# Quantum diffusion via an approximate semigroup property

Felipe Hernández<EMAIL_ADDRESS>
###### Abstract.

In this paper we introduce a new approach to the diffusive limit of the weakly
random Schrodinger equation, first studied by L. Erdos, M. Salmhofer, and H.T.
Yau. Our approach is based on a wavepacket decomposition of the evolution
operator, which allows us to interpret the Duhamel series as an integral over
piecewise linear paths. We relate the geometry of these paths to combinatorial
features of a diagrammatic expansion which allows us to express the error
terms in the expansion as an integral over paths that are exceptional in some
way. These error terms are bounded using geometric arguments. The main term is
then shown to have a semigroup property, which allows us to iteratively
increase the timescale of validity of an effective diffusion. This is the
first derivation of an effective diffusion equation from the random
Schrodinger equation that is valid in dimensions $d\geq 2$.

###### Contents

  1. 1 Introduction
  2. 2 More detailed outline of the proof
  3. 3 A sketch of the derivation of the path integral
  4. 4 The ladder approximation for ${\mathcal{E}}_{\tau}$
  5. 5 Iterating the path integral
  6. 6 Geometry and combinatorics of extended paths
  7. 7 Skeletons and diagrams
  8. 8 Colored operators
  9. 9 Constructing partitions from colorings
  10. 10 The diagrammatic expansion
  11. 11 Bounding the diffusive diagram contributions
  12. 12 Analysis of the ladder superoperator
  13. 13 The path integral
  14. 14 A first operator moment estimate
  15. 15 Interspersing the free evolution
  16. A The Boltzmann limit for short times
  17. B Using graphs, forests, and partitions to compute moments
  18. C Wavepackets and quantization
  19. D Elementary estimates for the linear Boltzmann equation

##  1\. Introduction

###  1.1. The kinetic limit for the Schödinger equation

In this paper, we study the equation

(1.1) $i\partial_{t}\psi=-\frac{1}{2}\Delta\psi+{\varepsilon}V\psi$

with a stationary random potential $V$. An example of a potential we will
consider is a mean- zero Gaussian random field with a smooth and compactly
supported two point correlation function

(1.2) ${\mathbf{E}\,}V(x)V(y)=R(x-y).$

with $R\in C_{c}^{\infty}({\mathbf{R}}^{d})$. Our approach works for more
general potentials that are stationary, have finite range of dependence, and
have bounded moments in $C^{k}$ (for $k>20d$, say).

The equation (1.1) is a simple model for wave propagation in a random
environment. It also has a more direct physical significance, as it models the
motion of a cold electron in a disordered environment [29]. We are interested
in this paper in the regime where the frequency of the initial condition
$\psi$ is comparable to the correlation length of the potential, which we
consider to be of unit scale. This regime is out of reach of both traditional
WKB-type semiclassical approximations, which are more appropriate for high-
frequency $\psi$, and of homogenization techniques, which are appropriate for
low-frequency $\psi$.

This regime was first rigorously studied by H. Spohn in [29] , who showed that
the spectral density
$\mu_{t}(p)={\mathbf{E}\,}|\psi_{t/{\varepsilon}^{2}}(p)|^{2}$ converges, in
the semiclassical limit ${\varepsilon}\to 0$, to a weak solution of a
spatially homogeneous kinetic equation

(1.3)
$\partial_{t}\mu(p)=\int\delta(|p|^{2}-|p^{\prime}|^{2}){\widehat{R}}(p-p^{\prime})[\mu(p^{\prime})-\mu(p)]\mathop{}\\!\mathrm{d}p^{\prime},$

where $R(x)$ is the two-point correlation function defined in (1.2). The term
$\delta(|p|^{2}-|p^{\prime}|^{2})$ enforces conservation of kinetic energy,
which is appropriate in the limit ${\varepsilon}\to 0$ since the potential
energy becomes negligible. The time scale between scattering events is on the
order ${\varepsilon}^{-2}$, which can be heuristically justified by using the
Born approximation of the solution of (1.1). Spohn’s technique for
demonstrating (1.3) was to write out the Duhamel expansion for the solution
$\psi_{t}(p)$ to (1.1) in momentum space, take an expectation of the quantity
$|\psi_{t}(p)|^{2}$ using the Wick rule for the expectation of a product of
Gaussian random variables, and separate terms into a main term and an error
term. The error terms are controlled by additional cancellations and the main
terms are compared to a series expansion for the solution of (1.3). Spohn’s
analysis of the Dyson series allowed him to control the solution up to times
$c{\varepsilon}^{-2}$ for some small constant $c>0$.

This proof technique has been used by many authors since to improve upon our
understanding of (1.1). Most notably, L. Erdös and H.T. Yau in a series of
works [17, 18] were able to improve the time scale to arbitrary kinetic times
of the form of the order $O({\varepsilon}^{-2})$ while also demonstrating the
weak convergence of the Wigner function

${\mathcal{W}_{\psi}}(x,p):=\int e^{iy\cdot
p}\overline{\psi}(x-y/2)\psi(x+y/2)\mathop{}\\!\mathrm{d}y$

to the solution of the linear Boltzmann equation

(1.4)
$\partial_{t}\rho+p\cdot\nabla_{x}\rho={\varepsilon}^{2}\int\delta(|p|^{2}-|p^{\prime}|^{2}){\widehat{R}}(p-p^{\prime})[\rho(x,p^{\prime})-\rho(x,p)]\mathop{}\\!\mathrm{d}p^{\prime}.$

Introducing the rescaled coordinates $T={\varepsilon}^{2}t$,
$X={\varepsilon}^{2}x$ along with the rescaled solution

$\rho^{\varepsilon}_{T}(X,p):=\rho_{{\varepsilon}^{-2}T}({\varepsilon}^{-2}x,p),$

the equation (1.4) can be written

$\partial_{T}\rho^{\varepsilon}+p\cdot\nabla_{X}\rho=\int\delta(|p|^{2}-|p^{\prime}|^{2}){\widehat{R}}(p-p^{\prime})[\rho^{\varepsilon}(X,p^{\prime})-\rho^{\varepsilon}(X,p)]\mathop{}\\!\mathrm{d}p^{\prime}.$

In an impressive sequence of refinements to this work, L. Erdos, M. Salmhofer
and H.T. Yau [14, 16, 15] were able to improve the timescale even further to
diffusive times ${\varepsilon}^{-2-\kappa}$ for some positive $\kappa>0$ (in
fact, one can take $\kappa=1/370$ when $d=3$). At this timescale a diffusion
equation emerges. The principle is that the momentum variable is no longer
relevant to the evolution because it becomes uniformly distributed over the
sphere within time $O({\varepsilon}^{-2})$, and all that remains of the
momentum information is the kinetic energy variable $e=|p|^{2}/2$. Moreover,
for diffusive times ${\varepsilon}^{-2-\kappa}$ the particle travels a
distance ${\varepsilon}^{-2-\kappa/2}$ so the diffusive length scale is
${\varepsilon}^{-2-\kappa/2}$.

For solutions $\rho$ of the linear Boltzmann equation (1.4), the particle
distribution $f$ defined by

(1.5)
$f_{T}(X,e)=\int_{|p|^{2}/2=e}\rho_{{\varepsilon}^{-2-\kappa}T}({\varepsilon}^{-2-\kappa/2}X,p)\mathop{}\\!\mathrm{d}{\mathcal{H}}^{n-1}(p)$

converges in the limit ${\varepsilon}\to 0$ to a solution of the diffusion
equation

(1.6) $\partial_{T}f_{T}=D_{e}\Delta_{X}f_{T},$

where $D_{e}$ is a diffusion coefficient depending on the energy $e$. See [16]
for more details on the limiting diffusion equation.

To reach the diffusive time scale in which the particle experiences infinitely
many scattering events, one must consider terms with ${\varepsilon}^{-c}$
collisions, which produces more than ${\varepsilon}^{-{\varepsilon}^{-c}}$
diagrams when one applies the Wick expansion. To deal with the explosion in
the number of terms, Erdos, Salmhofer, and Yau developed a resummation
technique to more accurately estimate the sizes of the terms and additionally
had to exploit intricate cancellations coming from the combinatorial features
of the diagrams considered.

###  1.2. Statement of the main result

In this paper, we provide an alternative derivation of the linear Boltzmann
equation which is also valid up to diffusive times but with a fundamentally
different approach. In our proof, we use a wavepacket decomposition of the
evolution operator. The wavepacket decomposition allows us to keep information
about the position and momentum of the particle simultaneously (up to the
limits imposed by the uncertainty principle), and we therefore express the
solution as an integral over piecewise linear paths in phase space.

To make the connection between operators and the linear Boltzmann equation, we
use the Weyl quantization $a\in
C^{\infty}({{\mathbf{R}}^{2d}})\mapsto\operatorname{Op}^{w}(a)\in\mathcal{B}(L^{2}({\mathbf{R}}^{d}))$
defined by

$\operatorname{Op}^{w}(a)f(x)=\int e^{i(x-y)\cdot
p}a((x+y)/2,p)f(y)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}p.$

The relationship between the Weyl quantization and the Wigner transform is
given by the identity

$\langle\operatorname{Op}^{w}(a)\psi,\psi\rangle=\int
a(x,p){\mathcal{W}_{\psi}}(x,p)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}p.$

In particular, applying this identity to the solution $\psi_{t}=e^{-itH}\psi$
to (1.1) where $H$ is the random Hamiltonian

$H=-\frac{1}{2}\Delta+{\varepsilon}V,$

we have

$\int
a(x,p){\mathbf{E}\,}{\mathcal{W}_{\psi_{t}}}(x,p)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}p={\mathbf{E}\,}\langle\operatorname{Op}^{w}(a)e^{-itH}\psi,e^{-itH}\psi\rangle=\langle{\mathbf{E}\,}e^{itH}\operatorname{Op}^{w}(a)e^{-itH}\psi,\psi\rangle.$

Therefore in order to answer questions about the weak convergence of
${\mathcal{W}_{\psi_{t}}}$, it suffices to study the quantum evolution channel

(1.7) ${\mathcal{E}}_{t}[A]:={\mathbf{E}\,}e^{itH}Ae^{-itH}$

applied to operators of the form $A=\operatorname{Op}^{w}(a)$ with
sufficiently regular symbols $a$. In particular, we will show that for
suitable observables $a_{0}$ and for times $t\leq{\varepsilon}^{-2-\kappa}$,
we have

$\|{\mathcal{E}}_{t}[\operatorname{Op}^{w}(a_{0})]-\operatorname{Op}^{w}(a_{t})\|_{op}=o(1),$

where $a_{t}$ solves the dual of the linear Boltzmann equation (1.4),

(1.8)
$\partial_{t}a-p\cdot\nabla_{x}a={\varepsilon}^{2}\int\delta(|p|^{2}-|p^{\prime}|^{2}){\widehat{R}}(p-p^{\prime})[a(x,p^{\prime})-a(x,p)]\mathop{}\\!\mathrm{d}p^{\prime}.$

A natural norm that we use on $a$ which also controls the operator norm of
$\operatorname{Op}^{w}(a)$ is the $C^{k}$ norm with $k=2d+1$ (see Appendix C
for a self-contained proof that the $C^{2d+1}$ norm of $a$ controls the
operator norm of $\operatorname{Op}^{w}(a)$). We will use a $C^{k}$ norm which
is rescaled to the appropriate length scales of the problem. Because the time
scale between scattering events is ${\varepsilon}^{-2}$, a natural spatial
length scale is ${\varepsilon}^{-1}$. For the rest of the paper we write
$r={\varepsilon}^{-1}$ for this length scale. This is the length scale of a
wavepacket that remains coherent between scattering events. Conversely the
natural length scale in momentum is ${\varepsilon}=r^{-1}$. This “microscopic”
scale is the one we use for the wavepacket decomposition of the operator
$e^{itH}$.

On the other hand, a natural “macroscopic” length scale of the problem is
${\varepsilon}^{-2}$, which is the distance that a particle with momentum
$O(1)$ travels between scatterings. A natural “macroscopic” length scale in
momentum is $O(1)$, which is the impulse applied to a particle in a typical
scattering event.

The following norm measures the smoothness of an observable at these length
scales:

$\|a\|_{C^{k}_{r,L}}:=\sum_{|\alpha_{x}|+|\alpha_{p}|\leq
k}\sup_{(x,p)}|(rL\partial_{x})^{\alpha_{x}}(r^{-1}L\partial_{p})^{\alpha_{p}}a(x,p)|.$

When $L=1$, this norm probes the microscopic smoothness of observables,
whereas when $L={\varepsilon}^{-1}$, the norm probes the macroscopic
smoothness.

We make one final comment before we state the main result of the paper, which
is that we will not treat the evolution of low-frequency modes. In dimension
$d=2$, the scattering cross section of a low frequency wave with momentum
$|p|\ll 1$ is still on the order ${\varepsilon}^{2}$ but the speed of travel
is only $|p|$, so the distance between typical scattering events is only
$|p|{\varepsilon}^{-2}$ rather than ${\varepsilon}^{-2}$. Because scattering
events are more closely spaced, the bounds coming from the geometric arguments
we use deteriorate and we make no attempt to understand what happens in this
regime. In higher dimensions the scattering cross section also shrinks with
momentum so that one could in principle first approximate the evolution of low
frequency modes by a free evolution with no potential and therefore recover
the result for all frequencies. We do not make this argument in this paper.

######  Theorem 1.1.

For each $d\geq 2$, there exists $\theta=\theta(d)>0$ and $\kappa=\kappa(d)>0$
such that the following holds. Let $V$ be an admissible potential as described
in Definition (B.2), and let $a_{0}\in C^{2d+1}({{\mathbf{R}}^{2d}})$ be a
classical observable supported away from zero momentum;

$\operatorname{supp}a_{0}\subset\\{(x,p)\in{{\mathbf{R}}^{2d}}\mid|p|\geq{\varepsilon}^{\theta(d)}\\}.$

Suppose moreover that $a_{t}$ solves (1.8) with initial condition $a_{0}$.
Then

(1.9)
$\|{\mathcal{E}}_{t}[\operatorname{Op}^{w}(a_{0})]-\operatorname{Op}^{w}(a_{t})\|_{op}\leq
C_{d}{\varepsilon}^{2+\kappa}t\|a_{0}\|_{C^{2d+1}_{{\varepsilon}^{-1},{\varepsilon}^{-0.5}}}.$

In particular, for arbitrary $\psi_{0}\in L^{2}({\mathbf{R}}^{d})$ and
$\psi_{t}$ solving (1.1) it follows that

(1.10)
$\int{\mathcal{W}_{\psi_{t}}}(x,p)a_{0}(x,p)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}p=\int{\mathcal{W}_{\psi_{0}}}(x,p)a_{t}(x,p)\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}p+O({\varepsilon}^{2+\kappa}t\|a_{0}\|_{C^{2d+1}_{{\varepsilon}^{-1},{\varepsilon}^{-0.5}}}\|\psi\|_{L^{2}}^{2}).$

To see how the diffusion equation (1.6) emerges as a scaling limit, we
consider observables of the form

$a_{0}(x,p)=\bar{a}({\varepsilon}^{2+\kappa/2}x,p)$

with $\bar{a}\in C^{2d+1}$. With this rescaling, we have

$\|a_{0}\|_{C^{2d+1}_{{\varepsilon}^{-1},{\varepsilon}^{-1}}}\leq\|\bar{a}\|_{C^{2d+1}}.$

In particular,
$\|a_{0}\|_{C^{2d+1}_{{\varepsilon}^{-1},{\varepsilon}^{-0.5}}}$ is bounded
uniformly in ${\varepsilon}$. Moreover, the solution $a_{t}$ solves (1.6).

One major difference between Theorem 1.1 and the main results of [16, 15],
apart from the very different approaches to the proof, is that our result
holds in dimension $d=2$. At first this may appear to be in contradiction with
the conjectured phenomenon of Anderson localization in $d=2$, but the
contradiction disappears when one compares the timescale
${\varepsilon}^{-2-\kappa}$ considered in this paper to the expected length
scale $e^{{\varepsilon}^{-2}}$ of localization in this dimension. Indeed, it
is expected that the particle exhibits diffusive behavior for an exponentially
long time before getting trapped by localization.

The exponent $\kappa(d)$ can in principle be extracted from the proof. However
in this paper we focus on demonstrating the new technique in its simplest form
and therefore do not attempt to optimize $\kappa(d)$. Perhaps with some
optimization one could obtain $\kappa(3)$ comparable to $1/370$, but the proof
we give yields a bound of the order $\kappa(3)\sim 10^{-6}$.

###  1.3. A heuristic sketch of the argument

####  1.3.1. The phase space path integral

The main idea behind the proof of Theorem 1.1 is to focus on justifying an
approximate semigroup property

(1.11) ${\mathcal{E}}_{2t}[A]\approx{\mathcal{E}}_{t}[{\mathcal{E}}_{t}[A]],$

for suitable operators $A$ including operators of the form
$A=\operatorname{Op}^{w}(a)$. Observe that the approximation (1.11) has the
following physical interpretation. Let

$\displaystyle H_{1}$ $\displaystyle=-\frac{1}{2}\Delta+{\varepsilon}V_{1}$
$\displaystyle H_{2}$ $\displaystyle=-\frac{1}{2}\Delta+{\varepsilon}V_{2}.$

be Hamiltonians with two independently sampled potentials, and observe that

${\mathcal{E}}_{t}\circ{\mathcal{E}}_{t}[A]={\mathbf{E}\,}e^{itH_{2}}e^{itH_{1}}Ae^{-itH_{1}}e^{-itH_{2}}.$

In other words, ${\mathcal{E}}_{t}\circ{\mathcal{E}}_{t}$ represents an
evolution with a potential that abruptly changes into an independently sampled
potential at time $t$. Although such a resampling of the potential drastically
changes the evolution of the wavefunction $\psi_{t}$ itself, we will see that
the effect on observables is minimal

To prove the approximate semigroup property we approximate the evolution
operator $e^{-itH}$ as an integral over piecewise linear paths in phase space,
representing the possible paths of a particle as it scatters. To decompose
phase space we use a family of wavepackets of the form

$\phi_{x,p}(y):=r^{-d/2}e^{iy\cdot p}\chi_{env}((x-y)/r),$

where $\chi_{env}\in C_{c}^{\infty}({\mathbf{R}}^{d})$ is a fixed envelope
normalized in $L^{2}$ and satisfying some additional conditions described in
Appendix C. The functions $\phi_{x,p}$ are localized in space to scale $r$ and
in momentum to scale $r^{-1}$. We use the notation $\xi=(x,p)$ and write
$\mathinner{|{\xi}\rangle}$ as a shorthand for the function $\phi_{\xi}$.

The use of a phase-space path integral already represents a departure from
previous approaches to the problem. Indeed, since the paper of Spohn [29] it
has been customary to write the terms of the Duhamel series expansion for
$e^{itH}$ in the Fourier basis. We will see that by using the spatial
localization of the particle we can more easily compute the expectation
appearing in the integrand without the need for a full Wick expansion (or
cumulant decomposition in the case of a non-Gaussian potential).

The free evolution of a wavepacket approximates the motion of a free classical
particle in the sense that

$e^{-it\Delta/2}\mathinner{|{(x,p)}\rangle}\approx
e^{it|p|^{2}/2}\mathinner{|{(x+tp,p)}\rangle}$

for $t\ll r^{2}$. For multiplication against the potential we use the identity

(1.12)
$V\mathinner{|{(x,p)}\rangle}=\int{\widehat{V_{x}}}(p^{\prime}-p)\mathinner{|{(x,p^{\prime})}\rangle}\mathop{}\\!\mathrm{d}p^{\prime},$

where $V_{x}$ is the potential $V$ multiplied by a cutoff near $y$ which is
$1$ in a ball large enough to contain the support of $\chi_{env}$,

$V_{x}(y)=b((y-x)/r)V(y).$

We can use these two identities to write an expansion of the evolution of a
wavepacket $e^{itH}\mathinner{|{(x,p)}\rangle}$ as an integral over paths in
which the phase space point travels in straight lines with an occasional
impulse from the potential causing discontinuities in the momentum variable.
We represent these piecewise linear paths as a tuple
$\omega=({\mathbf{s}},{\mathbf{p}},{\mathbf{y}})$ with
${\mathbf{s}}=(s_{0},\cdots,s_{k})$ being the sequence of times between the
scattering events (satisfying $\sum s_{j}=t$ and $s_{j}\geq 0$),
${\mathbf{p}}=(p_{0},p_{1}\cdots,p_{k})$ being the sequence of momentum
variables which we require to have the same magnitude
$|p_{j}|=|p_{j^{\prime}}|$ and with initial momentum $p_{0}=p$, and
${\mathbf{y}}=(y_{1},\cdots,y_{k})$ being the sequence of scattering locations
defined by

$\displaystyle y_{1}$ $\displaystyle=x+s_{0}p$ $\displaystyle y_{j+1}$
$\displaystyle=y_{j}+s_{j}p_{j}.$

An example of a path is depicted in Figure 1.

Figure 1. A sample scattering path of a particle with $2$ collisions. The
displacement between consecutive collisions is given by $s_{j}p_{j}$, where
$p_{j}\in{\mathbf{R}}^{d}$ is a momentum vector with constrained kinetic
energy $|p_{j}|^{2}/2$, and $s_{j}\geq 0$ is the time between collisions.

Each such path $\omega$ defines an operator $O_{\omega}$ which approximately
acts on wavepackets by

(1.13)
$O_{\omega}\mathinner{|{(x,p)}\rangle}=e^{i\varphi(\omega)}\prod_{j=1}^{k}{\widehat{V_{y_{j}}}}(p_{j}-p_{j-1})\mathinner{|{y_{k}+s_{k}p_{k},p_{k}}\rangle},$

where $\varphi(\omega)$ is a deterministic phase accumulated from the
stretches of free evolution. These phases do not matter for this sketch of the
proof. However, in the actual proof we use stationary phase to ensure that the
geometric constraints $y_{j+1}=y_{j}+s_{j}p_{j}$ are approximately satisfied
and to show that kinetic energy is approximately conserved. Then, at least
formally, we can write out the path integral for the evolution of a wavepacket
$\mathinner{|{\xi}\rangle}$ as

(1.14) $e^{itH}\mathinner{|{\xi}\rangle}=\int
O_{\omega}\mathinner{|{\xi}\rangle}\mathop{}\\!\mathrm{d}\omega.$

We apply this decomposition of the evolution to investigate the approximate
semigroup property for operators of the form

$A=\int_{{{\mathbf{R}}^{2d}}}a(\xi)\mathinner{|{\xi}\rangle}\mathinner{\langle{\xi}|}\mathop{}\\!\mathrm{d}\xi,$

which are local in phase space in the sense that

$A\mathinner{|{\xi}\rangle}\approx a(\xi)\mathinner{|{\xi}\rangle}.$

Using the path integral (1.14) in the definition of ${\mathcal{E}}_{t}[A]$ we
obtain

${\mathcal{E}}_{t}[A]=\int_{{{\mathbf{R}}^{2d}}}\iint{\mathbf{E}\,}O_{\omega^{\prime}}^{*}\mathinner{|{\xi}\rangle}\mathinner{\langle{\xi}|}O_{\omega}\mathop{}\\!\mathrm{d}\omega\mathop{}\\!\mathrm{d}\omega^{\prime}\mathop{}\\!\mathrm{d}\xi.$

What is important for this sketch of the proof is to simply investigate which
pairs of paths $\omega$ and $\omega^{\prime}$ have

${\mathbf{E}\,}O_{\omega^{\prime}}^{*}\mathinner{|{\xi}\rangle}\mathinner{\langle{\xi}|}O_{\omega}\not=0.$

In particular, we are interested in understanding for which sequence of
positions $y_{j},y^{\prime}_{j}$ and impulses $q_{j},q^{\prime}_{j}$ we have

${\mathbf{E}\,}\prod_{j=1}^{k}{\widehat{V_{y_{j}}}}(q_{j})\prod_{j^{\prime}=1}^{k^{\prime}}{\widehat{V_{y^{\prime}_{j^{\prime}}}}}^{*}(q^{\prime}_{j^{\prime}})\not=0.$

Using the fact that $V$ is real and therefore
${\widehat{V_{y}}}^{*}(q)={\widehat{V_{y}}}(-q)$, we can rewrite the
expectation above as

${\mathbf{E}\,}\prod_{b\in[k]\sqcup[k^{\prime}]}{\widehat{V_{y_{b}}}}(q_{b}),$

where $[k]\sqcup[k^{\prime}]$ is shorthand for the doubled index set
$[k]\times\\{0\\}\cup[k^{\prime}]\times\\{1\\}$ and the $q_{b}$ impulses are
reversed for $b=(j,1)$, so in particular
$q_{j,1}=-(p^{\prime}_{j}-p^{\prime}_{j-1})$. Because $V_{y}$ are localized,
the expectation above splits along a partition
$P({\mathbf{y}},{\mathbf{y}}^{\prime})\in{\mathcal{P}}([k]\sqcup[k^{\prime}])$
defined by the clusters of the collision locations $y_{b}$ (so that
$b\sim_{P}b^{\prime}$ when $|y_{b}-y_{b^{\prime}}|{\,\lesssim\,}r$). That is,
we have an identity of the form

${\mathbf{E}\,}\prod_{b\in[k]\sqcup[k^{\prime}]}{\widehat{V_{y_{b}}}}(q_{b})=\prod_{S\in
P({\mathbf{y}},{\mathbf{y}}^{\prime})}{\mathbf{E}\,}\prod_{b\in
S}{\widehat{V_{y_{b}}}}(q_{b}).$

Within each cluster the expectations is zero unless the sum of the impulses is
zero. This “conservation of momentum” condition is a consequence of the
stationarity of the potential and is made rigorous in Lemma B.3. In the case
of Gaussian potentials, this is a consequence of the Wick formula and the
identity

${\mathbf{E}\,}{\widehat{V}}(p){\widehat{V}}(q)={\widehat{R}}(p)\delta(p+q).$

We are led to the following purely geometric constraints on the pair of paths
$\omega,\omega^{\prime}$.

######  Definition 1.2.

Two paths $\omega,\omega^{\prime}$ are said to be _compatible_ if the
partition $P({\mathbf{y}},{\mathbf{y^{\prime}}})$ has no singletons and

$\sum_{b\in S}q_{b}=0$

for every $S\in P({\mathbf{y}},{\mathbf{y^{\prime}}})$.

####  1.3.2. Geometric characterization of the error term

This geometric notion of compatible paths is perhaps the most significant idea
in the proof. Indeed, the point is that we can usefully manipulate the path
integral before computing an expectation. In other words, we will first
decompose the path integral according to geometry and then allow the geometry
to dictate the combinatorics of the diagrams.

Now, we take a step back to appreciate which paths contribute to the error
term in the semigroup property. As the discussion above indicates, the
semigroup property compares the evolution ${\mathcal{E}}_{t}$ with a fixed
potential $V$ to the evolution ${\mathcal{E}}_{t/2}\circ{\mathcal{E}}_{t/2}$
during which the potential is refreshed from $V_{1}$ to $V_{2}$ at time $t/2$.
To keep track of which potential a scattering event sees we introduce the
collision times $t_{b}$, defined by $t_{0}=0$ and $t_{b+1}=t_{b}+s_{b}$. The
following condition suffices to ensure that the pair of paths
$(\omega,\omega^{\prime})$ has an expected amplitude that is unaffected by the
possibility of a time-dependent potential.

######  Definition 1.3.

Two paths $\omega,\omega^{\prime}$ are _time-consistent_ if
$t_{b}=t_{b^{\prime}}$ for all pairs of indices
$b,b^{\prime}\in[k]\sqcup[k^{\prime}]$ such that $y_{b}=y_{b^{\prime}}$. Pairs
$(\omega,\omega^{\prime})$ that are not time-consistent are said to be _time-
inconsistent_.

Observe that $\omega$ is compatible with itself and is also time-consistent.
We will show that in fact the paths $(\omega,\omega^{\prime})$ in which
$\omega=\omega^{\prime}$ or is otherwise a small perturbation of $\omega$ form
the bulk of the contribution to ${\mathcal{E}}_{t}$.

To understand the error term, we characterize pairs of paths
$(\omega,\omega^{\prime})$ which are compatible but time-inconsistent. A
simple way that a pair $(\omega,\omega^{\prime})$ could be time-inconsistent
is if either $\omega$ or $\omega^{\prime}$ have a recollision. A simple
example of a pair of compatible paths which are time-inconsistent due to a
recollision is depicted in Figure 2

Figure 2. An example of a pair of paths with a recollision. The paths $\omega$
and $\omega^{\prime}$ are depicted at the top left and top right, and the
collisions are colored according to the cluster in the partition
$P(\omega,\omega^{\prime})$. On the bottom, an abstract depiction of the
partition $P(\omega,\omega^{\prime})$. Notice that there is a red cluster with
$4$ collisions, two from $\omega$ and two from $\omega^{\prime}$.

There is another geometric feature which we call a “tube event” which occurs
when three collisions are collinear (in general, when they lie on narrow
tube). Tube events can also lead to time inconsistencies, as depicted in
Figure 3

Figure 3. An example of a tube event. Note that there are three collisions
lying on a line, but neither $\omega$ nor $\omega^{\prime}$ forms a
recollision. Nonetheless, the collisions are time-inconsistent because the
second collision of $\omega$ coincides with the fourth collision of
$\omega^{\prime}$, as shown in the diagram of $P(\omega,\omega^{\prime})$
below.

We note that the estimation of the contribution of these special geometric
features replaces the need for crossing estimates such as the ones studied in
[23]. Our diagrams are bounded using relatively simple-minded volumetric
considerations (essentially, after taking care of deterministic cancellations
in the integral, we use the triangle inequality and account for the
contribution of each degree of freedom). This simple-minded approach works
particularly well for subkinetic timescales
$\tau{\,\lesssim\,}{\varepsilon}^{-2+\kappa/2}$ in which one only needs
$k_{max}=O(\kappa^{-1})$ collisions in the series expansion to approximate
$e^{i\tau H}$ and therefore all combinatorial factors are bounded by a (very
large) absolute constant.

The general strategy of the proof therefore is as follows:

  1. (1)

Classify the geometric behaviors that can lead to time-inconsistencies.

  2. (2)

Partition the path integral into paths with bad behaviors and paths without
bad behaviors.

  3. (3)

Use geometric estimates to bound the operator norm of the contribution of the
bad paths.

The main new feature of this proof strategy is that the path integral is
partitioned before the expectation is computed. That is, we do not decompose
the expectation until we already have some information about the partition
$P(\omega,\omega^{\prime})$. This is in contrast to the traditional approach
used in [29, 16, 15, 18] which is summarized below.

  1. (1)

Expand the expectation using the Wick rule or a cumulant expansion.

  2. (2)

Partition the diagrams according to complexity by a combinatorial criterion.

  3. (3)

Use oscillatory integral estimates to bound the contributions of the bad
diagrams.

####  1.3.3. Reaching the diffusive timescale

To reach the diffusive timescale we prove a semigroup property of the form

${\mathcal{E}}_{N\tau}\approx{\mathcal{E}}_{\tau}^{N}$

where $\tau={\varepsilon}^{-2+\kappa/2}$ and $N={\varepsilon}^{-\kappa}$. The
challenge we face in trying to understand the evolution operator $e^{itH}$ for
times $t\sim{\varepsilon}^{-2-\delta}$ is that one needs to resolve at least
${\varepsilon}^{-\delta}$ collisions. This requires a path integral in a space
of dimension ${\varepsilon}^{-\delta}$. If we then try to use crude estimates
to bound the contribution of the terms in the Duhamel expansion we may lose a
factor of $C^{{\varepsilon}^{-\delta}}$. What we need to do is take into
account cancellations that occur between the terms in the Duhamel expansion.
In [16, 15] this is done by renormalizing the propagator. This is equivalent
to viewing $H=-\Delta/2+{\varepsilon}V$ not as a perturbation of the free
evolution $-\Delta/2$ but as a perturbation of
$-\Delta/2+{\varepsilon}^{2}\Theta$ where $\Theta$ is a multiplier operator
that takes into account the effect of immediate recollisions. The multiplier
$\Theta$ has a nonzero imaginary part so that $e^{-i\Delta/2+i\Theta}$ decays
exponentially in time. This exponential decay exactly matches the exponential
growth in the volume of the path inetgral. The value of $\Theta$ is also
chosen so that a precise cancellation occurs in diagrams with immediate
recollisions.

We take an alternative approach to resummation. The idea is that we first
write

$e^{itH}=e^{iN\tau H}=e^{i\tau H}\cdots e^{i\tau H},$

where $N\sim{\varepsilon}^{-\kappa}$ and
$\tau\sim{\varepsilon}^{-2+\kappa/2}$. Each of the terms $e^{i\tau H}$ is
expanded as a Duhamel series of $k_{max}$ terms. We then partition the
resulting path integral into pieces depending on geometric features of the
paths and decompose the expectation using this geometric information. When
this is done we resum the terms in the Duhamel series corresponding to
segments that do not have any geometrically special collisions. This can be
intepreted as a way of writing the evolution channel ${\mathcal{E}}_{N\tau}$
as a perturbation of the refreshed evolution channel
${\mathcal{E}}_{\tau}^{N}$. This seems to be a more general strategy for
deriving kinetic limits – the resummation procedure is dictated by the desired
semigroup structure.

Another important point of comparison concerns the diagrammatic expansion we
derive to reach the diffusive time scale. In both this paper and in [15, 16]
one expands the solution as a sum over diagrams which are stratified in some
way by combinatorial complexity. In [15, 16] the more complex diagrams contain
more opportunities to find decay via crossing estimates, which are nontrivial
bounds on oscillatory integrals. In this paper, we first split the path
integral itself according to geometric complexity and then bound the
combinatorial complexity of the diagrams associated to paths of a fixed
geometric complexity. The difference between these approaches is summarized in
Figure 4.

Geometric complexity of paths |  | Combinatorial complexity of diagrams
---|---|---
| 1 | 2 | 3 | 4
1 | ${\varepsilon}^{c}N^{C}$ | - | - | -
2 | ${\varepsilon}^{2c}N^{C}$ | ${\varepsilon}^{2c}N^{2C}$ | - | -
3 | ${\varepsilon}^{3c}N^{C}$ | ${\varepsilon}^{3c}N^{2C}$ | ${\varepsilon}^{3c}N^{3C}$ | -
4 | ${\varepsilon}^{4c}N^{C}$ | ${\varepsilon}^{4c}N^{2C}$ | ${\varepsilon}^{4c}N^{3C}$ | ${\varepsilon}^{4c}N^{4C}$
Figure 4. A cartoon of the contribution of various diagrams. Diagrams have a
combinatorial complexity, and there are at most $N^{Ck}$ diagrams having
complexity exactly $k$. Moreover paths have a geometric complexity, and the
volume of the paths with geometric complexity $k$ is ${\varepsilon}^{ck}$. The
approach taken in [15, 16] is to sort diagrams by combinatorial complexity and
then show that the only contributions to diagrams with high combinatorial
complexity also have high geometric complexity. In this paper, we first sort
the paths by geometric complexity and show that paths with low geometric
complexity only contribute to diagrams with low combinatorial complexity. In
summary, we sum along the rows of this table whereas previous works proceed by
summing over the columns. Note that as long as
$N\ll{\varepsilon}^{-c^{\prime}}$ the sum of the contributions is small.

####  1.3.4. More explanation of the diagrammatic expansion

To reach subkinetic times we used the following crude idea to verify the
approximate semigroup property: either the pair of paths
$(\omega,\omega^{\prime})$ has a nontrivial geometric event, or it does not.
If there is a nontrivial geometric event, we use the triangle inequality
inside the path integral and the geometric information about the event to pick
up a factor of ${\varepsilon}^{c}$, which is small enough to suppress the
large constant appearing from two inefficiencies in our argument. The first
inefficiency is to fail to take into account precise cancellations in the path
integral, which costs us a factor of $C^{k}$ where $k$ is the number of
collisions. The second inefficiency is the failure to take into account the
combinatorial constraints imposed on the collision partition. The constraints
come from “negative information” about the path – as an oversimplification, if
a collision index $b$ is not part of a tube event or a recollision event, then
it must form part of a ladder or anti-ladder. By failing to take into account
this information, we bound the number of partitions we must sum over by a
large combinatorial factor $(Ck_{max})^{k_{max}}$ rather than a factor that
depends on the precise geometric constraints on the path.

To reach diffusive times we must make our bounds more efficient on both
fronts. To perform our resummation, we introduce in Section 5 the notion of an
“extended path”, which is a path formed from $N$ segments each describing the
evolution of the particle on an interval of length $\tau$. An extended path is
a sequence of path segments with phase space points in between consecutive
segments,

$\Gamma=(\xi_{0},\omega_{1},\xi_{1},\xi_{2},\omega_{2},\xi_{3},\cdots,\xi_{2\ell-2},\omega_{\ell},\xi_{2\ell-1},\cdots,\xi_{2N-2},\omega_{N},\xi_{2N-1}).$

An example of an extended path is drawn in Figure 5.

Figure 5. A depiction of an extended path $\Gamma$. The segment $\omega_{j}$
describes a piecewise linear path between the endpoints $\xi_{2j-2}$ and
$\xi_{2j-1}$. The $\xi_{j}$ variables are phase space pairs $(x,p)$ describing
the position and momentum of the particle at the boundary of the path
segments.

Given an extended path we define an operator $O_{\Gamma}$ by

$O_{\Gamma}=\mathinner{|{\xi_{0}}\rangle}\mathinner{\langle{\xi_{2N-1}}|}\prod_{\ell=1}^{N}\mathinner{\langle{\xi_{2\ell-2}|O_{\omega_{\ell}}|\xi_{2\ell-1}}\rangle},$

so that including the sum over all possible collision numbers $0\leq k_{j}\leq
k_{max}$ of each segment in the integral, we have

$e^{iN\tau H}\approx\int O_{\Gamma}\mathop{}\\!\mathrm{d}\Gamma,$

where there is an error term in the approximation that is described in Section
3.

To write down the evolution channel ${\mathcal{E}}_{N\tau}$ we therefore
arrive at an integral of the form

(1.15)
${\mathcal{E}}_{N\tau}[A]\approx\int{\mathbf{E}\,}O_{\Gamma^{+}}^{*}AO_{\Gamma^{-}}\mathop{}\\!\mathrm{d}\Gamma^{+}\mathop{}\\!\mathrm{d}\Gamma^{-}.$

Before we take the expectation, we will split up the pairs of paths
$(\Gamma^{+},\Gamma^{-})$ according to their geometric properties. The key
result we will need is a description of the structure of the correlation
partition of the paths in terms of the geometric features. This is done in
Section 6, which characterizes the allowed partitions using ladders and anti-
ladders. Here we simply provide a quick sketch. An example of a ladder
partition on the disjoint union $[k]\sqcup[k]=[k]\times\\{+,-\\}$ is

$P_{lad}=\\{(j,+),(j,-)\\}_{j\in[k]}.$

An example of an anti-ladder partition on $[k]\sqcup[k]$ is the partition

$P_{anti}=\\{(j,+),(k+1-j,-)\\}_{j\in[k]}.$

The ladder and anti-ladder partitions are drawn in Figure 6

Figure 6. An example of a ladder and an anti-ladder partition with five rungs.

The main result of Section 6 is Lemma 6.16, which states that collisions that
are not part of a geometric feature (so-called “typical collisions”) form part
of either a ladder or an anti-ladder structure in the collision partition of
$(\Gamma^{+},\Gamma^{-})$. Figure 7 illustrates the main result in a special
case.

Figure 7. An example illustrating Lemma 6.16. On the top, a path that has a
single recollision event. At the bottom, an example of a collision partition
compatible with this single recollision event. Note that every collision that
is not part of the recollision is either part of a ladder or an anti-ladder.

The next step is to partition the path integral according to the geometric
information of the paths, which we encapsulate in a structure that we call a
“skeleton” ${\mathcal{F}}(\Gamma^{+},\Gamma^{-})$. Given a skeleton
${\mathcal{F}}$, Lemma 6.16 allows us to construct a set of partitions
${\mathcal{Q}}_{\mathcal{F}}$ such that for any pair of paths
$(\Gamma^{+},\Gamma^{-})$ with
${\mathcal{F}}(\Gamma^{+},\Gamma^{-})={\mathcal{F}}$, the collision partition
$P(\Gamma^{+},\Gamma^{-})\in{\mathcal{Q}}_{\mathcal{F}}$. In fact, we have the
stronger statement that for such pairs of paths,

(1.16)
${\mathbf{E}\,}O_{\Gamma^{+}}^{*}AO_{\Gamma^{-}}=\sum_{P\in{\mathcal{Q}}({\mathcal{F}})}{\mathbf{E}\,}_{P}O_{\Gamma^{+}}^{*}AO_{\Gamma^{-}},$

where the sum is over partitions of the collision indices of
$(\Gamma^{+},\Gamma^{-})$, and ${\mathbf{E}\,}_{P}$ is shorthand for a
splitting of the expectation along the partition $P$.

Writing ${\mathbf{1}}_{{\mathcal{F}}}(\Gamma^{+},\Gamma^{-})$ for the
indicator function that ${\mathcal{F}}(\Gamma^{+},\Gamma^{-})={\mathcal{F}}$,
we then attain the following decomposition for the path integral:

$\displaystyle{\mathcal{E}}_{N\tau}[A]$
$\displaystyle=\int{\mathbf{E}\,}O_{\Gamma^{+}}^{*}AO_{\Gamma^{-}}\mathop{}\\!\mathrm{d}\Gamma^{+}\mathop{}\\!\mathrm{d}\Gamma^{-}$
$\displaystyle=\sum_{{\mathcal{F}}}\int{\mathbf{1}}_{{\mathcal{F}}}(\Gamma^{+},\Gamma^{-}){\mathbf{E}\,}O_{\Gamma^{+}}^{*}AO_{\Gamma^{-}}\mathop{}\\!\mathrm{d}\Gamma^{+}\mathop{}\\!\mathrm{d}\Gamma^{-}$
$\displaystyle=\sum_{{\mathcal{F}}}\sum_{P\in{\mathcal{Q}}_{\mathcal{F}}}\int{\mathbf{1}}_{{\mathcal{F}}}(\Gamma^{+},\Gamma^{-}){\mathbf{E}\,}_{P}(O_{\Gamma^{+}})^{*}AO_{\Gamma^{-}}\mathop{}\\!\mathrm{d}\Gamma^{+}\mathop{}\\!\mathrm{d}\Gamma^{-}.$

The benefit of decomposing the path integral in this way is that the
expectation ${\mathbf{E}\,}_{P}$ splits in a known way. On the other hand
there is now the challenge of dealing with the indicator function
${\mathbf{1}}_{\mathcal{F}}$. The reason this indicator functions causes a
problem is not the discontinuity (this could be solved by using a smoother
partition of unity) but rather the global nature of the constraints. In
particular, ${\mathbf{1}}_{{\mathcal{F}}}(\Gamma^{+},\Gamma^{-})$ includes a
product of indicator functions for each negative constraint, that is each pair
of collisions that does not form a recollision or a tube event. The negative
constraints are needed to be able to apply Lemma 6.16, but they make it
difficult to exploit the cancellations needed. To get around this we use a
special form of the inclusion-exclusion principle that is tailored to this
purpose. In particular, in Section 7 we decompose the indicator function
${\mathbf{1}}_{\mathcal{F}}$ in the form

(1.17)
${\mathbf{1}}_{{\mathcal{F}}}=\sum_{{\mathcal{F}}^{\prime}\geq{\mathcal{F}}}G_{{\mathcal{F}},{\mathcal{F}}^{\prime}},$

where we impose a partial ordering $\leq$ on skeletons, and where
$G_{{\mathcal{F}},{\mathcal{F}}^{\prime}}$ is supported on the set of pairs
$(\Gamma^{+},\Gamma^{-})$ such that
${\mathcal{F}}(\Gamma^{+},\Gamma^{-})\geq{\mathcal{F}}^{\prime}$. In the
decomposition (1.17), the terms $G_{{\mathcal{F}},{\mathcal{F}}^{\prime}}$
depend only on the variables involving collisions that are in the support of
the skeleton ${\mathcal{F}}^{\prime}$ (that is, collisions involved in a
recollision, a cone event, or a tube event).

A challenge is to find a better way to handle the sum over partitions in
${\mathcal{Q}}_{\mathcal{F}}$. For this we introduce the concept of colored
operators. Given a “coloring” function $\chi$ which assigns a unique color to
each collision in $\Gamma$, we define the colored operator $O_{\Gamma}^{\chi}$
to be an analogue of $O_{\Gamma}$ which replaces each instance of the
potential $V$ with an appropriately chosen independent copy of $V$. Then given
a skeleton ${\mathcal{F}}$, we construct two sets of colors
$\Psi^{+}({\mathcal{F}})$ and $\Psi^{-}({\mathcal{F}})$ so that

$\sum_{P\in{\mathcal{Q}}({\mathcal{F}})}{\mathbf{E}\,}_{P}O_{\Gamma^{+}}^{*}AO_{\Gamma^{-}}=\sum_{\chi^{+}\in\Psi^{+}({\mathcal{F}})}\sum_{\chi^{-}\in\Psi^{-}({\mathcal{F}})}{\mathbf{E}\,}(O_{\Gamma^{+}}^{\chi^{+}})^{*}AO_{\Gamma^{-}}^{\chi^{-}}=:{\mathbf{E}\,}(O_{\Gamma^{+}}^{\Psi^{+}})^{*}AO_{\Gamma^{-}}^{\Psi^{-}}.$

The precise definition of colored operators is given in Section 8, and the
construction of colorings that reproduce the partition collection
${\mathcal{Q}}_{\mathcal{F}}$ is done in Section 9. The benefit of writing the
expectation in this way is that we can use the “operator Cauchy-Schwartz”
inequality

$\|{\mathbf{E}\,}X^{*}AY\|_{op}\leq\|A\|_{op}\|{\mathbf{E}\,}X^{*}X\|_{op}^{1/2}\|{\mathbf{E}\,}Y^{*}Y\|_{op}^{1/2}$

where $X$ and $Y$ are random operators, to simplify the estimation of the
contribution from paths with skeleton ${\mathcal{F}}$. The result of this
Cauchy-Schwartz procedure is depicted in Figure 8.

Figure 8. On the left, a partition with a collision coloring chosen so that
matched collisions have the same color. When the operator Cauchy-Schwartz
inequality is applied, a copy of the top and bottom rows are produced and
matched to each other, converting the anti-ladder portion of the partition
into a ladder (right).

More precisely, we will apply the operator Cauchy-Schwartz inequality to the
operator

$\displaystyle{\mathcal{E}}_{{\mathcal{F}},{\mathcal{F}}^{\prime}}[A]$
$\displaystyle:=\int
G_{{\mathcal{F}},{\mathcal{F}}^{\prime}}(\Gamma^{+},\Gamma^{-})\sum_{P\in{\mathcal{Q}}_{\mathcal{F}}}{\mathbf{E}\,}_{P}(O_{\Gamma^{+}})^{*}AO_{\Gamma^{-}}\mathop{}\\!\mathrm{d}\Gamma^{+}\mathop{}\\!\mathrm{d}\Gamma^{-}$
$\displaystyle=\int
G_{{\mathcal{F}},{\mathcal{F}}^{\prime}}(\Gamma^{+},\Gamma^{-}){\mathbf{E}\,}(O_{\Gamma^{+}}^{\Psi^{+}})^{*}AO_{\Gamma^{-}}^{\Psi^{-}}.\mathop{}\\!\mathrm{d}\Gamma^{+}\mathop{}\\!\mathrm{d}\Gamma^{-}.$

To do this we first split $G_{{\mathcal{F}},{\mathcal{F}}^{\prime}}$ as a
mixture of product functions of the form

(1.18) $G_{{\mathcal{F}},{\mathcal{F}}^{\prime}}(\Gamma^{+},\Gamma^{-})=\int
H(\theta)\chi_{{\mathcal{F}},{\mathcal{F}}^{\prime},\theta}^{+}(\omega)\chi_{{\mathcal{F}},{\mathcal{F}}^{\prime},\theta}^{-}(\omega^{\prime})\mathop{}\\!\mathrm{d}\theta.$

We also decompose the coloring sets
$\Psi^{\pm}({\mathcal{F}},{\mathcal{F}}^{\prime})$ into carefully chosen
components which are specified by a data structure called a scaffold, so we
decompose

$\Psi^{\pm}({\mathcal{F}},{\mathcal{F}}^{\prime})=\bigcup_{\operatorname{Scaff}\in{\mathcal{D}}^{\pm}({\mathcal{F}},{\mathcal{F}}^{\prime})}\Psi^{\pm}(\operatorname{Scaff}).$

Then by applying the operator Cauchy-Schwartz inequality we arrive at the
estimate

(1.19)
$\begin{split}\|{\mathcal{E}}_{{\mathcal{F}},{\mathcal{F}}^{\prime}}[A]\|_{op}&\leq\|A\|_{op}(\\#{\mathcal{D}}^{+}({\mathcal{F}},{\mathcal{F}}^{\prime}))\sum_{\operatorname{Scaff}\in{\mathcal{D}}^{+}({\mathcal{F}},{\mathcal{F}}^{\prime})}\big{\|}\int|H(\theta)|{\mathbf{E}\,}(X^{+}_{\operatorname{Scaff},\theta})^{*}(X^{+}_{\operatorname{Scaff},\theta})\mathop{}\\!\mathrm{d}\theta\big{\|}_{op}^{1/2}\\\
&\qquad(\\#{\mathcal{D}}^{-}({\mathcal{F}},{\mathcal{F}}^{\prime}))\sum_{\operatorname{Scaff}\in{\mathcal{D}}^{-}({\mathcal{F}},{\mathcal{F}}^{\prime})}\big{\|}\int|H(\theta)|{\mathbf{E}\,}(X^{-}_{\operatorname{Scaff},\theta})^{*}(X^{-}_{\operatorname{Scaff},\theta})\mathop{}\\!\mathrm{d}\theta\big{\|}_{op}^{1/2},\end{split}$

where

$X^{\pm}_{\operatorname{Scaff},\theta}:=\int\chi_{{\mathcal{F}},{\mathcal{F}}^{\prime},\theta}^{\pm}(\omega)\sum_{\psi\in\Psi^{\pm}(\operatorname{Scaff})}O_{\Gamma}^{\psi}\mathop{}\\!\mathrm{d}\Gamma.$

This calculation involving the Cauchy-Schwartz inequality is done more
carefully in Section 10.

The point of defining the scaffolds is that we can arrange that the operators
${\mathbf{E}\,}(X^{\pm}_{\operatorname{Scaff},\theta})^{*}(X^{\pm}_{\operatorname{Scaff},\theta})$
involve sums over partitions that are formed only from ladders, not anti-
ladders. The point is that ladder partitions have a semigroup structure in the
sense that the concatenation of two ladders is a ladder. We use this structure
to more easily use exact cancellations. More precisely, we use the fact that
the ladder partitions form a good approximation to the evolution
${\mathcal{E}}_{\tau}$ at subkinetic times and the fact that
${\mathcal{E}}_{\tau}$ is a contraction in operator norm in order to obtain
the bounds we need. These bounds are proven in Section 12.

We also point out that the use of Cauchy-Schwartz in this way is lossy, but it
only loses a factor of $N^{C\|{\mathcal{F}}\|}$. We can afford to lose this
factor because the operator norm appearing in the right hand side (1.19) will
have order ${\varepsilon}^{c\|{\mathcal{F}}\|}$. Roughly this is because we
obtain a factor of ${\varepsilon}$ for each special geometric event described
in ${\mathcal{F}}$. A careful argument is needed to ensure that one can obtain
additional factors of ${\varepsilon}$ for each recollision (say) in an
integral over paths containing multiple recollisions, and this is done in
Section 11.

###  1.4. An abbreviated review of related works

Here we point out some related works, making no attempt at providing a
complete review of the rich field of dynamics in random fields.

The rigorous study of the random Schrodinger equation began with the
previously mentioned work of H. Spohn [29]. As mentioned previously, Spohn’s
analysis was extended to the kinetic time in [18] and then to diffusive times
in [15, 16]. Each of these papers considers the convergence in mean of the
Wigner function to the solution of a kinetic equation. A natural question is
to understand the size of the fluctuations of the Wigner function. An analysis
was carried out by T. Chen in [8] which showed that in fact that the $r$-th
moments of the Wigner function are bounded for any $r<\infty$. Chen’s analysis
was later improved by M. Butz in [7]. We also point out the work of J.
Lukkarinen and H. Spohn in [24], which shows that the diagrammatic methods
applied to the Schrödinger equation can also be used to derive kinetic
equations for a random wave equation. See the review [3] for a more complete
discussion of the kinetic regime for waves in a random medium.

Other regimes of interest are the homogenization regime in which the
wavelength of the initial condition is substantially longer than the
decorrelation length scale of the potential. This was studied by G. Bal and N.
Zhang in [31], where a homogenized equation with a constant effective
potential is shown to describe the evolution of the average wave function.
This limit was further studied by T. Chen, T. Komorowski, and L. Ryzhik in
[9]. An entirely different approach to the study of the average wave function
in the kinetic regime was introduced by M. Duerinckx and C. Shirley in [12].
There the authors use ideas from spectral theory to understand the evolution
operator, and are able to show with this method that the average wave function
decays exponentially on the kinetic time scale.

The high frequency regime, in which the wavelength of the initial condition is
much shorter than the decorrelation length of the potential, was considered by
G. Bal, T. Komorowski, and L. Ryzhik in [2]. There the authors derive a
Fokker-Planck equation for the evolution of the Wigner function.

The study of the random Schrodinger equation falls into a larger body of work
of understanding the emergence of apparently irreversible phenomena from
reversible dynamics [28]. From this point of view, the random Schrodinger
equation is simply the one-particle quantum mechanical manifestation of a
larger phenomenon.

The classical version is the stochastic acceleration problem given by the
ordinary differential equation

$\ddot{x}=-{\varepsilon}\nabla V(x)$

where again $V$ is a stationary random potential. Diffusive behavior for the
stochastic acceleration problem was first demonstrated by H. Kesten and G.
Papanicolau in [20] for dimensions $d\geq 3$. For a special class of
potentials their argument was then applied to the two dimensional case by D.
Dürr, S. Goldstein, and J. Lebowitz in [13], and then T. Komorowski and L.
Ryzhik lifted the restriction on the potentials in [21]. The argument used by
Kesten and Papanicolau inspired the semigroup approach taken in this paper.
The connection is that Kesten and Papanicolau define a modified version of the
stochastic acceleration problem in which unwanted correlations are ruled out
by fiat. They then show that this modified evolution is unlikely to have
recollisions after all, and therefore is close to the original evolution. In a
similar way we define an evolution (the refreshed evolution
${\mathcal{E}}_{s}^{m}$) which removes unwanted correlations and use
properties of this evolution to study the true evolution ${\mathcal{E}}_{ms}$.
Although this is where the similarities end, it does seem that a further
unification of the proof techniques may be possible one day.

There are a number of other classical models of particles moving in a random
environment. A popular model is the Lorentz gas, in which a billiard travels
through ${\mathbf{R}}^{d}$ with some obstacles placed according to a Poisson
process. A pioneering paper in the study of the Lorentz gas is [4] where a
linear Boltzmann equation is derived at the Boltzmann-Grad limit of the model.
A review of this model is provided in [11]. We refer the reader also to some
exciting recent developments in this field [1, 22, 26]. It seems that the
classical models of particles moving in random environment contain many of the
same difficulties of understanding the quantum evolution. A deeper
understanding of the phase-space path integral may lead us to a better
understanding of the relationship between the classical and quantum problems.

The random Schrodinger equation is also closely related to wave-kinetic theory
in which one studies the evolution of random waves with a nonlinear
interaction (see [27] for a physically-motivated introduction to this theory).
A pioneering work in this field is the paper of Lukkarinen and Spohn [25], in
which a wave kinetic equation is derived for the nonlinear Schrodinger
equation for initial conditions that are perturbations of an equilibrium
state. In a series of works [19, 5, 10, 6] a wave kinetic equation was derived
for the nonlinear Schrodinger equation on a torus with more general initial
conditions. Independently, in [30] a wave kinetic equation was derived for the
Zakharov-Kuznetsov equation on ${\mathbf{R}}^{d}$ for $d\geq 2$. Each of these
works follows the traditional strategy of writing out a diagrammatic expansion
for the solution and finding sources of cancellation in the error terms and
comparing the main terms to a perturbative expansion of the kinetic equation.
It seems possible that the wavepacket decomposition used in this paper and the
approximate-semigroup argument could be used to make further progress in wave-
kinetic theory.

###  1.5. Acknowledgements

The author is very grateful to Lenya Ryzhik for years of support, advice, and
many clarifying discussions. The author also warmly thanks Minh-Binh Tran for
many helpful conversations about the paper. The author is supported by the
Fannie and John Hertz Foundation.

##  2\. More detailed outline of the proof

In this section we lay out the main lemmas used to prove Theorem 1.1.

The proof involves analysis of three time scales. The first time scale is the
time ${\varepsilon}^{-1.5}$ during which the particle is unlikely to scatter
at all and in particular is unlikely to experience more than one scattering
event. The main result we need from this time scale shows that the linear
Boltzmann equation agrees with the evolution ${\mathcal{E}}_{s}$ with an error
that is very small in operator norm. This calculation is standard and is
reproduced in Appendix A for the sake of completeness. The calculation only
involves two terms from the Duhamel expansion of $e^{-itH}$ so there are no
combinatorial difficulties. propositionshorttime There exists $\theta(d)>0$
such that the following holds: Let $a_{0}\in C^{2d+1}$ be an observable
supported on the set $\\{(x,p)||p|>{\varepsilon}^{\theta}\\}$, and suppose
that $a_{s}$ solves the linear Boltzmann equation (1.8). Then for
$\sigma\leq{\varepsilon}^{-1.5}$,

(2.1)
$\|\operatorname{Op}(a_{\sigma})-{\mathcal{E}}_{\sigma}[\operatorname{Op}(a_{0})]\|_{op}{\,\lesssim\,}{\varepsilon}^{2.1}\sigma\|a_{0}\|_{C^{2d+1}_{{\varepsilon}^{-1},{\varepsilon}^{-0.25}}}.$

To use Proposition 2 along with the semigroup approximation strategy, we need
the following regularity result for the short time evolution of the linear
Boltzmann equation.

######  Lemma 2.1.

There exists $\theta=\theta(d)>0$ such that the following holds: Let $a_{s}$
solve the linear Boltzmann equation (1.8) and
$\operatorname{supp}a_{0}\subset\\{(x,p)\mid|p|\geq{\varepsilon}^{\theta}\\}$.
Then for $s\leq{\varepsilon}^{-2.05}$,

$\|a_{s}\|_{C^{2d+1}_{{\varepsilon}^{-1},{\varepsilon}^{-0.25}}}\leq
C\|a_{0}\|_{C^{2d+1}_{{\varepsilon}^{-1},{\varepsilon}^{-0.5}}}.$

Lemma 2.1 is proven with a simple and suboptimal argument in Appendix D, where
we prove a slightly stronger version in Lemma D.1.

Using Proposition 2 and Lemma 2.1 we can prove that the “$\sigma$-refreshed”
evolution ${\mathcal{E}}_{\sigma}^{n}$ approximates the linear Boltzmann
equation up to a diffusive timescale.

######  Corollary 2.2.

For $\sigma={\varepsilon}^{-1.5}$ and $m\in{\mathbb{N}}$ such that
$m\sigma\leq{\varepsilon}^{-2.05}$, and $a_{s}$ solving (1.8),

(2.2)
$\|{\mathcal{E}}_{\sigma}^{m}[\operatorname{Op}(a_{0})]-\operatorname{Op}(a_{m\sigma})\|_{op}\leq
C{\varepsilon}^{2.1}m\sigma\|a_{0}\|_{C^{2d+1}_{{\varepsilon}^{-1},{\varepsilon}^{-0.5}}}.$

###### Proof.

We define the quantity

$F_{j}:=\sup_{\|a_{0}\|_{C^{2d+1}_{{\varepsilon}^{-1},{\varepsilon}^{-0.5}}}=1}\|{\mathcal{E}}_{\sigma}^{j}[\operatorname{Op}(a_{0})]-\operatorname{Op}(a_{j\sigma})\|_{op}.$

By Proposition 2, $F_{1}\leq C{\varepsilon}^{2.1}\sigma$. To obtain a bound
for $F_{j+1}$ from $F_{j}$ we write

$\|{\mathcal{E}}_{\sigma}^{m}[\operatorname{Op}(a)]-\operatorname{Op}(a_{m\sigma})\|_{op}\leq\|{\mathcal{E}}_{\sigma}[\operatorname{Op}(a_{(m-1)\sigma})]-\operatorname{Op}(a_{m\sigma})\|_{op}+\|{\mathcal{E}}_{\sigma}^{m}[\operatorname{Op}(a)]-{\mathcal{E}}_{\sigma}[\operatorname{Op}(a_{(m-1)\sigma})]\|_{op}.$

The first quantity is bounded using Proposition 2 and Lemma 2.1. The second
term is bounded by $F_{j}$ using the fact that ${\mathcal{E}}_{\sigma}$ is
linear and is a contraction in the operator norm:

$\|{\mathcal{E}}_{\sigma}[A]\|_{op}=\|{\mathbf{E}\,}e^{i\sigma H}Ae^{-i\sigma
H}\|_{op}\leq{\mathbf{E}\,}\|e^{i\sigma H}Ae^{-i\sigma H}\|_{op}=\|A\|_{op}.$

Therefore we obtain the bound

$F_{j+1}\leq F_{j}+C{\varepsilon}^{2.1}\sigma.$

In particular,

$F_{m}\leq C{\varepsilon}^{2.1}m\sigma,$

so (2.2) follows. ∎

The more substantial component of the proof of Theorem 1.1 is the approximate
semigroup property relating the “refreshed” evolution
${\mathcal{E}}_{\sigma}^{M}$ to the correct evolution channel
${\mathcal{E}}_{M\sigma}$. For the purposes of proving an approximate
semigroup property it is more convenient to work with the “wavepacket
quantization” defined by.

$\operatorname{Op}(a):=\int_{{\mathbf{R}}^{d}}\mathinner{|{\xi}\rangle}\mathinner{\langle{\xi}|}a(\xi)\mathop{}\\!\mathrm{d}\xi.$

In Appendix C we will show that the wavepacket quantization is close to the
Weyl quantization, in the sense that

$\|\operatorname{Op}(a)-\operatorname{Op}^{w}(a)\|_{op}\leq{\varepsilon}^{0.05}\|a\|_{C^{2d+1}_{{\varepsilon}^{-1},{\varepsilon}^{-0.1}}}.$

In general, we will be interested in operators of the form

$A=\int\mathinner{|{\xi}\rangle}\mathinner{\langle{\eta}|}a(\xi,\eta)\mathop{}\\!\mathrm{d}\xi$

with kernel $a(\xi,\eta)$ satisfying $|a(\xi,\eta)|\leq C\|A\|_{op}$ and
supported on near the diagonal. To quantify this we introduce the distance
$d_{r}$ on ${{\mathbf{R}}^{2d}}$ so that, writing $\xi=(\xi_{x},\xi_{p})$ and
$\eta=(\eta_{x},\eta_{p})$,

$d_{r}(\xi,\eta):=r^{-1}|\xi_{x}-\eta_{x}|+r|\xi_{p}-\eta_{p}|.$

More precisely, we are interested in families of _good operators_ , defined
below.

######  Definition 2.3 (Good operators).

An operator $A\in\mathcal{B}(L^{2}({\mathbf{R}}^{d}))$ is said to be
$(C_{1},C_{2},\delta)$-good if there exists a function
$a:{{\mathbf{R}}^{2d}}\times{{\mathbf{R}}^{2d}}\to{\mathbf{C}}$ supported in
the set

$\operatorname{supp}a\subset\\{(\xi,\eta)\in{{\mathbf{R}}^{2d}}\times{{\mathbf{R}}^{2d}}\mid
d_{r}(\xi,\eta)<C_{1},|\xi_{p}|\geq{\varepsilon}^{\theta}-C_{2}{\varepsilon}\\}$

such that

$\|A-\int
a(\xi,\eta)\mathinner{|{\xi}\rangle}\mathinner{\langle{\eta}|}\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}\eta\|_{op}\leq\delta.$

Note that the rank one projection onto a wavepacket
$\mathinner{|{\xi}\rangle}\mathinner{\langle{\xi}|}$ is (formally) a
$(0,0,0)$-good operator if $|\xi_{p}|\geq{\varepsilon}^{\theta}$, but its
Wigner transform has smoothness only at the microscopic scale $(r,r^{-1})$.
Similarly if $a\in C^{2d+1}$ is an observable supported on
$\\{(x,p)||p|\geq{\varepsilon}^{\theta}\\}$, then the wavepacket quantization
$\operatorname{Op}(a)$ is a $(0,0,0)$-good operator. Moreover, by (C.6) we
have that $\operatorname{Op}^{w}(a)$ is a $(0,0,{\varepsilon}^{1/2})$-good
operator if $a\in C^{2d+1}_{{\varepsilon}^{-1},{\varepsilon}^{-1/2}}$.

The first step in the proof of the approximate semigroup property is to verify
a semigroup property up to times ${\varepsilon}^{-2+\kappa/2}$.

######  Proposition 2.4.

If $A$ is a $(C_{1},C_{2},\delta)$-good operator with
$C_{1}\leq{\varepsilon}^{-0.1}$ and
$C_{2}\leq\frac{1}{2}{\varepsilon}^{\theta}$ and
${\varepsilon}^{-1.5}<s<{\varepsilon}^{-2+\kappa/2}$ then

$\|{\mathcal{E}}_{2s}[A]-{\mathcal{E}}_{s}^{2}[A]\|_{op}\leq{\varepsilon}^{2.1}s\|A\|_{op}+2\delta.$

and moreover ${\mathcal{E}}_{s}[A]$ is a
$(C_{1}+|\log{\varepsilon}|^{10},C_{2}-10^{3}{\varepsilon},\delta+{\varepsilon}^{100})$-good
operator.

Proposition 2.4 is proved by first comparing ${\mathcal{E}}_{s}$ to the
expectation over ladders, and then observing that the semigroup property holds
for ladders. The first step is done in Section 4 where we prove Proposition
4.10. The derivation of Proposition 2.4 from Proposition 4.10 is explained by
Lemma 12.2.

Using Proposition 2.4 we can prove a comparison result between the linear
Boltzmann equation and the quantum evolution for times up to
${\varepsilon}^{-2+\kappa/10}$.

######  Corollary 2.5.

If $A$ is a $(C_{1},C_{2},\delta)$-good operator with
$C_{1}\leq\frac{1}{2}{\varepsilon}^{-0.1}$ and
$C_{2}\leq{\varepsilon}^{\theta}$ then with $\sigma={\varepsilon}^{-1.5}$,

$\|{\mathcal{E}}_{\sigma}^{m}[A]-{\mathcal{E}}_{m\sigma}[A]\|_{op}\leq
C_{\kappa}{\varepsilon}^{2.1}\sigma\|A\|_{op}+2\delta+{\varepsilon}^{20}$

for $m$ such that $m\sigma\leq{\varepsilon}^{-2+\kappa/2}$.

###### Proof.

We perform an iteration, defining the error

$E_{m}:=\sup_{\begin{subarray}{c}\|A\|_{op}=1\\\ A\text{ is
}(C_{1}-K_{1}m,C_{2}+K_{2}m,\beta-\delta
m){-good}\end{subarray}}\|{\mathcal{E}}_{2^{m}\sigma}[A]-{\mathcal{E}}_{\sigma}^{2^{m}}[A]\|_{op},$

where we choose $C_{1}=\frac{1}{2}{\varepsilon}^{-0.1}$,
$K_{1}=|\log{\varepsilon}|^{10}$, $C_{2}=\frac{1}{4}{\varepsilon}^{0.1}$,
$K_{2}=10^{3}{\varepsilon}$, $\beta={\varepsilon}^{100}$, and
$\delta={\varepsilon}^{100}$. Let ${\mathcal{A}}_{m}$ be the class of
admissible operators in the supremum defining $E_{m}$. The significant point
about ${\mathcal{A}}_{m}$ is that ${\mathcal{E}}_{s}[A]\in{\mathcal{A}}_{m}$
when $A\in{\mathcal{A}}_{m+1}$. To find a recursion for $E_{m}$, we write

(2.3)
$\begin{split}\|{\mathcal{E}}_{2^{m+1}s}[A]-{\mathcal{E}}_{s}^{2^{m+1}}[A]\|_{op}&\leq\|{\mathcal{E}}_{2^{m+1}s}[A]-{\mathcal{E}}_{2^{m}s}^{2}[A]\|_{op}+\|{\mathcal{E}}_{2^{m}s}^{2}[A]-{\mathcal{E}}_{s}^{2^{m+1}}[A]\|_{op}\\\
&\leq\|{\mathcal{E}}_{2^{m+1}s}[A]-{\mathcal{E}}_{2^{m}s}^{2}[A]\|_{op}+\|{\mathcal{E}}_{2^{m}s}[{\mathcal{E}}_{2^{m}s}[A]-{\mathcal{E}}_{s}^{2^{m}}[A]]\|_{op}\\\
&\qquad\qquad+\|({\mathcal{E}}_{2^{m}s}-{\mathcal{E}}_{s}^{2^{m}}){\mathcal{E}}_{s}^{2^{m}}[A]\|_{op}.\end{split}$

Since ${\mathcal{E}}_{s}$ is a contraction in the operator norm, and since
${\mathcal{E}}_{s}$ maps ${\mathcal{A}}_{m+1}$ into ${\mathcal{A}}_{m}$ we
have by Proposition 2.4 that

$\|{\mathcal{E}}_{2^{m+1}s}[A]-{\mathcal{E}}_{s}^{2^{m+1}}[A]\|_{op}\leq(({\varepsilon}^{2.1}2^{m}s)\|A\|_{op}+2\delta)+E_{m}.$

Taking a supremum over $A$ we obtain the relation

$E_{m+1}\leq{\varepsilon}^{2.1}2^{m}\sigma+2E_{m}.$

Since $E_{0}=0$, we obtain

$E_{m}\leq m{\varepsilon}^{2.1}2^{m}\sigma.$

∎

The remaining ingredient needed to prove Theorem 1.1 is a semigroup property
that holds up to diffusive times. This is substantially more difficult than
establishing the semigroup property for subkinetic times because of the need
for resummation in the Duhamel series. The main result is the following.

######  Proposition 2.6.

There exists $\kappa=\kappa(d)$ such that the following holds: If $A$ is a
$({\varepsilon}^{-0.1},\frac{1}{2}{\varepsilon}^{0.1},\delta)$-good operator,
then with $\tau={\varepsilon}^{-2+\kappa/2}$ and
$N=\lfloor{\varepsilon}^{-\kappa}\rfloor$,

$\|{\mathcal{E}}_{N\tau}[A]-{\mathcal{E}}_{\tau}^{N}[A]\|_{op}\leq{\varepsilon}^{c\kappa}\|A\|_{op}+2\delta.$

Having sketched the argument proving the main result, we now outline the
remaining sections in the paper. In Section 3 we explain the phase space path
integral approximation we use throughout the paper. Then in Section 4 we
introduce the ladder superoperator ${\mathcal{L}}_{s}$ which is a main
character in the derivation of the approximate semigroup property. Section 4
contains the bulk of the proof of Proposition 2.4 and contains most of the
main ideas of the paper.

The remaining sections in the paper are dedicated to the proof of Proposition
2.6. In Section 5 we write down the path integral used to represent the
solution operator up to this time. Then in Section 6 we clarify the
relationship between the geometry of paths and the combinatorial features of
their collision partitions. Then in Section 7 we split up the path integral
according to the geometry of the paths. To exploit the combinatorial structure
of the correlation partition, we introduce the formalism of colored operators
in Section 8 and Section 9. Then in Section 10 we finally write out our
version of a “diagrammatic expansion” (which is different than previous
expansions in that the first term of the expansion for
${\mathcal{E}}_{N\tau}[A]$ is the refreshed evolution
${\mathcal{E}}_{\tau}^{N}[A]$). The diagrams are bounded in Section 11.

The remaining sections contain proofs of more technical results needed
throughout the argument, and are referenced as needed.

##  3\. A sketch of the derivation of the path integral

In this section we state the precise version of the phase-space path integral
alluded to in Section 1.3. The proofs of the assertions made in this section
are given in Sections 13, 14, and 15.

The first step is to write out an expansion for $e^{-isH}$ that is valid for
times $s\leq\tau:={\varepsilon}^{-2+\kappa/2}$ in terms of paths. More
precisely, a path $\omega=({\mathbf{s}},{\mathbf{p}},{\mathbf{y}})$ having $k$
collisions is a tuple containing a list of collision invervals
${\mathbf{s}}\in{\mathbf{R}}_{+}^{k+1}$ satisfying $\sum_{j=0}^{k}s_{j}=s$,
momenta ${\mathbf{p}}\in({\mathbf{R}}^{d})^{k+1}$, and collision locations
${\mathbf{y}}\in({\mathbf{R}}^{d})^{k}$. Each path $\omega$ is associated to
the operator $O_{\omega}$ defined by

${\widehat{O_{\omega}\psi}}=\delta_{p_{k}}{\widehat{\psi}}(p_{0})e^{i\varphi(\omega)}\prod_{j=1}^{k}{\widehat{V_{y_{j}}}}(p_{j}-p_{j-1}),$

where $\varphi(\omega)$ is the phase function

$\varphi(\omega)=\sum_{j=0}^{k}s_{j}|p_{j}|^{2}/2+\sum_{j=1}^{k}y_{j}\cdot(p_{j}-p_{j-1}).$

In Dirac notation, we express $O_{\omega}$ as

(3.1)
$O_{\omega}=\mathinner{|{p_{k}}\rangle}\mathinner{\langle{p_{0}}|}e^{i\varphi(\omega)}\prod_{j=1}^{k}{\widehat{V_{y_{j}}}}(p_{j}-p_{j-1}).$

Here $V_{y}(x)=V(x-y)\chi^{V}(x)$ is a localized and shifted version of the
potential, with localization $\chi^{V}$ having width $r$ and satisfying
$\int\chi^{V}=1$.

Let $\Omega_{k}(s)$ denote the space of paths with $k$ collisions and duration
$s$,

$\Omega_{k}(s)=\triangle_{k}(s)\times({\mathbf{R}}^{d})^{k+1}\times({\mathbf{R}}^{d})^{k},$

where $\triangle_{k}(s)\subset{\mathbf{R}}_{+}^{k+1}$ is the set of tuples of
time intervals summing to $s$,

$\triangle_{k}(s)=\\{{\mathbf{s}}=(s_{0},\cdots,s_{k})\in{\mathbf{R}}_{+}^{k+1}\mid\sum_{j=0}^{k}s_{j}=s\\}.$

We will see in Section 13 that the Duhamel expansion can be formally written

$e^{isH}=\sum_{k=0}^{\infty}T_{k}:=\sum_{k=0}^{\infty}\int_{\Omega_{k}(s)}O_{\omega}\mathop{}\\!\mathrm{d}\omega.$

In this integral there is no need for the collision locations ${\mathbf{y}}$
to have any relationship with the variables ${\mathbf{s}}$ and ${\mathbf{p}}$.
There is however significant cancellation due to the presence of the phase
$e^{i\varphi(\omega)}$. For example, by integrating by parts in the $p_{j}$
variables and using the identity,

$\displaystyle\partial_{p_{j}}\varphi(\omega)=y_{j}+s_{j}p_{j}-y_{j+1},$

we can reduce the path integral to paths which satisfy

$|y_{j+1}-(y_{j}+s_{j}p_{j})|\lessapprox r.$

Integration by parts in the $s_{j}$ is somewhat more delicate because of the
hard constraint $\sum s_{j}=\tau$. By decomposing this hard constraint as a
sum of softer constraints, we can impose a cutoff on the weaker conservation
of kinetic energy condition

$||p_{j}|^{2}/2-|p_{j^{\prime}}|^{2}/2|\lessapprox\max\\{|p_{j}|r^{-1},s_{j}^{-1},s_{j^{\prime}}^{-1}\\}.$

The integration by parts argument will allow us to construct a function
$\chi^{path}$ supported on the set of such “good” paths and for which

$T_{k}\approx\int_{\Omega_{k}(s)}\chi^{path}(\omega)O_{\omega}\mathop{}\\!\mathrm{d}\omega,$

with an error that is negligible in operator norm. To be more precise, given a
tolerance $\alpha$ (which we set to be $|\log{\varepsilon}|^{10}$), we define

(3.2)
$\begin{split}\Omega_{\alpha,k}(s)=\\{({\mathbf{s}},{\mathbf{p}},{\mathbf{y}})\in\Omega_{k}(s)\mid&|y_{j+1}-(y_{j}+s_{j}p_{j})|\leq\alpha
r\text{ for all }j\in[1,k-1]\text{ and }\\\
&||p_{j}|^{2}/2-|p_{j^{\prime}}|^{2}/2|\leq\alpha\max\\{s_{j^{\prime}}^{-1},s_{j}^{-1},|p_{j}|r^{-1},\alpha
r^{-2}\\}\text{ for all }j,j^{\prime}\in[0,k]\\}.\end{split}$

Within $\Omega_{\alpha,k}(s)$ we also define the subset

(3.3)
$\begin{split}\Omega_{\alpha,k}(t;\xi,\eta):=\\{({\mathbf{s}},{\mathbf{p}},{\mathbf{y}})\in\Omega_{\alpha,k}(t)\mid&|y_{1}-(\xi_{x}+s_{0}p_{0})|\leq\alpha
r,|\eta_{x}-(y_{k}+s_{k}p_{k})|\leq\alpha r,\\\ &|p_{0}-\xi_{p}|\leq\alpha
r^{-1},\text{ and }|p_{k}-\eta_{p}|\leq\alpha r^{-1}\\},\end{split}$

where $\xi,\eta\in{{\mathbf{R}}^{2d}}$ with $\xi=(\xi_{x},\xi_{p})$ and
$\eta=(\eta_{x},\eta_{p})$.

The following lemma is simply a careful application of integration by parts,
and is done in Section 13.

######  Lemma 3.1.

There exists a cutoff function $\chi\in
C^{\infty}(\Omega_{k}(\tau)\times{{\mathbf{R}}^{2d}}\times{{\mathbf{R}}^{2d}})$
supported in the set

$\operatorname{supp}\chi\subset\\{(\omega,\xi,\eta)\mid\omega\in\Omega_{\alpha,k}(\tau;\xi,\eta)\\}$

such that, with

(3.4)
$T_{k}^{\chi}(\tau):=(i{\varepsilon})^{k}\int_{\Omega_{k}(\tau)\times{{\mathbf{R}}^{2d}}\times{{\mathbf{R}}^{2d}}}\mathinner{|{\eta}\rangle}\mathinner{\langle{\xi}|}\chi(\omega,\xi,\eta)\mathinner{\langle{\eta|O_{\omega}|\xi}\rangle}\mathop{}\\!\mathrm{d}\omega\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}\eta,$

we have the approximation

(3.5)
$\|T_{k}^{\chi}-T_{k}\|_{op}\leq{\varepsilon}^{-C_{d}k}\|V\|_{C^{10d}}^{Ck}\exp(-c\alpha^{0.99}).$

The point of Lemma 3.1 is that it allows us to neglect the contribution of
“physically unreasonable paths” – those that either badly violate conservation
of kinetic energy or the transport constraints $y_{k+1}\approx
y_{k}+s_{k}p_{k}$.

We remark that Lemma 3.1 is deterministic in the sense that that the
conclusion holds for all potentials, and when we apply it we will simply need
moment bounds for the $C^{10d}$ norm of the potential (after being cutoff to a
ball of large radius). With the choice $\alpha=|\log{\varepsilon}|^{10}$, and
assuming that $\|V\|_{C^{10d}}\leq{\varepsilon}^{-100}$ (say), the right hand
side is still $O({\varepsilon}^{K})$ for any $K>0$.

Having given a description to the collision operators $T_{k}$, it remains to
estimate moments of the form ${\mathbf{E}\,}\|T_{k}^{\chi}\|_{op}^{M}$, for
which we use the moment method:

${\mathbf{E}\,}\|T_{k}^{\chi}\|_{op}^{2M}\leq{\mathbf{E}\,}\operatorname{tr}((T_{k}^{\chi})^{*}T_{k}^{\chi})^{M}.$

Note that this step is where the cutoff on the potential is crucial – without
the cutoff $\chi_{R}$ the trace above would be infinite. In Section 14 we
prove Lemma 14.1, which states that

$\Big{(}{\mathbf{E}\,}\|T_{k}^{\chi}(s)\|_{op}^{2M}\Big{)}^{1/M}\leq
R^{C/m}{\varepsilon}^{2}s\,\,(C(kM)^{C}|\log{\varepsilon}|^{10}).$

The presence of the factor $(km)^{C}$ makes this bound unsuitable for reaching
diffusive time scales. However this bound is good enough to approximate
$e^{isH}$ by $e^{-is\Delta/2}$ for times $s\leq{\varepsilon}^{-1.1}$ (say). We
use this result in Section 15 to define a modified operator
$T_{k,\rho}^{\chi}$ which involves a first short period of free evolution.
More precisely, given a time $\sigma>0$ we construct a function
$\rho_{\sigma}\geq 0$ that is supported on the interval $[\sigma 2\sigma]$, is
Gevrey regular, and satisfies $\int\rho_{\sigma}=1$. Then we define

${\widetilde{U}}_{\tau,\sigma}:=\int_{\sigma}^{2\sigma}\int_{\sigma}^{2\sigma}e^{is\Delta/2}e^{-i(\tau-
s-s^{\prime})H}e^{is^{\prime}\Delta/2}\rho_{\sigma}(s)\rho_{\sigma}(s^{\prime})\mathop{}\\!\mathrm{d}s\mathop{}\\!\mathrm{d}s^{\prime}.$

We will fix for the remainder of the paper $\sigma={\varepsilon}^{-1.5}$. In
Section 15 we use Lemma 3.1 to prove Lemma 15.1, which justifies the
approximation

$\|{\mathcal{E}}_{N\tau}[A]-{\mathbf{E}\,}({\widetilde{U}}_{\tau,\sigma}^{*})^{N}A{\widetilde{U}}_{\tau,\sigma}^{N}\|_{op}\leq{\varepsilon}^{0.2}.$

This will allow us to restrict the path integral to a space of paths which do
not have a collision too close to either endpoint,

$\Omega_{k,\alpha}(\tau,\sigma;\xi,\eta):=\\{(\omega,\xi,\eta)\in\Omega_{k,\alpha}(\tau;\xi,\eta)\mid
s_{0}\geq\sigma\text{ and }s_{k}\geq\sigma\\}.$

The operator ${\widetilde{U}}_{\tau,\sigma}$ also has a path integral
expansion in terms of collision operators $T_{k}^{\chi,\sigma}$, which in
addition to having the smooth cutoff $\chi$ have a cutoff enforcing that
$\omega\in\Omega_{k,\alpha}(\tau,\sigma;\xi,\eta)$.

Combining the above arguments we obtain the following approximation result for
the evolution operator ${\widetilde{U}}_{\tau,\sigma}$.

######  Proposition 3.2.

There exists a bounded smooth function $\chi_{\sigma}\in
C^{\infty}(\Omega_{k}(\tau)\times{\mathbf{R}}^{2d}\times{\mathbf{R}}^{2d})$
supported in the set

$\operatorname{supp}(\chi_{\sigma})\subset\Omega_{\alpha,k}(\tau,S;\xi,\eta):=\\{(\omega,\xi,\eta)\mid\omega\in\Omega_{k,\alpha}(\tau;\xi,\eta)\text{
and }s_{0}\geq{\varepsilon}^{-1.6},\text{ and
}s_{k}\geq{\varepsilon}^{-1.6}\\},$

and such that, with

(3.6)
$T_{k}^{\chi,\sigma}:=(i{\varepsilon})^{k}\int_{\Omega_{k}(\tau)\times{\mathbf{R}}^{2d}\times{\mathbf{R}}^{2d}}\mathinner{|{\eta}\rangle}\mathinner{\langle{\xi}|}\chi_{\sigma}(\omega,\xi,\eta)\mathinner{\langle{\eta|O_{\omega}|\xi}\rangle}\mathop{}\\!\mathrm{d}\omega\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}\eta,$

we have the Duhamel expansion

${\widetilde{U}}_{\tau,\sigma}:=\sum_{k=0}^{k_{max}}T_{k}^{\chi,\sigma}+R_{k_{max}}^{\chi,\sigma},$

where the remainder $R_{k_{max}}^{\chi,\rho}$ has the expression

(3.7)
$R_{k_{max}}^{\chi,\sigma}:=(i{\varepsilon})^{k}\int_{0}^{\tau}\mathop{}\\!\mathrm{d}s\int_{\Omega_{k_{max}}(s)\times{\mathbf{R}}^{2d}\times{\mathbf{R}}^{2d}}e^{i(\tau-s)H}\mathinner{|{\eta}\rangle}\mathinner{\langle{\xi}|}\chi_{\sigma}(\omega,\xi,\eta)\mathinner{\langle{\eta|O_{\omega}|\xi}\rangle}\mathop{}\\!\mathrm{d}\omega\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}\eta,$

and

(3.8) $\left({\mathbf{E}\,}\|e^{i\tau
H}-{\widetilde{U}}_{\tau,\sigma}\|_{op}^{{\varepsilon}^{-0.1}}\right)^{{\varepsilon}^{0.1}}\leq{\varepsilon}^{0.3}$

We use the operator
${\widetilde{U}}_{\tau,\sigma}=:U_{\tau,\sigma}+R_{k_{max}}^{\chi,\sigma}$ to
decompose $e^{iN\tau H}$ as follows:

$e^{iN\tau H}\approx U_{\tau,\sigma}^{N}+\sum_{j=1}^{N}e^{i(N-j)\tau
H}R_{k_{max}}^{\chi,\sigma}U_{\tau,\sigma}^{j-1}.$

Let ${\widetilde{{\mathcal{E}}}}_{N\tau}$ be the superoperator formed from the
main term,

${\widetilde{{\mathcal{E}}}}_{N\tau}[A]:={\mathbf{E}\,}(U_{\tau,\sigma}^{N})^{*}AU_{\tau,\sigma}^{N}.$

Let $R_{j}$ be the operator $R_{k_{max}}^{\chi,\sigma}U_{\tau,\sigma}^{j-1}$,
which also can be written
$R_{j}=\int_{0}^{\tau}e^{i(\tau-s)H}R_{j,s}\mathop{}\\!\mathrm{d}s$ with

(3.9)
$R_{j,s}:=\int_{\Omega_{k_{max}}(s)\times{\mathbf{R}}^{2d}\times{\mathbf{R}}^{2d}}\mathinner{|{\eta}\rangle}\mathinner{\langle{\xi}|}U_{\tau,\sigma}^{j-1}\chi_{\sigma}(\omega,\xi,\eta)\mathinner{\langle{\eta|O_{\omega}|\xi}\rangle}\mathop{}\\!\mathrm{d}\omega\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}\eta.$

Then by an application of the operator Cauchy-Schwarz inequality and the
triangle inequality we have the estimate

(3.10)
$\|{\mathcal{E}}_{N\tau}[A]-{\widetilde{{\mathcal{E}}}}_{N\tau}[A]\|_{op}\leq\|A\|_{op}({\varepsilon}^{0.3}+N\tau\max_{j\in[N]}\sup_{s\in[0,\tau]}\|{\mathbf{E}\,}R_{j,s}^{*}R_{j,s}\|_{op}).$

In the course of understanding the evolution
${\widetilde{{\mathcal{E}}}}_{N\tau}$ we will derive estimates that as a
byproduct prove

(3.11) $\max_{j\in[N]}\sup_{0\leq
s\leq\tau}\|{\mathbf{E}\,}R_{j,s}^{*}R_{j,s}\|_{op}\leq{\varepsilon}^{50}.$

In Section 10.1 we explain how this bound is obtained as a modification of the
argument used to control the diffusive diagrams.

##  4\. The ladder approximation for ${\mathcal{E}}_{\tau}$

In this section we sketch the proof that the evolution channel
${\mathcal{E}}_{s}$ is well approximated by a sum over ladder diagrams when
$s\leq\tau={\varepsilon}^{-2}N^{-\kappa/10}$. This is closely related to the
semigroup property which we will explore in a later section. The statement of
the main result of the section, Proposition 4.10, is given in Section 4.3
after some preliminary calculations which motivate the definition of the
ladder superoperator.

###  4.1. An introduction to the channel ${\mathcal{E}}_{s}$

For times $s\leq\tau={\varepsilon}^{-2}N^{-\kappa/10}$, we may use Lemma 3.1
to write

(4.1)
$e^{-isH}=\sum_{k=0}^{k_{max}}\int_{\Omega_{k}(s)}\chi_{S}(\omega,\xi,\eta)\mathinner{|{\eta}\rangle}\mathinner{\langle{\xi}|}\mathinner{\langle{\eta|O_{\omega}|\xi}\rangle}\mathop{}\\!\mathrm{d}\omega\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}\eta+E$

where $\chi_{S}(\omega,\xi,\eta)$ is a smooth function supported on the set
$\Omega_{\alpha,k}(s,\sigma;\xi,\eta)$ and the approximation is up to an error
$E$ that satisfies ${\mathbf{E}\,}\|E\|_{op}^{2}\leq{\varepsilon}^{0.2}$.

The operator $e^{isH}$ can similarly be expressed as an integral over paths,

(4.2)
$e^{isH}=(e^{-isH})^{*}=\sum_{k=0}^{k_{max}}\int_{\Omega_{k}(s)}\chi_{S}(\omega,\xi,\eta)\mathinner{|{\xi}\rangle}\mathinner{\langle{\eta}|}\mathinner{\langle{\eta|O_{\omega}|\xi}\rangle}^{*}\mathop{}\\!\mathrm{d}\omega\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}\eta+E^{*}.$

We now use (4.1) and (4.2) to write an expansion for ${\mathcal{E}}_{s}[A]$.
We will drop the summation over $k$ and handle the sum implicitly in the
integral over $\Omega(s)=\bigcup_{j=0}^{k_{max}}\Omega_{k}(s)$.

${\mathcal{E}}_{s}[A]\approx\int\mathinner{|{\xi_{0}^{-}}\rangle}\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\mathinner{\langle{\xi_{1}^{+}}|}\chi_{S}(\omega^{+},\xi_{0}^{+},\xi_{1}^{+})\chi_{S}(\omega^{-},\xi_{0}^{-},\xi_{1}^{-})\mathop{}\\!\mathrm{d}\bm{\omega}\mathop{}\\!\mathrm{d}\bm{\xi}.$

up to a remainder that is bounded by $O({\varepsilon}^{0.2}\|A\|_{op})$ in
operator norm.

To express the operator more compactly we introduce some notation. We write
$\bm{\Gamma}=(\xi_{0}^{+},\omega^{+},\xi_{1}^{+};\xi_{0}^{-},\omega^{-},\xi_{0}^{-})=(\Gamma^{+};\Gamma^{-})$
for the full path, and then define the path cutoff function

$\Xi(\bm{\Gamma}):=\chi_{S}(\omega^{+},\xi_{0}^{+},\xi_{1}^{+})\chi_{S}(\omega^{-},\xi_{0}^{-},\xi_{1}^{-}).$

Moreover, we will stack like terms to keep the integrand more organized. That
is we will write $\begin{smallmatrix}A\\\ B\end{smallmatrix}$ to mean the
product $AB$. With this notation,

${\mathcal{E}}_{s}[\operatorname{Op}(a)]\approx\int\mathinner{|{\xi_{0}^{-}}\rangle}\mathinner{\langle{\xi_{0}^{+}}|}\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma){\mathbf{E}\,}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}\mathop{}\\!\mathrm{d}\Gamma$

Given a path $\omega=({\mathbf{s}},{\mathbf{p}},{\mathbf{y}})\in\Omega_{k}$,
the random amplitude $\mathinner{\langle{\xi|O_{\omega}|\eta}\rangle}$ is
given by

$\mathinner{\langle{\xi|O_{\omega}|\eta}\rangle}=\mathinner{\langle{\xi|p_{0}}\rangle}\mathinner{\langle{p_{k}|\eta}\rangle}e^{i\varphi(\omega,\xi,\eta)}\prod_{j=1}^{k}{\widehat{V_{y_{j}}}}(p_{j}-p_{j-1}).$

Since $V$ is real and therefore ${\widehat{V}}(q)^{*}={\widehat{V}}(-q)$, the
term in the expectation can be written

(4.3)
${\mathbf{E}\,}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}=\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|p_{+,k_{+}}}\rangle}\mathinner{\langle{p_{+,0}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|p_{-,k_{-}}}\rangle}\mathinner{\langle{p_{-,0}|\xi_{0}^{-}}\rangle}\end{smallmatrix}e^{i(\varphi(\omega^{+})-\varphi(\omega^{-}))}{\mathbf{E}\,}\prod_{a\in
K}{\widehat{V_{y_{a}}}}(q_{a}),$

where

$K_{k_{+},k_{-}}=\\{(\ell,j)\mid\ell\in\\{+,-\\}\text{ and
}j\in[1,k_{\ell}]\\}$

and

$q_{(\ell,j)}=\begin{cases}p_{\ell,j}-p_{\ell,j-1},&\ell=+\\\
p_{\ell,j-1}-p_{\ell,j},&\ell=-.\end{cases}$

We write $X=X(\Gamma)$ for the collision set $X=\\{(y_{a},q_{a})\\}_{a\in
K_{(k_{1},k_{2})}}$.

To split up the expectation, let $P(X)=P({\mathbf{y}})\in{\mathcal{P}}(K)$ be
the finest partition such that $|y_{a}-y_{a^{\prime}}|\leq 2\alpha r$ implies
that $a$ and $a^{\prime}$ belong to the same set. Then

${\mathbf{E}\,}\prod_{a\in K}{\widehat{V_{y_{a}}}}(q_{a}),=\prod_{S\in
P(X)}{\mathbf{E}\,}\prod_{a\in S}{\widehat{V_{y_{a}}}}(q_{a}).$

For admissible potentials, Lemma B.3 implies that

(4.4) $\Big{|}{\mathbf{E}\,}\prod_{a\in
K}{\widehat{V_{y_{a}}}}(q_{a})\Big{|}\leq r^{-|K|d}\prod_{a\in
K}(1+|q_{a}|)^{-10d}\sum_{P^{\prime}\leq P({\mathbf{y}})}\prod_{S\in
P^{\prime}}(C_{V}|S|)^{2|S|}r^{d}b_{|S|r^{-1}}(\sum_{j\in S}q_{j}),$

with $b_{t}(x):=\exp(-c|t^{-1}x|^{0.99})$.

This quantity is only nonnegligible when $|\sum_{j\in S}q_{j}|\lessapprox
r^{-1}$ for each $S\in P({\mathbf{y}})$. This leads us to define the notion of
a $\beta$-complete collision set.

######  Definition 4.1 (Complete collision sets).

A collision set $X=\\{(y_{j},q_{j})\\}_{j=1}^{k}$ is $\beta$-complete if

(4.5) $\Big{|}\sum_{j\in S}q_{j}\Big{|}\leq\beta|S|r^{-1}$

holds for every $S\in Q({\mathbf{y}})$.

An early approximation we can make is to reduce the integration over paths
$\omega^{+}$ and $\omega^{-}$ to only paths which are $\beta$-complete for
$\beta=|\log{\varepsilon}|^{20}$.

######  Lemma 4.2.

Let $s\leq\tau$ and let $A$ be a band-limited operator with bandwidth at most
${\varepsilon}^{-0.5}$. Then for $\beta>|\log{\varepsilon}|^{20}$,

(4.6)
$\begin{split}\big{\|}\int\mathinner{|{\xi_{0}^{-}}\rangle}\mathinner{\langle{\xi_{0}^{+}}|}\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma){\mathbf{E}\,}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}(1-{\mathbf{1}}_{X(\Gamma)\text{
is
}\beta\text{-complete}})\mathop{}\\!\mathrm{d}\Gamma\big{\|}_{op}\leq{\varepsilon}^{100}\|A\|_{op}.\end{split}$

###### Proof.

The difference is an integral over paths which form $\beta$-incomplete
collision sets, and the norm of the integrand is at most
${\varepsilon}^{|\log{\varepsilon}|^{5}}$ for such paths. The volume of
integration for fixed $\xi_{0}^{+}$ or for fixed $\xi_{0}^{-}$ is only
${\varepsilon}^{-C}$ , so the result follows upon applying the Schur test. ∎

###  4.2. The structure of partitions from generic paths

The idea is that the main contribution to the channel ${\mathcal{E}}_{s}$
should come from paths $\bm{\Gamma}$ that are generic. We define generic paths
as those that do not have incidences.

In general incidences are any geometric feature of a path that can change its
correlation structure. The simplest type of incidence is a recollision.

######  Definition 4.3 (Recollisions).

A _recollision_ in a path $\omega\in\Omega_{k}$ is a pair
$a,a^{\prime}\in[1,k]$ such that $|y_{a}-y_{a^{\prime}}|\leq 2r$.

A special kind of recollision is an _immediate recollision_ , which satisfies
$a^{\prime}=a+1$ and $|p_{a+1}-p_{a-1}|\leq 10\alpha r^{-1}$. Let
$I^{imm}(\omega)\subset[1,k]$ be the set of all indices belonging to a
immediate recollision, and let $I^{rec}(\omega)$ be the set of indices
belonging to a recollision that is not an immediate recollision.

At this point we stop to observe that immediate recollisions do not
substantially alter the trajectory of a path.

An example of an immediate recollision in a path. Note that such a recollision
can only cause a small displacement in position on the order $O(r)$ and a
small perturbation in momentum on the order $O(r^{-1})$.

The first fact we prove is that the time betweeen recollisions cannot be too
large.

######  Lemma 4.4.

Let $\omega\in\Omega_{k,\alpha}(\tau,\sigma;\xi,\eta)$ be a path with
$|\xi_{p}|\geq r^{-1}\sigma^{-1}$. If $|y_{j}-y_{j+1}|\leq 2r$, then
$s_{j}\leq 10|p_{0}|^{-1}\alpha r$.

###### Proof.

First, the condition $|\xi_{0}|\geq r^{-1}\sigma^{-1}$ and the constraint
$|p_{0}-(\xi_{0})_{p}|\leq\alpha r^{-1}$ imply $|p_{0}|\leq 2\alpha
r^{-1}\sigma^{-1}$. Then, since $s_{0}\geq\sigma$

$||p_{j}|^{2}/2-|p_{0}|^{2}/2|\leq\alpha\max\\{s_{j}^{-1},|p_{0}|r^{-1}\\}.$

Assuming that $s_{j}\geq 10|p_{0}|^{-1}\alpha r$, it follows that

$||p_{j}|-|p_{0}||\leq\alpha 2\alpha r^{-1}.$

But the condition $|y_{j}-y_{j+1}|\leq 2r$ implies

$|s_{j}p_{j}|\leq 4\alpha r,$

so that $s_{j}\leq 4\alpha|p_{j}|^{-1}r\leq 10|p_{0}|^{-1}r$. ∎

To state the second fact, we introduce the notion of the collision time
$t_{a}$, simply defined by

$t_{\pm,j}:=\sum_{0\leq j^{\prime}<j}s_{\pm,j^{\prime}}.$

######  Lemma 4.5.

Let $\bm{\Gamma}$ be a $|\log{\varepsilon}|^{20}$-complete path, and suppose
that $\\{a,a+1\\}\in P(\bm{\Gamma})$ for every immediate recollision $a$. If
$a<a^{\prime}\in K(\bm{\Gamma})\setminus I^{imm}(\bm{\Gamma})$ are two
consecutive collisions when ignoring immediate recollisions, then

(4.7) $|y_{a^{\prime}}-(t_{a^{\prime}}-t_{a})p_{a}|\leq 4m^{2}\alpha^{20}r.$

###### Proof.

To prove this, first observe that $a^{\prime}=a+2m+1$ for some number $m$ of
immediate recollisions between $a$ and $a^{\prime}$, and

$y_{a+2m+1}-y_{a}=\sum_{j=0}^{m}y_{a+2j+1}-y_{a+2j}+\sum_{j=1}^{m}y_{a+2j}-y_{a+2j-1}.$

The latter terms are each bounded by $2r$ because $(a+2j-1,a+2j)$ are all
immediate recollisions. The former terms are well approximated by
$s_{a+2j}p_{a+2j}$, so we have

$|y_{a+2m+1}-y_{a}-\sum_{j=0}^{m}s_{a+2j}p_{a+2j}|\leq 2m\alpha r.$

Next we observe that, since $(a+2j-1,a+2j)$ forms a pair in $P(\bm{\Gamma})$
and $\bm{\Gamma})$ is $|\log{\varepsilon}|^{20}$-complete,
$|q_{a+2j-1}+q_{a+2j}|\leq 2|\log{\varepsilon}|^{20}r^{-1}$. Expanding the
definition of $q_{a+2j-1}$ and $q_{a+2j}$ it follows that

$|q_{a+2j-1}+q_{a+2j}|=|p_{a+2j-1}-p_{a+2j-2}+p_{a+2j}-p_{a+2j-1}|=|p_{a+2j}-p_{a+2(j-1)}|\leq
2|\log{\varepsilon}|^{20}r^{-1}$

for every $1\leq j\leq m$. In particular, $|p_{a+2j}-p_{a}|\leq
2|\log{\varepsilon}|^{20}mr^{-1}$ for each $a$, and therefore

$|y_{a+2m+1}-y_{a}-(\sum_{j=0}^{m}s_{a+2j})p_{a}|\leq 2m^{2}\alpha^{20}r.$

Finally, we observe that

$t_{a+2m+1}-t_{a}=\sum_{j=0}^{m}s_{a+2j}+\sum_{j=1}^{m}s_{a+2j-1}.$

The latter sum is bounded by $10m|p_{0}|^{-1}\alpha r$ by Lemma 4.4. ∎

Recollisions form just one type of incidence. It is possible that paths
$\omega^{+}$ and $\omega^{-}$ have a nontrivial collision structure even if
neither path has a recollision. Consider for example the paths

$\omega^{+}=({\mathbf{s}}^{+},{\mathbf{p}}^{+},{\mathbf{y}}^{+})=(((4,2,1,2,4),(v,-v,v,-v,v),(4v,2v,3v,v))\in\Omega_{0}(13,1)$

and

$\omega^{-}:=({\mathbf{s}}^{-},{\mathbf{p}}^{-},{\mathbf{y}}^{-})=((4,3,2,1,3),(v,-v,v,-v,v),(4v,v,3v,2v))\in\Omega_{0}(13,1)$

where $v\in{\mathbf{S}}^{d-1}$ is any unit vector. This example is depicted in
Figure 9. Then the collision partition associated to $\omega^{+}$ and
$\omega^{-}$ is given by

$\displaystyle
P=\\{\\{(+,1),(-,1)\\},\\{(+,2),(-,4)\\},\\{(+,3),(-,3)\\},\\{(+,4),(-,2)\\}\\}.$

Figure 9. Another example of a pair of paths with a nontrivial collision
partition and such that neither $\omega$ nor $\omega^{\prime}$ has a
recollision event. The paths are depicted with a slight downward drift to
clarify the order of the collisions. Note that this behavior is typical in one
dimension.

This partition has a nontrivial structure because the second collision of
$\omega^{-}$ correlates with the fourth collision of $\omega^{+}$ and vice-
versa. This behavior is not uncommon for paths that are constrained to one
dimension. The problem with the above example is that there are non-
consecutive collisions in $\Gamma$ which can be visited by a single path with
two collisions.

We need to introduce another type of incidence to prevent this behavior which
we call a _tube incidence_.

######  Definition 4.6 (Tube incidences).

A _tube incidence_ for a path $\omega\in\Omega_{\alpha,k}(s,S)$ is a pair
$(a,a^{\prime})\in[0,k]$ of collisions such that there exists a collision
$b\in I^{imm}(\omega)$ with $a<b<a^{\prime}$ and there exists a time
$s\in{\mathbf{R}}$ such that

(4.8) $|y_{a}+sp_{a}-y_{a^{\prime}}|\leq 100k^{4}\alpha^{20}r.$

Above we use the convention $y_{0}=y_{1}-s_{0}p_{0}$. We set
$I^{tube}(\omega)$ to be the set of pairs $(a,a^{\prime})$ that form a tube
incidence.

A tube incidence occurs when a particle scattering out of site $a$ can choose
to “skip” its next collision (possibly at $b$) and instead scatter at site
$a^{\prime}$.

The key idea is that the partition $P({\mathbf{y}})$ of a doubled path
$\bm{\Gamma}$ is severely constrained when neither $\omega^{+}$ nor
$\omega^{-}$ have an incidence. In particular, the partition must be a
generalized ladder. To define a generalized ladder we first define a ladder
partition.

######  Definition 4.7 (Ladder partitions).

Let $A$ and $B$ be two finite ordered sets with $|A|=|B|$. The _ladder
matching_ $P_{lad}\in{\mathcal{P}}(A\sqcup B)$ is the unique matching of the
form $P_{lad}=\\{\\{a,\varphi(a)\\}\\}_{a\in A}$ where $\varphi:A\to B$ is the
unique order-preserving bijection between $A$ and $B$.

We can now state the main result.

######  Lemma 4.8.

Let
$\bm{\Gamma}=(\xi_{0}^{+},\omega^{+},\xi_{1}^{+};\xi_{0}^{-},\omega^{-},\xi_{1}^{-})$
be a doubled path such that $X(\bm{\Gamma})$ is
$|\log{\varepsilon}|^{20}$-complete and
$d_{r}(\xi_{0}^{+},\xi_{0}^{-})\leq|\log{\varepsilon}|^{20}$. Then at least
one of the following holds:

  * •

One of $\omega^{+}$ or $\omega^{-}$ has an incidence. That is,

$I^{rec}(\omega^{+})\cup I^{tube}(\omega^{+})\cup I^{rec}(\omega^{-})\cup
I^{tube}(\omega^{-})\not=\varnothing.$

  * •

The partition $P(\bm{\Gamma})$ has a cell with more than two elements.

  * •

The partition $P(\bm{\Gamma})$ is a generalized ladder in the sense that (1)
$P$ saturates the set $I^{imm}(\omega^{+})\cup I^{imm}(\omega^{-})$ and (2)
The restriction $P|_{K(\bm{\Gamma})\setminus I^{imm}(\omega^{+})\setminus
I^{imm}(\omega^{-})}$ is a ladder partition on the set $K(\Gamma^{+})\setminus
I^{imm}(\omega^{+})\sqcup K(\Gamma^{-})\setminus I^{imm}(\omega^{-})$.

###### Proof.

We will assume that we are not in either of the first two cases, so that
$P(\bm{\Gamma})$ is a perfect matching of $K(\bm{\Gamma})$ and neither
$\omega^{+}$ nor $\omega^{-}$ has an incidence.

First we observe that every immediate recollision $(a,a+1)$ must also be a
cell in $P(\bm{\Gamma})$. Since $P(\bm{\Gamma})$ is assumed to be a perfect
matching, it follows that $P(\bm{\Gamma})$ saturates the set of immediate
recollisions.

It remains to show that $P|_{K(\bm{\Gamma})\setminus I^{imm}(\bm{\Gamma})}$
forms a ladder partition. Order the collisions in $K(\Gamma^{+})\setminus
I^{imm}(\Gamma^{+})$ as $(a_{1},a_{2},\cdots,a_{m})$, and the collisions of
$K(\Gamma^{-})\setminus I^{imm}$ as $(b_{1},b_{2},\cdots,b_{m^{\prime}})$
(with $b_{j+1}>b_{j}$ and $a_{j+1}>a_{j}$).We first observe that there are no
pairs $\\{a_{j},a_{j^{\prime}}\\}\in P(\bm{\Gamma})$ because then
$(j,j^{\prime})$ would form a recollision. Likewise
$\\{b_{j},b_{j^{\prime}}\\}\not\in P(\bm{\Gamma})$ for any $j,j^{\prime}$.
This shows that $m=m^{\prime}$. To show that $P(\bm{\Gamma})$ is a generalized
ladder, it suffices to check that $\\{a_{j},b_{j}\\}\in P(\bm{\Gamma})$ for
every $j\in[m]$. We prove this by induction on $j$.

The base case is that $j=1$. Choose $j_{1}$ such that
$\\{a_{1},b_{j_{1}}\\}\in P(\bm{\Gamma})$. then $|y_{a_{1}}-y_{b_{j_{1}}}|\leq
2r$. Using (4.7), we have

$\displaystyle|y_{a_{1}}-(y_{+,0}+t_{a_{1}}p_{+,0})|$ $\displaystyle\leq
4k^{2}\alpha^{20}r$ $\displaystyle|y_{b_{1}}-(y_{-,0}+t_{b_{1}}p_{-,0})|$
$\displaystyle\leq 4k^{2}\alpha^{20}r.$

It follows that for $s=t_{a_{1}}-t_{b_{1}}$,

$|y_{b_{j_{1}}}-(y_{+,0}+sp_{-,0}|\leq 10k^{2}\alpha^{20}r,$

so that either $j_{1}=1$ or else $j_{1}>1$ and therefore $(0,b_{j_{1}})$ is a
tube incidence for $\Gamma^{-}$.

The proof of the inductive step follows the same argument, with an additional
calculation to show that $|p_{a_{j}}-p_{b_{j}}|\leq
m|\log{\varepsilon}|^{20}r^{-1}$ if $\\{a_{j^{\prime}},b_{j^{\prime}}\\}\in
P(\bm{\Gamma})$ for all $j^{\prime}\leq j$. ∎

###  4.3. Ladders and the semigroup property

Now we state the main result of the section, which is that the expectation
appearing in ${\mathcal{E}}_{s}[A]$ is well approximated by simply summing
over the ladder matchings.

######  Definition 4.9 (Generalized ladders).

A generalized ladder partition on the set $[k_{+}]\sqcup[k_{-}]\\{(\pm,j)\mid
j\leq k_{\pm}\\}$ is a partition $P$ of $[k_{+}]\sqcup[k_{-}]$ such that there
exists a set $I_{+}^{imm}\subset[k_{+}]$ and $I_{-}^{imm}\subset[k_{-}]$ such
that $\\{(+,j),(+,j+1)\\}\in P$ for every $j\in I_{+}^{\operatorname{Imm}}$
and $\\{(-,j),(-,j+1)\\}\in P$ for every $j\in I_{-}^{\operatorname{Imm}}$,
and such that $P|_{[k_{+}]\sqcup[k_{-}]\setminus\\{+\\}\times
I_{+}^{\operatorname{Imm}}\setminus\\{-\\}\times I_{-}^{\operatorname{Imm}}}$
is a ladder partition on the set $([k_{+}]\setminus
I_{+}^{imm})\sqcup([k_{-}]\setminus I_{-}^{imm})$.

We set ${\mathcal{Q}}_{gl}\subset{\mathcal{P}}([k_{+}]\sqcup[k_{-}])$ to be
the set of generalized ladder partitions.

To relate partitions to the superoperator ${\mathcal{E}}_{s}$ we first define
the notion of a $P$-expectation. Given a partition
$P\in{\mathcal{P}}([k_{+}]\sqcup[k_{-}])$, $\omega^{+}\in\Omega_{k_{+}}$ and
$\omega^{-}\in\Omega_{k_{-}}$, we define

(4.9)
${\mathbf{E}\,}_{P}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}=\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|p_{+,k_{+}}}\rangle}\mathinner{\langle{p_{+,0}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|p_{-,k_{-}}}\rangle}\mathinner{\langle{p_{-,0}|\xi_{0}^{-}}\rangle}\end{smallmatrix}e^{i(\varphi(\omega^{+})-\varphi(\omega^{-}))}{\varepsilon}^{k_{+}+k_{-}}\prod_{S\in
P}{\mathbf{E}\,}\prod_{a\in S}{\widehat{V_{y_{a}}}}(q_{a}).$

We then define the ladder expectation to be the sum over all ladder
partitions,

${\mathbf{E}\,}_{lad}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}:=\sum_{P\in{\mathcal{Q}}_{rl}([k_{+}]\sqcup[k_{-}])}{\mathbf{E}\,}_{P}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}$

We are now ready to define the ladder superoperator ${\mathcal{L}}_{s}$.

(4.10)
${\mathcal{L}}_{s}[A]:=\int\mathinner{|{\xi_{0}^{-}}\rangle}\mathinner{\langle{\xi_{0}^{+}}|}\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma){\mathbf{E}\,}_{lad}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}\mathop{}\\!\mathrm{d}\Gamma.$

For convenience we define

The main result of this section is that the ladder superoperator
${\mathcal{L}}_{s}$ is a good approximation to the evolution
${\mathcal{E}}_{s}$.

######  Proposition 4.10.

Let $A$ be an operator with good support and let
$s\leq{\varepsilon}^{-2-\kappa/2}$. Then

$\|{\mathcal{E}}_{s}[A]-{\mathcal{L}}_{s}[A]\|_{op}\leq
C{\varepsilon}^{2.1}s\|A\|_{op}.$

Before we are ready to prove Proposition 4.10 we must first establish that the
sum over _all_ generalized ladders is the same as the sum over ther _correct_
generalized ladder in the case that $\bm{\Gamma}$ is a path with no
incidences.

######  Lemma 4.11.

Let $\bm{\Gamma}$ be a $|\log{\varepsilon}|^{20}$-complete path and suppose
that $d_{r}(\xi_{0}^{+},\xi_{0}^{-})\leq|\log{\varepsilon}^{-20}$, and
$|\xi_{p}|\geq r^{-1}\sigma^{-1}$. Suppose moreover that $\bm{\Gamma}$ has no
incidences and that $P(\bm{\Gamma})$ is a matching. Then

${\mathbf{E}\,}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}={\mathbf{E}\,}_{lad}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}.$

###### Proof.

By Lemma 4.8, it follows that for _some_ generalized ladder
$P\in{\mathcal{Q}}_{gl}([k_{+}]\sqcup[k_{-}])$,

${\mathbf{E}\,}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}={\mathbf{E}\,}_{P}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}.$

We will show that if $P^{\prime}\in{\mathcal{Q}}_{gl}([k_{+}]\sqcup[k_{-}])$
is another generalized ladder with $P^{\prime}\not=P$, then

${\mathbf{E}\,}_{P^{\prime}}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}=\prod_{S\in
P}{\mathbf{E}\,}\prod_{a\in S}{\widehat{V_{y_{a}}}}(q_{a})=0.$

Indeed, since $P^{\prime}\not=P$ there exists some
$a,b,b^{\prime}\in[k_{+}]\sqcup[k_{-}]$ such that $\\{a,b\\}\in
P=P(\bm{\Gamma})$ and $\\{a,b^{\prime}\\}\in P^{\prime}$ with $b\not=b$. But
since $\\{a,b^{\prime}\\}\not\in P(\bm{\Gamma})$, $|y_{a}-y_{b^{\prime}}|>2r$
so the expectation

${\mathbf{E}\,}{\widehat{V_{y_{a}}}}(q_{a}){\widehat{V_{y_{b}}}}(q_{b^{\prime}})=0$

vanishes. ∎

###  4.4. The proof of Proposition 4.10

The error ${\mathcal{E}}_{s}[A]-{\mathcal{L}}_{s}[A]$ can be written as a path
integral

${\mathcal{E}}_{s}[A]-{\mathcal{L}}_{s}[A]=\int\mathinner{|{\xi_{0}^{-}}\rangle}\mathinner{\langle{\xi_{0}^{+}}|}\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma)\big{(}{\mathbf{E}\,}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}-{\mathbf{E}\,}_{lad}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}\big{)}\mathop{}\\!\mathrm{d}\bm{\Gamma}.$

The argument of Lemma 4.2 still works to show that we can restrict the
integral to paths that are $|\log{\varepsilon}|^{20}$-complete. Moreover,
using the support condition on $A$ we can also restrict to the case that
$d_{r}(\xi_{1}^{+},\xi_{1}^{-})\leq|\log{\varepsilon}|^{20}$ and that
$|(\xi_{1}^{+})_{p}|\geq(r\sigma)^{-1}$.

Under these constraints, the only paths that contribute to the path integral
above are those for which either $P(\bm{\Gamma})$ has a cell of more than two
elements and paths which have some kind of incidence. Let
${\mathbf{1}}^{bad}(\bm{\Gamma})$ be the indicator function for such paths. We
decompose this function according to the exact partition $P(\bm{\Gamma})$ and
the exact incidence set $I^{rec}(\bm{\Gamma})$, $I^{tube}(\bm{\Gamma})$,

$\displaystyle{\mathbf{1}}^{bad}(\bm{\Gamma})$
$\displaystyle=\sum_{P\in{\mathcal{P}}([k_{+}]\sqcup[k_{-}])}\sum_{I^{rec},I^{tube}}{\mathbf{1}}(P(\bm{\Gamma})=P){\mathbf{1}}(I^{rec}(\bm{\Gamma})=I^{rec}){\mathbf{1}}(I^{tube}(\bm{\Gamma})=I^{tube})$
$\displaystyle=:\sum_{P\in{\mathcal{P}}([k_{+}]\sqcup[k_{-}])}\sum_{I^{rec},I^{tube}}{\mathbf{1}}_{P,I^{rec},I^{tube}}(\bm{\Gamma}).$

where the sum includes the constraint that either $I^{rec}\cup
I^{rec}\not=\varnothing$ or else $P$ has a cell with at least three elements.
Then we have the estimate

$\displaystyle\|{\mathcal{E}}_{s}[A]-{\mathcal{L}}_{s}[A]\|_{op}\leq\sum_{P,I^{rec},I^{tube}}\Big{\|}\int\mathinner{|{\xi_{0}^{-}}\rangle}\mathinner{\langle{\xi_{0}^{+}}|}$
$\displaystyle\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma){\mathbf{1}}_{P,I^{rec},I^{tube}}(\bm{\Gamma})$
$\displaystyle\big{(}{\mathbf{E}\,}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}-{\mathbf{E}\,}_{lad}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}\big{)}\mathop{}\\!\mathrm{d}\bm{\Gamma}\Big{\|}_{op}.$

Using the triangle inequality we bound

$\big{|}{\mathbf{E}\,}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}-{\mathbf{E}\,}_{lad}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}\big{|}\leq\sum_{Q\leq
P(\bm{\Gamma})}\big{|}{\mathbf{E}\,}_{Q}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}\big{|},$

and now applying the Schur test we estimate

(4.11)
$\begin{split}\|{\mathcal{E}}_{s}[A]-{\mathcal{L}}_{s}[A]\|_{op}\leq\sum_{k_{+},k_{-}}&\sum_{P,I^{rec},I^{tube}}\\\
&\sup_{\xi_{0}^{-}}\Big{|}\int\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma){\mathbf{1}}_{P,I^{rec},I^{tube}}(\bm{\Gamma})\sum_{Q\leq
P(\bm{\Gamma})}|{\mathbf{E}\,}_{Q}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}|\mathop{}\\!\mathrm{d}\omega^{\pm}\mathop{}\\!\mathrm{d}\xi_{1}^{\pm}\mathop{}\\!\mathrm{d}\xi_{0}^{-}\Big{|}^{1/2}\\\
&\times\sup_{\xi_{0}^{+}}\Big{|}\int\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma){\mathbf{1}}_{P,I^{rec},I^{tube}}(\bm{\Gamma})\sum_{Q\leq
P(\bm{\Gamma})}|{\mathbf{E}\,}_{Q}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}|\mathop{}\\!\mathrm{d}\omega^{\pm}\mathop{}\\!\mathrm{d}\xi_{1}^{\pm}\mathop{}\\!\mathrm{d}\xi_{0}^{+}\Big{|}^{1/2}.\end{split}$

The first step to bound the term
${\mathbf{E}\,}_{Q}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}$
appearing in the integrand. Then, expanding out the formula (4.9) we have

$\big{|}{\mathbf{E}\,}_{Q}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}\big{|}\leq\begin{smallmatrix}|\mathinner{\langle{\xi_{1}^{+}|p_{+,k_{+}}}\rangle}\mathinner{\langle{p_{+,0}|\xi_{0}^{+}}\rangle}|\\\
|\mathinner{\langle{\xi_{1}^{-}|p_{-,k_{-}}}\rangle}\mathinner{\langle{p_{-,0}|\xi_{0}^{-}}\rangle}|\end{smallmatrix}e^{i(\varphi(\omega^{+})-\varphi(\omega^{-}))}\prod_{S\in
P(\bm{\Gamma})\vee Q}{\mathbf{E}\,}\prod_{a\in
S}{\widehat{V_{y_{a}}}}(q_{a}).$

Now we use

$|\langle{p|\xi}\rangle|\leq Cr^{d/2}\exp(-c(r|(\xi)_{p}-p|)^{0.5})$

as well as Lemma B.3 to estimate

$\displaystyle\big{|}{\mathbf{E}\,}_{Q}\begin{smallmatrix}\mathinner{\langle{\xi_{1}^{+}|O_{\omega^{+}}|\xi_{0}^{+}}\rangle}\\\
\mathinner{\langle{\xi_{1}^{-}|O_{\omega^{-}}|\xi_{0}^{-}}\rangle}^{*}\end{smallmatrix}\big{|}\leq
Cr^{2d}{\varepsilon}^{k_{+}+k_{-}}$
$\displaystyle\begin{smallmatrix}\exp(-c(r|(\xi_{1}^{+})_{p}-p_{+,k_{+}}|)^{0.5})\exp(-c(r|(\xi_{0}^{+})_{p}-p_{+,0}|)^{0.5})\\\
\exp(-c(r|(\xi_{1}^{-})_{p}-p_{-,k_{-}}|)^{0.5})\exp(-c(r|(\xi_{0}^{-})_{p}-p_{-,0}|)^{0.5})\end{smallmatrix}$
$\displaystyle
r^{-d(k_{+}+k_{-})}(C(k_{+}+k_{-}))^{k_{+}+k_{-}}\prod_{Q^{\prime}\leq
P(\bm{\Gamma})\vee Q}\prod_{S^{\prime}\in
Q^{\prime}}r^{d}\exp\Big{(}-c\big{|}r|S|^{-1}\sum_{a\in
S}q_{a}\big{|}^{0.99}\Big{)}$
$\displaystyle\qquad\qquad\qquad\qquad\times\prod_{a\in[k_{+}]\sqcup[k_{-}]}(1+|q_{a}|)^{-20d}.$

We collect the important terms above in the function $E_{Q}(\bm{\Gamma})$,

(4.12)
$E_{Q}(\bm{\Gamma}):=r^{2d}r^{-d(k_{+}+k_{-})}{\varepsilon}^{k_{+}+k_{-}}\prod_{S\in
Q}r^{d}\exp(-c|\frac{r}{|S|}\sum_{a\in
S}q_{a}|^{0.99})\times\prod_{a}(1+|q_{a}|)^{-20d}.$

Because $k_{+},k_{-}\leq k_{max}$, the combinatorial factors contribute at
most an absolute constant. Therefore Proposition 4.10 reduces to the following
integral bound.

######  Lemma 4.12.

Let $A$ be an admissible operator, let $k_{+},k_{-}\in[k_{max}]$, let
$Q\in{\mathcal{P}}([k_{+}]\sqcup[k_{-}])$, and let $(P,I^{rec},I^{imm})$ be a
triple such that if $I^{rec}\cup I^{tube}=\varnothing$, then $P$ has a cell of
more than two elements. Then

(4.13)
$\sup_{\xi_{0}^{-}}\Big{|}\int\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma)E_{Q\vee
P}(\bm{\Gamma}){\mathbf{1}}_{P,I^{rec},I^{tube}}(\bm{\Gamma})\mathop{}\\!\mathrm{d}\omega^{\pm}\mathop{}\\!\mathrm{d}\xi_{1}^{\pm}\mathop{}\\!\mathrm{d}\xi_{0}^{-}\Big{|}\leq
C{\varepsilon}^{2.25}\tau.$

To estimate (4.13) we use the following simple lemma, which is just an
iterated application of Fubini’s theorem and the triangle inequality.

######  Lemma 4.13.

Let ${\mathcal{X}}=X_{1}\times\cdots\times X_{N}$ be the product of $N$
measure spaces $X_{j}$, and let ${\mathcal{X}}_{j}=X_{1}\times\cdots\times
X_{j}$ be the product of the first $j$ factors. Then for any positive
functions $F_{j}:{\mathcal{X}}_{j}\to{\mathbf{R}}^{+}$,

(4.14)
$\int_{{\mathcal{X}}}\prod_{j=1}^{N}F_{j}(x_{1},\cdots,x_{j})\mathop{}\\!\mathrm{d}X\leq\prod_{j=1}^{N}\sup_{x^{\prime}_{1},\dots,x^{\prime}_{j-1}}\int_{X_{j}}F_{j}(x^{\prime}_{1},\dots,x^{\prime}_{j-1},x_{j})\mathop{}\\!\mathrm{d}x_{j}.$

To apply Lemma 4.13 we need to order the variables in $\bm{\Gamma}$ and bound
the integrand as a product of functions constraining each variable in
$\bm{\Gamma}$ as a function only of variables that come earlier in the
ordering. The reader may find it useful to refer to Figure 10 for a quick
overview of the constraints on the variables.

Figure 10. Two typical collision pairs that can appear. On the left, a
recollision. The contribution from recollisions is heuristically counted as
follows: The $s_{0}$ variable has only the constraint $0\leq s_{0}\leq\tau$.
The $s_{1}$ variable is then constrained to $s_{1}{\,\lesssim\,}r|p_{1}|^{-1}$
so that $|y_{1}-y_{2}|\leq r$. Then $p_{1}$ is chosen from an annulus of width
$r^{-1}$ and radius $|p_{0}|$ and $p_{2}$ is effectively constrained by a
delta function in the momentum variables. The total contribution is at most
${\varepsilon}^{2}\tau{\,\lesssim\,}1$. On the right, a typical “rung”
collision forming a ladder. The time variable $s_{0}$ satisfies $0\leq
s_{0}\leq\tau$, but then $s_{0}^{\prime}$ is constrained by
$|s_{0}-s^{\prime}_{0}|{\,\lesssim\,}r|p_{0}|^{-1}$ so that
$|y_{1}-y_{1}^{\prime}|\leq r$. The momentum variable $p_{1}$ is again chosen
from an annulus of thickness $r^{-1}$ and width $|p_{0}|$, and
$p_{1}^{\prime}$ is constrained by a delta function to match $p_{1}$. Again
the contribution is bounded by ${\varepsilon}^{2}\tau$.

We use the following ordering of the variables:

(4.15)
$\bm{\Gamma}=(\xi_{0}^{+},p_{+,0},s_{+,0},y_{+,1},\cdots,y_{+,k_{+}},p_{+,k_{+}},s_{+,k_{+}},\xi_{1}^{+},\xi_{1}^{-},p_{-,k_{-}},s_{-,k_{-}},y_{-,k_{-}},\cdots,y_{-,1}p_{-,0},s_{-,0},\xi_{0}^{-}).$

Given a variable label $\lambda\in\\{{\mlq\mlq y\mrq\mrq}_{a},{\mlq\mlq
p\mrq\mrq}_{a},{\mlq\mlq s\mrq\mrq}_{a}\\}_{a\in
K(\bm{\Gamma})}\cup\\{{\mlq\mlq\xi\mrq\mrq}_{\ell}^{\pm}\\}_{\ell\in\\{0,1\\}}$,
we define the partial paths $\bm{\Gamma}_{<\lambda}$ to be the sequence of
variables preceding $\lambda$. Thus for example $\bm{\Gamma}_{<{\mlq\mlq
y\mrq\mrq}_{+,1}}=(\xi_{0}^{+},p_{+,0},s_{+,0})$. We also define a total
ordering $\leq^{\prime}$ on $[k_{+}]\sqcup[k_{-}]$ implied by the ordering of
the variables (14.3), in which $(+,j)\leq^{\prime}(-,j^{\prime})$ for any
$j,j^{\prime}$ and $(\pm,j)\leq(\pm,j^{\prime})$ when $\pm j\leq\pm
j^{\prime}$ (that is, the ordering is reversed for the negative indices, as
indicated in (14.3)).

The next step is to bound the integrand as a product of constraints assigned
to each variable as a function of the prior variables.
${\mathbf{1}}_{P,I^{rec},I^{tube}}(\bm{\Gamma})$. We will write out the
“standard term” in the integrand that does not use the indicator function as a
product of constraints of the $p$, $y$, and $\xi$ variables.

$|\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}||\Xi(\bm{\Gamma})|E_{Q\vee
P}(\bm{\Gamma})\leq{\varepsilon}^{k_{+}+k_{-}}F_{p}(\bm{\Gamma})F_{y}(\bm{\Gamma})F_{\xi}(\bm{\Gamma}).$

The constraints on momentum come from several sources. First there is a term
$\prod(1+|q_{a}|)^{-20d}$ ensuring that no impulse is too large. Second, there
is a term enforcing conservation of kinetic energy. Third, there are terms
ensuring that $p_{+,0}$ and $p_{-,k_{-}}$ match with $(\xi_{0}^{+})_{p}$ and
$(\xi_{1}^{-})_{p}$, respectively. Finally, there are the constraints
(approximate delta functions) coming from the expectation. We also take the
factor of $r^{2d}$ from (4.12) and distribute one factor of $r^{d}$ to each
$p_{+,0}$ and $p_{-,k_{-}}$:

$\displaystyle F_{p}(\bm{\Gamma}):=$
$\displaystyle\prod(1+|p_{a}-p_{a-1}|)^{-20d}\prod{\mathbf{1}}(||p_{\pm,j}|-|p_{\pm,0}||\leq\alpha\max\\{s_{\pm,j}^{-1},r^{-1}\\})$
$\displaystyle\times(r^{d}{\mathbf{1}}(|p_{+,0}-(\xi_{0}^{+})_{p}|\leq\alpha
r^{-1}))(r^{d}{\mathbf{1}}(|p_{-,{k_{-}}}-(\xi_{1}^{-})_{p}|\leq\alpha
r^{-1}))\prod_{S\in{Q\vee P}}r^{d}\exp(-ck_{max}^{-1}|r\sum_{a\in
S}q_{a}|^{0.99})$

The constraints in the position variables are determined by the compatibility
conditions $|y_{a+1}-(y_{a}+s_{a}p_{a})|\leq\alpha r$ and the compatibility of
the first and last collisions of $\omega^{+}$ and $\omega^{-}$ against the
boundaries $\xi_{0}^{+}$ and $\xi_{1}^{-}$. We take the factor of
$r^{-d(k_{+}+k_{-})}$ from (4.12) and distribute one $r^{-d}$ to each $y_{a}$
variable:

$\displaystyle F_{y}(\bm{\Gamma}):=$
$\displaystyle\prod(r^{-d}{\mathbf{1}}(|y_{a+1}-(y_{a}+s_{a}p_{a})|\leq\alpha
r))$
$\displaystyle\qquad\times(r^{-d}{\mathbf{1}}(|y_{+,1}-((\xi^{+}_{0})_{x}+s_{0}p_{0})|\leq\alpha
r))(r^{-d}{\mathbf{1}}(|y_{-,k_{-}}-((\xi^{-}_{1})_{x}-s_{-,k_{-}}p_{-,k_{-}})|)).$

The constraint on the $\xi$ variables comes from the compatibility with the
path, along with the support condition on $A$

$\displaystyle
F_{\xi}(\bm{\Gamma}):={\mathbf{1}}(d_{r}(\xi_{1}^{+},(y_{+,k_{+}}+s_{+,k_{+}}p_{+,k_{+}},p_{+,k_{+}}))\leq\alpha){\mathbf{1}}(d_{r}(\xi_{0}^{-},(y_{-,1}-s_{-,0}p_{-,0},p_{-,0}))\leq\alpha)(1+d_{r}(\xi_{1}^{+},\xi_{1}^{-}))^{-20d}.$

There are also indirect constraints on the $s$ variables coming indirectly
from the combination of the compatibility conditions
$|y_{a+1}-(y_{a}+s_{a}p_{a})|\leq\alpha r$ and constraints of the form
$|y_{a}-y_{b}|\leq 2kr$ for collisions $a\sim_{P}b$ of the same cell of the
partition $P(\bm{\Gamma})$. We also note that by Lemma 4.5, we have
$|y_{a^{\prime}}-(y_{a}+(t_{a^{\prime}}-t_{a})p_{a})|\leq C\alpha^{20}r$ when
$a$ and $a^{\prime}$ are collisions separated only by immediate recollisions.

We have no worry of “double-dipping” on the basic compatibility constraints
such as ${\mathbf{1}}(|y_{a+1}-(y_{a}+s_{a}p_{a})|\leq\alpha r)$ because for
example

$\operatorname{supp}(F_{p}(\bm{\Gamma})F_{y}(\bm{\Gamma})F_{\xi}(\bm{\Gamma}))\subset\\{|y_{a+1}-(y_{a}+s_{a}p_{a})|\leq\alpha
r\\},$

so we can freely apply extra such indicator functions where they are useful.

####  4.4.1. The “standard constraint” bounds

In this section we use the partitions $P$ and $Q$ to “assign constraints” to
each variable.

In particular, we write define functions
$f_{\lambda}(\bm{\Gamma}_{\leq\lambda})$ such that

$\displaystyle F_{p}(\bm{\Gamma})$ $\displaystyle\leq\prod f_{{\mlq\mlq
p\mrq\mrq},a}(\bm{\Gamma}_{\leq{\mlq\mlq p\mrq\mrq},a})$ $\displaystyle
F_{y}(\bm{\Gamma})$ $\displaystyle\leq\prod f_{{\mlq\mlq
y\mrq\mrq},a}(\bm{\Gamma}_{\leq{\mlq\mlq y\mrq\mrq},a})$ $\displaystyle
F_{\xi}(\bm{\Gamma})$ $\displaystyle\leq
f_{{\mlq\mlq\xi\mrq\mrq},1,+}(\bm{\Gamma}_{\leq{\mlq\mlq\xi\mrq\mrq},1,+})f_{{\mlq\mlq\xi\mrq\mrq},1,-}(\bm{\Gamma}_{\leq{\mlq\mlq\xi\mrq\mrq},1,-})f_{{\mlq\mlq\xi\mrq\mrq},0,-}(\bm{\Gamma}_{\leq{\mlq\mlq\xi\mrq\mrq},0,-}).$

These “standard constraint” functions can simply be read off of the
definitions of $F_{p}$, $F_{y}$, and $F_{\xi}$.

(4.16) $f_{{\mlq\mlq p\mrq\mrq},a}(\bm{\Gamma}_{\leq{\mlq\mlq
p\mrq\mrq},a}):=\begin{cases}r^{d}\exp(-ck_{max}^{-1}|r\sum_{a\in
S}q_{a}|^{0.99}),&a=\max_{\leq^{\prime}}S\text{ for some }S\in P\vee Q\\\
r^{d}{\mathbf{1}}(|p_{-,k_{-}}-(\xi_{1}^{-})_{p}|\leq\alpha
r^{-1}),&a=(-,k_{-})\\\
r^{d}{\mathbf{1}}(|p_{+,0}-(\xi_{0}^{+})_{p}|\leq\alpha r^{-1}),&a=(+,0)\\\
(1+|p_{a}-p_{a-1}|)^{-20d}{\mathbf{1}}(||p_{a}|-|p_{+,0}||\leq\alpha\max\\{|p_{0}|^{-1}s_{a}^{-1},r^{-1}\\}),&\text{
else}.\end{cases}$

The standard $y$ constraints are given by

(4.17) $f_{{\mlq\mlq y\mrq\mrq},a}(\bm{\Gamma}_{\leq{\mlq\mlq
y\mrq\mrq},a}):=\begin{cases}r^{-d}{\mathbf{1}}(|y_{+,1}-((\xi^{+}_{0})_{x}+s_{0}p_{0})|\leq\alpha
r),&a=(+,1)\\\
r^{-d}{\mathbf{1}}(|y_{-,k_{-}}-((\xi^{-}_{1})_{x}-s_{-,k_{-}}p_{-,k_{-}})|),&a=(-,k_{-})\\\
r^{-d}{\mathbf{1}}(|y_{+,j}-(y_{+,j-1}+s_{+,j-1}p_{+,j-1})|\leq\alpha
r),&a=(+,j)\text{ for }j>1\\\
r^{-d}{\mathbf{1}}(|y_{-,j+1}-(y_{-,j}+s_{-,j}p_{-,j})|\leq\alpha
r),&a=(-,j)\text{ for }j<k_{-}.\end{cases}$

Finally, the standard $\xi$ constraints are

$\displaystyle
f_{{\mlq\mlq\xi\mrq\mrq},1,+}(\bm{\Gamma}_{\leq{\mlq\mlq\xi\mrq\mrq},1,+})$
$\displaystyle:={\mathbf{1}}(d_{r}(\xi_{1}^{+},(y_{+,k_{+}}+s_{+,k_{+}}p_{+,k_{+}},p_{+,k_{+}}))\leq\alpha)$
$\displaystyle
f_{{\mlq\mlq\xi\mrq\mrq},1,-}(\bm{\Gamma}_{\leq{\mlq\mlq\xi\mrq\mrq},1,-})$
$\displaystyle:=(1+d_{r}(\xi_{1}^{+},\xi_{1}^{-}))^{-20d}$ $\displaystyle
f_{{\mlq\mlq\xi\mrq\mrq},0,-}(\bm{\Gamma}_{\leq{\mlq\mlq\xi\mrq\mrq},0,-})$
$\displaystyle:={\mathbf{1}}(d_{r}(\xi_{0}^{-},(y_{-,1}-s_{-,0}p_{-,0},p_{-,0}))\leq\alpha).$

The contributions from the ${\mlq\mlq\xi\mrq\mrq}$ and ${\mlq\mlq y\mrq\mrq}$
variables are easy to account for

######  Lemma 4.14 (Standard position and phase space bounds).

For any $\lambda\in\\{({\mlq\mlq
y\mrq\mrq},a),({\mlq\mlq\xi\mrq\mrq},\ell,\pm)\\}$

$\sup_{\bm{\Gamma}_{<\lambda}}\int
f_{\lambda}(\bm{\Gamma}_{\leq\lambda})\mathop{}\\!\mathrm{d}\Gamma_{\lambda}\leq
C.$

The momentum constraints are slightly more complicated.

######  Lemma 4.15 (Standard momentum bounds).

If $a=\max_{\leq^{\prime}}S$ for some $S\in P\vee Q$ or
$a\in\\{(-,k_{-}),(+,0)\\}$, then

$\sup_{\bm{\Gamma}_{<{\mlq\mlq p\mrq\mrq},a}}\int f_{{\mlq\mlq
p\mrq\mrq},a}(\bm{\Gamma}_{\leq{\mlq\mlq
p\mrq\mrq},a})\mathop{}\\!\mathrm{d}p_{a}\leq C.$

Otherwise, if $a\not\sim_{P}a+1$ is not the first collision of an immediate
recollision, then

(4.18) $\sup_{\bm{\Gamma}_{<{\mlq\mlq p\mrq\mrq},a}}\int f_{{\mlq\mlq
p\mrq\mrq},a}(\bm{\Gamma}_{\leq{\mlq\mlq
p\mrq\mrq},a})\mathop{}\\!\mathrm{d}p_{a}\leq
Cr^{-1}\min\\{|p_{0}|^{d-1},1\\}.$

###### Proof.

Only the second bound needs proof. Since $a+1\not\sim_{P}a$, it follows from
Lemma 4.4 that $s_{a}\gtrsim\alpha|p_{0}|^{-1}r$. Therefore
$|p_{0}|^{-1}s_{a}^{-1}{\,\lesssim\,}r^{-1}$, so $p_{a}$ is constrained to an
annulus of thickness $r^{-1}$ and radius $|p_{0}|$. This annulus has volume
$r^{-1}|p_{0}|^{d-1}$. If $|p_{0}|\gtrsim 1$ the additional factor
$(1+|p_{a}-p_{a-1}|)^{-20d}$ ensures that $p_{a}$ is essentially also confined
to a ball of unit radius. ∎

The immediate recollisions require some more detailed attention. If $a\sim
a+1$ is an immediate recollision, then we group the variables $(s_{a},p_{a})$
and use the following estimate.

######  Lemma 4.16.

For any $|p_{0}|>r^{-1}\sigma^{-1}\geq{\varepsilon}^{0.5}$ and
$q\in{\mathbf{R}}^{d}$,

(4.19)
$\begin{split}\int_{0}^{10\alpha|p_{0}|^{-1}r}\int_{{\mathbf{R}}^{d}}&{\mathbf{1}}(||p|^{2}/2-|p_{0}|^{2}/2|\leq\alpha
s^{-1})(1+|p-q|)^{-20d}\mathop{}\\!\mathrm{d}q\mathop{}\\!\mathrm{d}s\\\ &\leq
C\min\\{|p_{0}|^{-1},|p_{0}|^{d-2}\\}(1+\log({\varepsilon}^{-1})).\end{split}$

###### Proof.

We split the integral over $s$ into dyadic intervals $[2^{k},2^{k+1}]$ for
$k\in{\mathbb{Z}}$. On this interval, the variable $p$ is retricted to an
annulus of radius $|p_{0}|$ and width $|p_{0}|^{-1}2^{-k}$. Moreover,

Now consider the case $|p_{0}|\gtrsim 1$. In this case the factor
$(1+|p-q|)^{-20d}$ additionally restricts the integration over $p$ to a unit
ball, and the integration over $p_{0}$ produces a factor on the order
$\min\\{1,|p_{0}|^{-1}2^{-k}\\}$. Integrating over $s$ to produce a factor
$2^{k}$ and summing over $k$ such that $2^{k}\leq 20\alpha|p_{0}|^{-1}r$, we
obtain the bound

$\displaystyle\int_{0}^{10\alpha|p_{0}|^{-1}r}\int_{{\mathbf{R}}^{d}}$
$\displaystyle{\mathbf{1}}(||p|^{2}/2-|p_{0}|^{2}/2|\leq\alpha
s^{-1})(1+|p-q|)^{-20d}\mathop{}\\!\mathrm{d}q\mathop{}\\!\mathrm{d}s$
$\displaystyle\leq\sum_{\begin{subarray}{c}k\in{\mathbb{Z}}\\\
2^{k}<|p_{0}|^{-1}\end{subarray}}2^{k}+\sum_{\begin{subarray}{c}|p_{0}|^{-1}<2^{k}<20\alpha|p_{0}|^{-1}r\end{subarray}}|p_{0}|^{-1}$
$\displaystyle\leq|p_{0}|^{-1}(1+\log(20\alpha r)).$

The second case we consider is that $|p_{0}|\lesssim 1$. In this case, the
annulus of radius $|p_{0}|$ and width $|p_{0}|^{-1}2^{-k}$ has volume on the
order $|p_{0}|^{d-2}2^{-k}$. The bound then follows from integrating in $s$
and summming over $k$, as above. ∎

Conspicuously missing from the discussion above is the integration over the
time variables. For many of the time variables we simply use the constraint
$s_{a}\leq\tau$ to pick up a factor of $\tau$. Additional constraints come
from the partition $P$. Suppose that $a\leq^{\prime}b$ and $a\sim_{P}b$. Then
there is a constraint $|y_{b}-y_{a}|\leq 2r$, which coupled with the
constraint $|y_{b}-(y_{b-1}+s_{b-1}p_{b-1})|\leq\alpha r$ imposes a constraint
on $s_{b-1}$ in terms of the variables $y_{a}$,$y_{b-1}$, and $p_{b-1}$, which
are all in $\Gamma_{<{\mlq\mlq s\mrq\mrq},b}$. This constraint picks up a
factor of $|p_{0}|^{-1}r$ instead of $\tau$:

(4.20) $\sup_{\Gamma_{<{\mlq\mlq
s\mrq\mrq},b-1}}\int{\mathbf{1}}(|y_{a}-(y_{b-1}+s_{b-1}p_{b-1})|\leq\alpha
r)\mathop{}\\!\mathrm{d}s_{b-1}\leq|p_{0}|^{-1}r.$

####  4.4.2. The case that $P$ has a cluster

With just the bounds we have already proven, it is possible to obtain a good
estimate in the case that $P$ has a set of more than $2$ elements. This is the
simplest case, as suggested by Figure 11.

Figure 11. The case that the collision partition has a cluster. There is only
one full time degree of freedom contributing a factor of $\tau$, but three
factors of ${\varepsilon}$ from the potential. Such clusters therefore
contribute ${\varepsilon}^{3}\tau$ rather than ${\varepsilon}^{2}\tau$.

######  Lemma 4.17 (The cluster bound).

Let $A$ be an admissible operator, let $k_{+},k_{-}\in[k_{max}]$, let
$Q\in{\mathcal{P}}([k_{+}]\sqcup[k_{-}])$, and let
$P\in{\mathcal{P}}([k_{+}]\sqcup[k_{-}])$ be a partition having a cell of more
than two elements. Then

(4.21)
$\sup_{\xi_{0}^{-}}\Big{|}\int\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma)E_{Q\vee
P}(\bm{\Gamma}){\mathbf{1}}_{P}(\bm{\Gamma})\mathop{}\\!\mathrm{d}\omega^{\pm}\mathop{}\\!\mathrm{d}\xi_{1}^{\pm}\mathop{}\\!\mathrm{d}\xi_{0}^{-}\Big{|}\leq
C^{k_{+}+k_{-}}({\varepsilon}^{2}\tau)^{(k_{+}+k_{-})/2}\tau^{-1}.$

###### Proof.

We bound the standard part of the integrand by the product $F_{p}F_{y}F_{s}$
as described in the previous section and apply Lemma 4.13. The ${\mlq\mlq
y\mrq\mrq}$ and ${\mlq\mlq\xi\mrq\mrq}$ variables contribute a factor of
$C^{k_{+}+k_{-}}$. The product of the contributions from the $(s_{a},p_{a})$
pairs coming from immediate recollisions produces a factor of
$(C\log{\varepsilon}^{-1})^{n_{r}}$, where $n_{r}$ is the number of immediate
recollision clusters of $P$. To account for the ${\mlq\mlq s\mrq\mrq}$ and
${\mlq\mlq p\mrq\mrq}$ variables, let $P^{\prime}\subset P$ The ${\mlq\mlq
p\mrq\mrq}$ variables contribute a total of
$(Cr^{-1}\min\\{|p_{0}|^{d-1},1\\})^{|P|-n_{r}}$ by taking the product of the
integral over all $p_{a}$ with $a=\max_{\leq^{\prime}}S$ for each $S\in P$.
Then for each $s_{a}$ variable that is the first in its cluster (of which
there are $|P|$), we get a trivial factor of $\tau$. Each of the rest of the
$s_{a}$ variables contribute $|p_{0}|^{-1}r$ by (4.20). The product of all of
these factors gives

(4.22)
$\begin{split}\sup_{\xi_{0}^{-}}\Big{|}&\int\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma)E_{Q\vee
P}(\bm{\Gamma}){\mathbf{1}}_{P}(\bm{\Gamma})\mathop{}\\!\mathrm{d}\omega^{\pm}\mathop{}\\!\mathrm{d}\xi_{1}^{\pm}\mathop{}\\!\mathrm{d}\xi_{0}^{-}\Big{|}\\\
&\qquad\leq
C^{k_{+}+k_{-}}{\varepsilon}^{k_{+}+k_{-}}\tau^{|P|}(C\min\\{|p_{0}|^{d-2},|p_{0}|^{-1}\\})^{|P|-n_{r}}\end{split}$

The first and last factors are bounded by $C^{k_{+}+k_{-}}$. Then the fact
that $P$ is not a perfect matching implies $|P|<(k_{+}+k_{-})/2$ and since
$|P|$ is an integer in particular it follows that $|P|\leq(k_{+}+k_{-})/2-1$.
This proves (4.21). ∎

####  4.4.3. The recollision case

To complete the proof of Lemma 4.12 we need to find a way to use the
additional constraints coming from a recollision or tube incidence. A
simplified version of the argument is presented in Figure 12

Figure 12. A path with a recollision at indices $(2,5)$. On the left, the case
that $y_{4}$ is far from $y_{5}$. In this case the momentum variable $p_{4}$
is constrained to be approximately parallel to $y_{2}-y_{4}$. On the right,
the case that $y_{4}$ is close to $y_{5}$. In this case there is an additional
constraint on the time variable $s_{3}$.

Suppose that $(a,b)$ is a recollision occuring in $\omega^{+}$,
$\operatorname{sgn}(a)=\operatorname{sgn}(b)=+$. The idea is that a
recollision at $(a,b)$ typically enforces a strong constraint on the momentum
_before_ the collision at $b$. Indeed, if $|y_{a}-y_{b}|\leq 2r$ then
$|s_{b-1}p_{b-1}-(y_{a}-y_{b-1})|\leq 2r$, so in particular

$|p_{b-1}-z|\leq 2rs_{b-1}^{-1},$

with $z=(y_{a}-y_{b-1})s_{b-1}^{-1}$. If $|y_{a}-y_{b-1}|>10\alpha r$, then
$s_{b-1}\geq|y_{a}-y_{b-1}||p_{0}|^{-1}$. This is where the constraint on
$p_{b-1}$ comes from. On the other hand, if $|y_{a}-y_{b-1}|$ is small, then
there is a constraint on $s_{b-2}$ of exactly the same kind as (4.20). The
only additional subtlety to deal with is the possibility that $b-1$ is itself
a recollision or immediate recollision, in which case $p_{b-1}$ is localized
by the momentum constraint on a collision cluster and further localization is
not helpful. This is only a minor difficulty, and so we now prove

######  Lemma 4.18 (The recollision bound).

Let $A$ be an admissible operator, let $k_{+},k_{-}\in[k_{max}]$, let
$P,Q\in{\mathcal{P}}([k_{+}]\sqcup[k_{-}])$, and let $I^{rec}\not=\varnothing$
be a nonempty set of recollisions in $[k_{+}]$. Then

(4.23)
$\sup_{\xi_{0}^{-}}\Big{|}\int\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma)E_{Q}(\bm{\Gamma}){\mathbf{1}}_{P}(\bm{\Gamma}){\mathbf{1}}_{I^{rec}}(\bm{\Gamma})\mathop{}\\!\mathrm{d}\omega^{\pm}\mathop{}\\!\mathrm{d}\xi_{1}^{\pm}\mathop{}\\!\mathrm{d}\xi_{0}^{-}\Big{|}\leq
C^{k_{+}+k_{-}}({\varepsilon}^{2}\tau)^{(k_{+}+k_{-})/2}({\varepsilon}^{-1.5}\tau^{-1}).$

###### Proof.

Let $(a,a^{\prime})\in I^{rec}$ be the recollision with minimal $a^{\prime}$.
Then let $b<a^{\prime}$ be the first collision before $a^{\prime}$ that is not
an immediate recollision. Because $(a,a^{\prime})$ is not an immediate
recollision, it is clear that $b\not=a^{\prime}$. Moreover, because
$(a,a^{\prime})$ is minimal, the index $b$ is not of the form
$\max_{\leq^{\prime}}S$ for any $S\in Q$.

We bound the indicator function for a recollision as a sum of indicator
functions depending on the distance $|y_{b-1}-y_{b}|$

${\mathbf{1}}_{I^{rec}}(\bm{\Gamma})\leq{\mathbf{1}}(|y_{a}-y_{b}|\leq
2r){\mathbf{1}}(|y_{b-1}-y_{b}|\geq Kr)+{\mathbf{1}}(|y_{b-1}-y_{b}|\leq Kr),$

and use this to split (4.27) as a sum of two integrals each corresponding to a
different term.

For the first term we follow the proof of Lemma 4.17, with the modification
that we set

$f^{\prime}_{{\mlq\mlq p\mrq\mrq},b-1}(\bm{\Gamma}_{\leq{\mlq\mlq
p\mrq\mrq}_{b-1}})={\mathbf{1}}(|p_{b-1}-|p_{0}|\frac{y_{b-1}-y_{a}}{|y_{b-1}-y_{a}|}|\leq|p_{0}|K^{-1}).$

In this case $p_{b-1}$ is sampled from the intersection of an annulus with
thickness $r^{-1}$ and radius $|p_{0}|$ and a ball or radius $|p_{0}|K^{-1}$,
so that

$\sup_{\bm{\Gamma}_{<{\mlq\mlq p\mrq\mrq}_{b-1}}}\int f^{\prime}_{{\mlq\mlq
p\mrq\mrq},b-1}(\bm{\Gamma}_{\leq{\mlq\mlq
p\mrq\mrq}_{b-1}})\mathop{}\\!\mathrm{d}p_{b-1}\leq
Cr^{-1}\min\\{(K^{-1}|p_{0}|)^{d-1},1\\}.$

Applying this estimate in the place of the bound (4.18) along with the rest of
the argument that leads to (4.22) yields

(4.24)
$\begin{split}\sup_{\xi_{0}^{-}}\Big{|}&\int\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma)E_{Q\vee
P}(\bm{\Gamma}){\mathbf{1}}_{P}(\bm{\Gamma}){\mathbf{1}}(|y_{a}-y_{b}|\leq
2r){\mathbf{1}}(|y_{b-1}-y_{b}|\geq
Kr)\mathop{}\\!\mathrm{d}\omega^{\pm}\mathop{}\\!\mathrm{d}\xi_{1}^{\pm}\mathop{}\\!\mathrm{d}\xi_{0}^{-}\Big{|}\\\
&\qquad\leq
C^{k_{+}+k_{-}}{\varepsilon}^{k_{+}+k_{-}}\tau^{|P|}(C\min\\{|p_{0}|^{d-2},|p_{0}|^{-1}\\})^{|P|-n_{r}-1}(C\min\\{K^{1-d}|p_{0}|)^{d-2},|p_{0}|^{-1}\\})\end{split}$

The last factor is maximized when $|K^{1-d}||p_{0}|^{d-1}=|p_{0}|^{-1}$, which
occurs when $|p_{0}|=K$. In this case we obtain a savings of $K^{-1}$ over the
bound (4.22), and therefore conclude

(4.25)
$\begin{split}\sup_{\xi_{0}^{-}}\Big{|}&\int\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma)E_{Q\vee
P}(\bm{\Gamma}){\mathbf{1}}_{P}(\bm{\Gamma}){\mathbf{1}}(|y_{a}-y_{b}|\leq
2r){\mathbf{1}}(|y_{b-1}-y_{b}|\geq
Kr)\mathop{}\\!\mathrm{d}\omega^{\pm}\mathop{}\\!\mathrm{d}\xi_{1}^{\pm}\mathop{}\\!\mathrm{d}\xi_{0}^{-}\Big{|}\\\
&\qquad\leq
C^{k_{+}+k_{-}}{\varepsilon}^{k_{+}+k_{-}}\tau^{|P|}K^{-1}.\end{split}$

The second term to deal with is the integral involving
${\mathbf{1}}(|y_{b-1}-y_{b}|\leq Kr)$. In this case, we apply the bound
(4.20) to get a factor of $Kr$ instead of $\tau$ for the integration over the
variable $s_{b-2}$. Thus

(4.26)
$\begin{split}\sup_{\xi_{0}^{-}}\Big{|}&\int\mathinner{\langle{\xi_{1}^{-}|A|\xi_{1}^{+}}\rangle}\Xi(\Gamma)E_{Q\vee
P}(\bm{\Gamma}){\mathbf{1}}_{P}(\bm{\Gamma}){\mathbf{1}}(|y_{b-1}-y_{b}|\leq
Kr)\mathop{}\\!\mathrm{d}\omega^{\pm}\mathop{}\\!\mathrm{d}\xi_{1}^{\pm}\mathop{}\\!\mathrm{d}\xi_{0}^{-}\Big{|}\\\
&\qquad\leq
C^{k_{+}+k_{-}}{\varepsilon}^{k_{+}+k_{-}}\tau^{|P|}(Kr\tau^{-1}).\end{split}$

Choosing $K={\varepsilon}^{-0.5}$ yields the desired result. ∎

####  4.4.4. The tube incidence case

The final remaining case is that $\bm{\Gamma}$ does not have a recollision but
does have a tube incidence. Suppose that the first tube incidence occurs at
$(a,b)$. Then combining the tube incidence constraint (4.8) with the
compatibility condition $|y_{b}-y_{b-1}-s_{b-1}p_{b-1}|\leq 2r$, we conclude
that there exists $s\in{\mathbf{R}}$ such that

$|s_{b-1}p_{b-1}+(y_{b-1}-y_{a})+sp_{a}|\leq C\alpha^{20}r.$

In other words, the vector $s_{b-1}p_{b-1}$ lies on the tube with thickness
$\alpha^{20}r$, axis $p_{a}$, and passing through $y_{b-1}-y_{a}$. If $p_{a}$
is transverse to $p_{b-1}$ then this imposes a strong constraint on the time
variable $s_{b-1}$. On the other hand if $p_{a}$ is parallel with $p_{b-1}$
then this imposes a constraint on the momentum variable $p_{b-1}$. Both cases
are depicted in Figure 13

Figure 13. Paths with a tube incidence at $(1,5)$, so that
$|y_{5}-(y_{1}+sp_{1})|\lessapprox r$ for some $s\in{\mathbf{R}}$. On the
left, an example in which $p_{4}$ is transverse to $p_{1}$. In this case the
time variable $s_{4}$ is constrained so that $y_{5}$ can lie on the gray tube.
On the right, an example in which $p_{4}$ is approximately parallel to
$p_{1}$. In this case $s_{4}$ is much less constrained, but $p_{4}$ is much
more constrained. In either case there is a gain of a factor of (at least)
${\varepsilon}^{1/2}$.

######  Lemma 4.19 (The tube incidence bound).

Let $A$ be an admissible operator, let $k_{+},k_{-}\in[k_{max}]$, let
$P,Q\in{\mathcal{P}}([k_{+}]\sqcup[k_{-}])$, and let
$I^{tube}\not=\varnothing$ be a nonempty set of tube incidences in $[k_{+}]$,
and suppose moreover that $I^{rec}=\varnothing$ so that there are no
recollisions.

(4.27)