3,
2. Γ2(z) − z is boundedly invertible on (cid:98)H2, where
i(cid:98)K0 + i(cid:98)U + λ2(cid:98)V (Γ3 − z)−1(cid:98)V(cid:17)

(cid:16)
3. J − z is boundedly invertible on (cid:98)H⊥
In particular, J is boundedly invertible. Let Π2 be the projection onto Ker(P1(cid:98)K0) (cid:40) (cid:98)H2. If
Π2(cid:101)φ (cid:54)= 0 for some (cid:101)φ ∈ (cid:98)H2, then P2J −1(cid:101)φ (cid:54)= 0 and

Γ2(z) = P2

(4.3.15)

P2,

0 .

(cid:68)(cid:101)φ, P2J −1(cid:101)φ

(cid:69) ≥ g(cid:107)P2J −1(cid:101)φ(cid:107)2 > 0.

Re

(4.3.16)

Proof. We obtain this result by repeated applications of the Schur complement formula. As
observed above, we may restrict attention to the sectorial domain z ∈ N+. Fix z ∈ N+ and
consider the equation

 =



ζ

φ

Φ



0

(J − z)

−z

iP2(cid:98)K0P1 P2(i(cid:98)K0 + i(cid:98)U)P2 − z

iP1(cid:98)K0P2
iλP3(cid:98)VP2





ζ

φ

Φ

 =



 ,
(cid:101)ζ(cid:101)φ(cid:101)Φ

0

iλP2(cid:98)VP3
P3(cid:98)L0P3 − z

for (ζ, φ, Φ) ∈ (cid:98)H1 ⊕ (cid:98)H2 ⊕ (cid:98)H3 given ((cid:101)ζ,(cid:101)φ,(cid:101)Φ) ∈ (cid:98)H1 ⊕ (cid:98)H2 ⊕ (cid:98)H3. By the gap condition (4.1.4)

(4.3.17)

on B,

Re P3(cid:98)L0P3 = Re P3(i(cid:98)K0 + i(cid:98)U + B + iλ(cid:98)V)P3 ≥ 1

Therefore, Γ3 − z = P3(cid:98)L0P3 − z is boundedly invertible on (cid:98)H3 provided Re z < 1

P3.

T

T . For such

z, we may solve the third equation of (4.3.17) to obtain

Φ = (Γ3 − z)−1(cid:101)Φ − (Γ3 − z)−1 iλ(cid:98)Vφ.

Using the solution (4.3.18), we reduce the second equation of (4.3.17) to

[Γ2(z) − z] φ = (cid:101)φ − iP2(cid:98)K0ζ − iλP2(cid:98)V (Γ3 − z)−1(cid:101)Φ

(4.3.18)

(4.3.19)

39

with Γ2(z) as in (4.3.15). For ϕ ⊗ 1 ∈ (cid:98)H2 = L2(Zd\{0}; C⊗p), notice that ϕ ⊗ 1 = P2ϕ ⊗ 1

and Γ2 = P2Γ2, we have

Re(cid:104)ϕ ⊗ 1, Γ2(z) ϕ ⊗ 1(cid:105)(cid:98)H2

1
2

by,

(4.3.20)

T

T

T

≥ λ2

= λ2

= λ2

− Re z

− Re z

− Re z

ϕ ⊗ 1,

=

(Γ2(z) + Γ

= Re(cid:104)ϕ ⊗ 1, Γ2(z) ϕ ⊗ 1(cid:105)(cid:98)H
(cid:28)
(cid:69)(cid:98)H
= λ2(cid:68)
P3(Γ3 − z)−1(cid:98)V ϕ ⊗ 1, (Re B − Re z) P3(Γ3 − z)−1(cid:98)V ϕ ⊗ 1
(cid:18) 1
(cid:18) 1
(cid:18) 1

(cid:29)
(cid:98)H
†
2(z)) ϕ ⊗ 1
(cid:19)(cid:13)(cid:13)(cid:13)P3(Γ3 − z)−1(cid:98)V ϕ ⊗ 1
(cid:13)(cid:13)(cid:13)2(cid:98)H
(cid:19)(cid:13)(cid:13)(cid:13)(Γ3 − z)−1(cid:98)V ϕ ⊗ 1
(cid:13)(cid:13)(cid:13)2(cid:98)H3
(cid:13)(cid:13)(cid:13)(cid:13)2(cid:98)H3
(cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:16)
(cid:17)−1
B−1(cid:98)V ϕ ⊗ 1
B−1(Γ3 − z)
where the inverse of B is well defined since (cid:98)Vϕ ⊗ 1 ∈ (cid:98)H3 = RanP3. Furthermore, B−1 is
(cid:13)(cid:13) ≤ T . Thus B−1(Γ3−z) is bounded for z ∈ N+∩{Re z < 1
bounded on (cid:98)H3, with(cid:13)(cid:13)B−1P3
T }
(cid:13)(cid:13)(cid:13)B−1P3((cid:98)K0 + (cid:98)U + λ(cid:98)V)
(cid:13)(cid:13)(cid:13)(cid:98)H ≤ 1 +
(cid:13)(cid:13)(cid:13)B−1P3(Γ3 − z)P3
≤ 1 + T (2(cid:107)(cid:98)h(cid:107)∞ + 2(cid:107)u(cid:107)∞ + 2λ + |z|)
≤ 2 + γ + 4T ((cid:107)(cid:98)h(cid:107)∞ + (cid:107)u(cid:107)∞ + λ).
(cid:13)(cid:13)(cid:13)2(cid:98)H
(cid:19) (cid:13)(cid:13)(cid:13)B−1(cid:98)V ϕ ⊗ 1
(cid:18) 1
(cid:13)(cid:13)B−1(Γ3 − z)(cid:13)(cid:13)2(cid:98)H
(cid:80)
(cid:80)
(cid:18) 1
(cid:19)
(cid:16)
(cid:17)2
2 + γ + 4T ((cid:107)(cid:98)h(cid:107)∞ + (cid:107)u(cid:107)∞ + λ)
(cid:17)2 (cid:107)ϕ ⊗ 1(cid:107)2(cid:98)H2
(cid:16)

Re(cid:104)ϕ ⊗ 1, Γ2(z) ϕ ⊗ 1(cid:105)(cid:98)H2

λ2χ2(1 − T Re z)

2 + γ + 4T ((cid:107)(cid:98)h(cid:107)∞ + (cid:107)u(cid:107)∞ + λ)

(cid:13)(cid:13)(cid:13) + |z|(cid:13)(cid:13)(cid:13)B−1P3

(cid:13)(cid:13)(cid:13)

Putting (4.3.20), (4.3.21) and (4.1.6) together, we obtain

χ2|πσϕ(x)|2

σ∈Zp

x(cid:54)=0

− Re z

≥ λ2

− Re z

(4.3.21)

.

T

T

≥ λ2

=

T

40

Let

so that

have

(cid:18)

λ2χ2

(cid:16)
2 + γ + 4T ((cid:107)(cid:98)h(cid:107)∞ + (cid:107)u(cid:107)∞ + λ)
(cid:17)2 (1 − T c1) = 2c1. Then for z ∈ N+ ∩ {Re z ≤ c1}, we

(4.3.22)

(cid:17)2(cid:19) ,

c1 =

T

λ2χ2 + 2

(cid:16)

T

2+γ+4T ((cid:107)(cid:98)h(cid:107)∞+(cid:107)u(cid:107)∞+λ)

λ2χ2

Re Γ2(z) − Re z ≥ 2c1 − Re z ≥ c1,

implying that Γ2(z)−z is boundedly invertible. Thus, (4.3.19) can be solved on (cid:98)H2 to obtain
φ = (Γ2(z) − z)−1(cid:101)φ − (Γ2(z) − z)−1iP2(cid:98)K0ζ

(4.3.23)

− (Γ2(z) − z)−1iλP2(cid:98)V (Γ3 − z)−1(cid:101)Φ.

(4.3.24)

Now, the first equation of (4.3.17) reduces to the following

[Γ1(z) − z]ζ = (cid:101)ζ − iP1(cid:98)K0 (Γ2(z) − z)−1(cid:101)φ
where Γ1(z) = P1(cid:98)K0(Γ2(z) − z)−1P2(cid:98)K0P1. We will use the same strategy to show that
Γ1(z)− z is invertible. Take ζ = δ0⊗−→w ⊗ 1 ∈ (cid:98)H1. Recall, by definition of (cid:98)H1, that −→w ⊥ −→

− λP1(cid:98)K0(Γ2(z) − z)−1P2(cid:98)V (Γ3 − z)−1(cid:101)Φ,

(4.3.25)

1 .

Thus, by (4.3.23) and Corollary 4.3.5,

Re(cid:104)ζ, Γ1(z)ζ(cid:105)(cid:98)H1
(cid:68)

=

(cid:69)(cid:98)H
(Γ2(z) − z)−1(cid:98)K0ζ, (Re Γ2(z) − Re z) (Γ2(z) − z)−1 (cid:98)K0ζ
(cid:107)ζ(cid:107)2(cid:98)H.
(cid:107)Γ2(z) − z(cid:107)2(cid:98)H

c1c0

≥

For z ∈ N+ ∩ {Re z < 1

(cid:107)Γ2(z) − z(cid:107)(cid:98)H ≤ 2(cid:107)(cid:98)h(cid:107)∞ + 2(cid:107)u(cid:107)∞ + 4λ2(cid:13)(cid:13)(cid:13)(P3(cid:98)L0P3 − z)−1(cid:13)(cid:13)(cid:13)(cid:98)H3
(cid:19)−1

2T },
≤ 4(cid:107)(cid:98)h(cid:107)∞ + 4(cid:107)u(cid:107)∞ + 4λ2
= 4(cid:107)(cid:98)h(cid:107)∞ + 4(cid:107)u(cid:107)∞ + 8T λ2 + 2λ + (γ + 1)(2T )−1,

(cid:18) 1

− 1
2T

+ |z|

T

+ 2λ + (γ + 1) Re z

(4.3.26)

(4.3.27)

41

by (4.3.15) and (4.1.4). Putting (4.3.26) and (4.3.27) together, we obtain

Re(cid:104)ζ, Γ1(z)ζ(cid:105)(cid:98)H1
≥

(cid:16)

4(cid:107)(cid:98)h(cid:107)∞ + 4(cid:107)u(cid:107)∞ + 8T λ2 + 2λ + (γ + 1)(2T )−1(cid:17)2 (cid:107)ζ(cid:107)2(cid:98)H =: c2(cid:107)ζ(cid:107)2(cid:98)H.

c1c0

Therefore, Re Γ1(z) > Re z on (cid:98)H1 provided z ∈ N+ and Re z < min{c1, 1
2T , c2} =: g.
For such z it follows that Γ1(z) − z is boundedly invertible and (4.3.25) can be solved on
(cid:98)H1. Therefore, (4.3.17) is explicitly solvable on (cid:98)H = (cid:98)H1 ⊕ (cid:98)H2 ⊕ (cid:98)H3 and J − z is boundedly
invertible for all z ∈ {z : | Re z| < g}(cid:84)N+.
To prove the second part of Lemma 4.3.9, it is enough to solve J Ψ = (cid:101)Ψ for Ψ = (ζ, φ, Φ)
given (cid:101)Ψ = (0,(cid:101)φ, 0). The three equations are reduced to

iP1(cid:98)K0P2 φ = 0
iP2(cid:98)K0P1 ζ + P2(i(cid:98)K0 + i(cid:98)U)P2 φ + iλP2(cid:98)VP3 Φ =(cid:101)φ
iλP3(cid:98)VP2 φ + P3(cid:98)L0P3 Φ = 0

The first equation implies φ ∈ Ker(P1(cid:98)K0). Therefore, φ = Π2φ, where Π2 is the pro-
jection onto the kernel of P1(cid:98)K0. As derived in the general case, the second and the third

equations imply that

If ξ satisfies P1(cid:98)K0ξ = 0, then

(cid:68)

Applying Π2 to (4.3.28), we have

(cid:69)

(cid:68)

iP2(cid:98)K0 P1ζ + Γ2φ =(cid:101)φ.
ξ, (cid:98)K0 P1ζ
P1(cid:98)K0ξ, P1ζ
Π2Γ2Π2 φ = Π2(cid:101)φ.

=

Clearly, if Π2(cid:101)φ (cid:54)= 0, then φ = P2Ψ = P2J −1(cid:101)φ (cid:54)= 0. Notice that
(cid:68)
(cid:69)
Π2(cid:98)K0 P1ζ, Π2φ
(cid:68)(cid:101)φ, φ
(cid:69)

iP2(cid:98)K0 P1ζ + Γ2φ, φ

= 0. Equation (4.3.28) also implies that

(cid:68)

(cid:69)

= Re

Re

which completes the proof of (4.3.16).

42

(cid:69)

= 0. Therefore, Π2(cid:98)K0P1ζ = 0.

(4.3.28)

(cid:69)
(cid:68)(cid:98)K0 P1ζ, φ

(cid:68)(cid:98)K0 P1ζ, Π2φ
(cid:69)

=

=

= Re(cid:104)Γ2φ, φ(cid:105) ≥ 2c1 (cid:107)φ(cid:107)2 ≥ g (cid:107)φ(cid:107)2 > 0,

The spectral gap g of (cid:98)L0 has consequences for the dynamics of the semi-group.
Lemma 4.3.10. Let Q0 = orthogonal projection onto (cid:98)H0 = span δ0⊗−→
1 ⊗1 in L2((cid:99)M ; C⊗p).
Then e−t(cid:98)L0(1−Q0) is a contraction semi-group on ran(1−Q0), and for all sufficiently small


 > 0 there is C
 > 0 such that

≤ C
e−t(g−
)

(4.3.29)

Lemma 4.3.11. There is c0 > 0 such that

≤ c0|k|.
If |k| is sufficiently small, the spectrum of (cid:98)Lk consists of:
1. A non-degenerate eigenvalue E(k) contained in S0 = {z : |z| < c0|k|}.
2. The rest of the spectrum is contained in the half plane S1 = {z : Re z > g− c0|k|} such

(cid:13)(cid:13)(cid:13)e−t(cid:98)L0(1 − Q0)
(cid:13)(cid:13)(cid:13)(cid:98)Lk − (cid:98)L0

(cid:13)(cid:13)(cid:13)L2((cid:99)M ;C⊗p)
(cid:13)(cid:13)(cid:13)L2((cid:99)M ;C⊗p)

that S0 ∩ S1 = ∅.

Furthermore, E(k) is C2 in a neighborhood of 0,

Denote ∂j = ∂kj

and ϕ0 = 1√⊗p
(cid:68)
∂j(cid:98)K0ϕ0, P2J −1 ∂i(cid:98)K0ϕ0

∂i∂jE(0) =

1 ⊗ 1 for simplicity where ⊗p = p1 · p2 ··· pd, then

E(0) = 0, ∇E(0) = 0.
δ0 ⊗ −→

(cid:69)

+

(cid:69)
(cid:68)
∂i(cid:98)K0ϕ0, P2J −1 ∂j(cid:98)K0ϕ0
d×d =(cid:0)∂i∂jE(0)(cid:1)

(cid:1)

(4.3.30)

(4.3.31)

where P2,J and J −1 are given in (4.3.14) and Lemma 4.3.9.
Remark 4.3.12. Let D :=(cid:0)Di,j
symmetric. Furthermore, for any k ∈ Td, in view of the expression of ∂i(cid:98)K0 in (4.2.26),
0 (cid:54)=(cid:80)
i ki∂i(cid:98)K0ϕ0 ∈ (cid:96)2(Zd) ⊗ −→
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2

1 ⊗ 1. It is non-zero due the non-degeneracy of h. Therfore,

ki∂i(cid:98)K0ϕ0, P2J −1(cid:88)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)P2J −1(cid:88)

d×d. It is clear from (4.3.31) that D is

ki∂i(cid:98)K0ϕ0

ki∂i(cid:98)K0ϕ0

Re(cid:104)k, Dk(cid:105) = 2 Re

by (4.3.16) in Lemma 4.3.9,

(cid:42)(cid:88)

i

(cid:43)

i

> 2g

> 0.

i

43

In the next section, we will relate the matrix element of D with limits of diffusively scaled
moments. From the real valued moments, we will see that ∂i∂jE(0) ∈ R and then D is
(cid:52)
Similar to Lemma 4.3.10, dynamical information about the semi-group e−t(cid:98)Lk follows
positive definite.
from the spectral gap of (cid:98)Lk in Lemma 4.3.11:
(cid:13)(cid:13)(cid:13)L2((cid:99)M ;C⊗p)

Lemma 4.3.13. If 
 is sufficiently small, then there is C
 < ∞ such that

(cid:13)(cid:13)(cid:13)e−t(cid:98)Lk(1 − Qk)

≤ C
e−t(g−
−c0|k|)

for all sufficiently small k.

Notice that ⊗p = p1 · p2 ··· pd. The case where d = 1 and ⊗p = p1 = 1 is equivalent to
the free case considered in [23], where the above lemmas were proved. The proof follows

from the standard perturbation theory of analytic semi-groups—see for instance [11, 24].
There are no essential differences in the proof when ⊗p > 1. We omit the proofs for Lemma

4.3.10-Lemma 4.3.13 here. We only sketch the proofs for (4.3.30) and (4.3.31), which plays

the most important role for the explicit expression of the diffusion constant in the next

section.

Proof of (4.3.30) and (4.3.31). Write ∂j = ∂kj

eigenvalue of (cid:98)Lk, and the associated normalized eigenvector ϕk. Let Qk be the orthogonal
projection onto ϕk. Clearly E(0) = 0, ϕ0 = 1√⊗p

1 ⊗ 1 and (cid:98)L0ϕ0 = (cid:98)L†

for short. Let E(k) be the non-degenerate

0ϕ0 = 0. Since

δ0 ⊗ −→
(cid:98)Lkϕk = E(k)ϕk,

direct computation shows

∂j(cid:98)Lk ϕk + (cid:98)Lk∂jϕk = ∂jE(k)ϕk + E(k)∂jϕk
=⇒∂j(cid:98)L0ϕ0 + (cid:98)L0∂jϕ0 = ∂jE(0)ϕ0.

Notice that ∂j(cid:98)L0 = i∂j(cid:98)K0 maps (cid:98)H0 = ranQ0 to (cid:98)H2, therefore, Q0∂j(cid:98)L0 = 0 and

(cid:68)

ϕ0, ∂j(cid:98)L0ϕ0

(cid:68)

(cid:69)

+

ϕ0, (cid:98)L0∂jϕ0

∂jE(0) =

(cid:68)

(cid:69)

=

Q0ϕ0, ∂j(cid:98)L0ϕ0

(cid:69)

(cid:68)(cid:98)L†

+

0ϕ0, ∂jϕ0

44

(4.3.32)

(4.3.33)

(4.3.34)

(cid:69)

= 0.

Differentiating (4.3.33) again, we have

∂i∂j(cid:98)Lkϕk + ∂j(cid:98)Lk∂iϕk + ∂i(cid:98)Lk∂jϕk + (cid:98)Lk∂i∂jϕk

=∂i∂jE(k)ϕk + ∂jE(k)∂iϕk + ∂iE(k)∂jϕk + E(k)∂i∂jϕk.

(4.3.35)

∂j∂jE(0) =

Evaluating (4.3.35) at k = 0 and using ∇E(0) = 0, we have that

1 ⊗ 1 because of (4.2.26). Corollary 4.3.5

∂i∂j(cid:98)L0ϕ0 + ∂j(cid:98)L0∂iϕ0 + ∂i(cid:98)L0∂jϕ0 + (cid:98)L0∂i∂jϕ0 = ∂i∂jE(0)ϕ0.
0 and ∂j(cid:98)L0ϕ0 ∈ (cid:96)2(Zd\{0}) ⊗ −→
(cid:68)

We also have Q0∂i∂j(cid:98)L0 = 0 for the same reason as for Q0∂j(cid:98)L0. Notice that ∂j(cid:98)L0 =
i∂j(cid:98)K0 = −∂j(cid:98)L†
implies ∂j(cid:98)L0ϕ0 ∈ Ker(P1(cid:98)K0) = ran(Π2) (cid:40) (cid:98)H2. Therefore,
(cid:69)
ϕ0, ∂i(cid:98)L0∂jϕ0
(cid:68)
∂i(cid:98)K0 ϕ0, ∂jϕ0
(cid:69)
(cid:68)
∂j(cid:98)L0ϕ0 + (cid:98)L0∂jϕ0 = 0
i∂j(cid:98)K0ϕ0 + (cid:98)L0∂jϕ0 = 0

(cid:69)
(cid:68)
ϕ0, ∂j(cid:98)L0∂iϕ0
(cid:69)
(cid:68)
∂j(cid:98)K0 ϕ0, ∂iϕ0
(cid:68)
P2∂j(cid:98)K0 ϕ0, P2∂iϕ0

for P2∂iϕ0. Recall the block form of (cid:98)L0 in (4.3.12) and J in (4.3.14). The key fact ∂j(cid:98)K0ϕ0 =
Π2∂j(cid:98)K0ϕ0 ∈ (cid:98)H2 reduces Equation (4.3.36) to what we have considered in the second part

(4.3.36)

P2∂i(cid:98)K0 ϕ0, P2∂jϕ0

It remains to solve

(cid:69)

(cid:69)

i.e.,

+ i

+ i

+

=i

=i

.





 =



 .

0

0

−i∂j(cid:98)K0ϕ0

0

45



of Lemma 4.3.9:

∗
∗

0

0

0

0

0

0

0

iP1(cid:98)K0P2
0 iP2(cid:98)K0P1 P2(i(cid:98)K0 + i(cid:98)U)P2
iλP3(cid:98)VP2
P2∂jϕ0 = −iP2J −1 ∂j(cid:98)K0ϕ0,

iλP2(cid:98)VP3
P3(cid:98)L0P3

∗

0

0

P2∂jϕ0

As derived in Lemma 4.3.9:

where P2 is the projection onto (cid:98)H2. Therefore,
(cid:68)
∂j(cid:98)K0 ϕ0, −iP2J −1 ∂i(cid:98)K0ϕ0
(cid:69)
(cid:68)
∂j(cid:98)K0 ϕ0, P2J −1 ∂i(cid:98)K0ϕ0

∂j∂jE(0) =i

=

+

(cid:69)
(cid:68)

(cid:68)
∂i(cid:98)K0 ϕ0, −iP2J −1 ∂j(cid:98)K0ϕ0
∂i(cid:98)K0 ϕ0, P2J −1 ∂j(cid:98)K0ϕ0

(cid:69)

,

+ i

(cid:69)

which gives (4.3.31).

4.4 Proof of the main results

4.4.1 Central limit theorem
We first prove (4.1.12) for bounded continuous f and normalized ψ0 ∈ (cid:96)2(Zd). The extension
to quadratically bounded f follows from some standard arguments combining (4.1.12) for

bounded continuous f and diffusive scaling for second moments, Lemma 4.4.1. We refer

readers to Section 4.5 in [33] for more details about this extension. We omit the proof of the

extension here.

To prove (4.1.12) for bounded continuous f , it suffices, by Levy’s Continuity Theorem

and a limiting argument, to prove

(cid:88)

x∈Zd

lim
t→∞

t E(cid:16)|ψt(x)|2(cid:17)

ik· x√
e

− 1
2(cid:104)k, Dk(cid:105)

,

= e

(4.4.1)

where ψt(x) ∈ (cid:96)2(Zd) is the solution to Equation (4.1.1) with initial condition ψ0 ∈ (cid:96)2(Zd).
As pointed out in Section 4.2, [33], it is enough to establish Equation (4.4.1) for ψ0 ∈ (cid:96)1(Zd);
it then extends to all of ψ0 ∈ (cid:96)2(Zd) by a limiting argument. So throughout this section, we
assume that

(cid:107)ψ0(cid:107)(cid:96)2 = 1,

and (cid:107)ψ0(cid:107)(cid:96)1 :=

|ψ0(x)| < ∞.

(4.4.2)

(cid:88)

x∈Zd

We also denote for simplicity

ϕ0 := ϕ0(x, ω) =

δ0 ⊗ −→

1 ⊗ 1, Φk := Φk(x, ω) =

1√⊗p

√⊗p ·(cid:98)ρ0;k(x) ⊗ 1,

(4.4.3)

46

where

−→

1 ,(cid:98)ρ0;k(x) ∈ C⊗p are defined in (4.2.31). Recall that for any σ ∈ Zp
ψ0(x − n)ψ0(−n).

−→
1 = 1,

(cid:88)

πσ

πσ(cid:98)ρ0;0(x) =
(cid:42)

n∈pZd+σ

=

x∈Zd

i k√
t

e

ϕ0 , e

√
k/

−t(cid:98)L

(cid:43)
L2((cid:99)M ;C⊗p)

xE(cid:16)|ψt(x)|2(cid:17)

By (4.2.33), we have(cid:88)
Letting Qk denote the Riesz projection onto the eigenvector of (cid:98)Lk near zero, we have
(cid:33)
(cid:88)
(cid:33)

xE(cid:16)|ψt(x)|2(cid:17)

−t(cid:98)L
(cid:42)

(cid:43)
(cid:43)

(cid:42)
(cid:42)

t Φ k√
t

tQ k√
t

Φ k√
t

√
k/

√
k/

ϕ0 , e

ϕ0 , e

(cid:42)

i k√
t

e

x∈Zd

Φ k√
t

+

=

.

t

(cid:43)
(cid:43)

−t(cid:98)L
−t(cid:98)L

√
k/

t

(cid:32)
1 − Q k√
(cid:32)
1 − Q k√

t

t

ϕ0 , Q k√
t

Φ k√
t

+

ϕ0 , e

−tE( k√
t

)

=e

.

Φ k√
t

(4.4.4)

By Lemma 4.3.13, the second term in (4.4.4) is exponentially small in the large t limit,

(cid:32)

−t(cid:98)L

√
k/

t

ϕ0 , e

(cid:42)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:33)

1 − Q k√

t

Φ k√
t

(cid:43)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(1 − Q k√

t
−t(g−
−c

≤ C
e

√
k/

t

−t(cid:98)L
)e
|k|√

) · (cid:107)ϕ0(cid:107) ·

t

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) · (cid:107)ϕ0(cid:107) ·
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Φ k√

t

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Φ k√
(cid:13)(cid:13)(cid:13)(cid:13)(cid:13).

t

(4.4.5)

Direct computation shows that

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Φ k√

t

(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)2
L2((cid:99)M ;C⊗p)

lim
t→∞

(cid:96)2(Zd;C⊗p)

=(⊗p)(cid:13)(cid:13)(cid:98)ρ0;0
(cid:88)

≤(⊗p)

σ∈Zp

(cid:13)(cid:13)2
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:88)
(cid:88)

x∈Zd

(cid:88)
 (cid:88)

n∈pZd+σ

≤(⊗p)(cid:107)ψ0(cid:107)2
(cid:96)2

≤(⊗p)(cid:107)ψ0(cid:107)2

σ∈Zp
(cid:96)2 · (cid:107)ψ0(cid:107)2

n∈pZd+σ
(cid:96)1 < ∞.

|ψ0(−n)|

ψ0(x − n)ψ0(−n)

2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

2

47

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:42)

ϕ0 , e

(cid:32)

−t(cid:98)L k√

t

(cid:33)

Φ k√
t

1 − Q k√

t

(cid:43)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −→ 0 as t → ∞.
(cid:88)

Regarding the first term in (4.4.4), we have by Taylor’s formula,

=

1
2

∂j∂jE(0)

ki√
t

kj√
t

+ o(

1
t

) =

1
2t

∂j∂jE(0)kikj + o(

1
t

),

i,j

since E(0) = ∇E(0) = 0. Thus,

√

e−tE(k/

t) = e

−t 1
2t

i,j ∂j ∂j E(0)kikj + o(1) = e

(cid:80)

i,j ∂j ∂j E(0)kikj + o(1).

− 1
2

(4.4.6)

Therefore, in (4.4.5),

(cid:19)

(cid:18) k√

t

E

(cid:88)

i,j

(cid:80)

Direct computation shows that

(cid:104)ϕ0, Φ0(cid:105)

L2((cid:99)M ;C⊗p)

=

=

Thus,

(cid:68)
(cid:69)
1 ⊗ 1 , (cid:98)ρ0;0 ⊗ 1
δ0 ⊗ −→
(cid:88)
(cid:88)

L2((cid:99)M ;C⊗p)
ψ0(−n)ψ0(−n) = (cid:107)ψ0(cid:107)2

σ∈Zp

n∈pZd+σ

(cid:96)2(Zd;C)

Q0Φ0 = Projϕ0

Φ0 = (cid:104)ϕ0, Φ0(cid:105) · ϕ0
(cid:18)

(cid:107)ϕ0(cid:107)2 = ϕ0.
(cid:43)

(cid:19)(cid:42)

−tE

k√
t

Putting together everything, we have

xE(cid:16)|ψt(x)|2(cid:17)

i k√
t

e

(cid:88)

x∈Zd

lim
t→∞

= lim

t→∞ e
(cid:80)
− 1
i,j ∂j ∂j E(0)kikj (cid:104)ϕ0 , Q0Φ0(cid:105) = e
2

ϕ0 , Q k√
t

Φ k√
t

=e

− 1
2

(cid:80)

i,j ∂j ∂j E(0)kikj .

Therefore, (4.4.1) holds true with Di,j = ∂j∂jE(0) for any normalized ψ0 ∈ (cid:96)2(Zd).

4.4.2 Diffusive scaling and reality of the diffusion matrix

= 1.

(4.4.7)

We proceed to prove the diffusive scaling (4.1.13) under the assumption that

|ψ0(x)|2 = 1,

|x|2|ψ0(x)|2 < ∞.

(4.4.8)

(cid:88)

(cid:88)

x

x

48

(cid:88)

x

Similar to (4.4.2), it is enough to establish the results for xψ0 ∈ (cid:96)1(Zd); it then extends to
all of xψ0 ∈ (cid:96)2(Zd) by a limiting argument. We assume that

|x||ψ0(x)| < ∞.

(4.4.9)

We continue to use the notation in (4.4.3). Also, (cid:104)·, ·(cid:105) will stand for (cid:104)·, ·(cid:105)

otherwise specified. We also denote ∂i = ∂ki

As pointed out in Section 4.4 in [33],(cid:80)
, i = 1,··· , d for short.
x(1 + |x|2)|ψt(x)|2 ≤ eCt for each t > 0. Thus
(cid:88)

the second moments of the position

xixjE(cid:16)|ψt(x)|2(cid:17)

Mi,j(t) :=

(4.4.10)

L2((cid:99)M ;C⊗p)

unless

x∈Zd

are well defined and finite. The main task of this section is to show that Mi,j(t) ∼ Di,jt,
where Di,j = ∂i∂jE(0) are given in (4.3.31). More precisely,
Lemma 4.4.1. Let P2J −1 be as in Lemma 4.3.9 . Suppose the initial value ψ0 satisfies
(4.4.8), then for all 1 ≤ i, j ≤ d,

1
t

(cid:69)

(cid:68)

lim
t→∞

Mi,j(t) =

(cid:68)
As a consequence, ∂i∂jE(0) ∈ R and D =(cid:0)∂i∂jE(0)(cid:1)

∂j(cid:98)K0ϕ0, P2J −1 ∂i(cid:98)K0ϕ0
|x|2 E(cid:16)|ψt(x)|2(cid:17)

(cid:88)

+

lim
t→∞

1
t

x∈Zd

i=1

= 2

(cid:68)
d(cid:88)
eik·xE(cid:16)|ψt(x)|2(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)k=0

(cid:88)

x∈Zd

By (4.2.33), we have

Mi,j(t) = − ∂i∂j

∂i(cid:98)K0ϕ0, P2J −1 ∂j(cid:98)K0ϕ0

(cid:69)

= ∂i∂jE(0).

d×d is positive definite. In particular,

∂i(cid:98)K0ϕ0, P2J −1 ∂i(cid:98)K0ϕ0

(cid:69)

= tr D ∈ (0,∞).

(cid:68)
ϕ0, e−t(cid:98)Lk Φk

(cid:69)(cid:12)(cid:12)(cid:12)k=0

= − ∂i∂j

.

(4.4.11)

The following decomposition of Mi,j is essentially contained in [33]. We sketch the proof in

Appendix A for reader’s convenience.

49

5(cid:80)

n=1

Nn, where

Lemma 4.4.2. For all 1 ≤ i, j ≤ d and t ∈ R+, Mi,j =

N1 = −(cid:10)ϕ0 , ∂i∂jΦ0

(cid:11) ;

N2 =

N3 =

(4.4.12)

(cid:69)(cid:105)

ds;

∂j(cid:98)L†
0ϕ0 , e−s(cid:98)L0 (1 − Q0) ∂iΦ0

0

0

+

(cid:68)

(cid:90) t
(cid:69)
(cid:104)(cid:68)
∂i(cid:98)L†
0ϕ0 , e−s(cid:98)L0 (1 − Q0) ∂jΦ0
(cid:90) t
(cid:69)
(cid:68)
∂i∂j (cid:98)L†
0 ϕ0 , e−s(cid:98)L0 (1 − Q0) Φ0
(cid:90) s
(cid:90) t
(cid:69)
(cid:104)(cid:68)
∂i(cid:98)L†
0 ϕ0 , e−(s−r)(cid:98)L0(1 − Q0)∂j(cid:98)L0 e−r(cid:98)L0 (1 − Q0)Φ0
(cid:69)(cid:105)
(cid:68)
0 ϕ0 , e−(s−r)(cid:98)L0(1 − Q0)∂i(cid:98)L0 e−r(cid:98)L0 (1 − Q0)Φ0
∂j(cid:98)L†
(cid:90) s
(cid:90) t
(cid:69)
(cid:104)(cid:68)
0 ϕ0 , e−(s−r)(cid:98)L0(1 − Q0)∂j(cid:98)L0 Q0Φ0
∂i(cid:98)L†
(cid:69)(cid:105)
(cid:68)
0 ϕ0 , e−(s−r)(cid:98)L0(1 − Q0)∂i(cid:98)L0 Q0Φ0
∂j(cid:98)L†

dr ds.

ds;

+

+

0

0

0

0

;

;

(4.4.13)

(4.4.14)

(4.4.15)

dr ds (4.4.16)

(4.4.17)

(4.4.18)

Combining the above decomposition and the contraction property of e−t(cid:98)L0 in Lemma

N4 = −

N5 = −

4.3.10, we have the following convergence of Nn, which implies Lemma 4.4.1 immediately.

Lemma 4.4.3. Let Mi,j =

Nn be given as in Lemma 4.4.2. Then

5(cid:80)

n=1

lim
t→∞

lim
t→∞

1
t
1
t

(cid:68)

|Nn| = 0, n = 1,··· , 4.

∂i(cid:98)K0 ϕ0 , P2J −1∂j(cid:98)K0 ϕ0

N5 =

(cid:68)

(cid:69)
√⊗p ∂i∂j(cid:98)ρ0;k

+

∂j(cid:98)K0 ϕ0 , P2J −1∂i(cid:98)K0 ϕ0

(cid:12)(cid:12)k=0 ⊗ 1. Direct computation by

(4.4.20)

.

(4.4.19)

(cid:69)

Proof. Case n = 1: Note that ∂i∂jΦ0 =

(4.2.31) shows

πσ ∂i∂j(cid:98)ρ0;0(x) = − (cid:88)
(cid:69)(cid:12)(cid:12)(cid:12) =
1 ⊗ 1 , ∂i∂j(cid:98)ρ0;0 ⊗ 1

n∈pZd+σ

Therefore, by (4.4.12)

(cid:12)(cid:12)(cid:12)(cid:68)

|N1| =

δ0 ⊗ −→

ninjψ0(x − n)ψ0(−n).

(4.4.21)

(cid:88)

n∈Zd

ninj|ψ0(n)|2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88)

n∈Zd

|n|2|ψ0(n)|2.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

50

Clearly, |N1| is uniformly bounded in t by (4.4.8), which implies lim
t→∞

1

t|N1(t)| = 0.

Case n = 2: By (4.2.31) and the same computation as in (4.4.21), we have ∂jΦ0 =

∂j(cid:98)ρ0;k

(cid:12)(cid:12)k=0 ⊗ 1 with

πσ ∂j(cid:98)ρ0;0(x) = −i

(cid:88)

By (4.4.8), (4.4.9) and direct computation, we obtain

(cid:13)(cid:13)∂j(cid:98)ρ0;0

(cid:13)(cid:13)2

(cid:96)2(Zd;C⊗p)

nj ψ0(x − n)ψ0(−n).

(4.4.22)

n∈pZd+σ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
(cid:88)
(cid:88)

x∈Zd

(cid:88)
 (cid:88)

n∈pZd+σ

σ∈Zp
(cid:96)2 · (cid:107)xψ0(cid:107)2

n∈pZd+σ
(cid:96)1 < ∞.

≤ (cid:88)

σ∈Zp

≤(cid:107)ψ0(cid:107)2
(cid:96)2

≤(cid:107)ψ0(cid:107)2

2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

2

nj ψ0(x − n)ψ0(−n)

|n||ψ0(−n)|

=

(cid:13)(cid:13)(cid:13)∂j(cid:98)K0

(cid:13)(cid:13)(cid:13)L2((cid:99)M ;C⊗p)
(cid:13)(cid:13)(cid:13)L2((cid:99)M ;C⊗p)
By Lemma 4.2.10, (cid:13)(cid:13)(cid:13)∂j(cid:98)L0
By Lemma 4.3.10, we have(cid:90) t
(cid:12)(cid:12)(cid:12)(cid:68)
∂i(cid:98)L†
0ϕ0 , e−s(cid:98)L0 (1 − Q0) ∂jΦ0
(cid:90) t
≤(cid:13)(cid:13)(cid:13)∂i(cid:98)K0ϕ0
(cid:13)(cid:13)(cid:13) ·(cid:13)(cid:13)∂jΦ0
(cid:13)(cid:13) · C

e−s(g−
) ds
(cid:13)(cid:13) · C

≤(cid:107)(cid:98)h(cid:48)(cid:107)∞ · √⊗p ·(cid:13)(cid:13)∂j(cid:98)ρ0;0
< ∞.

0

0

g − 


≤ (cid:107)(cid:98)h(cid:48)(cid:107)∞.
(cid:69)(cid:12)(cid:12)(cid:12) ds

Therefore,

lim
t→∞

1

t|N2(t)| = 0.

Case n = 3: N3 can be estimated exact in the same way as N2. Again by Lemma 4.2.10,

we have

(cid:13)(cid:13)(cid:13)∂i∂j(cid:98)L0

(cid:13)(cid:13)(cid:13)L2((cid:99)M ;C⊗p)

(cid:13)(cid:13)(cid:13)∂i∂j(cid:98)K0

(cid:13)(cid:13)(cid:13)L2((cid:99)M ;C⊗p)

=

≤ (cid:107)(cid:98)h(cid:48)(cid:48)(cid:107)∞.

51

By Lemma 4.3.10, we have

|N3(t)| ≤ sup

sup

t

(cid:90) t

(cid:12)(cid:12)(cid:12)(cid:68)

∂i∂j (cid:98)L†
≤(cid:107)(cid:98)h(cid:48)(cid:48)(cid:107)∞ · √⊗p ·(cid:13)(cid:13)(cid:98)ρ0;0

0ϕ0 , e−s(cid:98)L0 (1 − Q0) Φ0
(cid:13)(cid:13)(cid:96)2 · C


< ∞,

0

t

g − 


(cid:69)(cid:12)(cid:12)(cid:12) ds

which gives lim
t→∞

Case n = 4: N4 can be estimated by applying Lemma 4.3.10 twice:

1

t|N3(t)| = 0.
(cid:90) s
(cid:90) t
(cid:68)
∂i(cid:98)L†
0 ϕ0 , e−(s−r)(cid:98)L0 (1 − Q0)∂j(cid:98)L0 e−r(cid:98)L0(1 − Q0) Φ0
(cid:13)(cid:13)(cid:13) · (cid:107)Φ0(cid:107) · C2
(cid:13)(cid:13)(cid:13) ·(cid:13)(cid:13)(cid:13)∂j(cid:98)L0
≤(cid:13)(cid:13)(cid:13)∂i(cid:98)K0ϕ0
(cid:18) 1
≤(cid:107)(cid:98)h(cid:48)(cid:107)2∞ · √⊗p ·(cid:13)(cid:13)(cid:98)ρ0;0
(cid:13)(cid:13)(cid:96)2 · C2

(cid:69)
e−(s−r)(g−
) e−r(g−
) dr ds

(cid:90) s

(cid:90) t

< ∞,


 sup
t

(cid:19)


 ·

sup

+

1

0

0

0

0

t

g − 


(g − 
)2

dr ds

1

and thus,

lim
t→∞

Case n = 5: It remains to estimate 1

t|N4(t)| = 0.
(cid:90) s
(cid:90) t
(cid:68)
0 ϕ0 , e−(s−r)(cid:98)L0 (1 − Q0)∂j(cid:98)L0 e−r(cid:98)L0 Q0 Φ0
∂i(cid:98)L†
(cid:18)1
(cid:28)
(cid:29)
(cid:90) t
∂i(cid:98)K0 ϕ0 ,

(cid:19)
e−(s−r)(cid:98)L0 dr ds
since ∂j(cid:98)K0ϕ0 ∈ ran(1 − Q0), (1 − Q0)∂j(cid:98)K0ϕ0 = ∂j(cid:98)K0ϕ0.
Since Re(cid:98)L0 ≥ 0, by a standard contour integral argument, the following formula was

t N5. Recall we obtained Q0Φ0 = ϕ0 in (4.4.7). This

∂j(cid:98)K0 ϕ0

1
t
= −

(cid:90) s

(cid:69)

dr ds

t

0

0

0

0

,

obtained in [23, 33]

(cid:90) t

(cid:90) s

0

0

lim
t→∞

1
t

Π2e−(s−r)(cid:98)L0Π2 dr ds = Π2

(cid:16)

(cid:17)−1
(1 − Q0)(cid:98)L0(1 − Q0)

Π2 = Π2J −1Π2,

where J −1 is as in Lemma 4.3.9 . Recall that ∂i(cid:98)K0 ∈ Ran(Π2) ⊆ Ran(P2). Thus
(cid:68)
∂j(cid:98)K0 ϕ0 , Π2J −1Π2∂i(cid:98)K0 ϕ0
(cid:69)
∂j(cid:98)K0 ϕ0 , P2J −1∂i(cid:98)K0 ϕ0

∂i(cid:98)K0 ϕ0 , Π2J −1Π2∂j(cid:98)K0 ϕ0
(cid:69)
∂i(cid:98)K0 ϕ0 , P2J −1∂j(cid:98)K0 ϕ0

(cid:69)
(cid:68)

(cid:68)
(cid:68)

lim
t→∞

N5 =

1
t

=

+

+

(4.4.23)

(cid:69)

=∂i∂jE(0),

52

where the last line follows from the formula of ∂i∂jE(0) in (4.3.31).

4.4.3 Limiting behavior of D(λ) for small λ

The following lemma can be found in [33].

It will be the main tool for us to study the

asymptotic behavior of D(λ).

Lemma 4.4.4 (Lemma D.1, [33]). Let A and R be bounded operators on a Hilbert space H.
If A is normal, Re A ≥ 0 and Re R ≥ c > 0, then for any φ, ψ ∈ H,

(cid:28)

(cid:16)

lim
η→0

φ,

(cid:17)−1

(cid:29)

η−1A + R

ψ

=

H

Πφ, (ΠRΠ)−1 Πψ

ranΠ

where Π = projection onto the kernel of A.

Remark 4.4.5. A similar statement holds for a family of bounded operators Rη such that
Re Rη ≥ c > 0 and limη→0 Rη = R0 in the strong operator topology and R0 ≥ c > 0, i.e.,

(cid:28)

(cid:16)

lim
η→0

φ,

(cid:17)−1

(cid:29)

ψ

=

H

η−1A + Rη

Πφ, (ΠR0Π)−1 Πψ

.

ranΠ

(cid:52)

In view of Lemma 4.3.9, we want to have the block form of the above lemma.

Lemma 4.4.6. Let A be a bounded self-adjoint operator on a Hilbert space H = H1 ⊕ H2
with the following block form:

 0 A2

†
2 A3

A

 , A

A =

†
3 = A3.

Let Π = projection onto the kernel of A, Π2 = projection onto the kernel of A2 and (cid:101)Π =

(4.4.24)

(cid:68)

(cid:68)

(cid:69)

(cid:69)

projection onto the kernel of Π2A3Π2. For any ϕ = Π2ϕ,

Πϕ = 0 if and only if (cid:101)Πϕ = 0.

53

Proof. For any ϕ ∈ H, direct application of Lemma 4.4.4 to I + i η−1A gives

(cid:68)

(cid:69)
ϕ, (I + i η−1A)−1ϕ

(cid:68)

(cid:69)
Πϕ, (Π I Π)−1Πϕ

lim
η→0

(4.4.25)
Let P1, P2 be the projection onto H1,H2 correspondingly and consider ϕ ∈ ran(Π2). By the
block form of A and Schur’s formula, we have

=

= (cid:107)Πϕ(cid:107)2.

(cid:68)

(cid:69)
ϕ, (I + i η−1A)−1ϕ

=

(cid:68)
(cid:68)

(cid:69)
(cid:69)
†
ϕ, (P2 + iη−1A3 + η−2A
2A2)−1 ϕ
†
2A2)−1Π2 ϕ
ϕ, Π2(P2 + iη−1A3 + η−2A

=

(4.4.26)
If we apply Schur’s formula one more time with respect to the decomposition H2 = ran(Π2)⊕
ran(Π⊥

†
2 = A2Π2 = 0, then we have

2 ) and notice that Π2A

.

(cid:29)

ϕ

(cid:28)

2

=

(cid:16)

(cid:68)

(cid:17)−1

(cid:16)
iη−1Π2A3Π2 + Π2 + (cid:101)A
†
2 + Π⊥
2A2Π⊥
2 A

Π⊥

ϕ,

Therefore, by Lemma 4.4.4 and Remark 4.4.5, we have that

η2Π⊥

ϕ, (I + i η−1A)−1ϕ

(cid:17)−1
where (cid:101)A = Π2A3Π⊥
0 on ran(Π2), which implies Re(Π2 + (cid:101)A) ≥ 1 > 0 on ran(Π2).

(cid:69)
2 A3Π⊥
2 + iηΠ⊥
(cid:68)
(cid:69)
(cid:29)
(cid:28)
ϕ, (I + i η−1A)−1ϕ
(cid:17)−1
(cid:16)
iη−1Π2A3Π2 + Π2 + (cid:101)A
(cid:42)(cid:101)Πϕ,
(cid:18)(cid:101)Π +(cid:101)Π Π2A3Π⊥
(cid:16)
(cid:17)−1
†
Π⊥
2A2Π⊥
2 A
where (cid:101)Π = projection onto the kernel of Π2A3Π2.
(cid:42)(cid:101)Πϕ,
(cid:17)−1

Putting (4.4.25) and (4.4.27) together, we have that
Π⊥

(cid:18)(cid:101)Π +(cid:101)Π Π2A3Π⊥

= lim
η→0

2 A3Π2(cid:101)Π

Π⊥

2 A3Π2(cid:101)Π

(cid:43)
(cid:19)−1 (cid:101)Πϕ
(cid:43)

(cid:19)−1 (cid:101)Πϕ

Π⊥
2 A

†
2A2Π⊥

2

lim
η→0

(cid:16)

ϕ,

2

2

=

ϕ

2

2 A3Π2. By (4.4.24), Re (cid:101)A ≥

2

,

(4.4.27)

= (cid:107)Πϕ(cid:107)2,

which completes the proof of Lemma 4.4.6.

Now we can proceed to prove (4.1.15) in Theorem 4.1.11. As showed in Equation (4.3.31)
and Lemma 4.4.1, the diffusion matrix D(λ) is independent of the initial condition ψ0 ∈
(cid:96)2(Zd). To study the asymptotic behavior of D(λ), it is enough to consider ψ0(x) = δ0,
where we assume the ballistic motion holds in (4.1.14).

54

(4.4.28)

Proof of (4.1.15). We are going to apply Lemma 4.4.6 to A acting on (cid:98)H1 ⊕ (cid:98)H2 given by:

A = P ((cid:98)K0 + (cid:98)U)P =

 0
P1(cid:98)K0P2
P2(cid:98)K0P1 P2((cid:98)K0 + (cid:98)U)P2

 ,

where (cid:98)Hi, Pi, i = 1, 2 are as in (4.3.11) and P = P1 + P2. Let Π = projection onto the kernel
of P ((cid:98)K0 + (cid:98)U)P , Π2 = projection onto the kernel of P1(cid:98)K0P2 and (cid:101)Π = projection onto the
kernel of Π2((cid:98)K0 + (cid:98)U)Π2.
Let (cid:101)φj = ∂j(cid:98)K0ϕ0, j = 1··· , d, which are given as in (4.2.26). Recall that (cid:101)φj ∈
Ker(P1(cid:98)K0P2), therefore (cid:101)φj = Π2(cid:101)φj. Let Mi,j be as in (4.4.10) and (cid:98)ρ0;k be as in (4.2.31).
(cid:28)(cid:101)φj,
When λ = 0, (cid:98)L0 = i((cid:98)K0 + (cid:98)U) is the unperturbed periodic operator on (cid:96)2(Zd; C⊗p).

By the decomposition in Lemma 4.4.2 at λ = 0, one can check that 2

P + η−1i((cid:98)K0 + (cid:98)U)

e−η t Mj,j(t) dt + O(η2).

(cid:17)−1(cid:101)φj

(cid:90) ∞

(4.4.29)

(cid:29)

= η3

(cid:16)

2

0

(cid:16)

(cid:29)

> 0.

− 2t

e

x2
j

x∈Zd

(4.4.30)

8
T 3

0

η3

0

= lim
η→0

(cid:90) ∞

(cid:90) ∞

dt ≥ c > 0.

(cid:28)(cid:101)φj,

(cid:13)(cid:13)(cid:13)Π(cid:101)φj

T (cid:88)

e−η t Mj,j(t) dt =

Put (4.4.25), (4.4.29) and (4.4.30) together, we have

Setting η = 2T−1 in (4.1.14), there is a c > 0 such that for all j and η small,

E(cid:16)|ψt(x)|2(cid:17)
(cid:13)(cid:13)(cid:13)2
(cid:17)−1(cid:101)φj
P + η−1i((cid:98)K0 + (cid:98)U)
Therefore, Π(cid:101)φj (cid:54)= 0 and Lemma 4.4.6 implies that (cid:101)Π(cid:101)φj (cid:54)= 0.
(cid:16)
i(cid:98)K0 + i(cid:98)U + λ2(cid:98)V (P3(cid:98)L0P3)−1(cid:98)V(cid:17)
Recall that (cid:98)L0 = i(cid:98)K0 + i(cid:98)U + iλ(cid:98)V + B and Γ2 = P2
as in (4.3.15). Let Rλ = Π2(cid:98)V(cid:16)
(cid:17)−1(cid:98)VΠ2.
Then Π2Γ2Π2 = i Π2P2((cid:98)K0 +(cid:98)U)P2Π2 + λ2Rλ and limλ→0 Rλ = R0 (in the strong operator
[33], the choice ψ0 = δ0 implies that Mj,j(0) = 0 and (cid:98)ρ0;0 = δ0 ⊗ −→
the simplified expressions of Mj,j(0) and(cid:98)ρ0;0. We need the correction term for small η. The

2This formula was obtained in [33], Section 4.7, where there is no error term O(η2). In
1 and the proof is
relatively simple. In the general p-periodic case, the initial condition δ0 no longer provides

(cid:17)−1(cid:98)VΠ2 and R0 = Π2(cid:98)V(cid:16)

P3(i(cid:98)K0 + i(cid:98)U)P3

P3(cid:98)L0P3

proof for the general case is essentially based on the same strategy for Lemma 4.4.3; we omit
the details here.

P2

55

(cid:69)

lim
λ→0

(cid:29)

= lim
λ→0

=

topology). Applying Lemma 4.4.4 (and Remark 4.4.5) to Π2Γ2Π2 on ran(Π2), we obtain
that, for any 1 ≤ i, j ≤ d,

λ2(cid:68)(cid:101)φi, (Π2Γ2Π2)−1 (cid:101)φj

(cid:28)(cid:101)φi,
(cid:17)−1 (cid:101)φj
(cid:16)
i λ−2Π2((cid:98)K0 + (cid:98)U)Π2 + Rλ
(cid:28)(cid:101)Π(cid:101)φi,
(cid:29)
(cid:16)(cid:101)ΠR0(cid:101)Π
(cid:17)−1 (cid:101)Π(cid:101)φj
(cid:28)(cid:101)Π(cid:101)φj,
(cid:29)
(cid:69)
In particular, limλ→0 λ2(cid:68)(cid:101)φj, (Π2Γ2Π2)−1 (cid:101)φj
(cid:16)(cid:101)ΠR0(cid:101)Π
(cid:17)−1 (cid:101)Π(cid:101)φj
(cid:28)(cid:101)Π(cid:101)φi,
(cid:17)−1 (cid:101)Π(cid:101)φi
(cid:16)(cid:101)ΠR0(cid:101)Π
(cid:17)−1 (cid:101)Π(cid:101)φj

By Lemma 4.3.9 and (4.3.31), we have

(cid:16)(cid:101)ΠR0(cid:101)Π

(cid:28)(cid:101)Π(cid:101)φj,

λ2∂i∂jE(0) =

ij)d×d. Then limλ→0 λ2D = D0 and(cid:10)k, D0k(cid:11) > 0 for any 0 (cid:54)= k ∈ Rd by

+

=: D0
ij.

> 0.

(cid:29)

lim
λ→0

Let D0 := (D0

.

=

(cid:29)

the same argument for D. As a consequence,

λ2 tr D = tr D0 > 0.

lim
λ→0

This completes the proof of Theorem 2.0.3.

56

CHAPTER 5

NUMERICAL ANALYSIS OF DIFFUSION IN MARKOV SCHR ¨ODINGER

EQUATIONS

Recall that we are interested in solving a one dimensional, time-dependent Schr¨odinger

equation of the form, i

∂ψt(n)

∂t

= Hαψt(n) + λvn(ω(t))ψt(n),

ψ0(n) = δ0(n).

(5.0.1)

Here, the unperturbed Hamiltonian Hα is given by either the dimer or trimmed Anderson

model and the time dependent potential vn(ω(t)) := ωn(t) is given by the “flip process”.
Briefly, the “flip process” is obtained by randomly selecting an initial state ω(0) ∈ {−1, 1}Z

and assigning independent and identical Poisson processes to each site. The disorder ω(t)

obtains time dependence by flipping the sign of ωn(t) once the Poisson process at n fires.

Another description of the “flip process” is given in Chapter 2.

For these two models we will numerically calculate the diffusion constant as a function

of the disorder, D(λ); with a particular interest in small values of λ. Section 5.1 outlines

the methods used to calculate D(λ), while Section 5.2 discusses the asymptotic behavior of

D(λ) for small λ. In the next chapter we will compare these results to those of the periodic

case.

5.1 Numerical Method

The time dependence of the Hamiltonian,

H(t, λ) = Hα + λv(ω(t))

(5.1.1)

is contained solely in the “flip process”, v(ω(t)). By restricting the possible arrival times of

each of the individual Poisson processes to the discrete set,

t ∈ ∆t · N := {∆t, 2∆t, . . . , N ∆t, . . .}, ∆t (cid:28) 1,

57

the evolution of the wave function from time t to t + ∆t is given by

ψt+∆t = e−iH(t)∆tψt.

(5.1.2)

Without knowing the eigenvalues of H(t), calculating the exponential in Equation (5.1.2)

can be extremely difficult and computationally expensive. To overcome this difficultly we

will approximate the exponential with the Cayley transform,

e−iH(t)∆t ≈ 1 − 1
1 + 1

2iH(t)∆t
2iH(t)∆t

.

(5.1.3)

Note, the operator on the right hand side of Equation (5.1.3) is unitary. Combining Equations

(5.1.2) and (5.1.3) yields the evolution equation,

(cid:18)

ψt+∆t =

1 +

iH(t)∆t

1
2

(cid:19)−1(cid:18)

1 − 1
2

(cid:19)

iH(t)∆t

ψt.

(5.1.4)

For a given value of λ, we solve Equation (5.0.1), by numerically solving the the matrix

equation (5.1.4), for 32 randomly generated initial disorder configurations ω(0). Each trial

begins with ψ0 = δ0 and is initially confined to a lattice of length 300. To minimize effects

from the boundary the lattice is expanded by adding 150 sites to each end once the probability
of finding the particle at either boundary becomes greater than 10−30.

Once the wave function ψt is known for each of the 32 trials, we calculate the second

moment as a function of time, (cid:68)

(cid:69)

X2
t

:=

(cid:88)

n

|n|2E(|ψt(n)|2).

(5.1.5)

Here E(|ψt(n)|2) denotes the average probability density over the 32 trials. This averaging
removes any dependence on any specific realization of Hα and v(ω(t)). Equipped with (cid:104)X2
t (cid:105),
we calculate the diffusion constant as a function of time and disorder,

D(λ, t) :=

(cid:104)X2
t (cid:105).

1
t

(5.1.6)

Finally, to remove time dependence we average D(λ, t) over the tail end of each trial, i.e.,

D(λ) =

1
T

dt(cid:48) D(λ, t(cid:48)).

(5.1.7)

(cid:90)

[t,t+T ]

58

5.2 Results

5.2.1 Dimer Model

For the dimer model we take the two site energies as εa = 1, εb = 0. This choice ensures

superdiffusive scaling in the absence of disorder,

(cid:68)

X2
t

(cid:69) ∼ t3/2.

(5.2.1)

Figure 5.1 shows the diffusion constant as a function of time for small values of λ. In figure 5.2

the average diffusion constant, Equation (5.1.7), is plotted as a function of λ. Importantly,

these two figures show that the addition of disorder leads to diffusion and in the small λ
limit the diffusion constant scales like λ−1.093.

Figure 5.1: The diffusion constant as a function of time and disorder strength.

5.2.2 Trimmed Anderson Model

For the trimmed Anderson model we support the static random potential of Hα on Γ = 2Z.

In this case the second moment scales subdiffusively, see figure 2.3. This system diffuses for

59

10-210-110010110210310410510-210-1100101102103Figure 5.2: The average diffusion constant as a function of the disorder strength for the
dimer model.

Figure 5.3: The average diffusion constant as a function of the disorder strength for the
trimmed Anderson model.

60

-6-4-2024602468-8-6-4-202468-3-2-101234any value of disorder. Plotting the average value of the diffusion constant vs disorder shows

that in the small λ limit the diffusion constant scales like λ1.186.

61

CHAPTER 6

CONCLUSIONS AND CONJECTURES

In light of the present work, it is natural to wonder whether a result similar to those presented

in Theorem 4.1.11 hold for a general ergodic/deterministic potential U . In particular,

1. Under what assumptions will a potential, U , cause solutions to the Markov Schr¨odinger

equation,

i∂tψt(x) = H0ψt(x) + U ψt(x) + λVω(t)ψt(x),

ψ0(x) ∈ (cid:96)2(Zd),

(6.0.1)

to display diffusive propagation over large time scales?

2. If diffusive propagation holds, what is the asymptotic behavior of the diffusion constant

in the small λ limit?

Based on the behavior of the (6.0.1) with U ≡ 0 [23], U a random potential leading

to localization [33], U periodic [34], and the anomalous cases (Chapter 5), we make the

following conjecture:

Conjecture 6.0.1. For any bounded potential, U , and any coupling constant, λ > 0, there

exist positive, finite upper and lower diffusion constants, D(U, λ) and D(U, λ), such that the

solutions to equation (6.0.1) satisfy

D(U, λ) := lim inf
t→∞

1
t

(cid:88)

x∈Zd

|x|2E(cid:16)|ψt(x)|2(cid:17) ≤ lim sup

t→∞

|x|2E(cid:16)|ψt(x)|2(cid:17)

(cid:88)

x∈Zd

1
t

=: D(U, λ).

Furthermore, if

(cid:88)

x∈Zd

lim
t→∞

1
tγ

|x|2|(cid:104)δx, e−i(H0+U )δ0(cid:105)|2 ∈ (0,∞),

γ ∈ [1, 2],

then

D(U, λ) ∼ O(λ2−2γ)

D(U, λ) ∼ O(λ2−2γ).

and

62

(6.0.2)

(6.0.3)

(6.0.4)

Remark 6.0.2. We have limited the transport exponent in the asymptotic formula (6.0.4)

to the range [1, 2]. This lower bound is suggested by the trimmed Anderson model, where
γ = 0.07795 and D ∼ λ1.186 instead of D ∼ λ1.8441 which (6.0.4) would predict.
(cid:52)

We end with a conjecture about the almost Mathiue operator on (cid:96)2(Z),

(Hg,α

θ ψt)(x) = ψt(x + 1) + ψt(x − 1) + 2g cos 2π(θ + xα)ψt(x),

(6.0.5)

with parameters g ∈ R and θ, α ∈ [0, 1]:

Conjecture 6.0.3. For almost every θ, α ∈ [0, 1], the AMO-Markovian equation has a
diffusion constant D(g, λ) ∈ (0,∞) which is a smooth function for all (g, λ) ∈ R × R+.
Moreover, D(g, λ) ∼ O(λ2) for all |g| > 1 and D(g, λ) ∼ O(λ−2) for all |g| < 1.

63

APPENDICES

64

APPENDIX A

DECOMPOSITION OF THE SECOND MOMENTS AND THE PROOF OF

LEMMA 4.4.2

The following facts will be used to simplify the expression of the second order partial deriva-

tive. Note that (cid:98)L0 ϕ0 = (cid:98)L†

0 ϕ0 = 0, implies that e−t(cid:98)L0 and e

−t(cid:98)L†

0 act trivially on ϕ0 for

any t, i.e.,

and

e−t(cid:98)L0ϕ0 = e

−t(cid:98)L†

0ϕ0 = ϕ0

(A.0.1)

(A.0.2)

On the other hand, recall the formula for differentiating a semi-group,

−t(cid:98)L†

0Q0 = Q0.

e−t(cid:98)L0Q0 = e
(cid:90) t
(cid:17)

= −

(cid:16)

e−t(cid:98)Lk

∂j

e−(t−s)(cid:98)Lk ∂j(cid:98)Lk e−s(cid:98)Lk ds.

By (4.2.20) and (4.2.28), we have ∂j(cid:98)L0 = i ∂j(cid:98)K0 = −∂j (cid:98)L†
to (cid:98)H2, we also have that

0

(A.0.3)

0. Because ∂j(cid:98)K0 maps (cid:98)H0 ⊕ (cid:98)H1

Q0∂j(cid:98)L0 = Q0∂j(cid:98)L†

0 = 0;

∂i∂j(cid:98)L0 = i∂i∂j(cid:98)K0 = −∂i∂j(cid:98)L†
0 and Q0∂i∂j(cid:98)L0 = Q0∂i∂j(cid:98)L†
(cid:69)(cid:12)(cid:12)(cid:12)k=0
(cid:69)

0 = 0.

(cid:28)

(cid:29)
(cid:29)

(cid:16)

∂je−t(cid:98)L0

(cid:17)

|k=0

∂jΦ0

−

ϕ0 ,

|k=0

∂iΦ0

Direct computation from (4.4.11) gives

(cid:68)
ϕ0, e−t(cid:98)Lk Φk
= −(cid:68)
Mi,j(t) = − ∂i∂j
ϕ0 , e−t(cid:98)L0 ∂i∂jΦ0
(cid:28)
(cid:16)
(cid:17)
∂ie−t(cid:98)L0
(cid:28)
(cid:17)
(cid:16)
∂i∂je−t(cid:98)L0

ϕ0 ,

ϕ0 ,

−

−

|k=0

Φ0

.

65

(A.0.4)

(A.0.5)

(A.0.6)

(A.0.7)

(A.0.8)

(cid:29)

Clearly, (A.0.6) gives the expression for N1 in (4.4.12). Now let’s proceed to simplify the

expression in (A.0.7). By the differential formula (A.0.3), we obtain

(cid:28)

ϕ0 ,

(cid:16)

∂i e−t(cid:98)Lk

(cid:17)

|k=0

∂jΦ0

=

ϕ0 ,

(cid:29)

(cid:28)

−

(cid:18)
(cid:19)
(cid:90) t
e−(t−s)(cid:98)L0 ∂i(cid:98)L0 e−s(cid:98)L0 ds
(cid:90) t
(cid:68)
(cid:69)
ϕ0 , ∂i(cid:98)L0 e−s(cid:98)L0 (1 − Q0) ∂jΦ0
(cid:68)
ϕ0 , ∂i(cid:98)L0 e−s(cid:98)L0 Q0 ∂jΦ0

(cid:69)

0

0

(cid:29)

∂jΦ0

ds,

= 0. This gives the

= −

where we use the fact by (A.0.2) that

expression for N2 in (4.4.13).

(cid:16)

e−t(cid:98)Lk

(cid:17)(cid:12)(cid:12)(cid:12)k=0

∂i∂j

Simplifying (A.0.8) requires applying (A.0.3) twice. Differentiating (A.0.3) again yields,

(cid:28)

Therefore,

−

ϕ0 ,

(cid:16)

(cid:17)
∂i∂je−t(cid:98)Lk
(cid:90) t−s
(cid:90) t
(cid:90) s
(cid:90) t
(cid:68)

−

0

0

|k=0

−

0

0

∂j(cid:98)L0 e−s(cid:98)L0 ds
(cid:19)

0

0

0

+

+

(cid:19)

= −

(cid:18)(cid:90) s

e−(t−s−r)(cid:98)L0 ∂i(cid:98)L0 e−r(cid:98)L0 dr

e−(t−s)(cid:98)L0 ∂i∂j(cid:98)L0 e−s(cid:98)L0 ds
(cid:18)(cid:90) t−s
e−(t−s)(cid:98)L0 ∂j(cid:98)L0
(cid:90) t

(cid:90) t
(cid:90) t
(cid:90) t
(cid:29)
ϕ0 , e−(t−s)(cid:98)L0 ∂i∂j (cid:98)L0 e−s(cid:98)L0 Φ0
(cid:69)
(cid:68)
ϕ0 , e−(t−s−r)(cid:98)L0 ∂i(cid:98)L0 e−r(cid:98)L0∂j(cid:98)L0 e−s(cid:98)L0 Φ0
(cid:69)
ϕ0 , e−(t−s)(cid:98)L0 ∂j(cid:98)L0 e−(s−r)(cid:98)L0 ∂i(cid:98)L0 e−r(cid:98)L0 Φ0

e−(s−r)(cid:98)L0 ∂i(cid:98)L0 e−r(cid:98)L0 dr
(cid:69)

dr ds.

(cid:68)

dr ds

Φ0

ds

=

0

0

0

ds.

(A.0.9)

(A.0.10)

(A.0.11)

.

(cid:69)

(cid:68)

The expression on the right hand side of (A.0.9) leads to N3 in (4.4.14) since

ϕ0 , e−(t−s)(cid:98)L0 ∂i∂j (cid:98)L0 e−s(cid:98)L0 Φ0

∂i∂j (cid:98)L†

0ϕ0 , e−s(cid:98)L0 (1 − Q0) Φ0

(cid:69)

(cid:68)

=

Expressions for (A.0.10) and (A.0.11) follow from (A.0.2) and (A.0.4) by direct compu-

tations. For (A.0.10) we have,

(cid:90) t

(cid:90) t−s

(cid:68)

0

0

−

(cid:69)
ϕ0 , e−(t−s−r)(cid:98)L0 ∂i(cid:98)L0 e−r(cid:98)L0∂j(cid:98)L0 e−s(cid:98)L0 Φ0
(cid:90) t
(cid:90) s
(cid:104)(cid:68)
(cid:68)
(cid:69)(cid:105)
∂i(cid:98)L†
0 ϕ0 , e−(s−r)(cid:98)L0(1 − Q0)∂j(cid:98)L0 Q0Φ0

∂i(cid:98)L†
0 ϕ0 , e−(s−r)(cid:98)L0(1 − Q0)∂j(cid:98)L0 e−r(cid:98)L0 (1 − Q0)Φ0

dr ds.

dr ds

0

0

= −

+

(cid:69)

(A.0.12)

(A.0.13)

66

Similarily, for (A.0.11),

(cid:90) t

(cid:90) s

(cid:68)

0

0

−

(cid:69)

ϕ0 , e−(t−s)(cid:98)L0 ∂j(cid:98)L0 e−(s−r)(cid:98)L0 ∂i(cid:98)L0 e−r(cid:98)L0 Φ0
(cid:90) s
(cid:90) t
(cid:104)(cid:68)
(cid:69)(cid:105)
(cid:68)
0 ϕ0 , e−(s−r)(cid:98)L0(1 − Q0)∂i(cid:98)L0 Q0Φ0
∂j(cid:98)L†

∂j(cid:98)L†
0 ϕ0 , e−(s−r)(cid:98)L0(1 − Q0)∂i(cid:98)L0 e−r(cid:98)L0 (1 − Q0)Φ0

dr ds.

dr ds

0

0

= −

+

(cid:69)

(A.0.14)

(A.0.15)

Clearly,

N4 = (A.0.12) + (A.0.14), N5 = (A.0.13) + (A.0.15).

(A.0.16)

This completes the proof of Lemma 5.2.

67

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