# Lieb–Thirring-type inequalities for random Schrödinger operators with
complex potentials

Jean-Claude Cuenin  Department of Mathematical Sciences, Loughborough
University, Loughborough, Leicestershire LE11 3TU, United Kingdom
<EMAIL_ADDRESS>and  Konstantin Merz  Institut für Analysis und Algebra,
Technische Universität Braunschweig, Universitätsplatz 2, 38106 Braunschweig,
Germany, and Department of Mathematics, Graduate School of Science, Osaka
University, Toyonaka, Osaka 560-0043, Japan<EMAIL_ADDRESS>
(Date: August 27, 2023)

###### Abstract.

We review some results and proofs on eigenvalue bounds for random Schrödinger
operators with complex-valued potentials. We also include new Schatten norm
estimates for the resolvent and use them to obtain bounds for sums of
eigenvalues.

The second-named author thanks Neal Bez and Yutaka Terasawa for the
organization and invitation to the RIMS Symposium on Harmonic Analysis and
Nonlinear Partial Differential Equations, where parts of this note were
presented. This work was supported by the Research Institute for Mathematical
Sciences, an International Joint Usage/Research Center located in Kyoto
University. Support by the Engineering & Physical Sciences Research Council
[grant numberEP/X011488/1] (J.-C. C.) and by the PRIME programme of the German
Academic Exchange Service (DAAD) with funds from the German Federal Ministry
of Education and Research (BMBF) (K.M.) is acknowledged. We thank Haruya
Mizutani for helpful discussions.

###### Contents

  1. 1 Introduction
    1. 1.1 Eigenvalue estimates for Schrödinger operators with complex potentials
    2. 1.2 Eigenvalue estimates for Schrödinger operators with random complex potentials
    3. 1.3 Ideas of the proof of Theorems 1–2
      1. 1.3.1 On the proof of Theorem 1
      2. 1.3.2 On the proof of Theorem 3
      3. 1.3.3 On the proof of Theorem 2
  2. 2 Eigenvalue sums
    1. 2.1 Alternative estimate for $\|\mathcal{V}_{M}^{(\omega)}\|$
    2. 2.2 Bounds for singular values of $\mathcal{V}_{M}^{(\omega)}$
    3. 2.3 Eigenvalue sums of random Schrödinger operators

##  1\. Introduction

###  1.1. Eigenvalue estimates for Schrödinger operators with complex
potentials

Schrödinger operators with complex-valued potentials arise in the analysis of
scattering resonances, open or damped quantum systems, etc. For decaying
potentials $V:\mathbb{R}^{d}\to\mathbb{C}$ that are not too singular,
$-\Delta-V$ can be realized as an m-sectorial operator in
$L^{2}(\mathbb{R}^{d})$, with essential spectrum $[0,\infty)$; see, e.g., [13,
Proposition B.2]111In the present situation, all definitions of essential
spectrum coincide.. We are interested in the discrete eigenvalues of finite
algebraic multiplicity, $\sigma_{\rm disc}(-\Delta-V)$, which can only
accumulate at $[0,\infty)$. Natural quantities of interest are, e.g., the
number and distribution of eigenvalues of $-\Delta-V$. For complex-valued $V$,
the analysis of these quantities yields results that are surprisingly
different from those in the case of real-valued $V$. We refer to [14, Section
5.13] and the references therein for a survey.

In the first part of this note, we review estimates for single eigenvalues of
$-\Delta-V$ which only depend on $\|V\|_{q}$. In the second part, we prove new
estimates for sums of eigenvalues. As in previous studies on this subject, the
decisive feature of $V$ is its decay. Hence, for the rest of the paper, we
will always assume that $V$ is bounded.

To motivate the sought-after estimates, let us consider the situation of real-
valued $V$ for a moment. In this case, the Keller–Lieb–Thirring inequality

$\displaystyle\sum_{\lambda_{j}\in\sigma_{\rm
disc}(-\Delta-V)}|\lambda_{j}|^{q-d/2}\lesssim_{d,q}\int_{\mathbb{R}^{d}}V(x)_{+}^{q}\,dx,\quad\begin{cases}q\in[1,\infty),&\
d=1\\\ q\in(1,\infty),&\ d=2\\\ q\in[d/2,\infty),&\ d\geq 3\end{cases}$ (1)

quantifies the eigenvalue accumulation at zero when $q>d/2$ and the number of
eigenvalues when $q=d/2$ and $d\geq 3$. Here and below, we write $A\lesssim B$
for two non-negative quantities $A,B\geq 0$ to indicate that there is a
constant $C>0$ such that $A\leq CB$. The dependence of the constant on fixed
parameters like $d$ and $q$ is usually omitted. The notation $A\sim B$ means
$A\lesssim B\lesssim A$. Inequality (1) for single eigenvalues was first
proved by Keller [17] and later generalized to eigenvalue sums by Lieb and
Thirring [21], who used this inequality to give a simple proof of stability of
quantum matter. Inequality (1) is interesting because it is invariant under
spatial rescalings and recovers Weyl’s law in the strong-coupling limit, i.e.,
when one replaces $V$ with $\kappa V$, and considers the limit
$\kappa\to\infty$.

A natural question to ask is how much of (1) survives when $V$ is complex-
valued. A first answer in the context of single eigenvalues was given by
Abramov, Aslanyan, and Davies [1]. For $d=1$, they showed

$\displaystyle|z|^{1/2}\leq\frac{1}{2}\int_{\mathbb{R}}|V|\,dx,\quad
z\in\sigma_{\rm disc}(-\Delta-V).$ (2)

Shortly afterwards, Frank, Laptev, Lieb, and Seiringer [15] considered the
higher-dimensional case and $V\in L^{q}(\mathbb{R}^{d})$, $q\geq d/2$, and
showed

$\displaystyle\sum_{z_{j}\in\sigma_{\rm
disc}(-\Delta-V),\,|\operatorname{Im}z_{j}|\geq\kappa\operatorname{Re}z_{j}}|z_{j}|^{q-\frac{d}{2}}\lesssim\left(1+\kappa^{-1}\right)^{q}\int_{\mathbb{R}^{d}}|V|^{q}.$
(3)

While this inequality shows that (1) continues to hold for eigenvalues of
$-\Delta-V$ outside any fixed sector in the complex plane, (3) deteriorates
for eigenvalues close to $[0,\infty)$. For individual eigenvalues, Laptev and
Safronov [19, p. 31] made the following

######  Conjecture 1.

Let $d\geq 2$, $q\in(d/2,d]$. Then, any eigenvalue
$z\in\mathbb{C}\setminus[0,\infty)$ of $-\Delta-V$ satisfies

$\displaystyle|z|^{q-\frac{d}{2}}\lesssim_{d,q}\int_{\mathbb{R}^{d}}|V|^{q}.$
(4)

Laptev and Safronov were careful to state this conjecture only for $q\leq d$
because they were aware of the real-valued Wigner–Neumann potential [23, p.
223]. This is a radial potential which oscillates and decays like $|x|^{-1}$,
i.e., it almost belongs to $L^{d}(\mathbb{R}^{d})$, and generates an
eigenvalue embedded at $1$. Bögli [2] proved that the range of $q$ would be
best possible for radial potentials. Frank and Simon [16] proved Conjecture 1
for radial potentials when $q<d$. Disregarding the endpoint $q=d$, these two
results provide a complete answer to the Laptev–Safronov conjecture for radial
potentials. For general potentials, Frank [12] proved

$\displaystyle|z|^{q-d/2}\lesssim\int_{\mathbb{R}^{d}}|V(x)|^{q}\,dx,\quad
z\in\sigma_{\rm disc}(-\Delta-V),\ q\leq\frac{d+1}{2}.$ (5)

The starting point to prove (5) is the Birman–Schwinger principle, i.e., the
statement that

$\displaystyle z\in\sigma_{\rm disc}(-\Delta-V)\Leftrightarrow 1\in\sigma_{\rm
disc}(BS(z)),\quad BS(z):=|V|^{1/2}(-\Delta-z)^{-1}V^{1/2},$ (6)

with $V^{1/2}=V/|V|^{1/2}$. In particular, if $z\in\sigma_{\rm
disc}(-\Delta-V)$, then the spectral radius ${\rm spr}(BS(z))\geq 1$. Thus, to
get an eigenvalue estimate, it suffices to bound ${\rm spr}(BS(z))$ from above
by a negative power of $|z|$. A simple estimate is ${\rm
spr}(BS(z))\leq\|BS(z)\|$. By Hölder’s inequality and the homogeneity of
$-\Delta$,

$\displaystyle\begin{split}1\leq{\rm
spr}(BS(z))\leq\|BS(z)\|&\leq\|V\|_{q}\|(-\Delta-z)^{-1}\|_{p\to
p^{\prime}}\\\ &=|z|^{d/(2q)-1}\|V\|_{q}\|(-\Delta-z/|z|)^{-1}\|_{p\to
p^{\prime}},\end{split}$ (7)

where $1/q=1/p-1/p^{\prime}$ and $p^{\prime}=(1-1/p)^{-1}$ for $1\leq
p\leq\infty$. Kenig, Ruiz, and Sogge [18] showed
$\|(-\Delta-z/|z|)^{-1}\|_{p\to p^{\prime}}\lesssim_{p,d}1$ for all
$z\in\mathbb{C}\setminus[0,\infty)$ if and only if $q\in[d/2,(d+1)/2]$. Thus,
(5) follows. Let us comment on the necessity of $q\in[d/2,(d+1)/2]$. The lower
bound is a consequence of Sobolev’s inequality. We explain the necessity of
$q\leq(d+1)/2$. By Stone’s formula and the spectral theorem or Plancherel’s
theorem, $\operatorname{Im}(-\Delta-1+i0)^{-1}$ is proportional to the
operator $\mathcal{E}\mathcal{E}^{*}$, i.e., the composition of the Fourier
restriction operator

$\displaystyle\mathcal{S}(\mathbb{R}^{d})\ni
f\mapsto(\mathcal{E}^{*}f)(\xi):=\int_{\mathbb{R}^{d}}dx\,\mathrm{e}^{-2\pi
ix\cdot\xi}f(x)\Big{|}_{\xi\in\mathbb{S}^{d-1}}\in
L^{\infty}(\mathbb{S}^{d-1})$ (8)

and the Fourier extension operator

$\displaystyle L^{\infty}(\mathbb{S}^{d-1})\ni
g\mapsto(\mathcal{E}g)(x):=\int_{\mathbb{S}^{d-1}}d\sigma(\xi)\mathrm{e}^{2\pi
ix\cdot\xi}g(\xi)\in\mathcal{S}^{\prime}(\mathbb{R}^{d}),$ (9)

with the Leray surface measure $\sigma$ on $\mathbb{S}^{d-1}$. In the
following, we will also write
$\hat{f}(\xi):=(\mathcal{F}f)(\xi):=\int_{\mathbb{R}^{d}}\mathrm{e}^{-2\pi
ix\cdot\xi}f(x)\,dx$ for the Fourier transform of $f$ and
$(f)^{\vee}(x)=\int_{\mathbb{R}^{d}}\mathrm{e}^{2\pi ix\cdot\xi}f(\xi)\,d\xi$
for the inverse Fourier transform. Thus, to investigate necessary conditions
for the $L^{p}\to L^{p^{\prime}}$-boundedness of $(-\Delta-z)^{-1}$, it
suffices to investigate necessary conditions for $p$ on the
$L^{p}(\mathbb{R}^{d})\to L^{2}(\mathbb{S}^{d-1})$-boundedness of
$\mathcal{E}^{*}$. On the one hand, by the Riemann–Lebesgue lemma,
$\|\mathcal{E}^{*}\|_{L^{1}(\mathbb{R}^{d})\to
L^{2}(\mathbb{S}^{d-1})}\lesssim 1$. On the other hand, since the Fourier
transform maps $L^{2}(\mathbb{R}^{d})$ unitarily into itself, a Fourier
restriction of $L^{2}(\mathbb{R}^{d})$-functions is meaningless, since
$L^{2}(\mathbb{R}^{d})$-functions belong to an equivalence class of functions
within which its members are allowed to differ from each other on Lebesgue
measure zero sets, i.e., in particular on hypersurfaces like
$\mathbb{S}^{d-1}$. Considering the Fourier transform of an indicator function
on a rectangular box $[0,\varepsilon^{-1}]^{d-1}\times[0,\varepsilon^{-2}]$
and exploiting the quadratic curvature of the sphere, one observes $p\leq
2(d+1)/(d+3)$ is a necessary condition for $\mathcal{E}^{*}$ to be
$L^{p}(\mathbb{R}^{d})\to L^{2}(\mathbb{S}^{d-1})$-bounded—this is known as
Knapp’s example [27, 24]. In particular, Knapp’s example shows the necessity
of $q\leq(d+1)/2$ in the resolvent bound of Kenig, Ruiz, and Sogge, and led to
the belief that Frank’s bound (5) is optimal. Inspired by Knapp’s
construction, Bögli and the first author [3] succeeded to construct potentials
$V\in L^{q}(\mathbb{R}^{d})$ with $q>(d+1)/2$ for which (4) is violated.

###  1.2. Eigenvalue estimates for Schrödinger operators with random complex
potentials

While the results [2, 16, 12, 3] provide a complete answer to the
Laptev–Safronov conjecture 1, it is interesting to investigate if the range of
exponents in (5) can be increased for “generic” $V$. Here, we investigate how
strong one must perturb or destroy the counterexample of [2, 3] to recover a
Keller–Lieb–Thirring inequality. A rather strong way to do so is to take this
counterexample, or, more generally, any given deterministic potential,
decomposing its support into boxes of side length $h>0$, and multiplying it on
every $h$-cube with a mean-zero random variable. More precisely, for
independently and identically distributed, mean-zero Gaussian or symmetric
Bernoulli random variables $(\omega_{j})_{j\in h\mathbb{Z}^{d}}$, we consider
the Anderson randomization of $V$, given by

$\displaystyle V_{\omega}(x)=\sum_{j\in
h\mathbb{Z}^{d}}\omega_{j}V(x)\mathbf{1}_{Q}((x-j)/h),\quad Q=[0,1)^{d}.$ (10)

We denote the product measure associated to the $\omega_{j}$ by $\mathbf{P}$
and the expectation by $\mathbf{E}$.

Our results in [9], Theorems 1–2 below, say that Frank’s eigenvalue estimate
(5) continues to hold for random potentials that are allowed to decay almost
twice as slowly as their deterministic counterparts. We state this result in
two forms. The first form involves compactly supported potentials with
$\operatorname{supp}V\subseteq B(R)$. We write $\langle x\rangle:=2+|x|$ for
$x\in\mathbb{R}^{d}$, $B(x,R):=\\{y\in\mathbb{R}^{d}:\,|x-y|<R\\}$, and $B(R)$
for a ball with radius $R$ and unspecified center.

######  Theorem 1 ([9, Theorem 1]).

There exist constants $M_{0},c>0$ such that the following holds. For any
$R,\lambda>0$, $0<h<R$, $|\varepsilon|\ll\lambda$, $q\leq d+1$, for any $V\in
L^{q}(\mathbb{R}^{d})$ with $\operatorname{supp}V\subseteq B(R)$, and for any
$M\geq M_{0}$, each eigenvalue $z=(\lambda+\mathrm{i}\varepsilon)^{2}$ of
$-\Delta-V_{\omega}$ satisfies

$\displaystyle\frac{\lambda^{2-\frac{d}{q}}}{\langle\lambda
h\rangle^{d/2}(\ln\langle\lambda R\rangle)^{7/2}}\leq
M\|V\|_{L^{q}(\mathbb{R}^{d})},$ (11)

except for $\omega$ in a set of measure at most $\exp(-cM^{2})$.

While compactly supported potentials belong to any $L^{q}$-space, the point of
(11) is the weak (logarithmic) dependence on $R$. In contrast, the
deterministic estimate (5) and Hölder’s inequality yield a power law, i.e.,

$\displaystyle\lambda^{\frac{2}{d+1}}\lesssim
R^{d(\frac{2}{d+1}-\frac{1}{q})}\|V\|_{L^{q}},\quad q\geq\frac{d+1}{2}.$ (12)

In the next theorem, we consider $V\in L^{q}$, which are not assumed to be
compactly supported anymore.

######  Theorem 2 ([9, Theorem 3]).

For any $q<d+1$, there exist constants $M_{0},c>0$ such that the following
holds. For any $h,\lambda>0$, $|\varepsilon|\ll\lambda$, for any $V\in
L^{q}(\mathbb{R}^{d})$ and for any $M\geq M_{0}$, each eigenvalue
$z=(\lambda+\mathrm{i}\varepsilon)^{2}$ of $-\Delta-V_{\omega}$ satisfies

$\displaystyle\frac{\lambda^{2-\frac{d}{q}}}{\langle\lambda
h\rangle^{d/2}(\ln\langle\lambda h\rangle)^{2}}\leq
M\|V\|_{L^{q}(\mathbb{R}^{d})},$ (13)

except for $\omega$ in a set of measure at most $\exp(-cM^{2})$.

In the next Subsection, we explain the strategy of the proof of Theorems 1–2.
In Section 2, we prove new Schatten norm estimates (Theorem 5) and apply them
to conclude bounds for eigenvalue sums of $-\Delta-V_{\omega}$ (Theorem 7).

###  1.3. Ideas of the proof of Theorems 1–2

By scaling, it suffices to consider eigenvalues of the form
$z=(1+i\varepsilon)^{2}$. Moreover, by (3), it suffices to consider
$|\varepsilon|\ll 1$. Again, the first step is to use the Birman–Schwinger
principle. For technical reasons, we consider the spectral radius, using
Gelfand’s formula

$\displaystyle{\rm spr}(BS(z))=\lim_{n\to\infty}\|BS(z)^{n}\|^{1/n},$ (14)

instead of $\|BS(z)\|$. By the decay of $V$ and the uncertainty principle, the
resolvents in

$\displaystyle|V|^{1/2}R(z)VR(z)V...$ (15)

are expected to be smoothed out in Fourier space. Indeed, by the convolution
theorem, we get, for any two balls $B(x_{1},R_{1})$ and $B(x_{2},R_{2})$,

$\displaystyle\mathbf{1}_{B(x_{1},R_{1})}R(z)\mathbf{1}_{B(x_{2},R_{2})}=\mathbf{1}_{B(x_{1},R_{1})}\mathcal{F}^{-1}\left(\frac{1}{|\xi|^{2}-z}\ast\gamma_{\delta}\right)\mathcal{F}\mathbf{1}_{B(x_{2},R_{2})},$
(16)

where $\delta^{-1}\geq R_{1}+R_{2}+|x_{1}-x_{2}|$ and
$\gamma\in\mathcal{S}(\mathbb{R}^{d})$ is a Schwartz function with
$\check{\gamma}\in C_{c}^{\infty}$. To simplify the following notation, we
denote, for $\delta>0$, by $C^{(\delta)}(D)$ any Fourier multiplier whose
symbol obeys

$\displaystyle|C^{(\delta)}(\xi)|\leq(||\xi|^{2}-|z||+\delta)^{-1/2},\quad
z=(1+i\varepsilon)^{2},\ |\varepsilon|\ll 1.$ (17)

We now explain some details of the proofs of Theorems 1 and 2.

####  1.3.1. On the proof of Theorem 1

In the situation of Theorem 1, all resolvents appearing in Gelfand’s formula
for ${\rm spr}(BS(z))$ can be replaced with $C^{(\delta)}(D)^{2}$ with
$\delta^{-1}\geq 2R$. Thus, to bound ${\rm spr}(BS(z))$, it suffices to
estimate the operator norm of the “elementary operators”
$C^{(\delta)}(D)V_{\omega}C^{(\delta^{\prime})}(D)$ for suitable
$\delta,\delta^{\prime}>0$. By elliptic estimates, it suffices to consider
those frequencies $|\xi|$ for which $|\xi|\sim|z|\sim 1$. By the spectral
theorem or Plancherel’s theorem, we have, for any
$F:\mathbb{R}_{+}\to\mathbb{C}$,

$\displaystyle
F(|D|)=\int_{0}^{\infty}d\lambda\,F(\lambda)\mathcal{E}_{\lambda}\mathcal{E}_{\lambda}^{*},$
(18)

where we denoted $M_{\lambda}:=\\{\xi\in\mathbb{R}^{d}:\,|\xi|=\lambda\\}$,
the Fourier extension and restriction operators

$\displaystyle\mathcal{E}_{\lambda}:L^{2}(M_{\lambda},d\sigma_{\lambda})\to
L^{\infty}(\mathbb{R}^{d}),\quad(\mathcal{E}_{\lambda}g)(x)=\int_{M_{\lambda}}\mathrm{e}^{2\pi
ix\cdot\xi}g(\xi)\,d\sigma_{\lambda}(\xi),$

and the Leray surface measure $\sigma_{\lambda}$ on $M_{\lambda}$. Thus, by
the Cauchy–Schwarz inequality,

$\displaystyle\begin{split}&\|C^{(\delta)}(D)\mathbf{1}_{|D|\in[1/2,2]}V_{\omega}C^{(\delta^{\prime})}(D)\mathbf{1}_{|D|\in[1/2,2]}\|\\\
&\quad\lesssim\ln^{\frac{1}{2}}\frac{1}{\delta}\ln^{\frac{1}{2}}\frac{1}{\delta^{\prime}}\sup_{\lambda,\lambda^{\prime}\in[1/2,2]}\|\mathcal{E}_{\lambda}^{*}V_{\omega}\mathcal{E}_{\lambda^{\prime}}\|_{L^{2}(M_{\lambda^{\prime}})\to
L^{2}(M_{\lambda})},\end{split}$ (19)

see also [9, Lemma 18]. Thus, it suffices to estimate
$\mathcal{E}_{\lambda}^{*}V_{\omega}\mathcal{E}_{\lambda^{\prime}}$. The main
technical result in [9] is the following

######  Theorem 3 ([9, Lemma 15]).

Let $q<d+1$, $R\geq h$, and $V_{\omega}(x)$ be defined as in (10) with
$\operatorname{supp}V_{\omega}\subseteq B(R)$. Then,

$\displaystyle\sup_{\lambda,\lambda^{\prime}\in[1/2,2]}\mathbf{E}\|\mathcal{E}_{\lambda}^{*}V_{\omega}\mathcal{E}_{\lambda^{\prime}}\|_{L^{2}(M_{\lambda^{\prime}})\to
L^{2}(M_{\lambda})}\lesssim\langle h\rangle^{\frac{d}{2}}(\ln\langle
R\rangle)^{\frac{1}{2}}\left(\ln\langle h\rangle+\ln\langle
R\rangle\right)^{2}\|V\|_{q}.$ (20)

In particular, there are constants $M_{0},c>0$ such that for any $M\geq
M_{0}$, the estimate

$\displaystyle\sup_{\lambda,\lambda^{\prime}\in[1/2,2]}\|\mathcal{E}_{\lambda}^{*}V_{\omega}\mathcal{E}_{\lambda^{\prime}}\|_{L^{2}(M_{\lambda^{\prime}})\to
L^{2}(M_{\lambda})}\lesssim M\langle h\rangle^{\frac{d}{2}}(\ln\langle
R\rangle)^{\frac{1}{2}}\left(\ln\langle h\rangle+\ln\langle
R\rangle\right)^{2}\|V\|_{q}$ (21)

holds for all $\omega$ outside a set of measure at most
$\mathrm{e}^{-cM^{2}}$.

Estimate (21) follows from (20) and the Gaussian tail bound

$\displaystyle\mathbf{P}(\|X\|>t)\leq\exp\left(-\frac{ct^{2}}{(\mathbf{E}\|X\|)^{2}}\right)$
(22)

for Banach space valued random variables $X$ with symmetric Bernoulli or
Gaussian distribution222We are not aware of tail bounds for Banach space
valued random variables with more general sub-Gaussian distributions. This is
the reason for our assumption on the distribution of the $\omega_{j}$ in
Theorems 1–3, see, e.g., [9, Lemma 12] and the reference [20] therein.
Plugging (21) into (19) and combining it with the previous strategy allows to
conclude the proof of Theorem 1.

####  1.3.2. On the proof of Theorem 3

The proof of Theorem 3 uses techniques of Bourgain [4, 5] in the context of
scattering theory of random lattice Schrödinger operators with long-range
potentials. The basic ingredient to prove Theorem 3 is the square root
cancellation in the estimate

$\displaystyle\mathbf{E}\sup_{j\leq N}\sum_{i=1}^{j}|X_{i}|\lesssim\sqrt{\log
N}\sup_{j\leq N}\sqrt{\sum_{i=1}^{j}\|X_{i}\|_{\psi_{2}}^{2}}$ (23)

for a sequence of symmetric Bernoulli or Gaussian random variables
$\\{X_{i}\\}_{i\in\mathbb{N}}$. This is stated in [9, Section 4], where we
refer to [28] for a reference of the definition of the $\psi_{2}$-norm and the
statement and proof of (23). Our goal is to apply (23) to estimate the right-
hand side of

$\displaystyle\mathbf{E}\|\mathcal{E}^{*}V_{\omega}\mathcal{E}\|=\mathbf{E}\sup_{g,g^{\prime}\in
L^{2}(M),\|g\|,\|g^{\prime}\|\leq
1}|\langle\mathcal{E}g,V_{\omega}\mathcal{E}g^{\prime}\rangle|.$ (24)

Since $L^{2}(M)$ has infinite cardinality, (23) cannot be applied directly.
However, some reductions are possible. Recall that we consider a problem where
both the frequencies and the positions are localized; indeed $|\xi|\sim 1$ and
$|x|\leq R$. By the uncertainty principle, we expect locally constant
properties in position and frequency on the respective reciprocal scales,
i.e., on the unit scale in position space, and on the scale $R^{-1}$ in
frequency space. In particular, we expect that we can discretize the problem
in position space on the unit scale and in frequency space on the scale
$R^{-1}$, and consider

$\displaystyle\begin{split}&\mathbf{E}\sup_{g,g^{\prime}\in\ell^{2}(M\cap\Lambda_{R}^{*}),\|g\|,\|g^{\prime}\|\leq
1}|\langle\mathcal{E}_{\rm d}g,v_{\omega}\mathcal{E}_{\rm
d}g^{\prime}\rangle|\end{split}$ (25)

instead of (24), for $v_{\omega}=v_{\omega}(n)=\omega_{n}v_{n}$, $n\in
h\mathbb{Z}^{d}$, discrete versions $\mathcal{E}_{\rm d}^{*}$ and
$\mathcal{E}_{\rm d}$ of the Fourier restriction and extension operators, and
a $1/R$-net $\Lambda_{R}^{*}\cap M$ of points on $M$. Following Bourgain, we
replace the supremum over the set $\\{\mathcal{E}_{\rm
d}g:\,\|g\|_{\ell^{2}(M\cap\Lambda_{R}^{*})}\leq 1\\}$ on the right-hand side
of (25) by a supremum over a finite set up to an “entropy” error. Importantly,
we can prove suitable bounds for the logarithm of the cardinality of this
finite set, called “entropy” [28, p. 170]. The idea to achieve this
replacement is as follows. Consider the set $\\{\mathcal{E}_{\rm
d}g:\,\|g\|_{\ell^{2}(M\cap\Lambda_{R}^{*})}\leq 1\\}$ and cover it with
disjoint dyadic boxes of side length $2^{-k}$, $k\in\mathbb{N}_{0}$, in the
$\ell^{\infty}(\mathbb{Z}^{d}\cap B(R))$-metric. We collect the centers of
these boxes in the set $Z_{k}:=\\{\xi_{k}(j)\\}_{j\in\mathbb{N}_{0}}$. Then,
to approximate any given $\mathcal{E}_{\rm d}g\in\ell^{\infty}$ with
$\|g\|_{\ell^{2}(M\cap\Lambda_{R}^{*})}\leq 1$, we start from the origin, walk
to the center $\xi_{0}(j_{0})$ of a box with side length $2^{0}$, which is
closest to $\mathcal{E}_{\rm d}g$, then walk to the center $\xi_{1}(j_{1})$ of
a box with side length $2^{-1}$, which is closest to $\mathcal{E}_{\rm d}g$,
etc. In this way, we can, for any $g\in\ell^{2}(M\cap\Lambda_{R}^{*})$ with
$\|g\|_{2}\leq 1$, construct a chain of centers $\\{\xi_{k}\\}$ of boxes of
side length $2^{-k}$ converging to $\mathcal{E}_{\rm d}g$. Formally, this is
just the telescoping series

$\displaystyle\begin{split}\mathcal{E}_{\rm d}g&=\mathcal{E}_{\rm
d}g-\xi_{0}(j_{0})+\xi_{0}(j_{0})=\mathcal{E}_{\rm
d}g-\xi_{0}(j_{0})+\xi_{0}(j_{0})-\xi_{1}(j_{1})+\xi_{1}(j_{1})=...\\\
&=:\mathcal{E}_{\rm d}g-\xi_{0}(j_{0})+\sum_{k\geq
0}\xi^{(k)},\quad\xi^{(k)}\in\mathcal{F}_{k}\subseteq
Z_{k}-Z_{k+1},\end{split}$ (26)

with the property that $\|\xi^{(k)}\|_{\ell^{\infty}}\lesssim 2^{-k}$,
$\xi^{(k)}\in\mathcal{F}_{k}$. To apply (23), we thus need to compute the
cardinality of the subset $\mathcal{F}_{k}$ of all difference vectors between
boxes with centers in $Z_{k}$ and $Z_{k+1}$ which are as close to each other
as possible. To that end, we use the “dual Sudakov inequality”, first proved
by Pajor and Tomczak–Jaegermann [22] and later, in a simpler fashion, by
Bourgain, Milman, and Lindenstrauss [6, Proposition 4.2]. It says that
$\ln|Z_{k}|\lesssim\ln(\langle R\rangle)4^{k}$. Thus,

$\displaystyle\ln|\mathcal{F}_{k}|\lesssim\ln(\langle R\rangle)4^{k}.$ (27)

A rigorous implementation of these ideas requires an averaging over
translations. Writing $\mathrm{e}(x\cdot\xi):=\mathrm{e}^{2\pi ix\cdot\xi}$,
we showed in the proof of [9, Lemma 15] that Formula (26) takes the form

$\displaystyle\sum_{\nu}e((x_{i}+y)(\eta_{\nu}+\tau))g(\eta_{\nu}+\tau)=\sum_{k\geq
0}\xi_{i}^{(k)},\quad\xi^{(k)}\in\mathcal{F}_{k},$ (28)

with $\\{x_{i}\\}$ and $\\{\eta_{\nu}\\}$ constituting a $1$-net in
$\mathbb{R}^{d}$ and a $R^{-1}$-net $\Lambda_{R}^{*}$ of $M$, respectively,
$y\in B(0,10)$, $\tau,\tau^{\prime}\in M\cap B(0,10/R)$, and

$\displaystyle\|\xi^{(k)}\|_{\ell^{\infty}}\lesssim
2^{-k}R^{\frac{d-1}{2}}\|g(\eta_{\nu}+\tau)\|_{\ell_{\nu}^{2}},\quad\|\xi^{(k)}\|_{\ell^{2(d+1)/(d-1)}}\lesssim
R^{\frac{d-1}{2}}\|g(\eta_{\nu}+\tau)\|_{\ell_{\nu}^{2}}.$ (29)

Thus, we can bound the right-hand side of (24) by

$\displaystyle\mathbf{E}\sup_{g,g^{\prime}\in
L^{2}(M),\|g\|,\|g^{\prime}\|\leq
1}|\langle\mathcal{E}^{*}V_{\omega}\mathcal{E}g,g^{\prime}\rangle|\leq\sum_{k,k^{\prime}\in\mathbb{Z}_{+}}\int\mathbf{E}\max_{(\xi,\xi^{\prime})\in\mathcal{F}_{k}\times\mathcal{F}_{k^{\prime}}}|X_{\xi,\xi^{\prime}}|\,dy\,d\tau\,d\tau^{\prime},$
(30)

where the dependence of

$\displaystyle
X_{\xi,\xi^{\prime}}=X_{\xi,\xi^{\prime}}(y,\tau,\tau^{\prime}):=\sum_{j\in
h\mathbb{Z}^{d}}\omega_{j}\sum_{i}\overline{V(x_{i}+y)\xi_{i}}\xi_{i}^{\prime}$
(31)

on the variables $y\in B(0,10)$, $\tau,\tau^{\prime}\in M\cap B(0,10/R)$ is
suppressed. Combining (23) with the dual Sudakov estimate (27) and applying
Hölder’s inequality repeatedly gives the probabilistic bound

$\displaystyle\int\mathbf{E}\max_{(\xi,\xi^{\prime})\in\mathcal{F}_{k}\times\mathcal{F}_{k^{\prime}}}|X_{\xi,\xi^{\prime}}|\,dy\,d\tau\,d\tau^{\prime}\lesssim(\ln\langle
R\rangle)^{1/2}h^{d/2}\|V\|_{L^{q}(\mathbb{R}^{d})}.$ (32)

On the other hand, the Riemann–Lebesgue lemma and repeated use of Hölder’s
inequality gives the deterministic bound

$\displaystyle\int\max_{(\xi,\xi^{\prime})\in\mathcal{F}_{k}\times\mathcal{F}_{k^{\prime}}}|X_{\xi,\xi^{\prime}}|dy\,d\tau\,d\tau^{\prime}\lesssim
R^{d/q^{\prime}}2^{-k-k^{\prime}}\|V\|_{L^{q}(\mathbb{R}^{d})}.$ (33)

Interpolating between these two bounds yields (20).

####  1.3.3. On the proof of Theorem 2

Even if $V$ is not necessarily compactly supported anymore, we expect the
decay of $V$ to smoothen out the resolvents in Gelfand’s formula (14). To
quantify the decay of $V$, we apply a horizontal dyadic decomposition (see,
e.g., [26, Theorem 6.6]), which is reminiscent to the definition of Lorentz
spaces. More precisely, we write

$\displaystyle V=\sum_{i\in\mathbb{Z}_{+}}V_{i},\quad
V_{i}=V\mathbf{1}_{H_{i}\geq|V|\geq H_{i+1}},\quad
H_{i}=\inf\\{t>0:\,|\\{|V|>t\\}|\leq 2^{i-1}\\}.$ (34)

The widths of the supports of $V_{i}$ are approximately $2^{i}$ and
$\|H_{i}2^{i/q}\|_{\ell^{r}_{i}(\mathbb{Z}_{+})}\sim\|V\|_{L^{q,r}}$, where
$L^{q,r}$ denotes a Lorentz space. As explained in (16), the smoothing of the
resolvents using two spatial cut-offs like
$\mathbf{1}_{\Omega_{1}}(x)R(z)\mathbf{1}_{\Omega_{2}}(x)$ depends on the
diameter of the $\Omega_{j}$ and their relative distance. While the sizes of
the $\operatorname{supp}(V_{i})$ are known, their diameter and relative
distance is not accessible yet. To control these properties, we apply a sparse
decomposition to each $\operatorname{supp}(V_{i})$. We say that for
$N\in\mathbb{N}$, $\gamma,R>0$, a family $\\{B(x_{k},R)\\}_{k=1}^{N}$ is
called $\gamma$-sparse if the centers $x_{k}$ are $(RN)^{\gamma}$-separated.
For fixed $K\gg 1$ and $\gamma>0$, whose values will be determined at the end
of the argument, we let $K_{i}=\mathcal{O}(K2^{i/K})$ be the number of sparse
families used to cover $\operatorname{supp}(V_{i})$, we let
$N_{i}=\mathcal{O}(2^{i})$ be the number of balls inside any sparse family,
and we let $R_{i}=\mathcal{O}(2^{i\gamma^{K}})$ be the radius of the balls.
Then, according to Tao [25, Lemma 3.3] (see also [7, Section 2.2]),

$\displaystyle V_{i}=\sum_{j=1}^{K_{i}}\sum_{k=1}^{N_{i}}V_{ijk},$ (35)

where, for fixed $i,j$, the $V_{ijk}$ are supported on a sparse collection of
balls $\\{B(x_{k},R_{i})\\}_{k=1}^{N_{i}}$. Plugging the decomposition (35)
into Gelfand’s formula (14) allows us to proceed similarly as in the case of
compactly supported potentials. We refer to [9, Sections 7.2–7.4] for details.

##  2\. Eigenvalue sums

Being able to locate a region in the complex plane for which we can say where
all eigenvalues of $-\Delta-V_{\omega}$ are located, it is natural to consider
the accumulation of these eigenvalues. This will lead us to estimate the
singular values of

$\displaystyle\mathcal{V}_{M_{\lambda}}^{(\omega)}:=\mathcal{E}_{\lambda}^{*}V_{\omega}\mathcal{E}_{\lambda},\quad\lambda>0.$
(36)

That is, we will estimate $\|\mathcal{V}_{M}^{(\omega)}\|_{\mathcal{S}^{p}}$
for some $p\geq 1$, with the $p$-th Schatten norm
$\|T\|_{\mathcal{S}^{p}(\mathcal{H})}^{p}:=\sum_{n\geq 1}s_{n}(T)^{p}$ and the
singular values $\\{s_{n}(T)\\}_{n\in\mathbb{N}}$ of some compact operator $T$
on a Hilbert space $\mathcal{H}$. We also write
$\|T\|_{\mathcal{S}^{p,\infty}(\mathcal{H})}:=\sup_{m}s_{m}(T)m^{\frac{1}{p}}$
for the weak $p$-th Schatten norm. For the sake of simplicity, and, because
our methods are not likely to yield optimal results for $L^{q}$-potentials, we
will only consider compactly supported or pointwise decaying potentials in
this section.

Our approach to bound $s_{k}(\mathcal{V}_{M_{\lambda}}^{(\omega)})$ consists
of two steps. First, we adapt the estimate (20) for
$\mathbf{E}\|\mathcal{V}_{M}^{(\omega)}\|$. Then, using an observation
recorded, e.g., in [11, (3.4.13)], we obtain bounds for all other singular
values of $\mathcal{V}_{M}^{(\omega)}$ by essentially comparing them to the
singular values of powers of the Laplace–Beltrami operator $-\Delta_{M}$; see
Theorem 5 for the final result. As an application, we prove estimates for sums
of eigenvalues of $-\Delta-V_{\omega}$ (Theorem 7).

###  2.1. Alternative estimate for $\|\mathcal{V}_{M}^{(\omega)}\|$

We first give an alternative bound for $\|\mathcal{V}_{M}^{(\omega)}\|$ to
that in Theorem 3, which is suitable for pointwise decaying potentials. Recall
(30) with

$\displaystyle
X_{\xi,\xi^{\prime}}=X_{\xi,\xi^{\prime}}(y,\tau,\tau^{\prime})=\sum_{j\in
h\mathbb{Z}^{d}}\omega_{j}\sum_{i}\overline{V(x_{i}+y)\xi_{i}}\xi_{i}^{\prime},$

$\\{x_{i}\\}$ constituting a $1$-net in $\mathbb{R}^{d}$, $y\in B(0,10)$, and

$\displaystyle\|\xi^{(k)}\|_{\ell^{\infty}}\lesssim
2^{-k}R^{\frac{d-1}{2}}\|g(\eta_{\nu}+\tau)\|_{\ell_{\nu}^{2}},$ (37)

where $\tau\in M\cap B(0,10/R)$ and the points $\eta_{\nu}$ form an $1/R$-net
$\Lambda_{R}^{*}$ of $M$. For the proof of the following proposition, we also
use

$\displaystyle|\\{j\in h\mathbb{Z}^{d}:\,x_{i}+y\in[j,j+h)^{d}\\}|$
$\displaystyle=1\quad\mbox{for each }i$ (38)

and

$\displaystyle|\\{i:\,x_{i}+y\in[j,j+h)^{d}\\}|$ $\displaystyle\leq
h^{d}\quad\mbox{for each }j\in h\mathbb{Z}^{d}.$ (39)

######  Proposition 4.

Let $R\geq h>0$ and $V\in L^{\infty}(\mathbb{R}^{d})$ with
$\operatorname{supp}V\subseteq B(R)$. Then,

$\displaystyle\int\mathbf{E}\max_{\mathcal{F}_{k}\times\mathcal{F}_{k^{\prime}}}|X_{\xi,\xi^{\prime}}|dyd\tau
d\tau^{\prime}\lesssim R^{1/2}(\ln\langle
R\rangle)^{1/2}h^{d/2}\|V\|_{\infty},$ (40)
$\displaystyle\int\max_{\mathcal{F}_{k}\times\mathcal{F}_{k^{\prime}}}|X_{\xi,\xi^{\prime}}|dyd\tau
d\tau^{\prime}\lesssim R\,2^{-k-k^{\prime}}\|V\|_{\infty}.$ (41)

Consequently,

$\displaystyle\begin{split}\mathbf{E}\|\mathcal{V}_{M}^{(\omega)}\|\lesssim\sum_{k,k^{\prime}}\int\mathbf{E}\max_{\mathcal{F}_{k}\times\mathcal{F}_{k^{\prime}}}|X_{\xi,\xi^{\prime}}|dyd\tau
d\tau^{\prime}\lesssim R^{1/2}\langle h\rangle^{d/2}(\ln\langle
R\rangle)^{5/2}\|V\|_{\infty},\end{split}$ (42)

and, for all $\varepsilon>0$ and $V\in\langle
x\rangle^{-(1/2+\varepsilon)}L^{\infty}$,

$\displaystyle\begin{split}\mathbf{E}\|\mathcal{V}_{M}^{(\omega)}\|\lesssim_{\varepsilon}\langle
h\rangle^{d/2}\|\langle
x\rangle^{\frac{1}{2}+\varepsilon}V\|_{\infty}.\end{split}$ (43)

###### Proof.

Formula (43) follows from (42) by a dyadic decomposition. In turn, (42)
follows from (40)–(41). To prove these inequalities, we follow [9] and assume
$\|V\|_{\infty}=1$ without loss of generality. We first prove (40). For $\ln
N:=\ln(\langle R\rangle)\cdot\max\\{4^{k},4^{k^{\prime}}\\}$, Formula (23)
yields

$\displaystyle\mathbf{E}\max_{\mathcal{F}_{k}\times\mathcal{F}_{k^{\prime}}}|X_{\xi,\xi^{\prime}}|\lesssim\sqrt{\ln
N}\left(\sum_{j\in
h\mathbb{Z}^{d}}\left|\sum_{i}\overline{V(x_{i}+y)\xi_{i}}\,\xi_{i}^{\prime}\right|^{2}\right)^{1/2}.$
(44)

By $\operatorname{supp}V\subseteq B(R)$ and Cauchy–Schwarz,

$\displaystyle\left|\sum_{i}\overline{V(x_{i}+y)\xi_{i}}\,\xi_{i}^{\prime}\right|\lesssim\sum_{|x_{i}|\lesssim
R}|\xi_{i}||\xi_{i}^{\prime}|\leq\left(\sum_{|x_{i}|\lesssim
R}|\xi_{i}|^{2}\right)^{1/2}\left(\sum_{|x_{i}|\lesssim
R}|\xi_{i}^{\prime}|^{2}\right)^{1/2}.$ (45)

Thus,

$\displaystyle\begin{split}\mathbf{E}\max_{\mathcal{F}_{k}\times\mathcal{F}_{k^{\prime}}}|X_{\xi,\xi^{\prime}}|&\lesssim\sqrt{\ln
N}\left(\sum_{j\in h\mathbb{Z}^{d}}\left(\sum_{|x_{i}|\lesssim
R}|\xi_{i}|^{2}\right)\left(\sum_{|x_{i}^{\prime}|\lesssim
R}|\xi_{i}^{\prime}|^{2}\right)\right)^{1/2}\\\ &\lesssim\sqrt{\ln
N}\left\|\sum_{|x_{i}|<R}|\xi_{i}|^{2}\right\|_{\ell_{j}^{1}}^{1/2}\left\|\sum_{|x_{i}|<R}|\xi_{i}^{\prime}|^{2}\right\|_{\ell_{j}^{\infty}}^{1/2}\\\
&\lesssim\sqrt{\ln
N}\|\xi_{i}\|_{\ell_{i}^{\infty}}\,\|\xi_{i}^{\prime}\|_{\ell_{i}^{2}(|x_{i}|<R)}\cdot
h^{d/2}.\end{split}$ (46)

To proceed, we estimate $\|\xi_{i}\|_{\ell_{i}^{2}(|x_{i}|<R)}$. Recall that
the $\xi_{i}$ are defined via
$\sum_{\nu}e((x_{i}+y)(\eta_{\nu}+\tau))g(\eta_{\nu}+\tau)=\sum_{k\geq
0}\xi_{i}^{(k)}$, i.e., they belong to the range of the discrete Fourier
restriction operator. Let
$\phi_{R}(\xi)=R^{d}\phi(R\xi)\in\mathcal{S}(\mathbb{R}^{d})$ such that
$\operatorname{supp}\phi_{R}\subseteq B(1)$ and
$\hat{\phi}\geq\mathbf{1}_{B(0,1)}$. Then, by the discrete Stein–Tomas theorem
[10, Proposition 1.29 and (1.18)] (see also [9, Section 3]),

$\displaystyle\begin{split}&\|\sum_{\mu\in\Lambda_{R}^{*}}a(\mu)e(\mu\cdot
x)\|_{L^{2}(|x|<R)}\leq\|\sum_{\mu\in\Lambda_{R}^{*}}a(\mu)\hat{\phi}_{R}(x)e(\mu\cdot
x)\|_{L^{2}(\mathbb{R}^{d})}\\\
&\quad=\|\sum_{\mu\in\Lambda_{R}^{*}}a(\mu)\phi_{R}(\xi+\nu)\|_{L^{2}(\mathbb{R}^{d})}\leq\|a(\mu)\|_{\ell_{\mu}^{2}(\Lambda_{R}^{*})}\|\phi_{R}\|_{L^{2}(\mathbb{R}^{d})}\\\
&\quad\lesssim
R^{d/2}\|a(\mu)\|_{\ell_{\mu}^{2}(\Lambda_{R}^{*})}.\end{split}$ (47)

Combining (46) with (47), we get, by symmetry between $k$ and $k^{\prime}$,

$\displaystyle\int_{B(0,10)}dy\,\iint_{B(0,10/R)^{2}}d\tau\,d\tau^{\prime}\,\mathbf{E}\max_{\mathcal{F}_{k}\times\mathcal{F}_{k^{\prime}}}|X_{\xi,\xi^{\prime}}|\lesssim\sqrt{\ln\langle
R\rangle}h^{d/2}R^{1/2}.$ (48)

This concludes the probabilistic bound (40).

We now prove the deterministic bound (41). By Hölder’s inequality,

$\displaystyle|X_{\xi,\xi^{\prime}}|\lesssim\|V(x_{i}+y)\|_{\ell_{i}^{1}}\|\xi\|_{\infty}\|\xi^{\prime}\|_{\infty}\lesssim
R^{2d-1}2^{-k-k^{\prime}}.$ (49)

Therefore,

$\displaystyle\int_{B(0,10)}dy\,\iint_{(B(0,10/R)\cap M)\times(B(0,10/R)\cap
M)}d\tau\,d\tau^{\prime}\,|X_{\xi,\xi^{\prime}}|\lesssim R2^{-k-k^{\prime}}.$
(50)

Combining (48) and (50) gives

$\displaystyle\int
dy\,d\tau\,d\tau^{\prime}\mathbf{E}|X_{\xi,\xi^{\prime}}|\lesssim\min\\{\sqrt{\ln\langle
R\rangle}h^{d/2}R^{1/2},R2^{-k-k^{\prime}}\\}$ (51)

and therefore, using [9, Lemma 25],

$\displaystyle\begin{split}\mathbf{E}\|\mathcal{E}^{*}V_{\omega}\mathcal{E}\|&\lesssim
R\sum_{k,k^{\prime}}\min\\{2^{-k-k^{\prime}},R^{-\frac{1}{2}}\ln(\langle
R\rangle)^{\frac{1}{2}}h^{\frac{d}{2}}\\}\lesssim R^{\frac{1}{2}}(\ln\langle
R\rangle)^{\frac{5}{2}}\langle h\rangle^{\frac{d}{2}}.\end{split}$ (52)

The proof is concluded. ∎

###  2.2. Bounds for singular values of $\mathcal{V}_{M}^{(\omega)}$

We now prove bounds for the singular values
$s_{k}(\mathcal{V}_{M}^{(\omega)})$ of $\mathcal{V}_{M}^{(\omega)}$.

######  Theorem 5.

Let $R\geq h>0$ and $\nu\in(0,d-1]$. Then, for all $\lambda>0$ and $V\in
L^{\infty}$ with $\operatorname{supp}V\subseteq B(R)$,

$\displaystyle\mathbf{E}\|\mathcal{E}_{\lambda}^{*}V_{\omega}\mathcal{E}_{\lambda}\|_{\mathcal{S}^{\frac{d-1}{\nu},\infty}}$
$\displaystyle\lesssim(\lambda R)^{\frac{1}{2}+\nu}\sqrt{\ln\langle\lambda
R\rangle}\langle\lambda h\rangle^{\frac{d}{2}}(\ln\langle\lambda
R\rangle+\ln\langle\lambda h\rangle)^{2}\lambda^{-d}\|V\|_{\infty}.$ (53)

Consequently, for all $\varepsilon>0$, $R_{0}\geq 1$ and $\lambda>R_{0}^{-1}$,

$\displaystyle\mathbf{E}\|\mathcal{E}_{\lambda}^{*}V_{\omega}\mathcal{E}_{\lambda}\|_{\mathcal{S}^{(d-1)/\nu,\infty}}\lesssim_{\varepsilon}R_{0}^{\varepsilon}\langle\lambda
h\rangle^{d/2}\lambda^{1/2+\nu+\varepsilon-d}\|\langle
x\rangle^{1/2+\varepsilon+\nu}V\|_{\infty}.$ (54)

###### Proof.

Formula (54) follows from (53) by a dyadic decomposition and using that
$\ln(\langle\lambda R\rangle)^{5/2}\lesssim(\lambda
R)^{\varepsilon}\,R_{0}^{\varepsilon}$ since $R\geq 1$ without loss of
generality. To prove (53), it suffices to consider $\lambda=1$ by scaling; see
also [8, Proposition 4.1, Lemma 4.2]. Using the Weyl asymptotic for the
Laplace–Beltrami operator $-\Delta_{M}$, we get, for any $\nu>0$,

$\displaystyle\begin{split}s_{k}(\mathcal{V}_{M}^{(\omega)})&\leq
s_{k}(\langle-\Delta_{M}\rangle^{-\nu/4})^{2}\cdot\|\langle-\Delta_{M}\rangle^{\nu/4}\mathcal{E}^{*}V_{\omega}\mathcal{E}\langle-\Delta_{M}\rangle^{\nu/4}\|\\\
&\lesssim
k^{-\nu/(d-1)}\|\langle-\Delta_{M}\rangle^{\nu/4}\mathcal{E}^{*}V_{\omega}\mathcal{E}\langle-\Delta_{M}\rangle^{\nu/4}\|.\end{split}$
(55)

To bound the operator norm on the right-hand side, we proceed as in Section
2.1. By an integration by parts,

$\displaystyle\mathcal{E}\langle-\Delta_{M}\rangle^{\nu/4}=:\mathcal{E}_{a},$
(56)

where $\mathcal{E}_{a}$ is a modulated Fourier extension operator, which acts
as

$\displaystyle(\mathcal{E}_{a}g)(x)=\int_{M}d\sigma(\xi)e(x\cdot\xi)a(x,\xi)g(\xi),$
(57)

and $a(x,\xi)$ is a smooth symbol, which satisfies

$\displaystyle\|a(x,\cdot)\|_{C^{N}}\lesssim_{N}\langle x\rangle^{\nu/2},\quad
x\in\mathbb{R}^{d},\ N\in\mathbb{N}_{0}.$ (58)

By inspection of the proof of Proposition 4, we see that (53) would follow
from

$\displaystyle\|\sum_{\mu\in\Lambda_{R}^{*}}b(\mu)\cdot a(x,\mu)\cdot
e(\mu\cdot x)\|_{L^{2}(|x|<R)}\lesssim
R^{d/2+\nu/2}\|b(\mu)\|_{\ell_{\mu}^{2}(\Lambda_{R}^{*})}.$ (59)

When $a(x,\xi)=c(x)d(\xi)$ factorizes, we proceed as in (47) and obtain

$\displaystyle\begin{split}&\|\sum_{\mu\in\Lambda_{R}^{*}}b(\mu)a(x,\mu)e(\mu\cdot
x)\|_{L_{x}^{2}(B(R))}\leq\|\sum_{\mu\in\Lambda_{R}^{*}}b(\mu)c(x)d(\mu)\hat{\phi}_{R}e(\mu\cdot
x)\|_{L_{x}^{2}(B(R))}\\\
&\quad\leq\|c\|_{L^{\infty}(B(R))}\|\sum_{\mu\in\Lambda_{R}^{*}}b(\mu)d(\mu)\phi_{R}(\xi+\mu)\|_{L_{\xi}^{2}(\mathbb{R}^{d})}\\\
&\quad\leq
R^{\frac{\nu}{2}}\|b(\mu)d(\mu)\|_{\ell^{2}(\Lambda_{R}^{*})}\|\phi_{R}\|_{L^{2}(\mathbb{R}^{d})}\lesssim
R^{\frac{d+\nu}{2}}\|b(\mu)\|_{\ell_{\mu}^{2}(\Lambda_{R}^{*})}\|d(\mu)\|_{\ell_{\mu}^{\infty}},\end{split}$
(60)

as desired. For general $a(x,\mu)$, we decompose

$\displaystyle a(x,\xi)=\sum_{y\in\mathbb{Z}^{d}}\hat{a}(x,y)e(-\xi\cdot y)$
(61)

into a Fourier series with

$\displaystyle\hat{a}(x,y):=2^{-d}\int_{[-1,1]^{d}}a(x,\xi)e(\xi\cdot
y)\,d\xi$ (62)

obeying

$\displaystyle\sup_{x\in B(R)}|\hat{a}(x,y)|\lesssim R^{\nu/2}\langle
y\rangle^{-N}.$ (63)

Thus,

$\displaystyle\begin{split}&\|\sum_{\mu\in\Lambda_{R}^{*}}b(\mu)a(x,\mu)e(\mu\cdot
x)\|_{L_{x}^{2}(B(R))}\\\
&\quad=\|\sum_{\mu\in\Lambda_{R}^{*}}\sum_{y\in\mathbb{Z}^{d}}b(\mu)e(\mu\cdot(x-y))\hat{a}(x,y)\hat{\phi}_{R}\|_{L_{x}^{2}(B(R))}\\\
&\quad\leq\sum_{y\in\mathbb{Z}^{d}}\sup_{x\in
B(R)}|\hat{a}(x,y)|\sup_{y\in\mathbb{Z}^{d}}\|\sum_{\mu\in\Lambda_{R}^{*}}b(\mu)e(-\mu\cdot
y)\phi_{R}(\xi+\mu)\|_{L_{\xi}^{2}(\mathbb{R}^{d})}\\\ &\quad\lesssim
R^{d/2+\nu/2}\sup_{y\in\mathbb{Z}^{d}}\|b(\mu)e(-\mu\cdot
y)\|_{\ell_{\mu}^{2}(\Lambda_{R}^{*})}\leq
R^{d/2+\nu/2}\|b(\mu)\|_{\ell^{2}(\Lambda_{R}^{*})}.\end{split}$ (64)

This concludes the proof of (59) and thereby that of (53) and Theorem 5. ∎

We remark that for deterministic $V$, the proof leading to (53) gives

$\displaystyle\|\mathcal{E}^{*}V\mathcal{E}\|_{\mathcal{S}^{(d-1)/\nu,\infty}}$
$\displaystyle\lesssim\|\langle
x\rangle^{1+\nu+\varepsilon}V\|_{\infty},\quad\nu\in(0,d-1],\ \varepsilon>0.$
(65)

Up to replacing the weak with the strong Schatten norm and the additional
power $\langle x\rangle^{\varepsilon}$, this bound coincides with [29,
Proposition 8.1.3].

###  2.3. Eigenvalue sums of random Schrödinger operators

We now apply the Schatten bound in Theorem 4 to estimate sums of eigenvalues
of $-\Delta-V_{\omega}$, where $\operatorname{supp}V_{\omega}\subseteq B(R)$
for some $R>0$. Let $\delta(z):=\operatorname{dist}(z,[0,\infty))$ be the
distance between $z\in\mathbb{C}$ and the essential spectrum of $-\Delta-
V_{\omega}$. We will use the following abstract result by Frank.

######  Theorem 6 ([13, Theorem 3]).

Let $\sigma>0$, $p\geq 1$. Let $K(z)$, $z\in\mathbb{C}\setminus[0,\infty)$, be
an analytic family of operators satisfying $\|K(z)\|_{p}\leq M|z|^{-\sigma}$
for all $z\in\mathbb{C}\setminus[0,\infty)$ and some $M\geq 0$. Let
$z_{j}\in\mathbb{C}\setminus[0,\infty)$ be the eigenvalues of $1+K$ of finite
type, repeated according to their algebraic multiplicity. Then, for all
$\varepsilon>0$,

$\displaystyle\sum_{j}\delta(z_{j})|z_{j}|^{-\frac{1}{2}+\frac{1}{2}(2p\sigma-1+\varepsilon)_{+}}\lesssim_{\sigma,p,\varepsilon}M^{\frac{1}{2\sigma}(1+(2p\sigma-1+\varepsilon)_{+})}.$
(66)

In the following, we only consider eigenvalues $z$ with
$R_{0}^{-1}\leq|z|^{1/2}\leq h^{-1}$ for any fixed $R_{0}\geq 1$ and
$|\operatorname{Im}z|\ll 1$. The reason for the restriction $|z|^{1/2}\leq
h^{-1}$ is that the randomization becomes ineffective for frequencies that are
significantly larger than the reciprocal randomization length scale. In this
case, the estimate (53) implies for any $\varepsilon>0$,

$\displaystyle\begin{split}\mathbf{E}\|\mathcal{E}_{\lambda}^{*}V_{\omega}\mathcal{E}_{\lambda}\|_{\mathcal{S}^{(d-1)/(\nu-\varepsilon)}}&\leq\mathbf{E}\|\mathcal{E}_{\lambda}^{*}V_{\omega}\mathcal{E}_{\lambda}\|_{\mathcal{S}^{(d-1)/\nu,\infty}}\\\
&\lesssim_{\varepsilon,\nu,d}R_{0}^{\varepsilon}\lambda^{1/2+\nu+\varepsilon-d}\|\langle
x\rangle^{1/2+\nu+\varepsilon}V\|_{\infty}.\end{split}$ (67)

We choose $\nu=2\varepsilon$. Repeating the previous arguments that led to
(19), we get, for any $\varepsilon\in(0,d-1]$,

$\displaystyle\begin{split}&\mathbf{E}\|C_{z}^{(\frac{1}{R})}(D)V_{\omega}C_{z}^{(\frac{1}{R})}(D)\|_{\mathcal{S}^{\frac{d-1}{\varepsilon}}}\lesssim
R_{0}^{\varepsilon}|z|^{-3/4+3\varepsilon/2}\|\langle
x\rangle^{1/2+3\varepsilon}V\|_{\infty},\end{split}$ (68)

where $C_{z}^{(\delta)}(D)$ is any Fourier multiplier whose symbol satisfies

$\displaystyle|C_{z}^{(\delta)}(\xi)|\leq(||\xi|^{2}-|z||+\delta)^{-1/2}.$
(69)

We now use Theorem 6 to bound eigenvalue sums of $-\Delta-V_{\omega}$. To that
end, we use
$\mathbf{1}_{B(R)}R(z)\mathbf{1}_{B(R)}=\mathbf{1}_{B(R)}\tilde{R}(z)\mathbf{1}_{B(R)}$,
where $\tilde{R}(z)$ is a smoothed out resolvents with the symbol of
$|\tilde{R}(z)|^{1/2}$ obeying the bound (69), and the resolvent identity

$\displaystyle\begin{split}&(-\Delta-V_{\omega}-z)^{-1}\\\
&\quad=R(z)+R(z)V_{\omega}R(z)+R(z)V_{\omega}\tilde{R}(z)^{\frac{1}{2}}(1-K(z))^{-1}|\tilde{R}(z)|^{1/2}V_{\omega}R(z),\end{split}$
(70)

where

$\displaystyle
K(z):=|\tilde{R}(z)|^{\frac{1}{2}}V_{\omega}\tilde{R}(z)^{\frac{1}{2}}.$ (71)

Thus, the eigenvalues $z$ of $-\Delta-V_{\omega}$ are those points for which
$1-K(z)$ fails to be invertible. Using Theorem 6 with

$\displaystyle
p=\frac{d-1}{\varepsilon}\quad\text{and}\quad\sigma=\frac{3}{4}-\frac{3\varepsilon}{2}>0,$
(72)

and the bound (68) for $\|K(z)\|$ yields the following

######  Theorem 7.

Let $\varepsilon\in(0,1/2)$, $h>0$, and $V\in\langle
x\rangle^{-\frac{1}{2}-3\varepsilon}L^{\infty}(\mathbb{R}^{d})$. Let
$\\{z_{j}\\}\subseteq\mathbb{C}\setminus[0,\infty)$ denote the set of discrete
eigenvalues of finite algebraic multiplicities of $-\Delta-V_{\omega}$,
repeated according to their algebraic multiplicity. Then, there are constants
$M_{0},c>0$ such that for all $R_{0}\geq 1$, and $M\geq M_{0}$,

$\displaystyle\begin{split}&\sum_{R_{0}^{-1/2}\leq|z_{j}|\leq
h^{-1/2}}\delta(z_{j})|z_{j}|^{-\frac{1}{2}+\frac{1}{2}(\frac{3(d-1)}{2\varepsilon}-3d+2+\varepsilon)_{+}}\\\
&\quad\lesssim_{d,\nu,\varepsilon}(R_{0}^{\varepsilon}\|\langle
x\rangle^{1/2+3\varepsilon}V\|_{\infty})^{\frac{1}{3/2-3\varepsilon}(1+(\frac{3(d-1)}{2\varepsilon}-3d+2+\varepsilon)_{+})}\end{split}$
(73)

holds for all $\omega$ outside a set of measure at most
$\mathrm{e}^{-cM^{2}}$. In particular, for these $\omega$, there are
$c_{1}=c_{1}(R_{0},\varepsilon,d)>0$ and $c_{2}=c_{2}(\varepsilon,d)>1$ such
that

$\displaystyle\sum_{R_{0}^{-1/2}\leq|z_{j}|\leq h^{-1/2}}\delta(z_{j})\leq
c_{1}\|\langle x\rangle^{1/2+3\varepsilon}V\|_{\infty}^{c_{2}}.$ (74)

Note that there is $\varepsilon_{d}>0$ such that
$\frac{3(d-1)}{2\varepsilon}-3d+2+\varepsilon>0$ if and only if
$\varepsilon<\varepsilon_{d}$.

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AWAKE Collaboration

# White Paper: AWAKE, Plasma Wakefield Acceleration of Electron Bunches for
Near and Long Term Particle Physics Applications

P. Muggli<EMAIL_ADDRESS>Max Planck Institute for Physics, Munich, Germany

##  I Executive summary

Plasma-based accelerators have made remarkable progress over the last two
decades. Their unique characteristics make them tools that can revolutionize
fields of science and applications. AWAKE takes advantage of the availability
of high-energy, relativistic proton bunches to drive large amplitude
wakefields ($\sim$GV/m) in a single plasma over distances sufficient to
produce hundreds of GeV to TeV electron bunches. Hundreds of GeV bunches with
$\mathcal{O}{(10^{9})}$ electrons could replace current electron sources based
on the generation of secondary particles, thereby extending in the next decade
the reach of dark photon search experiments. Collisions between
$\mathcal{O}{(50\,GeV)}$ electrons and protons with LHC energy would allow
studying _low x_ processes, processes with high cross-section, making studies
interesting even with low luminosity.

Bunches of TeV electrons with TeV protons would allow for center-of-mass
energy of 9 TeV, 30 times higher than at HERA. Such a very-high energy ep/eA
collider, would be smaller and give access to physics beyond the reach of
colliders using other electron acceleration schemes.

Based on experimental and numerical simulation results obtained in the early
phase of AWAKE (Run 1), that demonstrated the basic features of the
accelerator scheme, we developed a clear science roadmap that, at the
beginning of the next decade (end of Run 2), would put AWAKE in a position to
propose first particle physics experiments with 50-200 GeV electron bunches.
This raodmap based on a plasma for self-modulation of the p+ bunch and one for
acceleration of the electron bunch, and on external injection in order to
preserve the quality of the bunch that is accelerated (charge, energy spread,
emittance). The roadmap includes:

  * •

2021-22: Demonstrating seeding of the self-modulation (SM) of the p+ bunch
with an electron bunch;

  * •

2023-25: Introducing a step in the plasma density at a location during the SM
growth for wakefields to maintain large amplitude over long plasma distances;

  * •

2028-30: External injection in the accelerator plasma of an electron bunch
with parameters suitable full blowout of plasma electrons, loading of the
wakefields, and matching to the plasma focusing force;

  * •

2021-…: Development of a plasma source that can be extended to hundreds to
thousands of meters in length and with sub-%density uniformity. This is the
main R&D required for the application of this acceleration scheme to HEP.

Besides following this roadmap, AWAKE also studies many aspects of plasma-
based accelerators that are common to all schemes: external injection,
emittance preservation, hosing instability, development of suitable
diagnostics, development of simulation tools, and others. We plan to develop
and use a 150 MeV X-band linac as injector and explore even more advanced
schemes, such as a laser wakefield accelerator (LWFA) injector.

AWAKE is an open and international collaboration that would greatly benefit
from the involvement of US groups, in particular on the topics of diagnostics
for plasma waves and for accelerated electron bunches.

##  II Introduction

Synchrotons such as the the SPS and the LHC at CERN routinely produce
relativistic (400 GeV, 7 TeV) proton (p+) bunches that also carry large
amounts of energy (19, 112 kJ). The availability of these drivers makes it in
principle possible to drive large amplitude wakefields (GV/m) in plasma over
very long distances. Therefore an externally-injected electron witness bunch
can in principle reach very large energies, hundreds of GeVs [1] (Fig. 1 top,
1 GV/m over 200 m) to TeVs [2] (Fig. 1 bottom) in a single plasma hundreds-of-
meters to kilometers long.

Figure 1:  Top: the estimated maximum witness energy $W_{max}(z)$ for the
different accelerator parameters, showing that an energy gain $>$200 GeV
(blue, brown lines) can be reached in $<$300 m with a 400 GeV p+ bunch from
the SPS as driver. From [1]. Bottom: the maximum energy gain of electrons
$W_{e}$ versus propagation distance for a plasma with density step and a 7 TeV
LHC p+ bunch as driver. From [2].

Based on this potential, the international AWAKE Collaboration is developing
this acceleration scheme for applications to high-energy physics such as dark
photon searches and QED in collisions between high-energy electrons and
photons in the next decade [3] and, in the longer term, to a very-high-energy
ep/eA collider [4].

Following the scaling of plasma wakefield accelerators [5], these drive p+
bunches are too long ($\sim$10 cm) to directly drive these wakefields. They
must self-modulate [6] in a first plasma to be transformed into a train of
bunches with period $\sim$1 mm to resonantly drive wakefields with GV/m
amplitude in a second plasma [7]. Experiments with the SPS p+ bunch as driver
have already demonstrated the occurrence of self-modulation [8, 9], as well as
many of its characteristics [10, 11, 12, 13, 14, 15]. They have also
demonstrated that externally-injected test electrons can be accelerated (in
the self-modulator) from $\sim$19 MeV to $\sim$2 GeV [16, 17] over less than
10 m [12].

Based on the results, a scientific roaddmap was drawn [7, 18] to demonstrate
the potential of this acceleration scheme for HEP applications by the end of
the decade. The main ingredients of this roadmap are the use of two plasmas
for SM and acceleration, and external injection of a relativistic electron
bunch to demonstrate that a bunch with quality sufficient for first
application can be produced: charge, energy and energy spread, and emittance.
The major steps in this roadmap include: the seeding of SM with an electron
bunch; the use of a step in the plasma density so that wakefields maintain a
large amplitude past saturation of SM; the external injection of an electron
bunch with parameters suitable to reach blow-out of plasma electrons, to load
the wakefields and to be matched to the plasma focusing force; and to develop
plasma sources whose length can be extended and is scalable for large energy
gain.

##  III Background

The occurrence of SM was observed (fig. 2, top) [8, 9]. Modulation of the
bunch density is at the period of the wakefields in the plasma of electron
density $n_{e0}$: $\tau_{M}\cong 1/f_{pe}$,
$f_{pe}=\left(n_{e0}e^{2}/\epsilon_{0}m_{e}\right)^{1/2}/2\pi$ [8]. The
amplitude of the wakefields grows both along the plasma and along the bunch.
Their amplitude reaches hundreds of MV/m, much larger than their initial value
($<$10 MV/m) [9]. Most importantly, the predicted instability [6] (or SMI) can
be seeded and made reproducible and controllable using a relativistic
ionization front (RIF) placed within the p+ bunch (fig. 2, bottom) [10]. Self-
modulation is seeded (SSM) only when the amplitude of the wakefields exceed a
threshold value of $\sim$4 MV/m ($n_{e0}=10^{14}$ cm-3). Reproducibility and
control of the instability are necessary for deterministic injection of an
electron bunch into the accelerating _and_ focusing phase of the wakefields.

Figure 2: (a) Time-resolved, “stitched” image (11 series of $\sim$10 images or
events) of the self-modulated proton bunch with the relativistic ionization
front (RIF) $t_{RIF}=125\,ps$ (0.5 rms bunch length) ahead of the bunch
center, $n_{e0}\cong 1.81\times 10^{14}$ cm-3. The RIF is at $t=0$ (not
visible), time reference signal visible every 50 ps at the bottom of the
image. (b) The rms phase variation between the modulation along the bunch and
the time reference signal. Variations are limited to a small relative fraction
($<$7%) of the $2\pi$ phase of the period of the wakefields. From [10].

Measurements show that the wakefields have a phase velocity smaller than the
velocity of the p+ bunch [11], as expected from theory and simulations [19,
20]. Protons defocused by the SM process provide details on the development of
SM as well as on the fine scale evolution of the modulation period along the
plasma [14]. The SM saturates a distance between 3 and 5 m into the plasma
[12]. Side-injected, test electrons with an energy of $\sim$19 MeV were
accelerated to $\sim$2 GeV in less than 10 m of plasma with density
$n_{e0}\cong 7\times 10^{14}$ cm-3 (Fig.3) [16]. The SM is in general well
understood and there is excellent agreement between experimental and
simulation results [21, 14].

Figure 3: Measurement of the highest peak energies $\mu_{E}$ achieved at
different plasma densities $n_{pe}$, with and without a gradient in the plasma
density. The error bars arise from the position–energy conversion. The
gradients chosen are those that were observed to maximize the energy gain.
Acceleration to 2.0$\pm$0.1 GeV is achieved with a plasma density of
$n_{e0}\cong 6.6\times 10^{14}$ cm-3 with a density difference of $\pm$2.2%
$\pm$0.1% over 10 m. From [16].

##  IV Roadmap towards particle physics applications

Figure 4:  Roadmap for the AWAKE project towards application to HEP in the
next decade. Years in green/yellow are years with/without SPS beam available.

Based on the state-of-the-art, a clear science roadmap (Fig. 4) can be drawn
for experiments that should lead by the end of the decade to a demonstrator
for a device with a 10 m-long accelerator producing a $\sim$10 GeV electron
bunch with $\sim$100 pC charge and energy spread (%-level) and normalized
emittance (1-10 mm-mrad), sufficient for early HEP applications [3]. The
milestones along the roadmap are clear. They rely on developing a system with
two plasmas: the first one for SM of the p+ bunch and the second one for
acceleration of the electron bunch in the wakefields driven by the self-
modulated p+ bunch (Fig. 5). This program was discussed in Ref. [7] and is
constantly and further developed. The roadmap includes four main milestones.

Figure 5:  Schematic of the experimental setup for the AWAKE program. The
first milestones focus on the self-modulator (first) plasma. The last two
milestones include two plasmas $\sim$10 m-long each. Applications would use a
similar set-up, with a much longer second, accelerator plasma.

###  IV.1 1\. Demonstration of SM seeding with an electron bunch

Seeding of SM with a RIF [10] leaves the front of the p+ bunch not modulated.
However, a very long accelerator plasma is likely to be pre-ionized. This
could result in the SMI of the front of the bunch in the plasma of the
accelerator section. Wakefields driven by this front SMI could interfere with
the modulated bunch train emerging from the self-modulator. In contrast, an
electron bunch placed ahead of the p+ bunch drive wakefields that can seed SM
(eSSM)of the entire p+ bunch. Preliminary results show that e-SSM is possible
[15]. Studies of e-SSM started in 2021 and will continue in 2022.

###  IV.2 2\. Plasma density step

Numerical simulation results [6, 2] show that with a constant plasma density
along the self-modulator the amplitude of the wakefields decreases after
saturation of SM, because of the continuous evolution of the beam/plasma
interaction. They also show that introducing a plasma density step at a
location along the growth of the SM allows for the wakefields to maintain an
amplitude close to their saturation value over a long distance [2, 22] (Fig.
6). The plasma source, based on ionization of a rubidium vapor by a short
laser pulse (RIF) [23, 24] allows a temperature and thus density step to be
imposed. We are designing a source with variable step position and height for
experiments in 2023-25 aimed at demonstrating the effect of the density on the
SM and on its seeding.

Figure 6: The maximum longitudinal wakefield $E_{z,max}$ versus propagation
distance z, with an LHC p+ bunch as driver, without (thin red line) and with
(thick red line) a plasma density step. The density step freezes the amplitude
of the wakefields close to their value near saturation ($\sim$1 GeV/m). From
[2].

###  IV.3 3\. External injection of an electron bunch

Acceleration preserving bunch quality requires injection of a low energy
($\sim$150 MeV) electron bunch on the axis of the accelerator plasma driven by
the self-modulated p+ bunch emerging from the self-modulator plasma. The
length of the gap between the two plasmas must be minimized to decrease the
evolution of the self-modulated p+ bunch along the gap. This evolution leads
to wakefields with lower amplitude in the accelerator plasma than at the exit
of the self-modulator plasma. The parameters of the injected electron bunch
must be suitable to drive wakefields in the blowout regime and to load the
wakefields in order to preserve the emittance of the bunch and maintain a low
energy spread [25]. The bunch must be matched to the focusing force of the
plasma ion column to avoid deleterious effects caused by betatron oscillations
of the bunch envelope. Both the in-coming (matched) beam beta-function
($\beta_{0}\cong$5 mm, 150 MeV) and out-coming one ($\beta_{0}\cong$40 cm, 10
GeV) are much shorter than the plasma length ($\geq$10 m). If not matched,
variation of the phase advance of the beam over the plasma length due to small
beam and plasma parameter variations from event to event would lead to
mismatch of the beam to the following (magnetic) optics. We note here that
even though the accelerator operates at relatively low plasma density
(typically $n_{e0}=7\times 10^{14}$ cm-3), beam parameters and transverse
alignment tolerance are rather challenging to meet [26, 27].

The injector of the current design consists of an original S-band photo-
injector gun, X-band linac and transfer line combination to produce $\sim$100
pC, $\sim$200 fs, 150 MeV electron bunches with a beta-function of $\sim$5 mm
at the plasma entrance. However, we are also investigating other accelerator
schemes, such as a LWFA injector, or a foil injector.

###  IV.4 4\. Development of long accelerator plasma source

Plasma creation using laser ionization limits the plasma length to a few tens
of meters due to depletion of the laser pulse energy and to other optical
constraints. We are therefore investigating helicon and discharge sources for
the accelerator plasma. The main challenge for the accelerator plasma is the
density uniformity of $<$0.5% imposed by the resonant driving of the
wakefields.

The helicon source is a modular, weakly magnetized RF-discharge system whose
unit cell consists of an RF antenna and of two magnetic coils (Fig. 7).
Stacking of unit cells can therefore in principle produce very long plasmas.
Initial experiments have shown that densities required for AWAKE can be
produced [28].

Figure 7: Schematic of the 1 m long prototype module PROMETHEUS-A for the
plasma wakeﬁeld accelerator experiment AWAKE. The magnetic ﬁeld coils are
adjusted to produce a ﬁeld as homogeneous as possible, while providing access
to the radial ports of the tube for diagnostic purposes. From [28].

The direct-current discharge plasma source (DPS) is a simpler source. However
it requires stacking of multiple discharges each with start/end electrodes to
reach long distances. Demonstration SM experiments with a 10 m-long DPS are
tentatively planed for early 2023.

We are developing diagnostics to measure the plasma density uniformity and
reproducibility of both sources. If ready, one of these sources could become
the source for the accelerator plasma for the first external injection
experiments, currently planned with a laser-ionized rubidium vapor source [7,
18].

Early application of the acceleration scheme [3] may require an accelerator
plasma tens to hundreds of meters long [1], whereas eA collider applications
[4] will require kilometers-long plasmas.

The development of long, scalable plasma sources is the main R&D topic for the
application of the acceleration scheme.

##  V Physics cases, applications

Applications for high-energy electron bunches that can be produced by the
AWAKE acceleration scheme are described in Refs [3, 4, 18]. They all use a
layout similar of that of Fig. 5.

These applications use 50-200 GeV electron that can be produced with p+
bunches from the SPS (400 GeV) as driver and electrons at the TeV energy scale
that can be produced with the TeV p+ bunch of the LHC as driver of wakefields.
In general, the p+ bunch can be used only once to drive wakefields, which
limits the repetition rate of the electron acceleration process to that of the
p+ bunch production. This leads to a significantly lower luminosity than with
other electron production schemes. However, high-gradient acceleration permit
a shorter accelerator length and the scheme can produce high-energy electrons
with available or spent p+ beams, offering unique opportunities in particle
physics.

Using the SPS p+ bunch as driver electrons with energies up to $\sim$200 GeV
can be produced. These find applications in dark photon searches on fixed
target in the beam dump mode ($\sim$109 electrons per bunch). The larger
number of electrons produced at 50 GeV than with current secondary production
processes would allow for larger reach in coupling strength and dark photon
mass than other other existing or planned experiments (Fig. 8).

Figure 8: Limits on dark photon production decaying to an $e^{+}e^{-}$ pair in
terms of the mixing strength, $\epsilon$, and dark photon mass,
$m_{A^{\prime}}$, from previous measurements (light grey shading). The
expected sensitivity for the NA64 experiment is shown for a range of electrons
on target, $10^{10}-10^{13}$. Expectations from other potential experiments
are shown as colored lines. Expected limits are also shown for $10^{15}$
(orange line) or $10^{16}$ (green line) electrons of 50 GeV (“AWAKE50”) on
target and $10^{16}$ (blue line) electrons of 1 TeV (“AWAKE1k”) on target
provided to an experiment using the future AWAKE accelerator scheme. From Ref.
[29].

High-energy electron bunches could be used to investigate non-linear QED in
electron-photon collisions. These experiments usually have a repetition rate
limited by that of the high-power laser (typically 1-10 Hz). The low
repetition rate of the p+ bunch and thus of the acceleration process may not
be a limitation.

Much higher energy, TeV-range electrons produced with an LHC p+ bunch could be
used for lower luminosity measurements in electron-proton or electron-ion
collisions.

##  VI Synergies with other concepts

The p+-driven plasma wakefield accelerator addresses may issues common to all
plasma-based concepts. This includes the challenges linked to external
injection, which will have to be met at the entrance of every stage of a
plasma-based accelerator concept. Reaching transverse and time alignment
tolerances requires extremely reproducible beam optics. Misalignment can
degrade acceleration performances, but can also induce hose instability.
Therefore studying the hose instability, understanding tolerances for his
development is thus also essential for beam quality preservation in all
accelerator schemes.

Development of plasma sources with sufficient reproducibility and
characteristic is essential for all plasma-based accelerator concepts.

The interaction with, and the collaboration of US partners and institutions
would be a great benefit. We note here that the p+-driven concept could also
be developed with beams available at the Brookhaven National Laboratory and
also find applications there. Moreover, the near-term applications proposed
here can be essential to increase the credibility of all plasma-based
accelerator schemes for application to particle physics.

##  VII Conclusions

Proton-driven plasma wakefield acceleration has the potential to produce very
high energy electron bunches for a number of HEP applications. Proton bunches
suitable to drive the accelerator are available today. The concept is
developed in the context of an international collaboration that would benefit
from US contributions. The roadmap and mid- and long-term applications to HEP
described in this white paper are based on an active experimental, simulation
and theory program. The main R&D required for applications is on plasma
sources scalable to very long lengths (hundreds thousands of meters) with
sub-% level plasma density uniformity. The program addresses many issues
common to all plasma-based acceleration concepts. Applications to collisions
with fixed targets and p+ bunches relax the electron bunch parameters (in
particular emittance) and operation even with luminosity lower than that
required for example for an e-e+ is attractive. Therefore, first applications
to HEP could be become possible early in the next decade.

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