DIFFUSION FOR MARKOV WAVE EQUATIONS
By
Bernard Clark Musselman II

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Applied Mathematics
2012

ABSTRACT
DIFFUSION FOR MARKOV WAVE EQUATIONS
By
Bernard Clark Musselman II
We consider the long time evolution of solutions to a Schr¨dinger-type wave equation on
o
a lattice, with a divergence-form, Markov, random generator. We show that solutions to this
problem diffuse. That is, the amplitude converges to the solution of a diffusion equation, in
the diffusive scaling limit.
Additionally, we expand upon a similar result due to Kang and Schenker for a MarkovSchr¨dinger wave equation by computing higher moments of position, also in the diffusive
o
scaling limit.

In memory of my father, Dr. Bernard C. Musselman.

iii

ACKNOWLEDGMENT

First and foremost, I wish to thank my advisor, Jeffrey Schenker. Without his patience
and depth of knowledge, this work would not be possible. His generosity in supporting my
graduate work made the task of completing this thesis much more manageable. I am also
deeply indebted to my wife, Alexandria, for her unwavering encouragement and faith in my
abilities. I am most thankful to my mother for her unconditional support; to John Theakston
for interesting conversations related to this work; to Shari Ultman for encouragement early
on, when I needed it the most; to Liam Finlay for giving me an outlet and for putting things
in perspective; to Dr. Hill for giving me the opportunity, year after year, to be a part of
something really important; to my students for giving me purpose; to P.T. for teaching me
to live in the moment; and to Tom Bellsky, Matt St. Peter, and Sara Vredevoogd for their
friendship and collaboration. Additionally, I wish to thank Luke Williams and Dr. Weil for
their help in preparing this document. I am also grateful to the Erwin Schr¨dinger Institute
o
for their support and hospitality.

iv

TABLE OF CONTENTS

List of Figures

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vii

Chapter 1

Introduction to Diffusion for Markov Wave Equations . . . . .

1

Chapter 2

A Mathematical Characterization of Diffusion . . . . . . . . . .

4

Chapter 3 Diffusion for a Markov, Divergence-form Generator .
3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Flip Process . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . .
3.4 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 A More Appropriate Problem . . . . . . . . . . . . . . . .
3.4.2 The Resolvent of the Generator . . . . . . . . . . . . . . .
3.4.3 The Substantive Part of the Integral . . . . . . . . . . . .
3.4.4 Natural Projections and the Schur Complement Formula .
3.4.5 In Search of a Lower Bound . . . . . . . . . . . . . . . . .
3.4.6 Bounding the Resolvent . . . . . . . . . . . . . . . . . . .
3.4.7 The Diffusive-Scaling Limit . . . . . . . . . . . . . . . . .

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25

Chapter 4 Higher Moments for a Markov-Schr¨dinger
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4.1 Statement of the Problem . . . . . . . . . . . . . . . .
4.2 Diffusion for a Markov-Schr¨dinger Wave Equation . .
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4.3 Equivalence of Models . . . . . . . . . . . . . . . . . .
4.4 A Ballistic Upper Bound . . . . . . . . . . . . . . . . .
4.5 Moments by Analytic Continuation . . . . . . . . . . .
4.6 Convergence and Analyticity . . . . . . . . . . . . . . .

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37

Appendices . . . . . . . . . . . . . . . . . . . .
Appendix A The Numerical Range of a Linear
Appendix B The Markov Generator . . . . . .
Appendix C Another Realization of Dk . . . .
Appendix D Inversion of Linear Operators . .
Appendix E The Schur Complement Formula

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vi

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64

LIST OF FIGURES

Figure 3.1

ˆ
The contour Γ in (3.7) and the numerical range of iL√ηk + B. . . .

14

Figure 4.1

The contour Γ and the perturbed numerical range. . . . . . . . . . .

40

Figure B.1

The contour Γ and the numerical range of the Markov Generator. .

49

vii

Chapter 1
Introduction to Diffusion for Markov
Wave Equations
In the classic study of deterministic partial differential equations, the phenomena of wave
propagation and diffusion are treated separately. The derivations of the heat and wave
equations are distinct and rely on observations of different physical behavior. Important
properties such as the maximum principle, regularity and the domain of dependence, also
called the wave cone [2], have no analog on the other side. While both models have existence/uniqueness theorems, the methods of proof are vastly different and a deep understanding of one does not necessarily provide any intuition about the other. Waves in nature,
however, do not satisfy such a distinction. Indeed, a vibrating guitar string will eventually
come to rest as do the waves on water’s surface, in the absence of wind or further vibration.
It is not the immediate goal here to include every aspect of nature in a particular wave
model. Instead, we show that when noise or disorder is included, one can see a departure
from the classic understanding of wave propagation. The resulting more natural model allows
1

for the diffusion of wave packets. Specifically, we show that for two particular examples, if
the environment through which a wave packet propagates is governed by a Markov process,
then the wave packet will eventually diffuse.
Related to the above, the first question we address is “How does one detect diffusion?”.
In chapter 2, we develop a mathematical characterization of diffusion by beginning with the
diffusion equation. After all, whatever characterization we decide upon must be consistent
with the classic diffusion equation. From this, we derive a natural scaling and define diffusion
for a wave equation to be the convergence (under this scaling-limit) of its square-amplitude
to the solution of a diffusion equation.
In chapter 3 and with this understanding of diffusion, we discuss a model with a divergenceform generator that includes randomness. This begins by studying the semi-group generated
by a Markov process and its underlying probability measure. This semi-group gives rise to
a generator which encapsulates the process in a maximally-dissipative operator on a Hilbert
space. A few further assumptions on the generator give us enough leverage to control the
spectrum of the overall problem: the Markov generator together with the divergence-form
generator. We then demonstrate the (slightly weakened) diffusion criterion by computing
the diffusive-scaling limit of norm-squared solutions.
The weakening of this criterion is essential to our method. (A procedure for strengthening
this weakened criterion, as well as a complete example, is given in chapter 4.) By averaging
the diffusive criterion over all realizations of the disorder, instead of computing the criterion
for arbitrary realizations, we reformulate it in terms of a particular matrix element of the
complete generator. This reduces the problem to controlling the spectrum. That is, we
write the semi-group in terms of its resolvent by way of the holomorphic functional calculus.
2

We control the resolvent, in the diffusive scaling limit, by controlling its constituent parts
individually. This choice of a decomposition is motivated in a natural way by the generator
itself; we use the Schur complement (see Appendix E) according to the projections onto the
kernel and range.
In chapter 4, we elaborate on a similar problem addressed by Kang and Schenker [3].
In their paper, the weakened criterion is demonstrated for a Markov-Schr¨dinger equation.
o
We provide further evidence of diffusion by showing that higher moments of position also
possess a diffusive scaling limit and that this limit is a derivative of the heat kernel. We do
this by analytically continuing the diffusion criterion and recognizing that higher moments
are merely derivatives in the complex variable. Convergence then follows from the theory
of complex variables. It should be noted that this procedure will apply to the problem
considered in chapter 3 if sufficient control of the perturbed spectrum can be obtained. This
is a topic for further study.

3

Chapter 2

A Mathematical Characterization of
Diffusion

To detect diffusion in solutions to a Markov wave equation, we require a rigorous characterization of diffusion. For consistency, this characterization must also be a property of the
classic heat equation. That shall be our starting point.

Consider the solution to the classic heat equation with Dirac initial data:


 ∂t u(x, t) = ∆u(x, t), (x, t) ∈ Rd × R+
x ∈ Rd


 u(x, 0) = δ (x),
0
which we may write explicitly as

u(x, t) =

1
(2πt)d/2
4

|x|2
− 4t
e
.

,

The function x → cu(x, t) is then a probability density function on Rd , where we let
−1
c=

Rd

u(x, t) dx

= 2d/2

be the normalizing constant. For p ∈ N, the pth moment of position is given by

Rd

|x|p cu(x, t) dx =

∞
r2
cωd
p+d−1 e− 4t dr,
r
(2πt)d/2 0

where we have switched to polar coordinates. Here, ωd is the surface area of the unit ball in
Rd . We then see that the integrand on the right-hand side obtains its maximum, regardless
√
of the value of p, when the position r is proportional to t. This leads us to consider the
diffusive scaling


 t → t/η

 x → x/√η

,

in the small η limit, as in reference [3].

The models we wish to study are on the lattice, so we must find a way to apply this
∞
scaling in a discretized context. To this end, we use a mollifier h ∈ Cc (Rd ) with

h dx = 1

which we convolve with a lattice function to obtain a smooth approximation. Ultimately, our
characterization of diffusion should be independent of the choice of h. We may accomplish
this by using a Fourier transform. To see this, suppose ψt ∈ 2 (Zd ) satisfies


d
 ∂t ψt (x) = H
ω(t) ψt (x), x ∈ Z , t ∈ R+
,

 ψ =δ
0
0
5

(2.1)

a wave equation with a random, time-dependent generator. We say ψt exhibits diffusion
(see [3]) if

h ∗ |ψt |2 (x) =

h(x − ξ)|ψt (ξ)|2
ξ∈Zd

satisfies

1
√
h ∗ |ψt/η |2 (x/ η) φ(x) dx η→0
−→
η d/2 Rd

1
Rd (πDt)d/2

|x|2
− Dt
e
φ(x) dx

(2.2)

for all suitable test functions φ, and some D > 0. That is, under the diffusive scaling,
h ∗ |ψt |2 converges weakly to a solution of the heat equation. Using the Fourier transform,
we see that (2.2) is satisfied if
√
e−i ηk·x |ψt/η (x)|2

η→0 e−Dt|k|2 .
−→

x∈Zd
This is the characterization we seek.
Note that the disorder parameter here is suppressed. The solution ψt ∈ 2 (Zd ) depends
implicitly on which realization of the disorder actually occurs. The method we establish in
chapter 3 and Kang and Schenker discuss in [3] require us to weaken this condition to
√
e−i ηk·x E |ψt/η (x)|2

η→0 e−Dt|k|2 ,
−→

(2.3)

x∈Zd
where we have averaged over all realizations of the disorder. Later, we will establish the
existence of a diffusive-scaling limit to higher moments of position (see chapter 4).

6

Chapter 3
Diffusion for a Markov,
Divergence-form Generator
Here we demonstrate that the amplitude of the solution to a wave equation with a Markov,
Divergence-form generator satisfies the diffusion characterization (2.3). We begin with the
assumptions necessary to precisely state the wave model under consideration. These include
the construction of a Markov generator and differential operators on the lattice.
Having stated the theorem, we prove it in several steps. First, an equivalent problem is
derived which is more appropriate for the diffusion characterization (2.3). We then transform
(2.3) into a statement about the holomorphic functional calculus of a particular matrix
element of the generator. We then reduce the integral in the functional calculus to its
substantive part in the diffusive scaling limit. The remainder is then dissected by way of
the Schur Complement Formula and projections which are natural to the problem. The
components are then controlled by a spectral analysis and we are then free to compute the
diffusive scaling limit, establishing the theorem.
7

3.1

Assumptions

For the purposes of this chapter, we assume that we are given a probability space (Ω, µ) and
a Markov Generator1 B with domain D(B) ⊆ L2 (Ω). We assume that the numerical range2
of B is sectoral:

N (B) ⊆ {z = x + iy ∈ C : x ≥ 0, |y| ≤ mx},

for some m > 0, and that B satisfies a gap condition. That is, if we restrict B to it’s range,
then the numerical range of this restriction is bounded away from zero:

Re ψ, Bψ ≥

1
||ψ||2
T

for some T > 0 and all ψ ∈ Rng(B). Also, assume that there are µ-measure preserving maps
σx : Ω → Ω, for each x ∈ Zd , such that σx ◦ σy = σx+y . These maps shift the process by x
and will be used as part of a Fourier transform3 which partially diagonalizes the generator
for the overall problem.
Next, we construct the generator for the wave model. We start with “differential operators” on the lattice (finite difference operators) and include a function of a random variable
so that we may include the Markov process in the generator.
Let Ed be the space of directed edges connecting nearest neighbors in Zd and
2 (Zd ) → 2 (Ed ) be the discrete gradient
given by

† f (x) =

:

f (x, e) = f (x + e) − f (x). It’s adjoint is

(f (x + e, −e) − f (x, e)). Suppose that θ : Ed × Ω → R is positive,
e

1 See

appendix B for the complete construction of B.
appendix A for the definition of the numerical range and its implications.
3 See section 3.4.1 for the Fourier transform and the operator to be partially diagonalized.
2 See

8

bounded, non-constant, and translation covariant: θ(x, e, ω) = θ(x − ξ, e, σξ (ω)). That is, θ
¯
is invariant under shifts of the process in Zd . Further assume that ||θ − θ|| 2 d
=0
L (E ×Ω)
¯
where θ is the average

¯
θ(e) :=

θ(0, e, ω)dµ(ω),

θ(x, e, ω)dµ(ω) =
Ω

Ω

independent of x since θ is translation covariant and σx is µ-measure preserving. Lastly,
assume that θ is constant across directions on a given edge: θ(x, e, ω) = θ(x + e, −e, ω).
We are now able to state the initial-value problem under consideration and make the goal
of this chapter explicit. Before doing so, we give a brief example of such a Markov process.

3.2

The Flip Process

A particular example of the Markov process constructed above is the so-called “flip process”,
similar to [3]. For this process, we envision each (non-directed) edge in Ed as the site for a
˜
process which takes values in {1, 2}. Let Ed be the space of edges, irrespective of direction,
˜
connecting nearest neighbors in Zd . Then, the probability space is Ω = Ed ⊗ {1, 2}. Now,
˜
suppose that for each site (x, e) ∈ Ed , 0 ≤ t1 (x, e) ≤ t2 (x, e) ≤ . . . is a collection of random
times given by independent, identically distributed Poisson processes. At each of these times,
the process at the corresponding site will change sign. The Markov process ω is then a point
in the path space Ω[0,∞) . With this process, we may choose θ to be point evaluation of the
process at the given edge. That is, θ(x, e, ω(t)) ∈ {1, 2}, the value the process takes at time
t on the edge (x, e).
9

3.3

Statement of the Problem

The goal is to show that mean-squared solutions to −i∂t ψt =

†θ
ω(t) ψt on the lattice,

diffuse. That is, under the diffusive scaling limit,


 x → x/√η

as η → 0+ ,

(3.1)


 t → t/η
the quantity E(|ψt |2 ) converges to the solution of a heat equation. A criterion for diffusion
was derived in chapter 2, however, we use an equivalent statement. We establish the criterion
on the Fourier transform side as this allows for the partial diagonalization of the key generator
for the problem. The task before us is stated in the following theorem.

Theorem 1. If ψt ∈ 2 (Zd ) is a solution to the discrete Schr¨dinger initial-value problem
o


d
 i∂t ψt (x) = † θ
ω(t) ψt (x), t > 0, x ∈ Z
,

 ψ (x) = δ (x),
d
x∈Z
0
0

(3.2)

then there exists a symmetric matrix D such that

lim
η→0+

√
ei ηk·x E |ψt/η (x)|2 = e−t k,Dk
x∈Zd

for k ∈ Td .
10

(3.3)

3.4

Proof of the Theorem

First, we derive an equivalent problem which is more appropriate to our goal. We then
reformulate (3.3) in terms of a particular matrix element of the resolvent of the semigroup
generator by using a Feynman-Kac-Pillet formula. A symmetry in the new formulation
allows us to bound the resolvent and take the limit.

3.4.1

A More Appropriate Problem

Since the diffusion criterion (3.3) requires only information about the amplitude of the solution, a linear problem for |ψt |2 is more suitable. We will use a random density matrix
ρ(x, y) = ψt (x)ψt (y)∗ so that |ψt (x)|2 = ρt (x, x). It follows that ρt defined in this way
satisfies


 i∂t ρt (x, y) = L(ω)ρt (x, y),

 ρ (x, y) = δ (x) ⊗ δ (y)
0
0
0

where L(ω) =

t > 0, (x, y) ∈ Zd × Zd
(x, y) ∈ Zd × Zd

†
†
x θx,ω x − y θy,ω y . For each ω ∈ Ω, L(ω) is then a bounded, symmetric

operator on 2 (Zd × Zd ). Now, the Fourier transform

e−ik·ξ f (x − ξ, −ξ, σξ (ω)),

Ff (x, ω, k) =
ξ∈Zd

11

(3.4)

acting on the augmented space L2 (Zd × Zd × Ω), partially diagonalizes L. That is, the
ˆ
transformed operator, which we will refer to as Lk , depends on k as a parameter.

ˆ
Lk ψ(x, ω) =

θ(x, e, ω) (ψ(x, ω) − ψ(x + e, ω))

2
e

θ(0, e, ω) ψ(x, ω) − e−ik·e ψ(x − e, σe (ω))

−2
e

In light of this observation, we will now operate solely on the space L2 (Zd × Ω) for arbitrary
k ∈ Td .

3.4.2

The Resolvent of the Generator

To reduce the problem to a resolvent analysis, we will need a Feynman-Kac-Pillet formula [6],
derived as follows. The conditional expectation we wish to understand can be differentiated
and thus can be seen as the solution to an initial-value problem. The derivative invokes
both the Markov generator and the wave model generator. So it follows that the conditional
expectation – that is, the solution to the initial value problem – can be written as an
exponential of this operator.

1
(E(ρt+h : ω(t + h) = α) − E(ρt : ω(t) = α))
lim
h→0+ h
1
=
lim
(E(ρt+h : ω(t + h) = α) − E(ρt : ω(t + h) = α))
h→0+ h
1
+ lim
(E(ρt : ω(t + h) = α) − E(ρt : ω(t) = α))
+h
h→0

∂t E(ρt : ω(t) = α) =

= E(∂t ρt : ω(t) = α) − BE(ρt : ω(t) = α)
= (−iL(α) − B)E(ρt : ω(t) = α))
12

That B is the derivative of a conditional expectation is illustrated in appendix B. This is
the deterministic Cauchy problem with exponential solution

E(ρt : ω(t) = α)) = e−t(iL(α)+B) ρ0 ,

since the initial data is assumed to be non-random. Integrating over α ∈ Ω, we may now
express the left-hand side of (3.3) as a particular matrix element of the semigroup generated
by the Markov process and the revised problem.

E(ρt (x, y)) =
=

Ω

e−t(iL(α)+B) ρ0 ⊗ 1(x, y) dµ(α)

δx ⊗ δy ⊗ 1, e−t(iL+B) ρ0 ⊗ 1 2 d d
L (Z ×Z ×Ω)

(3.5)

Using the unitarity of the Fourier transform, we will exploit the partial diagonalization of
L. That is, we are fortunate that the matrix element under consideration is the one which
ˆ
corresponds to the function δ0 ⊗ 1 on Zd × Ω and that this function is in the kernel of Lk .
ˆ
This will allow us to pick projections according to the kernel of Lk , allowing us to decompose
ˆ
Lk in a natural way. This is the subject of section 3.4.4. For a brief tutorial on these ideas,
see section E.2. With this in mind, and using the fact that Fρ0 = δ0 , we write:

E(ρt (x, x)) =
=
=
=

δx ⊗ δx ⊗ 1, e−t(iL+B) ρ0 ⊗ 1

L2 (Zd ×Zd ×Ω)
Fδx ⊗ δx ⊗ 1, Fe−t(iL+B) F † Fρ0 ⊗ 1 2 d
L (Z ×Ω×Td )
ˆ
eik·x δ0 ⊗ 1, e−t(iLk +B) δ0 ⊗ 1 2 d
L (Z ×Ω×Td )
ˆ
e−ik·x δ0 ⊗ 1, e−t(iLk +B) δ0 ⊗ 1 2 d
d (k)
L (Z ×Ω)
Td

13

(3.6)

Im(z)

δ−

Γ1

||L||
ˆ
N (iL√ηk + B)

Γ2

Re(z)

−η
−||L||

Γ3

ˆ
Figure 3.1: The contour Γ in (3.7) and the numerical range of iL√ηk + B.

and we interpret the exponential of an unbounded operator by using the holomorphic functional calculus in [5]. This will allow us to write the semigroup in terms of the resolvent of
its generator. Under the diffusive scaling (3.1), we arrive at

e

ˆ
−(t/η)(iL√ηk +B)

=

1
1
− 1 tz
e η
dz.
ˆ
2πi Γ
z − (iL√ηk + B)
14

(3.7)

Our motivation for the particular choice of the contour Γ = Γ1 ∪ Γ2 ∪ Γ3 ,

Γ1 := {z = x + iy ∈ C

: y = 1 + ||L|| + cot(δ − )(x + η)

Γ2 := {z = −η + iy ∈ C : |y| ≤ 1 + ||L||
Γ3 := {z = x + iy ∈ C
with

}
}

: y = −1 − ||L|| + cot(δ − )(x + η) }

∈ (0, δ), subject to the constraints in [5], is as follows. By bounding the resolvent

in terms of the distance to the numerical range, we will show that in the small η limit,
the integral along Γ1 ∪ Γ3 vanishes. The substantive part of the integral is then along Γ2 .
We will show that this contribution is exactly what was stated in (3.3). To this end, we
apply the functional calculus (3.7) to (3.6). In doing so, we have reduced the problem to
understanding the limiting behavior of one matrix element of the resolvent.

√
ei ηk·x E |ψt/η (x)|2
x∈Zd
√
i ηk·x
e

=
x∈Zd
=

Td

Td

ˆ
−(t/η)(iL˜ +B)
˜
k
e−ik·x δ0 ⊗ 1, e
δ0 ⊗ 1

˜
d (k)
L2 (Zd ×Ω)

√
ˆ
˜
−(t/η)(iL˜ +B)
˜
k
ei( ηk−k)·x δ0 ⊗ 1, e
δ0 ⊗ 1 2 d
d (k)
L (Z ×Ω)

x∈Zd
ˆ
−(t/η)(iL˜ +B)
√
˜
˜
k
=
δ0 ( ηk − k) δ0 ⊗ 1, e
δ0 ⊗ 1 2 d
d (k)
d
L (Z ×Ω)
T
ˆ
−(t/η)(iL√ηk +B)
δ0 ⊗ 1, e
δ0 ⊗ 1
=
L2 (Zd ×Ω)
1
1
− 1 tz
= −
e η
δ0 ⊗ 1,
δ0 ⊗ 1 dz
ˆ
2πi Γ
iL√ηk + B − z

We proceed by showing that the contribution to the integral from the unbounded portion of
15

the contour is small.

3.4.3

The Substantive Part of the Integral

Given our sectorality assumption on the Markov generator B, our choice of the contours
Γ1 and Γ3 , and the above lemma, it is easy to see the integral over Γ1 ∪ Γ3 vanishes. For
ˆ
z = x + iy ∈ Γ1 , the distance from z to N (iL√ηk + B) is at least 1. Let

= 1 + ||L|| and

m = cot(δ − ). Then y = + m(x + η) for −η < x < ∞.
1
1
−tz
e η
dz
ˆ
2πi Γ
z − (iL√ηk + B)
1
=

≤

≤

t
−(1 + im) ∞ − η (x+iy)
1
e
dx
ˆ
2πi
x + iy − (iL√ηk + B)
−η
t
1 + m2 ∞ − η x
1
e
dx
ˆ
2π
dist(x + iy, N (iL√ηk + B))
−η
t
1 + m2 ∞ − η x
e
dx = O(η)
2π
−η

Likewise for Γ3 . We then have
√
ei ηk·x E |ψt/η (x)|2
x∈Zd

= −

1
2πi 1 Γ
η 2

e−tw

δ0 ⊗ 1,

(3.8)
η
δ ⊗1
√ + B − ηw 0
ˆ
iL ηk

dw + O(η)

after the substitution z = ηw.
To proceed, we wish to take the small η limit of the resolvent. It is unlikely that this
limit exists as a bounded operator, given that heuristically, L is a second order derivative.
16

Indeed, L will map large, slowly varying functions to functions with small norm. However,
we need not address the entire resolvent. We will be satisfied with the particular matrix
element in (3.8). We continue by dissecting the resolvent according to projections which are
natural to the generator.

3.4.4

Natural Projections and the Schur Complement Formula

Let P0 be the orthogonal projection on L2 (Ω) to non-random functions: P0 f = Ω f dµ.
⊥
Then P0 is the projection onto mean-zero functions. Also, let Q0 = (δ0 ⊗ 1) δ0 ⊗ 1, · .
We will use these projections with the Schur complement (see appendix E.2) to estimate
the resolvent in (3.8). These projections are the natural choice for the resolvent in question,
ˆ
given that Ker(L0 ) = Rng(Q0 ), Ker(B) = Rng(P0 ), and the gap condition on the Markov
⊥
generator gives us that B −1 P0 is norm bounded by 1/T (see appendix D).
A first iteration of the Schur complement formula yields

η
δ0 ⊗ 1, Q0
Q δ ⊗1
(3.9)
ˆ √ + B − ηw 0 0
iL ηk


ˆ
L√ηk
1
= δ0 ⊗ 1, − w + Q0 √ Q⊥ 
0
ˆ
η
iQ⊥ L√ηk Q⊥ + BQ⊥ − ηwQ⊥
0
0
0
0


−1
ˆ√
L ηk
Q⊥ √ Q 
δ0 ⊗ 1
0
0
η


ˆ
L√ηk
1
= − w + Bigg δ0 ⊗ 1, Q0 √ Q⊥ 
0
ˆ
η
iQ⊥ L√ηk Q⊥ + BQ⊥ − ηwQ⊥
0
0
0
0


−1
ˆ
L√ηk
Q⊥ √ Q  δ ⊗ 1
.
0
0
0
η

The last equality follows from the fact the operator has a one-dimensional domain and range
17

and thus may be treated as scalar multiplication. The factor on either side of the resolvent
in (3.9) will play a special role (see (3.12) below and appendix C). This leads us to define
ˆ
L√ηk
fη,k := P0 √ δ0 ⊗ 1
η
√
ηk
¯
−i 2 ·x −2iθ
= e
√
η

(3.10)
√
sin
|e|=1

ηk
· e (δe − δ−e ).
2

We now apply the Schur complement a second time. In this instance, we apply it to
the resolvent in (3.9) according to the projection P0 . We will then take the limit of the
⊥
P0 · · · P0 term in (3.9) and later we will use this to compute the other three terms, P0 · · · P0 ,
⊥
⊥
⊥
P0 · · · P0 , and P0 · · · P0 . A second iteration of the Schur complement gives us:

P0

1

P0
ˆ
iQ⊥ L√ηk Q⊥ + BQ⊥ − ηwQ⊥
0
0
0
0

(3.11)

ˆ
= iP0 Q⊥ L√ηk Q⊥ P0 − ηwQ⊥ P0
0
0
0
⊥
ˆ
+ (P0 Q⊥ L√ηk P0 )
0

1

ˆ
(P ⊥ L√ηk Q⊥ P0 )
0
⊥ L√ P ⊥ + BP ⊥ − ηwP ⊥ 0
ˆ
iP0
ηk 0
0
0

−1
.

Note that the validity of these two applications of the Schur complement formula hinge on
the inversion in the right-hand side of (3.11). Indeed, the inner-most resolvent on the right
hand side of (3.11) is norm bounded by 1/T , so by the Schur complement formula, the left
hand side is a bounded operator if the right hand side is invertible. Therefore, to proceed,
we must find a lower bound for operator being inverted on the right hand side of (3.11).
It is important to note that we need not compute the limit of (3.11) in its entirety. We
need only compute the limit of the particular matrix element fη,k , (P0 · · · P0 )fη,k because
18


ˆ
L√ηk
of the factor Q⊥ √ Q0  in (3.9).
0
η


3.4.5

In Search of a Lower Bound

For simplicity, we define

⊥ˆ
C√ηk = (P0 L√ηk Q⊥ P0 )
0
⊥
⊥ˆ
⊥
⊥
Fη,k (w) = iP0 L√ηk P0 + BP0 − ηwP0
1
†
ˆ
C√ηk .
Mη,k (w) = iP0 Q⊥ L√ηk Q⊥ P0 − ηwQ⊥ P0 + C√
0
0
0
ηk F
η,k
A reasonable next step might be to show that Re Mη,k (w) is bounded below, away from
zero, uniformly in η. We could then take the limit of a uniformly bounded sequence of
operators. In attempting to compute this lower bound, we find instead that Re Mη,k (w) is
bounded below by another operator, which we will call R√ηk/2 , and that this operator has
spectrum near zero. While the ideal lower bound does not exist, we may show nonetheless
−1/2
−1/2
1/2
Re Mη,k (w)R√
is uniformly bounded below on the range of R√
. To
that R√
ηk/2
ηk/2
ηk/2
−1/2
−1/2
show that R√
M (w)R√
is appropriate for our problem, we must also show that
ηk/2 η,k
ηk/2
−1/2
fη,k is in the domain of R√
. This compels us to find a more explicit representation for
ηk/2
R√ηk/2 .
As a first step to this goal, define

⊥ˆ
⊥ˆ
Dk := (B −1 P0 Lk P0 Q⊥ )† (B −1 P0 Lk P0 Q⊥ )
0
0
19

(3.12)

for k ∈ Td , a bounded operator on L2 (Zd × Ω). This operator has the more explicit form

Dk = 8χ N + 8χ N + 4χ
0
k

(δe − eik·e δ−e ) δe − eik·e δ−e , · ,

(3.13)

e

the derivation of which is the subject of Appendix C. Here ∆N is the “Neumann Laplacian”
0
N ψ(x) = (1 − δ (x))
0
0

(ψ(x) − ψ(x + e))
x+e=0
|e|=1

and ∆N is it’s Gauge transform:
k
N = e−ik·X
k

N eik·X
0
ψ(x) − eik·e ψ(x + e) .

N ψ(x) = (1 − δ (x))
0
k
x+e=0
|e|=1

We shall soon see that D√ηk is a lower bound for the operator in question. However fη,k ,
the function which forms the key matrix element, also varies with η. This makes some of
the required calculations difficult. To remedy this, we use a Gauge transform to “push” the
k and η dependence from the function to the operator. To see this, we first transform the
finite-rank part of Dk ,
i k ·x
4χe 2

i k ·X
(δe − eik·e δ−e ) δe − eik·e δ−e , ψ = 4χ
(δe − δ−e ) δe − δ−e , e 2 ψ
|e|=1
|e|=1

20

so that we may write

Dk = 8χe

−i k ·X
2

N
−k/2 +

N +1
(δe − δ−e ) δe − δ−e , ·
k/2 2
e

i k ·X
e 2 .

We then obtain a lower bound as follows. For |k| < 2π , 2 cos k · e > 1 and
3
2

ψ,

N
−k/2 +
ψ ∗ (x)

=
x=0

2 cos
x+e=0
|e|=1

ψ ∗ (x)

≥
x=0
ψ,

N
+
−k/2

e

−i k ·e
2

i k ·e
+e 2

ψ(x + e)

x+e=0
|e|=1

x=0

and thus

2ψ(x) −

ψ ∗ (x)

≥

=

N
k/2 ψ

k
· e ψ(x) − 2 cos
2

k
· e ψ(x + e)
2

(ψ(x) − ψ(x + e))
x+e=0
|e|=1

Nψ
0

N ≥
k/2

k
k
N and D ≥ e−i 2 ·X R ei 2 ·X =: R
0
k
k/2 , where we have
0

defined

R0 = 8χ N + 4χ
0

(δe − δ−e ) δe − δ−e , · .

(3.14)

e

Notice that the factor δe − δ−e appears in (3.14). Also, the same function is present in
the definition (3.10) of fη,k which was inspired by the P0 · · · P0 term of (3.9). With the
−1/2
following lemma, we may conclude that fη,k is in the domain of R√
which means that
ηk/2
−1/2
−1/2
R√
M (w)R√
is indeed appropriate for our problem.
ηk/2 η,k
ηk/2
21

Lemma 2. Let A ≥ 0 on a Hilbert space H and let f ∈ H. Then, for α > 0, f ∈
−1/2
D Aα

where Aα = A + αf f, · .

Proof. First note that the following three statements are equivalent. Here Q A−1 is the
α
form domain of A−1 .
α
−1/2
f ∈ D Aα
f ∈ Q A−1
α
lim f, (Aα + λ)−1 f < ∞
λ→0+
Using the resolvent identity we see that

(A + λ)−1 = (Aα + λ)−1 + (Aα + λ)−1 αf f, · (A + λ)−1
f, (A + λ)−1 f

=

f, (Aα + λ)−1 f + α f, (Aα + λ)−1 f f, (A + λ)−1 f

and

f, (Aα + λ)−1 f

=

1
1
≤
1
α
+α
f,(A+λ)−1 f

since A ≥ 0 and thus (A + λ)−1 > 0.

−1/2
By the above lemma, δe − δ−e is in the domain of R0
. From this, it follows that
22

−1/2
fη,k is in the domain of R √ . Indeed,
ηk
2
−1/2
ϕη,k := R √ fη,k
ηk
2
¯
−2iθ
=
sin
√
η
|e|=1

√
ηk
ηk
−i 2 ·X −1/2
·e e
R0
(δe − δ−e )
2

√

and

−1/2
ϕ0,k := lim R √ fη,k
ηk
η→0
2
−1/2
¯
(k · e) R0
(δe − δ−e ).
= −iθ
|e|=1
Now, it remains to examine the invertibility of Mη,k (w).

3.4.6

Bounding the Resolvent

1
Recall that for w ∈ η Γ2 , Re w = −1 and note that
†
Re Mη,k ≥ Re C√

1
ηk F

η,k

C√ηk

1
Re Fη,k ≥
T
D√ηk = (B −1 C√ηk )† (B −1 C√ηk ).
23

−1/2
Suppose f ∈ D D√
ηk

†
Re f, C√

with |f | = 1. It follows that

1
η,k

C√ηk f

=

C√ηk f, Re

=

ηk F

C√ηk f,

=

≥

=
≥
=

1
Fη,k

C√ηk f

1
1
ReFη,k
C√ηk f
†
Fη,k
F
η,k

1
C√ηk f, ReFη,k
C√ηk f
Fη,k
Fη,k
2
1
1
√ f
C ηk
T Fη,k
2
1
1
−1 C√ f
B
ηk
T B −1 Fη,k
−2
1
B −1 Fη,k
B −1 C√ηk f 2
T
1
B −1 Fη,k −2 f, D√ηk f
T
1

> c0 f, D√ηk f

where we have defined

1
lim B −1 Fη,k −2
2T η→0+
1
⊥ˆ
⊥
⊥
⊥
=
lim iB −1 P0 L√ηk P0 + P0 − ηwB −1 P0 −2
2T η→0+

c0 =

=

1
⊥ˆ ⊥
⊥
iB −1 P0 L0 P0 + P0 −2 .
2T

24

Thus, we have shown

Re

†
C√

1
ηk F

η,k

C√ηk

≥ c0 D√ηk ≥ c0 R √ηk ,
2




1
−1/2 †
−1/2
Re R √ C√
C√ηk R √  ≥ c0 ,
ηk
ηk F
ηk
η,k
2
2
and finally

1/2
1/2
R √ M −1 R √
ηk
ηk
e
2

≤

1
.
c0

With this bound, we may compute the small η limit of the P0 · · · P0 term in (3.9).

3.4.7

The Diffusive-Scaling Limit




−1/2
−1/2
Recall that fη,k ∈ D R √  and we have defined ϕη,k = R √ fη,k . The P0 · · · P0
ηk
ηk
2
2
term in the inner product in (3.9) is then



ˆ
ˆ
L√ηk
L√ηk
1
δ0 ⊗ 1, Q0 √ Q⊥  P0
P Q⊥ √ Q0 
0
0
⊥ L√ Q⊥ + BQ⊥ − ηwQ⊥ 0
ˆ
η
η
iQ0
ηk 0
0
0


δ0 ⊗ 1
=

=

=

fη,k , P0

1

P0 fη,k
ˆ
iQ⊥ L√ηk Q⊥ + BQ⊥ − ηwQ⊥
0
0
0
0

1/2
1/2
R √ ϕη,k , Mη,k (w)−1 R √
ηk
ηk
2
2
1/2
1/2
ϕη,k , R √ Mη,k (w)−1 R √
ηk
ηk
2
2

ϕη,k
ϕη,k
25

which tends to

1
1/2
ˆ
ˆ ⊥
(P ⊥ L Q⊥ P )
ϕ0,k , R0
(P0 Q⊥ L0 P0 )
0
⊥ L P ⊥ + BP ⊥ 0 0 0 0
ˆ
iP0 0 0
0

−1

1/2
R0 ϕ0,k

⊥ ⊥
in the small η limit. It is now a simple matter to compute the P0 · · · P0 , P0 · · · P0 , and
⊥
⊥
P0 · · · P0 terms of (3.9).
⊥
Using the Schur complement formula, the P0 · · · P0 term in the inner product in (3.9)
is



ˆ
ˆ
L√ηk
L√ηk
1
δ0 ⊗ 1, Q0 √ Q⊥  P0
P ⊥ Q⊥ √ Q0 
0
0
⊥ L√ Q⊥ + BQ⊥ − ηwQ⊥ 0
η
η
iQ0 ˆ ηk 0
0
0


δ0 ⊗ 1
=

ˆ
L√
⊥ √ ηk δ ⊗ 1
fη,k , P0
P0
ˆ
η 0
iQ⊥ L√ηk Q⊥ + BQ⊥ − ηwQ⊥
0
0
0
0

=

⊥
ˆ
fη,k , −Mη,k (w)−1 (iP0 Q⊥ L√ηk P0 )
0

1

ˆ
L√ηk
√
⊥ˆ
⊥
⊥
⊥
η
iP0 L√ηk P0 + BP0 − ηwP0
1

δ0 ⊗ 1

which tends to

f0,k ,

† 1
C0
C
F0,0 0

−1
ˆ ⊥
(iP0 Q⊥ L0 P0 )
0

1
⊥ L P ⊥ + BP ⊥
iP0 ˆ 0 0
0

θ(0, e, ω)(k · e)δe

2i
e

ˆ
since δ0 ⊗ 1 ∈ Ker(L0 ) and
ˆ
ˆ
ˆ
L√ηk
L√ηk − L0
lim
δ0 ⊗ 1 = −2i
θ(0, e, ω)(k · e)δe .
√ δ0 ⊗ 1 = lim
√
η
η
η→0+
η→0+
e
26

⊥
The P0 · · · P0 term in the inner product in (3.9) is then



ˆ
ˆ
L√ηk
L√ηk
1
⊥
δ0 ⊗ 1, Q0 √ Q⊥  P0
P Q⊥ √ Q0 
0
0
⊥ L√ Q⊥ + BQ⊥ − ηwQ⊥ 0
ˆ
η
η
iQ0
ηk 0
0
0


δ0 ⊗ 1
=

=

ˆ
L√ηk
1
⊥
P0 fη,k
√ δ0 ⊗ 1, P0
ˆ
η
iQ⊥ L√ηk Q⊥ + BQ⊥ − ηwQ⊥
0
0
0
0
ˆ
L√ηk
1
⊥ˆ
(iP0 L√ηk Q⊥ P0 )Mη,k (w)−1
√ δ0 ⊗ 1, −
0
⊥ L√ P ⊥ + BP ⊥ − ηwP ⊥
ˆ
η
iP0
ηk 0
0
0
fη,k

which tends to

1
† 1
⊥ˆ
θ(0, e, ω)(k · e)δe ,
2i
(iP0 L0 Q⊥ P0 ) C0
C
0
⊥ˆ ⊥
⊥
F0,0 0
iP0 L0 P0 + BP0
e

−1
f0,k

.

⊥
⊥
Lastly, we compute the limit of the P0 · · · P0 term.
1
P⊥ =
⊥ L√ Q⊥ + BQ⊥ − ηwQ⊥ 0
iQ0 ˆ ηk 0
0
0
1
1
ˆ
−
(iP ⊥ L√ηk Q⊥ P0 )
0
⊥ L√ P ⊥ + BP ⊥ − ηwP ⊥ iP ⊥ L√ P ⊥ + BP ⊥ − ηwP ⊥ 0
ˆ
ˆ
iP0
ηk 0
ηk 0
0
0
0
0
0
1
⊥
ˆ
·Mη,k (w)−1 (iP0 Q⊥ L√ηk P0 )
0
⊥ˆ
⊥
⊥
⊥
iP0 L√ηk P0 + BP0 − ηwP0
⊥
P0

27

and the appropriate matrix element tends to

θ(0, e, ω)(k · e)δe

2i

,

e
1
1
ˆ
−
(iP ⊥ L Q⊥ P )
⊥ L P ⊥ + BP ⊥ iP ⊥ L P ⊥ + BP ⊥ 0 0 0 0
ˆ
iP0 ˆ 0 0
0
0
0 0 0
−1
1
† 1
ˆ ⊥
· C0
2i
θ(0, e, ω)(k · e)δe
C0
(iP0 Q⊥ L0 P0 )
0
⊥ˆ ⊥
⊥
F0,0
iP0 L0 P0 + BP0
e

In the above, we have computed

Jη,k (w) :=





ˆ
ˆ
L√ηk
L√ηk
1
Q⊥ √ Q  δ ⊗ 1
δ0 ⊗ 1, Q0 √ Q⊥ 
0
0
0
0
⊥ L√ Q⊥ + BQ⊥ − ηwQ⊥
ˆ
η
η
iQ0
ηk 0
0
0
28

.

and

Jk :=

lim Jη,k (w)
η→0+

(3.15)

1
1/2
ˆ
ˆ ⊥
(P ⊥ L Q⊥ P )
=
ϕ0,k , D0
(P0 Q⊥ L0 P0 )
0
⊥ L P ⊥ + BP ⊥ 0 0 0 0
ˆ
iP0 0 0
0
−1
1
† 1
ˆ ⊥
+ f0,k , C0
C0
(iP0 Q⊥ L0 P0 )
0
⊥ˆ ⊥
⊥
F0,0
iP0 L0 P0 + BP0

−1

1/2
D0 ϕ0,k

θ(0, e, ω)(k · e)δe

2i
e
+

θ(0, e, ω)(k · e)δe ,

2i
e

† 1
C
C0
F0,0 0

1
ˆ
(iP ⊥ L Q⊥ P )
⊥ L P ⊥ + BP ⊥ 0 0 0 0
ˆ
iP0 0 0
0

−1
f0,k

1
1
−
⊥ L P ⊥ + BP ⊥ iP ⊥ L P ⊥ + BP ⊥
ˆ
iP0 ˆ 0 0
e
0
0 0 0
0
−1
1
† 1
⊥ˆ
ˆ ⊥
·(iP0 L0 Q⊥ P0 ) C0
C0
(iP0 Q⊥ L0 P0 )
0
0
⊥ L P ⊥ + BP ⊥
F0,0
iP0 ˆ 0 0
0

+

2i

θ(0, e, ω)(k · e)δe ,

θ(0, e, ω)(k · e)δe

2i
e

Although it is not obvious from (3.15) above, Jk is of the form k, Dk

Cd

with D symmetric.

To see this, we must compute Jk a different way.
Let D(η) be the matrix of time-averaged second moments:

(η)
Di,j := −η 2

∞
0

xi xj E |ψt (x)|2 e−ηt dt.
x∈Zd

This is just a time-averaged second derivative, evaluated at k = 0, of the left-hand side of
the diffusion criterion (3.3). It is clear that this matrix is symmetric. Now by using the
29

Feynman-Kac-Pillet formula (3.5), we see that

(η)
Di,j = η

δ0 ⊗ 1, 


∂ ∂
1

δ0 ⊗ 1 .
ˆ
∂ki ∂kj iL√ + B + η
ηk
k=0

But, by computing the derivatives, we see that k, D(η) k is the inner-product on the right
hand side of (3.9), which we have already named Jη,k . Thus, Jk is of the form k, Dk with
D symmetric.

All that remains is to compute the limit of the integral in (3.8).

−

1
1
e−tw
dw
1Γ
2πi
−w + Jη,k
η 2

=−

1
e−tw
2πi 1 Γ
η 2

1
1
−
−w + Jη,k −w + Jk

dw −

1
1
e−tw
dw
2πi 1 Γ
−w + Jk
η 2

1 t
For the first integral, the difference decays like c/w2 so 2π ce2 is an integrable upper
w
bound for the integrand. Thus, by the Lebesgue convergence theorem, the first integral goes
to zero as η → 0.

We will use the residue theorem to compute the final integral. The path

γ = w = −1 + Reiθ : −

π
π
,
<θ<
2
2

R=

1 + ||L||
,
η

1
together with η Γ2 , forms a closed loop. For η sufficiently small, this path encloses Jk . The
residue at this pole is e−tJk .
30

On the curve γ,

1
1
e−tw
dw
2πi γ
w − Jk

π
1
1
2 −t(−1+Reiθ )
iReiθ dθ
=
e
iθ − J
2πi − π
−1 + Re
k
2
π
1
R
2 −t(−1+R cos θ)
≤
e
dθ
2π − π
−1 + Reiθ − Jk
2
t π
ce
2 −tR cos θ
e
dθ,
≤
2π − π
2

which goes to zero as η → 0+ by the bounded convergence theorem.
Therefore,

lim
η→0+

Jk is of the form

√
ei ηk·x E |ψt/η (x)|2 = e−tJk ,
x∈Zd

e1 ,e2 (k · e1 )(k · e2 )De1 ,e2 as seen above, and the theorem is proven.

31

Chapter 4
Higher Moments for a
Markov-Schr¨dinger Equation
o

4.1

Statement of the Problem

The problem we consider here is that of computing moments of the position variable with
respect to the probability density function E(|ψt |2 ) on Zd . Here ψt is the solution to a
Markov-Schr¨dinger equation on the lattice and the expectation is an average over all realo
izations of the process. Specifically, ψt ∈ 2 (Zd ) satisfies


 i∂ ψ (x) =
 t t


y


 ψ (x) = δ (x),
 0
0

hω (x, y, t)ψt (y), x ∈ Zd , t > 0
(4.1)
x ∈ Zd

with ω a Markov process and hw (·, ·, t) ∈ 2 (Zd × Zd ). By way of a gauge transform, this
problem is seen to be equivalent to the wave model in [3] and their result is crucial to the one
presented here. We begin by giving the necessary details for diffusion in [3] and by showing
32

the equivalence of the two models. A few further assumptions will be made, giving us a
ballistic upper bound on solutions thus allowing for the computation of moments.

4.2

Diffusion for a Markov-Schr¨dinger Wave Equao
tion

Here we will provide the minimum details from [3] to define the original problem and state
their main result. For further details, the reader should consult [3].
˜
˜
˜
|x|2 |h(x)| < ∞ and h(−x) = h(x)∗ for all
x∈Zd
˜
x ∈ Zd . Further suppose that for k ∈ Rd , h(x) = 0 and k · x = 0 for some x ∈ Zd . We may
˜
Suppose h is a function on Zd with

˜
then define T ψ(x) =

˜
˜
y h(x, y)ψ(y). The assumptions on h imply that T is a bounded,

self-adjoint operator from 2 (Zd ) to itself.
Let λ > 0 be constant, t → ω(t) be a Markov process on a probability space (Ω, µ), and
for x ∈ Zd , suppose vx : Ω → R is measurable. Given |ψ0 | 2 d = 1, the problem under
(Z )
˜
consideration is that of finding ψt ∈ 2 (Zd ) for t > 0, such that

˜
˜
˜˜
i∂t ψt (x) = T ψt (x) + λvx (ω(t))ψt (x), x ∈ Zd .

(4.2)

In the case when ψ0 = δ0 , their main result is

lim
τ →∞

k
−i √ ·x
τ E(|ψ (x)|2 ) = e−t
e
τt

i,j Di,j (λ)ki kj ,

(4.3)

x∈Zd

for k in the torus Td and D = D(λ) a positive definite matrix. The result presented here
is that we may differentiate (4.3) term-by-term, at k = 0, which will allow us to compute
33

higher moments in the diffusive scaling limit. We do this by analytically continuing (4.3) in
the variable k to a neighborhood of the origin in Cd . But first, we must draw an equivalence
between the models (4.1) and (4.2).

4.3

Equivalence of Models

The result in [3] that is crucial to the result here, is stated in terms of the solution to (4.2).
To derive the bounds we need to compute moments, we would prefer to work in terms of
(4.1). These two are easily seen to be equivalent. Moreover, the corresponding probability
˜
density functions, and hence moments, are equal: E(|ψt |2 ) = E(|ψt |2 ). To see this, define
the following gauge transformation.
t
φω (x, t) = λ

vx (ω(s)) ds
0
˜
hω (x, y, t) = eiφω (x,t) h(x, y)e−iφω (y,t)

(4.4)

˜
ψt (x) = eiφω (x,t) ψt (x)

˜
Now hω in (4.1) is fully defined and hω inherits the assumptions placed on h in [3]. We
˜
then see that ψt satisfies (4.1) if and only if ψt is a solution to (4.2). It follows that the
probability density functions are equal, as claimed. For the extended result here, we require
˜
˜
some additional assumptions on h and ψ0 . Namely, those conditions that will guarantee a
ballistic upper bound on solutions to (4.2) and equivalently to (4.1). This is the topic of the
next section.
34

4.4

A Ballistic Upper Bound

˜
˜
We assume the following decay conditions on h and the initial condition ψ0 . Equivalent
assumptions on hω and ψ0 are then inherited and are also stated here.

eµ|x−y| |hω (x, y, t)| =
y

˜
eµ|x−y| |h(x, y)| ≤ A

(4.5)

y
˜
c0 := sup eµ|x| |ψ0 (x)| = sup eµ|x| |ψ0 (x)| < ∞
x
x

To compute a ballistic upper bound, we first integrate equation (4.1) and iterate the result
to arrive at
∞
rn−1 (ω)
t r1
(ω)
(ω)
Hr . . . Hrn ψ0 drn · · · dr1 ,
···
Hr ψ0 dr1 +
ψt (x) = ψ0 (x) +
1
1
0
0
n=2 0 0
t

(ω)
where Ht ψ(x) =

y hω (x, y, t)ψt (y). Now we multiply by the exponential

eµ|x| ψt (x) = eµ|x| ψ0 (x) +
∞

t

+
n=2 0

r1
0

t
0

eµ|x| Hr1 ψ0 dr1

···

rn−1
eµ|x| Hr1 . . . Hrn ψ0 drn · · · dr1
0

and notice that if we write

eµ|x| ≤ eµ|x−y1 | eµ|y1 −y2 | · · · eµ|yn | ,
35

and understand that Hrj provides a sum over yj ∈ Zd , then the decay conditions (4.5) give
us

eµ|x| |ψt (x)| ≤ c0 +

∞

t
0

c0 A dr1 +

t

n=2 0

r1

rn−1

···

0

0

c0 An drn · · · dr1 .

But this is just the integral over the simplex Γn (t) = {(r1 , . . . , rn ) : rn ≤ rn−1 ≤ · · · ≤
n
r1 ≤ t} which has Rn -Lebesgue measure t . This gives us
n!
eµ|x| |ψt (x)| ≤ c0 + c0 At +

∞

c0 A n

n=2

tn
= c0 eAt
n!

and |ψt (x)| ≤ c0 eAt−µ|x| is the bound we seek.

4.5

Moments by Analytic Continuation

To show that (4.3) can be analytically continued, we require some notation. For t > 0,
τ

1, and z ∈ Cd with |z| < µ, define

Ft,τ (z) =

z·x
√
e τ E(|ψτ t (x)|2 )

and

(4.6)

x∈Zd
t i,j Di,j (λ)zi zj
Ft (z) = e
,

where ψt is a solution to (4.1) and Di,j (λ) is given by (4.3). Equation (4.3) is the motivation
for these definitions since, in this notation, the main result in [3] is that limτ →∞ Ft,τ (−ik) =
Ft (−ik) for all k ∈ Td .
To begin, we wish to show that limτ →∞ Ft,τ ≡ Ft . Since Ft is clearly analytic, this is a
36

result of the identity theorem if we first show that limτ →∞ Ft,τ is analytic. Moreover, we
also wish to show that the derivatives of Ft,τ converge to the corresponding derivatives of
Ft . From this, it is a simple matter to compute any moment of the position. One may just
differentiate (4.3) with respect to k ∈ Td and substitute k = 0.
That these derivatives converge appropriately, is a consequence of the fact that the analytic functions (on regions in C with the sup norm) form a closed subset of the continuous
functions (see [1]). Therefore, it remains only to show that Ft,τ and limτ →∞ Ft,τ are
analytic.
For the remainder of this chapter, when we discuss convergence or analyticity, it is understood that the domain under consideration is {z ∈ Cd : |z| < µ0 }, for some µ0 ∈ (0, µ).
This extra space between µ0 and µ will allow us to conclude uniform convergence for the
series rather than mere point-wise convergence.

4.6

Convergence and Analyticity

First we show that the series in (4.6) converges uniformly and absolutely. We then show
that (4.6) and its limit are analytic. This will complete the argument laid out in section 4.5.
Let ωd be the surface area of the unit ball in Rd .
Lemma 1. The series in (4.6) is uniformly and absolutely convergent with

x∈Zd

|z||x|
√
ω (d − 1)!
e τ E(|ψτ t (x)|2 ) ≤ c0 eAτ t d
.
(µ − µ0 )d

Moreover, Ft,τ defined in (4.6) is analytic.
37

Proof. The partial sums of the series are analytic and approximate Ft,τ . So, to show analyticity, we need only show uniform convergence of the series. Since
µ0 |x|
√
e τ E(|ψτ t (x)|2 ),

z·x
√
e τ E(|ψτ t (x)|2 ) ≤
x∈Zd

x∈Zd

µ0 |x|
√
e τ E(|ψτ t (x)|2 ) converges.
x∈Zd

it suffices to show

For all x0 ∈ Zd , there exists a unique closed unit hypercube Cx0 ⊆ Rd with x0 , the
unique point in the hypercube furthest from 0. The union of all such cubes is Rd . Although
they are not disjoint, the intersection of any two cubes has Lebesgue measure zero. For
k > 0, it follows that

e−k|x0 | = e−k|x0 |

e−k|x| dx,

dx ≤
Cx 0

C x0

and we sum over x0 ∈ Zd to arrive at

e−k|x| ≤
x∈Zd

ω (d − 1)!
.
e−k|x| dx = d
kd
Rd

From section 4.4, we have |ψt (x)|2 ≤ c0 eAt−µ|x| . Note that |ψt (x)| ≤ 1 since the
solution generator is unitary and the initial condition is a unit vector. With this, we may
38

conclude that the series converges absolutely and with constant bound as follows.
|z||x|
√
e τ |ψτ t (x)|2 ≤
x∈Zd

eµ0 |x| c0 eAτ t−µ|x|

x∈Zd
= c0 eAτ t

e−(µ−µ0 )|x|

x∈Zd
ω (d − 1)!
≤ c0 eAτ t d
(µ − µ0 )d

Lemma 2.

lim F
is analytic.
τ →∞ t,τ

Proof. From [3],
z·x
√
e τ |ψτ t (x)|2 =

δ0 ⊗ 1, e

ˆ
−tτ L−iz/√τ

δ0 ⊗ 1

x∈Zd
and so it suffices to show

lim
δ ⊗ 1, e
τ →∞ 0

ˆ
−tτ L−iz/√τ
δ0 ⊗ 1

√
ˆ
ˆ
is analytic. As L−iz/√τ is a perturbation of L0 on the order of 1/ τ , the estimates
ˆ
on the spectral gap in [3] hold. In particular, E(z) is an isolated eigenvalue of Lz with
√
|E(z)| < c/ τ . The rest of the numerical range is contained in a sector of the form

Σ+ =

c
z = x + iy ∈ C : x > δλ − √ , |y| ≤ mx .
τ

We may then choose a contour Γ = Γ1 ∪ Γ2 , with index 1 around the numerical range.
39

Im(z)

Γ2

E

−iz
√
τ

Σ+

Γ1
c
√
τ

c
− 1 δλ − √ τ
τ 3

Re(z)

Figure 4.1: The contour Γ and the perturbed numerical range.

40

√
Γ1 is the circle centered at the origin, with radius c/ τ . Then we choose Γ2 winding once
around Σ+ , maintaining a distance at least on the order of τ −1/3 from Σ+ . We will use
the fact that

dist(τ Γ2 , τ Σ+ ) = O τ − τ 2/3

ˆ
and so an integral of the resolvent of τ L−iz/√τ over τ Γ2 is small (see Appendix A). With
this contour, we may use the holomorphic functional calculus (see [5]).

δ0 ⊗ 1, e

ˆ
−tτ L−iz/√τ

δ0 ⊗ 1

=

1
e−tτ w
2πi Γ

δ0 ⊗ 1,

1
δ0 ⊗ 1
ˆ
w − L−iz/√τ

dw

To make use of the resolvent estimate above and to avoid an asymptotic integral, we make
a linear change of variables to arrive at

1
e−tw
2πi τ Γ
=

1
e−tw
2πi τ Γ
1

δ0 ⊗ 1,

δ0 ⊗ 1,

1
δ0 ⊗ 1
ˆ
w − τ L−iz/√τ

1
δ0 ⊗ 1
ˆ
w − τ L−iz/√τ

dw + O

dw
1
τ

,

and the resulting integral over τ Γ1 is an exponential times a Riesz projection. Indeed, if
ˆ
Ψ(z) is a unit vector satisfying Lz Ψ(z) = E(z)Ψ(z) then

Qz =

1
1
dw
ˆ
2πi Γ w − Lz
1

41

ˆ
is the projection onto the span of Ψ(z) and Lz Qz = E(z)Qz . It follows that
z·x
√
√
e τ |ψτ t (x)|2 = e−tτ E(−iz/ τ ) δ0 ⊗ 1, Q−iz/√τ δ0 ⊗ 1 + O
x∈Zd

1
τ

.

It is clear that δ0 ⊗ 1, Q−iz/√τ δ0 ⊗ 1 tends to 1. From [3] we have an expansion for
E(z) near zero with E(0) = 0 and

lim
τ →∞

E(0) = 0 from which we may conclude
z·x
√
e τ |ψτ t (x)|2 = e−th(z)

x∈Zd

√
with h(z) = limτ →∞ τ E(−iz/ τ ) holomorphic.

42

APPENDICES

43

Appendix A

The Numerical Range of a Linear
Operator

Definition 1. The numerical range [8] of an operator on a Hilbert space is defined by

N (A) = { x, Ax : x ∈ D(A), |x| = 1}.

Lemma 2. Suppose H is a Hilbert space, D(A) ⊆ H is dense, and A : D(A) → H. Then
the norm of the resolvent is bounded by the inverse of the distance to the numerical range:

1
1
≤
.
z−A
dist(z, N (A))

Proof. Suppose z ∈ C such that dist(z, N (A)) > 0 and let ψ ∈ H be a unit vector. Define
44

−1
1
1
c = z−A ψ
and ϕ = c z−A ψ.1 Then
dist(z, N (A)) ≤ | ϕ, (z − A)ϕ | = c2

1
ψ, ψ
z−A

≤c

and the result follows by inverting the inequality and taking the supremum over {|ψ| =
1}.
As a consequence of this lemma, the spectrum of an operator is contained in the closure
of its numerical range. Indeed, if dist(z, N (A)) > 0 then Ker(z − A) = {0} and z ∈ ρ(A).
So, if z ∈ σ(A), then dist(z, N (A)) = 0 and z ∈ Clo N (A).

1 At

this point, one should verify that ϕ is well-defined; that the domain of the resolvent is the
entire Hilbert space. To do this, it suffices to show that the resolvent is bounded. One approach,
and perhaps the simplest, is to define it to be so. In [7] for example, the resolvent set is defined
to be those complex numbers such that the resolvent is a bounded operator, defined on the entire
space. Alternatively, [4] defines the resolvent set such that the resolvent need only be a bijection
on H. Then, the closed graph theorem asserts that the resolvent is bounded.

45

Appendix B
The Markov Generator
Here we construct the semigroup and generator corresponding to a Markov process. The
generator will be an unbounded operator, defined on a subset of L2 -functions on a probability
space. This semigroup structure encapsulates the process in the generator, allowing us to
reduce the problem of understanding the Markov dynamics (in an averaged sense, see 2.3)
to a spectral analysis of the generator. We may focus on the generator itself, and not the
process it represents, because of a Feynman-Kac formula due to Pillet [6]. This formula
allows us to write the amplitude of a wave-form in terms of a particular matrix element of
the overall generator – the generator for the wave model together with the Markov generator.
Hence, a detailed study of the Markov process and its underlying probability space is limited
to this section, allowing us to concentrate our focus in chapter 3 solely on the generator.
We begin by defining the underlying probability space, a path-space for the Markov process, and the appropriate semigroups. Several assumptions are placed on these spaces so that
we indeed have a semigroup, and that this semigroup is a strongly continuous contraction.
With this, we may write down its generator and, with a few additional assumptions, derive
46

the key aspects which are required in chapter 3. The development below is nearly identical to [3], except that we construct a collection of processes, independent and identically
distributed, and indexed by the space through which our wave solution will propagate.

Let Ed denote the space of directed edges between nearest-neighbor pairs in Zd . That is,
(x, e) ∈ Ed if x ∈ Zd and e is a unit vector with x + e ∈ Zd . We call the points x and x + e
in Zd , nearest neighbors. Each of these directed edges may be thought of as a site at which
a Markov process runs.

For each (x, e) ∈ Ed , we will construct a Markov process as follows. It is assumed that
the collection of processes constructed in this way are i.i.d. Let (Ω, µ) be a probability space
and suppose we have a collection {Pα : α ∈ Ω} of probability measures on the path space
P = Ω[0,∞) and that each measure Pα is supported on those processes which start at α.
That is,

Pα ({ω(·) ∈ P : ω(0) = α}) = 0.

Further, we assume that each path ω(·) ∈ P is right-continuous – Pα -a.s. With these
probability measures in mind, we assume that µ is invariant:

Ω

Pα (ω(t) ∈ A) dµ(α) = µ(A)

for measurable sets A ⊆ Ω.

−1
Let St be the backward shift on P, St ω(·) = ω(· + t), so that St (A) = {ω(·) : ω(· + t) ∈
47

A} for measureable sets A ⊆ P. Finally, we suppose that the Markov property holds:

P

−1
Pω(t) (A) dPα (ω(·)) = Pα (St (A)).

We then define

St f (α) = Eα (f (ω(t))),

St : L2 (Ω) → L2 (Ω),

and the above assumptions imply that {St }t≥0 is a strongly continuous, contraction semi†
group on L2 (Ω). It follows that the adjoint St is also a strongly continuous, contraction
semigroup given by

†
St f (α) = E(f (ω(0))|ω(t) = α).
†
Let B denote the generator of St ,
1
†
Bψ = − lim
St ψ − ψ
t→0+ t
†
so that e−tB = St . The generator B is defined on D(B) – those L2 (Ω) functions for which
the limit exists. The exponential of an unbounded operator may be interpreted using the
holomorphic functional calculus [5]:

e−tB =

1
e−tz
dz.
2πi Γ z − B

To ensure convergence of the functional calculus, and to further control the spectrum of the

48

Im(z)

N (B)

Γ

Re(z)

Figure B.1: The contour Γ and the numerical range of the Markov Generator.
overall generator in chapter 3, we assume that the numerical range (see appendix A) of B
is sectoral. That is, if z = x + iy ∈ N (B), then |y| ≤ mx for some m ≥ 0. This condition is
easily satisfied if B happens to be self-adjoint.

49

Appendix C

Another Realization of Dk

The operator Dk plays a special role in showing diffusion for the divergence-form model in
chapter 3. Here we derive another useful formulation for Dk .

For k ∈ Cd , let Dk be the bounded, self-adjoint operator defined in section 3.4.4 by
⊥ˆ
⊥ˆ
Dk = (B −1 P0 Lk P0 Q⊥ )† (B −1 P0 Lk P0 Q⊥ ).
0
0

(C.1)

At the present time, we are only concerned with k ∈ Td , however the full generality of
k ∈ Cd will be useful when computing higher moments of position.

Consider (x, e) ∈ Ed , a directed edge connecting nearest neighbors in Zd . The opposing
direction on the same edge is given by (x + e, −e). Due to translation covariance, θ does not
distinguish between these two directions. Indeed, θ(x, e, ω) = θ(x + e, −e, ω). Moreover, if
50

(x, e) and (x , e ) are distinct edges,

¯
¯
B −1 (θ(x, e, ω) − θ), B −1 (θ(x , e , ω) − θ)

L2 (Ω)

¯
¯
(B −1 (θ(x, e, ω) − θ))∗ B −1 (θ(x , e , ω) − θ)dµ(ω)

=
Ω

¯
B −1 (θ(x , e , ω) − θ)dµ(ω)

¯
(B −1 (θ(x, e, ω) − θ))∗ dµ(ω)

=

Ω

Ω
=0

since the Markov processes on each edge are independent and the set of mean-zero functions
are invariant under B. Thus, for any pair of edges (x, e) and (x , e ) we may write

¯
¯
B −1 (θ(x, e, ω) − θ), B −1 (θ(x , e , ω) − θ)

= χ(δx (x )δe (e ) + δx (x + e )δe (−e ))
L2 (Ω)

¯
where χ = ||B −1 (θ(x, e, ·)− θ)||2 2
. Note that χ is independent of (x, e) since the Markov
L (Ω)
processes on edges in Ed are i.i.d. We will use this equality to evaluate matrix elements of
Dk . Let ϕ ∈ 2 (Zd ) with ϕ = P0 Q⊥ ϕ. That is, ϕ is non-random and ϕ(0) = 0. Also, define
0
⊥
˜
¯
θ(x, e, ω) := P0 θ(x, e, ω) = θ(x, e, ω) − θ. Matrix elements of Dk are evaluated as follows.

51

⊥ˆ
= ||B −1 P0 Lk ϕ||2 2 d
L2 (Zd ×Ω)
L (Z ×Ω)

ϕ, Dk ϕ
=
x∈Zd

Ω

⊥ˆ
⊥ˆ
(B −1 P0 Lk ϕ(x, ω))∗ B −1 P0 Lk ϕ(x, ω) dµ(ω)

= 4
x∈Zd

Ω
∗

⊥
B −1 P0
e


θ(x, e, ω) (ϕ(x) − ϕ(x + e)) + θ(0, e, ω) e−ik·e ϕ(x − e) − ϕ(x)

B −1 P ⊥
0

θ(x, e , ω) ϕ(x) − ϕ(x + e ) + θ(0, e , ω) e−ik·e ϕ(x − e ) − ϕ(x)



e
dµ(ω)

We now may evaluate the integral by using the definition of χ. That is, by making use of
the assumption that the Markov processes at each site are i.i.d.

52



ϕ, Dk ϕ

L2 (Zd ×Ω)

= 4

dµ(ω)
Ω

x,e,e
˜
˜
B −1 θ(x, e, ω)(ϕ∗ (x) − ϕ∗ (x + e)) · B −1 θ(x, e , ω)(ϕ(x) − ϕ(x + e ))
˜
˜
+ B −1 θ(x, e, ω)(ϕ∗ (x) − ϕ∗ (x + e)) · B −1 θ(0, e , ω)(e−ik·e ϕ(x − e ) − ϕ(x))
¯
˜
˜
+ B −1 θ(0, e, ω)(eik·e ϕ∗ (x − e) − ϕ∗ (x)) · B −1 θ(x, e , ω)(ϕ(x) − ϕ(x + e ))
¯
˜
˜
+ B −1 θ(0, e, ω)(eik·e ϕ∗ (x − e) − ϕ∗ (x)) · B −1 θ(0, e , ω)(e−ik·e ϕ(x − e ) − ϕ(x))
= 4χ
x,e,e
(ϕ∗ (x) − ϕ∗ (x + e))(ϕ(x) − ϕ(x + e ))δe (e )
+ (ϕ∗ (x) − ϕ∗ (x + e))(e−ik·e ϕ(x − e ) − ϕ(x))(δ0 (x)δe (e ) + δ (x)δe (−e ))
e
¯
+ (eik·e ϕ∗ (x − e) − ϕ∗ (x))(ϕ(x) − ϕ(x + e ))(δ0 (x)δe (e ) + δe (x)δe (−e ))
¯
+ (eik·e ϕ∗ (x − e) − ϕ∗ (x))(e−ik·e ϕ(x − e ) − ϕ(x))δe (e )

53

Evaluating the Kronecker delta functions give us

=

(ϕ∗ (x) − ϕ∗ (x + e))(ϕ(x) − ϕ(x + e))

4χ
x,e

e−ik·e ϕ∗ (e)ϕ(−e) + ϕ∗ (−e)ϕ(−e)

−4χ
e

¯
eik·e ϕ∗ (−e)ϕ(e) + ϕ∗ (e)ϕ(e)

−4χ
e

¯
(eik·e ϕ∗ (x − e) − ϕ∗ (x))(e−ik·e ϕ(x − e) − ϕ(x)).

+4χ
x,e
In the case when k is real,

ϕ, Dk ϕ

=
L2 (Zd ×Ω)

ϕ∗ (x)

8χ

[ϕ(x) − ϕ(x + e)]
e

x
ϕ∗ (x)

+8χ
x

ϕ(x) − eik·e ϕ(x + e)
e

ϕ∗ (e) e−ik·e ϕ(−e) + ϕ(e)

−8χ
e

and we will examine each of these three terms separately. The first term is

ϕ∗ (x)

8χ

e

|x|>1

e

|ϕ(e)|2

+8χ
e
= 8χ ϕ,

ϕ∗ (e)

[ϕ(x) − ϕ(x + e)] + 8χ

N ϕ + 8χ
0

|ϕ(e)|2
e

54

ϕ(e) − ϕ(e + e )
e =−e

where

N is the “Neumann Laplacian”:
0
N ψ(x) = (1 − δ (x))
0
0

(ψ(x) − ψ(y))
y=0
|x−y|=1

This new “Laplacian” is just the traditional discrete Laplacian with Neumann boundary
conditions near zero. With the substitution e = y − x, the second term is then

ϕ∗ (x)e−ik·x

8χ
|x|>1

eik·x ϕ(x) − eik·y ϕ(y)
y∈Zd
|x−y|=1

ϕ∗ (e)e−ik·e

+8χ

eik·e ϕ(e) − eik·y ϕ(y)

e

y=0
|e−y|=1
|ϕ(e)|2

+8χ
e
= 8χ ϕ,

where

N ϕ + 8χ
k

|ϕ(e)|2
e

N is the Gauge transformation of the Neumann Laplacian,
k
N = e−ik·X
k

N eik·X
0
ψ(x) − eik·(y−x) ψ(y)

N ψ(x) = (1 − δ (x))
0
k
y=0
|x−y|=1

ψ(x) − eik·e ψ(x + e)

= (1 − δ0 (x))
x+e=0
|e|=1

Lastly, we take the third term along with the non-Laplacian terms from above. These
55

may be written as:

δe − eik·e δ−e

4χ ϕ,

δe − eik·e δ−e , · ϕ .

e
Combining these three terms, we have our alternate representation of Dk :

Dk = 8χ N + 8χ N + 4χ
0
k

(δe − eik·e δ−e ) δe − eik·e δ−e , · .

(C.2)

e

Note that we have only shown the equivalence of diagonal matrix elements of the operators
in (C.1) and (C.2). However, each of these operators is non-negative. By applying the
polarization identity [7] to the operator’s square root, one sees that this is sufficient to
conclude equality of (C.1) and (C.2). Indeed, if A ≥ 0 on a Hilbert Space, and x, Ax = 0
for all x ∈ D(A), then the polarization identity says

x, Ay

=

A1/2 x, A1/2 y

1
( A1/2 (x + y) 2 − A1/2 (x − y) 2 + i A1/2 (x − iy) 2 − i A1/2 (x + iy) 2 )
4
1
=
( x + y, A(x + y) − x − y, A(x − y) + i x − iy, A(x − iy)
4

=

−i x + iy, A(x + iy) )
= 0.

56

Appendix D
Inversion of Linear Operators
Lemma 3. Suppose A is a linear operator on a Hilbert space and

= inf z∈N (A) |z| > 0.

Then A is boundedly invertible and A−1 ≤ 1/ .

Proof. It is clear that A is invertible, since zero is not in the numerical range, thus not in
the spectrum. Suppose A−1 > 1/ . Then there is a unit vector y, in the domain of A−1 ,
such that |A−1 y| > 1/ . Let x = A−1 y so that 1/|x| < .

x
x
,A
|x|
|x|

≤

1
|x|

x
,y
|x|

<

This contradicts our assumption on the numerical range of A.

Indeed, what we have shown is that if an invertible operator fails to be boundedly invertible, then zero is in the closure of the numerical range. Another conclusion we may
draw is that, to show an operator is boundedly invertible, it suffices to show that its real
(or imaginary) part is bounded away from zero. By the real part of an operator, we mean
x, Re(A)x = Re x, Ax and we may extend this definition, by way of the polarization
57

identity, to x, Re(A)y . The imaginary part is then Im(A) = −i(A − Re(A)) as expected.

58

Appendix E
The Schur Complement Formula
The Schur complement [9] is a generalization of the notion of the determinate of a 2x2
matrix, in the case when the entries do not commute. A formal statement of this fact is
given here and the proof is given in section E.1. The formula is particularly useful when
used in conjunction with projections that are natural to the operator being inverted, as
demonstrated in section E.2.

Lemma 4. (The Schur Complement Formula) Suppose A, B, C and D are linear operators


 A B 
from a vector space to itself and that D is invertible. Then 
 is invertible if and
C D
only if (A − BD−1 C) is invertible. In the affirmative case, we also have


−1

 A B 
(E.1)


C D


−1 C)−1
−1 C)−1 BD−1
(A − BD
−(A − BD


=
.
−1 C(A − BD−1 C)−1 D−1 + D−1 C(A − BD−1 C)−1 BD−1
−D
59

Definition 5. Equation (E.1) is known as the Schur complement formula whereas (A −
BD−1 C)−1 is known as the Schur complement.

E.1

Proof and a Corollary

Proof. If (A − BD−1 C)−1 exists, then we may define


X=

(A − BD−1 C)−1

−(A − BD−1 C)−1 BD−1

−D−1 C(A − BD−1 C)−1 D−1 + D−1 C(A − BD−1 C)−1 BD−1





and use matrix multiplication to conclude








 A B 
 A B 
X =X
 = I.


C D
C D
Conversely, we may factor









−1
−1 C 0
I
0 
 A B   I BD
  A − BD


=



−1 C I
C D
0
I
0
D
D
to conclude




A − BD−1 C
0

−1 

−1
−1
0   I BD
I
0 
  A B 
=
 

 ,
−1 C I
D
0
I
C D
D




which is clearly invertible.
60

Corollary 6. In the affirmative case, if A is also invertible, then


−1

 A B 


C D



=

(A − BD−1 C)−1

−(A − BD−1 C)−1 BD−1

−D−1 C(A − BD−1 C)−1



(D − CA−1 B)−1



.



 D C 
Proof. Apply the lemma to 
 and take the (1,1) matrix element.
B A

E.2

Using the Schur Complement Formula

To see the utility of the Schur complement formula, consider the following. Let η > 0 and
suppose L and B are linear maps from a Hilbert space to itself. Suppose further that L
is bounded and self-adjoint and that the numerical range of B satisfies a gap condition:
N (B) ⊆ {0} ∪ {Rez ≥ c} for some c > 0. Let P be the projection onto the kernel of B and
suppose that the range of P ⊥ is invariant under both B and B † . When we write BP ⊥ , it
is understood that we mean the restriction to the range of B, and thus, BP ⊥ is invertible.
This is exactly the scenario that we encounter in section 3.4.4.
We may now identify the operator iL + B + η with its block matrix form:


P (iL + B + η)P ⊥





 P (iL + B + η)P
 Pψ 
(iL + B + η)ψ ∼ 


P ⊥ (iL + B + η)P P ⊥ (iL + B + η)P ⊥
P ⊥ψ



⊥
iP LP
 iP LP + ηP
 Pψ 
= 

.
⊥ LP
⊥ (iL + B + η)P ⊥
⊥ψ
iP
P
P
61

The Schur complement applies here since

Re(P ⊥ (iL + B + η)P ⊥ ) > Re(P ⊥ BP ⊥ ) ≥ c

and P ⊥ (iL + B + η)P ⊥ is invertible. The Schur complement is then:

P

1
P =
iL + B + η

iP LP + ηP − iP LP ⊥

−1
1
iP ⊥ LP
.
iP ⊥ LP ⊥ + P ⊥ BP ⊥ + ηP ⊥

If another appropriate projection is chosen, we may apply the Schur complement a second
time, further reducing the operator in question into its constitute parts. While the resulting
equations are more cumbersome, the difficulty in analyzing the original operator is greater
than the sum of the difficulties of its parts.

62

BIBLIOGRAPHY

63

BIBLIOGRAPHY
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Mathematics. American Mathematical Society, Providence, Rhode Island, first edition,
1998.
[3] Yang Kang and Jeffrey Schenker. Diffusion of wave packets in a markov random potential.
Journal of Statistical Physics, 134:1005–1022, 1005.
[4] Peter D. Lax. Functional Analysis. Pure and Applied Mathematics. John Wiley & Sons,
Inc., eighth edition, 2002.
[5] Klaus-Jochen Engel & Rainer Nagel. One-Parameter Semigroups for Linear Evolution
Equations, volume 194 of Graduate Studies in Mathematics. Springer, 2000.
[6] Claude-Alain Pillet. Some results on the quantum dynamics of a particle in a markovian
potential. Communications in Mathematical Physics, 102(2):237–254, 1985.
[7] Gerald Teschl. Mathematical Methods in Quantum Mechanics: With Applications to
Schr¨dinger Operators, volume 99 of Graduate Studies in Mathematics. American Matho
ematical Society, 2009.
[8] Lloyd Trefethen and David Bau. Numerical Linear Algebra. Society for Industrial and
Applied Mathematics, 1997.
[9] Fuzhen Zhang. The Schur Complement and its Applications, volume 4 of Numerical
Methods and Algorithms. Springer, 2005.

64