# Averaging along degenerate flows on the annulus

Jonathan Ben-Artzi  and  Baptiste Morisse  School of Mathematics, Cardiff
University, Cardiff CF24 4AG, Wales, UK<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Periodic flows on the annulus are considered. For flows that degenerate (i.e.
the flow becomes arbitrarily slow along some flow lines) a convergence rate
for the averaging of functions along the flow is obtained. This rate – which
is slower than the rate obtained for flows that do not degenerate (i.e. when
there is a spectral gap) – holds on an appropriate functional subspace. The
main ingredient is an estimate of the density of the spectrum of the generator
near zero.

MSC (2010): 37A30 (primary); 35P20, 35B40

Keywords: degenerate flows; density of states

Acknowledgements: The authors acknowledge support from Fellowship EP/N020154/1
of the Engineering and Physical Sciences Research Council (EPSRC).

Date: .

###### Contents

  1. 1 Introduction
  2. 2 Spectral analysis and fibered operators
  3. 3 Toy example
  4. 4 Flow in an annulus: Proof of Theorem 1.1

##  1\. Introduction

In this paper we study self-adjoint operators of the form

$A=-i\varphi(m)\frac{\partial}{\partial\theta}\qquad\text{acting in
}L^{2}(\mathcal{A})$ (1.1)

where $\mathcal{A}=[0,1]_{m}\times\mathbb{S}^{1}_{\theta}$ is an annulus and
where $\varphi\geq 0$ is continuous. The operator $A$ is the generator of a
periodic flow in $\mathcal{A}$ and $\varphi$ measures the speed of the flow
along the $m$-fiber. We are interested in obtaining a rate at which the time
averages

$P^{T}f:=\frac{1}{2T}\int_{-T}^{T}e^{itA}f\,\mathrm{d}t=\frac{1}{2T}\int_{-T}^{T}f(m,t\varphi(m))\,\mathrm{d}t$

converge to the spatial average

$Pf:=\int_{\mathbb{S}^{1}}f(m,\theta)\,\mathrm{d}\theta$ (1.2)

as $T\to+\infty$ in the degenerate case where $\min_{m\in[0,1]}\varphi(m)=0$.
In this case there cannot be a uniform rate on $L^{2}(\mathcal{A})$ and our
main task is to identify a subspace $\mathcal{X}\subset L^{2}(\mathcal{A})$ on
which a uniform rate _does_ hold.

###  1.1. Main result

Our main result is:

###### Theorem 1.1.

Assume that there exist $m_{0},c,\alpha>0$ such that $\varphi(m)=cm^{\alpha}$
on $[0,m_{0})$ and that $\varphi$ is continuous and bounded uniformly away
from $0$ on $[m_{0},1]$. Then there exists a subspace
$\dot{\mathcal{H}}^{s,\gamma}_{0}\subset L^{2}(\mathcal{A})$ (defined below in
(4.3)111Roughly speaking, $\dot{\mathcal{H}}^{s,\gamma}_{0}$ consists of
functions that have $\gamma$ derivatives in $\theta$, whose $\dot{H}^{\gamma}$
norm is $(s-\frac{1}{2})$–Hölder continuous in $m$ and are constant along the
fiber $m=0$.) and a constant $C>0$ such that the following uniform rate holds:

$\|P^{T}-P\|_{\dot{\mathcal{H}}^{s,\gamma}_{0}\to L^{2}(\mathcal{A})}\leq
CT^{-\frac{s}{s+\alpha}},\qquad\forall T>1,$

where $s>1/2$ and $\gamma\geq 0$ satisfy $\gamma+s/\alpha>1/2$, and where $C$
does not depend on $T$.

#### _Strategy of the proof_

The strong convergence (i.e without a rate) $P^{T}\to P$ follows immediately
from von Neumann’s ergodic theorem [vN32]. Von Neumann’s idea was to use the
spectral theorem to write $A=\int_{\mathbb{R}}\lambda\,\mathrm{d}E(\lambda)$,
where $\\{E(\lambda)\\}_{\lambda\in\mathbb{R}}$ is the resolution of the
identity of $A$. This leads to

$\displaystyle(P^{T}-P)f$
$\displaystyle=\frac{1}{2T}\int_{-T}^{T}e^{itA}f\,\mathrm{d}t-Pf=\frac{1}{2T}\int_{-T}^{T}\int_{\mathbb{R}}e^{it\lambda}\,\mathrm{d}E(\lambda)f\,\mathrm{d}t-Pf$
$\displaystyle=\frac{1}{2T}\int_{-T}^{T}\int_{\mathbb{R}\setminus\\{0\\}}e^{it\lambda}\,\mathrm{d}E(\lambda)f\,\mathrm{d}t=\int_{\mathbb{R}\setminus\\{0\\}}\frac{\sin
T\lambda}{T\lambda}\,\mathrm{d}E(\lambda)f.$ (1.3)

This last expression tends to $0$ as $T\to+\infty$. It is well-known that a
spectral gap leads to the rate $T^{-1}$. However here, owing to the assumption
that the flow degenerates, there is no spectral gap. In [BAM19] we showed that
a rate may be extracted if there exists a subspace $\mathcal{X}\subset
L^{2}(\mathcal{A})$ and some $r>0$ such that the density of states (DoS) has a
bound of the form

$\left|\frac{\mathrm{d}}{\mathrm{d}\lambda}\left(E(\lambda)f,g\right)_{L^{2}(\mathcal{A})}\right|\leq\psi(\lambda)\|f\|_{\mathcal{X}}\|g\|_{\mathcal{X}},\qquad\forall
f,g\in\mathcal{X},\,\forall\lambda\in(-r,r)\setminus\\{0\\},$ (1.4)

where $\psi\in L^{1}(-r,r)$ is strictly positive a.e. on $(-r,r)$. This then
allows one to replace the integration $\mathrm{d}E(\lambda)$ in (1.3) with
$\frac{\mathrm{d}E(\lambda)}{\mathrm{d}\lambda}\mathrm{d}\lambda$ and to apply
the estimate (4.9) near $\lambda=0$. Therefore, to prove Theorem 1.1 the main
task is to obtain an estimate of the form (4.9), which involves identifying an
appropriate subspace $\mathcal{X}$. This is achieved thanks to the observation
that the operator $A$ is unitarily equivalent to the multiplication operator
$\varphi k$ via a Fourier transform in $\theta$ (where $k\in\mathbb{Z}$ is the
Fourier conjugate of $\theta$).

To obtain an estimate of the DoS as in (4.9), the first step is to understand
the structure of the spectrum. Here we take the point of view that $A$ is
fibered in $m$, composed of the one-dimensional operators

$A(m)=-i\varphi(m)\frac{\mathrm{d}}{\mathrm{d}\theta}\qquad\forall m\in[0,1],$
(1.5)

acting on the circle $\mathbb{S}^{1}$. This point of view is common in the
context of the Euler equations, see [Cox14] for instance. The main difficulty
is now evident: the spectrum of each fiber $A(m)$ is discrete, while the
spectrum of $A$ may have discrete, absolutely continuous and singular
continuous parts.

###  1.2. Organization of the paper

In Section 2 we discuss in detail various properties of _fibered_ self-adjoint
operators. In Section 3 we present a toy model which mimics some of the
behavior of the operator $A$ in order to gain a better intuition. Then, in
Section 4 we turn our attention to the flow in an annulus, first proving a
bound on the DoS (Theorem 4.7) and then obtaining a rate for the associated
ergodic theorem, proving Theorem 1.1.

##  2\. Spectral analysis and fibered operators

In this section we recall some properties of self-adjoint operators and of
self-adjoint fibered operators, and prove some results which are not readily
available in the standard literature. Our discussion remains as general as
possible, working with abstract self-adjoint operators in abstract Hilbert
spaces. Our flow operator defined above is a special example and does not
appear in this section.

###  2.1. Resolution of the identity

Since the spectral theorem plays an essential role in our proof, it is
worthwhile recalling the definition of the resolution of the identity of a
self-adjoint operator.

Let $\mathcal{H}$ be some Hilbert space, and let
$H:D(H)\subset\mathcal{H}\to\mathcal{H}$ be a self-adjoint operator. Its
associated spectral family $\\{E(\lambda)\\}_{\lambda\in\mathbb{R}}$ is a
family of projection operators in $\mathcal{H}$ with the property that, for
each $\lambda\in\mathbb{R}$, the subspace
$\mathcal{H}^{\lambda}=E(\lambda)\mathcal{H}$ is the largest closed subspace
such that

  1. (1)

$\mathcal{H}^{\lambda}$ _reduces_ $H$, namely, $HE(\lambda)g=E(\lambda)Hg$ for
every $g\in D(H)$. In particular, if $g\in D(H)$ then also $E(\lambda)g\in
D(H)$.

  2. (2)

$(Hu,u)_{\mathcal{H}}\leq\lambda(u,u)_{\mathcal{H}}$ for every
$u\in\mathcal{H}^{\lambda}\cap D(H)$.

Given any $f,g\in\mathcal{H}$ the spectral family defines a complex function
of bounded variation on the real line, given by

$\mathbb{R}\ni\lambda\mapsto(E(\lambda)f,g)_{\mathcal{H}}\in\mathbb{C}.$

It is well-known that such a function gives rise to a complex measure
(depending on $f,g$) called the _spectral measure_. Recall the following
useful fact:

###### Proposition 2.1 ([Kat95, X-§1.2, Theorem 1.5]).

Let $U\subset\mathbb{R}$ be open. The set of $f,g\in\mathcal{H}$ for which the
spectral measure is absolutely continuous in $U$ with respect to the Lebesgue
measure forms a closed subspace $\mathcal{AC}_{U}\subset\mathcal{H}$. This
subspace is referred to as the _absolutely continuous subspace of $H$ on $U$_.

###  2.2. The density of states

Let $\mathcal{AC}_{U}\subset\mathcal{H}$ be the absolutely continuous subspace
of $H$ on $U$ and let $\lambda\in U$. If there exists a subspace
$\mathcal{X}\subset\mathcal{AC}_{U}$ equipped with a stronger norm such that
the bilinear form
$\frac{\mathrm{d}}{\mathrm{d}\lambda}(E(\lambda)\cdot,\cdot)_{\mathcal{H}}:\mathcal{X}\times\mathcal{X}\to\mathbb{C}$
is bounded then it induces a bounded operator
$B(\lambda):\mathcal{X}\to\mathcal{X}^{*}$ defined via

$\left<B(\lambda)f,g\right>_{(\mathcal{X}^{*},\mathcal{X})}=\frac{\mathrm{d}}{\mathrm{d}\lambda}(E(\lambda)f,g)_{\mathcal{H}},\quad
f,g\in\mathcal{X},$

where $\left<\cdot,\cdot\right>_{(\mathcal{X}^{*},\mathcal{X})}$ is the
$(\mathcal{X}^{*},\mathcal{X})$ dual-space pairing, and $\mathcal{X}^{*}$ is
the dual of $\mathcal{X}$ with respect to the inner-product on $\mathcal{H}$.

###### Definition 2.2.

We refer to both the bilinear form
$\frac{\mathrm{d}}{\mathrm{d}\lambda}(E(\lambda)\cdot,\cdot)_{\mathcal{H}}$
and the operator $B(\lambda)$ as the _density of states (DoS)_ of the operator
$H$ at $\lambda$.

In physics, the DoS at $\lambda$ represents the number possible states a
system can attain at the energy level $\lambda$.

###  2.3. Fibered operators

We follow the notation of [RS78, p. 283]. Let $\mathcal{H}^{\prime}$ be a
Hilbert space and let $(M,\mathrm{d}\mu)$ be a measure space. Let
$A(\cdot):M\to\mathcal{L}^{s.a.}(\mathcal{H}^{\prime})$ be a measurable
function taking values in the space of self-adjoint operators (not necessarily
bounded) on $\mathcal{H}^{\prime}$ (with appropriate domains). Let
$\mathcal{H}=\int_{M}^{\oplus}\mathcal{H}^{\prime}$ and let
$A=\int_{M}^{\oplus}A(m)\mathrm{d}\mu(m)$. For an element $f\in\mathcal{H}$,
we denote its fibers as $f_{m}\in\mathcal{H}^{\prime}$ so that

$f=\int_{M}^{\oplus}f_{m}\,\mathrm{d}\mu(m).$ (2.1)

We denote the resolution of the identity of $A$ by
$\\{E(\lambda)\\}_{\lambda\in\mathbb{R}}$ and of $A(m)$ by
$\\{E_{m}(\lambda)\\}_{\lambda\in\mathbb{R}}$.

###### Lemma 2.3.

The resolutions of the identity also satisfy the natural decomposition

$E(\lambda)=\int_{M}^{\oplus}E_{m}(\lambda)\,\mathrm{d}\mu(m).$

###### Proof.

By standard functional calculus, we apply the characteristic function
$\mathbbm{1}_{(-\infty,\lambda_{0}]}$ to

$\int_{\mathbb{R}}\lambda\,\mathrm{d}E(\lambda)=A=\int_{M}^{\oplus}A(m)\,\mathrm{d}\mu(m)=\int_{M}^{\oplus}\int_{\mathbb{R}}\lambda\,\mathrm{d}E_{m}(\lambda)\,\mathrm{d}\mu(m)$

to obtain the assertion of the lemma (at the point $\lambda_{0}$). ∎

It is well-known that since all $A(m)$ are self-adjoint, so is $A$, with
spectrum $\sigma(A)$ characterized as follows:

$\lambda\in\sigma(A)\qquad\Leftrightarrow\qquad\forall\varepsilon>0,\quad\mu\left(\left\\{m\;:\;\sigma(A(m))\cap(\lambda-\varepsilon,\lambda+\varepsilon)\neq\emptyset\right\\}\right)>0.$
(2.2)

An immediate consequence of this is:

$\sigma(A)\subset\overline{\bigcup_{m}\sigma(A(m))}.$ (2.3)

Let us mention some other important consequences. First, the following
characterization of eigenvalues:

$\mu\left(\left\\{m\;:\;\lambda\text{ is an eigenvalue of
}A(m)\right\\}\right)>0\qquad\Rightarrow\qquad\lambda\text{ is an eigenvalue
of }A.$ (2.4)

Hence, if there exist real numbers $E_{j}$ such that
$\cup_{m}\sigma(A(m))=\\{E_{1},\dots,E_{k}\\}$ with
$\mu(\\{m\;:\;\sigma(A(m))=E_{j}\\})>0$ for all $j$, then
$\sigma(A)=\\{E_{1},\dots,E_{k}\\}$. If, on the other hand, for all $m\in M$
the spectrum of $A(m)$ is purely absolutely continuous then so is the spectrum
of $A$.

###### Proposition 2.4.

Assume that the measure $\mathrm{d}\mu$ is the Borel measure associated to
some given topology and that $M$ is compact. Assume that
$\Sigma:M\to\operatorname{cl}(\mathbb{R})=\\{$closed subsets in
$\mathbb{R}\\}$ given by $m\mapsto\sigma(A(m))\subset\mathbb{R}$ is continuous
(we take the Hausdorff distance on $\operatorname{cl}(\mathbb{R})$). Then
$\sigma(A)=\cup_{m}\sigma(A(m))$.

###### Proof.

Since $M$ is compact and $\Sigma$ is continuous, we have that
$\cup_{m}\sigma(A(m))=\overline{\cup_{m}\sigma(A(m))}$. Hence, considering
(2.3) we only need to prove that $\sigma(A)\supset\cup_{m}\sigma(A(m))$.
Suppose that $\lambda\in\cup_{m}\sigma(A(m))=\cup_{m}\Sigma(m)$. In particular
there exists some $m_{0}\in M$ such that $\lambda\in\Sigma(m_{0})$. By the
continuity of $\Sigma$, for each $\varepsilon>0$ there exists an open
neighborhood $U$ of $m_{0}$ such that
$\Sigma(m)\cap(\lambda-\varepsilon,\lambda+\varepsilon)\neq\emptyset$ for all
$m\in U$, which, by (2.2), implies that $\lambda\in\sigma(A)$ (note that $U$
has positive measure). ∎

##  3\. Toy example

We view the flow operator (1.1) as fibered in the direction orthogonal to the
flow. That is, we view it as $A=\int_{[0,1]}^{\oplus}A(m)\,\mathrm{d}m$ where
$A(m)$ are one-dimensional flow operators, see (1.5). For small values of $m$,
the (discrete) spectrum of the individual fibers concentrates near
$\lambda=0$, thereby making the analysis of the (continuous) spectrum of $A$
near $\lambda=0$ difficult. To improve our intuition, here we study a
simplified problem.

Consider the following fibered problem. Let
$\left(M,\mathrm{d}\mu\right)=\left([0,1],\mathrm{d}m\right)$ with the usual
Borel $\sigma$-algebra and suppose that for all $m\in[0,1]$ the spectrum of
$A(m)$ in the energy band $I:=(a,b)\subset\mathbb{R}$ is given by a single
eigenvalue $\mathcal{E}(m)$ depending on $m$ with multiplicity $1$. Suppose
that $\mathcal{E}:[0,1]\to I$ is measurable (it actually has to be due to the
definition of fibered operators). As before, we let
$\mathcal{H}=\int_{[0,1]}^{\oplus}\mathcal{H}^{\prime}$ and
$A=\int_{[0,1]}^{\oplus}A(m)\mathrm{d}m$. We know that

  * •

$\sigma(A)\cap I=\operatorname{ess\,ran}(\mathcal{E})$ and if $\mathcal{E}$ is
continuous then $\operatorname{ess\,ran}$ may be replaced by
$\operatorname{ran}$ (see Proposition 2.4).

  * •

If $\mathcal{E}$ is constant on some open set $U$ and equal to
$\overline{\mathcal{E}}$ there, then $\overline{\mathcal{E}}$ is an eigenvalue
of $A$ (see (2.4)).

These two statements combined indicate that one can easily construct examples
with eigenvalues embedded in the essential spectrum (or at its boundary).
Hence even in this simple example, the absolute continuity of the spectrum of
$A$ depends on the nature of the function $\mathcal{E}$. Let us assume that
there are no embedded eigenvalues:

###### Assumption A1.

$\mathcal{E}\in C^{1}([0,1])$ and is not constant on sets of positive measure.

###### Definition 3.1.

Let $\lambda\in\operatorname{ran}(\mathcal{E})$ and denote
$M_{\lambda}:=\mathcal{E}^{-1}(\lambda)\subset[0,1]$. We say that $\lambda$ is
a regular value if for any $m\in M_{\lambda}$, $\mathcal{E}^{\prime}(m)\neq
0$. We also define the set

$\sigma_{\mathrm{reg},I}(A):=\left\\{\lambda\in
I\;:\;\lambda\in\operatorname{ran}(\mathcal{E})\text{ is a regular
value}\right\\}.$

###### Lemma 3.2.

$\sigma_{\mathrm{reg},I}(A)$ is an open subset of $I$.

###### Proof.

The claim is almost trivial, as the condition of being a regular point is an
open condition. We only note that any point in the set
$I\cap\partial(\operatorname{ran}(\mathcal{E}))$ cannot be a regular value
since $I$ is open. ∎

The central computation in the proof of the following theorem will appear in
the proof of our main theorem in the next section, and therefore it is useful
to understand it in this simplified toy model.

###### Theorem 3.3.

Suppose that $\mathcal{E}$ is as above and satisfies Assumption A1. Let
$\lambda\in\sigma_{\mathrm{reg},I}(A)$. Then the DoS of $A$ at $\lambda$
satisfies the estimate

$\left|\frac{\mathrm{d}}{\mathrm{d}\lambda}(E(\lambda)f,g)_{\mathcal{H}}\right|\leq\left(\sum_{m\in
M_{\lambda}}\frac{1}{|\mathcal{E}^{\prime}(m)|}\right)\,\|f\|_{\mathcal{H}}\|g\|_{\mathcal{H}},\qquad\forall
f,g\in\mathcal{H}.$ (3.1)

###### Proof.

Since $M=[0,1]$ is compact, $\mathcal{E}\in C^{1}$, and $\lambda$ is a regular
value, $M_{\lambda}$ is a finite set and we denote its elements
$M_{\lambda}=\\{m_{1},\dots,m_{k}\\}$. Recall that for any $m\in[0,1]$, the
spectrum of $A(m)$ in $I=(a,b)$ is made up of a single eigenvalue
$\mathcal{E}(m)$. Hence if $\lambda\in I$ and $\mathcal{E}(m)\leq\lambda$ then
the projection operator $E_{m}(\lambda)$ may be represented as
$P_{m}(\mathcal{E}(m))+E_{m}(a)$ where $P_{m}(\mathcal{E}(m))$ is the
projection onto the eigenspace in $\mathcal{H}^{\prime}$ (the $m$th “copy”)
corresponding to the eigenvalue $\mathcal{E}(m)$. If $\lambda\in I$ and
$\mathcal{E}(m)>\lambda$ then the projection operator $E_{m}(\lambda)$ is
equal to the projection operator $E_{m}(a)$. Letting $f,g\in\mathcal{H}$, with
fibers $f_{m},g_{m}\in\mathcal{H}^{\prime}$ (as in (2.1)), we have

$\displaystyle(E(\lambda)f,g)_{\mathcal{H}}$
$\displaystyle=\int_{[0,1]}\left(E_{m}(\lambda)f_{m},g_{m}\right)_{\mathcal{H}^{\prime}}\,\mathrm{d}m$
$\displaystyle=\int_{\\{m\;:\;\mathcal{E}(m)\leq\lambda\\}}\left(P_{m}(\mathcal{E}(m))f_{m},g_{m}\right)_{\mathcal{H}^{\prime}}\,\mathrm{d}m+\int_{[0,1]}\left(E_{m}(a)f_{m},g_{m}\right)_{\mathcal{H}^{\prime}}\,\mathrm{d}m.$

Differentiating in $\lambda$ the second term on the right hand side is
eliminated, and one is left with

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\lambda}(E(\lambda)f,g)_{\mathcal{H}}$
$\displaystyle=\lim_{h\downarrow
0}\left[\frac{1}{h}\int_{\\{m\;:\;\lambda<\mathcal{E}(m)\leq\lambda+h\\}}\left(P_{m}(\mathcal{E}(m))f_{m},g_{m}\right)_{\mathcal{H}^{\prime}}\,\mathrm{d}m\right]$
$\displaystyle=\sum_{i,\;\mathcal{E}^{\prime}(m_{i})>0}\lim_{h\downarrow
0}\left[\frac{1}{h}\int_{(m_{i},m_{i}+\frac{h}{\mathcal{E}^{\prime}(m_{i})}]}\left(P_{m}(\mathcal{E}(m))f_{m},g_{m}\right)_{\mathcal{H}^{\prime}}\,\mathrm{d}m\right]$
$\displaystyle\quad+\sum_{i,\;\mathcal{E}^{\prime}(m_{i})<0}\lim_{h\downarrow
0}\left[\frac{1}{h}\int_{(m_{i}+\frac{h}{\mathcal{E}^{\prime}(m_{i})},m_{i}]}\left(P_{m}(\mathcal{E}(m))f_{m},g_{m}\right)_{\mathcal{H}^{\prime}}\,\mathrm{d}m\right].$

Making the change of variables
$\eta_{i}=\frac{h}{|\mathcal{E}^{\prime}(m_{i})|}$ we have

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\lambda}(E(\lambda)f,g)_{\mathcal{H}}$
$\displaystyle=\sum_{i,\;\mathcal{E}^{\prime}(m_{i})>0}\frac{1}{\mathcal{E}^{\prime}(m_{i})}\lim_{\eta_{i}\downarrow
0}\left[\frac{1}{\eta_{i}}\int_{(m_{i},m_{i}+\eta_{i}]}\left(P_{m}(\mathcal{E}(m))f_{m},g_{m}\right)_{\mathcal{H}^{\prime}}\,\mathrm{d}m\right]$
$\displaystyle\quad+\sum_{i,\;\mathcal{E}^{\prime}(m_{i})<0}\frac{1}{|\mathcal{E}^{\prime}(m_{i})|}\lim_{\eta_{i}\downarrow
0}\left[\frac{1}{\eta_{i}}\int_{(m_{i}-\eta_{i},m_{i}]}\left(P_{m}(\mathcal{E}(m))f_{m},g_{m}\right)_{\mathcal{H}^{\prime}}\,\mathrm{d}m\right]$
$\displaystyle=\sum_{i=1}^{k}\frac{1}{|\mathcal{E}^{\prime}(m_{i})|}\left(P_{m_{i}}(\mathcal{E}(m_{i}))f_{m_{i}},g_{m_{i}}\right)_{\mathcal{H}^{\prime}}.$
(3.2)

Note that one must be careful if one of the $m_{i}$ is $0$ (resp. $1$) and
$\mathcal{E}^{\prime}(0)<0$ (resp. $\mathcal{E}^{\prime}(1)>0$) as then the
above argument requires a slight adjustment. However the same conclusion
holds. A simple use of the Cauchy-Schwartz inequality and the orthogonality
properties of the projection operators leads to the desired estimate (3.1). ∎

##  4\. Flow in an annulus: Proof of Theorem 1.1

###  4.1. Description and assumptions

We are now ready to study our main object of interest, a flow in an annulus.
We consider a steady flow in an annulus
$\mathcal{A}=[0,1]\times\mathbb{S}^{1}$. The variables in $[0,1]$ and
$\mathbb{S}^{1}$ shall be denoted $m$ and $\theta$, respectively. For each
$m\in(0,1]$ corresponds the self-adjoint operator

$A(m)=-i\varphi(m)\frac{\mathrm{d}}{\mathrm{d}\theta}:H^{1}(\mathbb{S}^{1})\subset
L^{2}(\mathbb{S}^{1})\to L^{2}(\mathbb{S}^{1})$

generating a flow along the circle $\mathbb{S}^{1}$ with speed $\varphi(m)>0$.
Its spectrum is given by

$\sigma(A(m))=\varphi(m)\mathbb{Z}.$

Furthermore, if $\varphi$ is continuous on $[0,1]$ the spectrum of
$A=\int_{[0,1]}^{\oplus}A(m)\,\mathrm{d}m$ is given by the expression

$\sigma(A)=\left(\bigcup_{-k\in\mathbb{N}}\Big{[}\varphi_{\max}k,\varphi_{\min}k\Big{]}\right)\cup\left(\bigcup_{k\in\mathbb{N}\cup\\{0\\}}\Big{[}\varphi_{\min}k,\varphi_{\max}k\Big{]}\right)$

where $\varphi_{\min}:=\min_{[0,1]}\varphi(m)$ and
$\varphi_{\max}:=\max_{[0,1]}\varphi(m)$. For more details on how to derive
this expression see for instance [Cox14]. We assume that the flow degenerates
at the fiber $m=0$, so that $\sigma(A)=\mathbb{R}$ (there is no spectral gap):

###### Assumption A2.

Assume that there exists $\alpha>0$ such that $\varphi(m)=m^{\alpha}$.

###### Remark 4.1.

This assumption can be weakened, as only the behavior of $\varphi$ near its
zeros is important. For instance, all our arguments below could be modified to
handle a case where there exist $m_{0},c,\alpha>0$ such that $\varphi(m)\sim
cm^{\alpha}$ on $(0,m_{0})$ and $\varphi$ is continuous and positive on
$[m_{0},1]$. This can then be extended to situations with several zeros, as
long as $\varphi$ vanishes like $m^{\alpha_{i}}$ for some $\alpha_{i}>0$ near
each zero. Furthermore, the requirement that $\varphi$ be continuous can also
be weakened. However these are technicalities that we do not pursue at
present.

In the toy model presented in Section 3, to each energy level $\lambda$ there
corresponded finitely many fibers $M_{\lambda}$. Now the situation is more
delicate: to each energy level correspond infinitely many fibers.
Contributions to the spectrum at energy level $\lambda>0$222Here we choose to
consider positive energy levels $\lambda>0$ and consequently $k\in\mathbb{N}$.
We could have chosen $\lambda<0$ and then $-k\in\mathbb{N}$. will come from
all $m_{\lambda,k}\in(0,1]$ such that $\lambda=k{m_{\lambda,k}^{\alpha}}$
where $k\in\mathbb{N}$. That is,

$m_{\lambda,k}=\left(\frac{\lambda}{k}\right)^{1/\alpha},\qquad
k\in\mathbb{N},\,\lambda>0,$

and conversely, each fiber $m\in(0,1]$ will contribute to the discrete energy
levels

$\lambda_{m,k}=km^{\alpha},\qquad k\in\mathbb{N},\,m\in(0,1].$

###  4.2. Functional setting

In order to prove Theorem 1.1, we must first obtain an estimate on the DoS as
in (4.9). One of the crucial ingredients in such an estimate is the
identification of a functional subspace $\mathcal{X}\subset
L^{2}(\mathcal{A})$ on which the estimate holds. Here we define this space,
denoted $\mathcal{H}^{s,\gamma}_{0}$, and discuss some of its properties.

We consider functions $f\in L^{2}(\mathcal{A})$ as fibered in $m$, as in
(2.1):

$f=\int_{[0,1]}^{\oplus}f_{m}\,\mathrm{d}m\qquad\text{with}\qquad f_{m}\in
L^{2}(\mathbb{S}).$

This is equivalent to considering $f$ in the space of functions
$L^{2}\left([0,1];L^{2}(\mathbb{S})\right)$. In the $\theta$ variable we
consider Sobolev regularity on the circle $\mathbb{S}$. For each function
$f_{m}\in L^{2}(\mathbb{S})$, we define the Fourier series of $f_{m}$ by

$\widehat{f_{m}}(k):=\int_{[0,1]}f_{m}(\theta)e^{-2\pi
ik\theta}\,\mathrm{d}\theta.$

Since we are ultimately interested in the rate of convergence to the average,
it is sensible to work with functions whose average along each fiber is zero:

$L^{2}_{0}(\mathbb{S})=\left\\{g\in
L^{2}(\mathbb{S})\,:\,\widehat{g}(0)=0\right\\}.$

###### Remark 4.2.

There is no loss of generality in considering $L^{2}_{0}(\mathbb{S})$ on each
fiber. Indeed, for a general function $f\in L^{2}(\mathcal{A})$ each of the
fibers $(f-Pf)_{m}$ of $f-Pf$ belongs to $L^{2}_{0}\left(\mathbb{S}\right)$
(here $P$ is as defined in (1.2)). As $Pf$ is invariant under the flow of the
Hamiltonian, we can safely replace $f$ by $f-Pf$ in all the following.

We consider now standard homogeneous Sobolev spaces on
$L^{2}_{0}(\mathbb{S})$. These are defined as

$\dot{H}^{\gamma}:=\left\\{g\in
L^{2}_{0}(\mathbb{S})\,:\,\|g\|_{\dot{H}^{\gamma}}^{2}:=\sum_{k\in\mathbb{Z}}|k|^{2\gamma}|\widehat{g}(k)|^{2}<+\infty\right\\},\qquad\gamma\geq
0.$

We associate to these spaces the norm $\|\cdot\|_{\dot{H}^{\gamma}}$.

Regarding the regularity with respect to the $m$ variable in the closed
interval $[0,1]$, we consider $f$ as a function of
$L^{2}\left([0,1];\dot{H}^{\gamma}\right)$. Such spaces of functions and
associated fractional Sobolev regularity have been extensively studied by
Lions and Magenes in [LM72]. In the following we use the framework and the
results of [Sim90] (see Theorem 10.2 in [LM72] for the equivalence of the
viewpoints, as Lions and Magenes define fractional Sobolev spaces on an
interval by interpolation spaces).

We define then, for $s\in(0,1)$ and $\gamma\geq 0$, the space

$\dot{\mathcal{H}}^{s,\gamma}:=\left\\{f\in
L^{2}\left([0,1];\dot{H}^{\gamma}\right)\,:\,\frac{\|f_{m}-f_{m^{\prime}}\|_{\dot{H}^{\gamma}}}{|m-m^{\prime}|^{1/2+s}}\in
L^{2}\left[0,1]\times[0,1]\right)\right\\}.$

This is in fact
$\dot{\mathcal{H}}^{s,\gamma}=W^{s,2}\left([0,1];\dot{H}^{\gamma}\right)$ in
the framework of [Sim90]. We associate to this space the usual norm defined by

$\|f\|_{s,\gamma}^{2}=\|f\|_{L^{2}\left([0,1];\dot{H}^{\gamma}\right)}^{2}+\iint_{[0,1]\times[0,1]}\frac{\|f_{m}-f_{m^{\prime}}\|^{2}_{\dot{H}^{\gamma}}}{|m-m^{\prime}|^{1+2s}}\,\mathrm{d}m\,\mathrm{d}m^{\prime}.$
(4.1)

One important result is the following Sobolev embedding-type theorem:

###### Proposition 4.3.

Let $s>1/2$. Then any $f\in\dot{\mathcal{H}}^{s,\gamma}$ is $(s-1/2)$-Hölder
continuous with respect to the $m$ variable. That is, there is a constant
$C(s)$ (independent of $f$ and $\gamma$) such that

$\left\|f_{m}-f_{m^{\prime}}\right\|_{\dot{H}^{\gamma}}\leq
C(s)|m-m^{\prime}|^{s-1/2},\qquad\forall m,m^{\prime}\in[0,1],\,\forall
f\in\dot{\mathcal{H}}^{s,\gamma}.$ (4.2)

For a proof of this result, we refer to the paper of Simon [Sim90] and in
particular Corollary 26 therein as
$\dot{\mathcal{H}}^{s,\gamma}=W^{s,2}\left([0,1];\dot{H}^{\gamma}\right)$ and
Hölder continuity is called Lipschitz continuity.

We may now define the space in which we will work:

###### Definition 4.4.

Let $s>1/2,\gamma\geq 0$. We define the subspace
$\dot{\mathcal{H}}^{s,\gamma}_{0}\subset\dot{\mathcal{H}}^{s,\gamma}$ to be
the set of functions that vanish along $\\{m=0\\}$:

$\dot{\mathcal{H}}^{s,\gamma}_{0}:=\left\\{f\in\dot{\mathcal{H}}^{s,\gamma}\,:\,f_{0}=0\right\\}.$
(4.3)

We equip this subspace with the norm $\left\|\cdot\right\|_{s,\gamma}$ defined
above in (4.1).

We now make some important observations regarding this $L^{2}$ subspace.
Thanks to the Hölder continuity result of Proposition 4.3, functions in
$\dot{\mathcal{H}}^{s,\gamma}_{0}$ decay as $m\downarrow 0$:

###### Proposition 4.5.

For all $f\in\dot{\mathcal{H}}^{s,\gamma}_{0}$, there holds

$\|f_{m}\|_{\dot{H}^{\gamma}}\leq
C(s)\left\|f\right\|_{s,\gamma}m^{s-1/2},\qquad\forall m\in[0,1],$ (4.4)

where $C(s)>0$ is the same constant appearing in (4.2) and which does not
depend on $\gamma$.

###### Proof.

This follows immediately from the inequality (4.2) together with the fact that
$f_{0}=0$ since $f\in\dot{\mathcal{H}}^{s,\gamma}_{0}$. ∎

Next, we state a result on the Hölder regularity of the Fourier coefficients
of functions in $\dot{\mathcal{H}}^{s,\gamma}$, which is just a corollary of
the Hölder continuity of $f$ stated in Proposition 4.3:

###### Proposition 4.6.

For any $f\in\dot{\mathcal{H}}^{s,\gamma}_{0}$, there holds

$\left|\widehat{f}_{m}(k)-\widehat{f}_{m^{\prime}}(k)\right|\leq
C\|f\|_{s,\gamma}|k|^{-\gamma}|m-m^{\prime}|^{s-1/2},\qquad\forall
m,m^{\prime}\in[0,1],\,\forall k\in\mathbb{Z},$ (4.5)

where the constant $C>0$ does not depend on $f$, $k$ or $m$.

###### Proof.

We readily compute

$\displaystyle\left|\widehat{f}_{m}(k)-\widehat{f}_{m^{\prime}}(k)\right|^{2}$
$\displaystyle=|k|^{-2\gamma}|k|^{2\gamma}\left|\widehat{f}_{m}(k)-\widehat{f}_{m^{\prime}}(k)\right|^{2}$
$\displaystyle\leq|k|^{-2\gamma}\sum_{k^{\prime}\in\mathbb{Z}}|k^{\prime}|^{2\gamma}\left|\widehat{f}_{m}(k^{\prime})-\widehat{f}_{m^{\prime}}(k^{\prime})\right|^{2}\leq|k|^{-2\gamma}\|f_{m}-f_{m^{\prime}}\|_{\dot{H}^{\gamma}}^{2}.$

By the Hölder continuity (4.2) of $f_{m}$ in the $\dot{H}^{\gamma}$ norm, we
obtain (4.5). ∎

###  4.3. Density of states estimate

The main ingredient in the proof of Theorem 1.1 is an estimate of the DoS near
$0$. This is stated as follows:

###### Theorem 4.7.

Let $A=-i\varphi(m)\frac{\partial}{\partial\theta}$ satisfy Assumption A2.
Further assume that $s>1/2$ and $\gamma\geq 0$ satisfy the constraint

$\gamma+s/\alpha>1/2.$ (4.6)

Then there exists $r>0$ such that the DoS of $A$ satisfies

$\left|\frac{\mathrm{d}}{\mathrm{d}\lambda}(E(\lambda)f,g)_{L^{2}(\mathcal{A})}\right|\leq
C|\lambda|^{\frac{2s}{\alpha}-1}\left\|f\right\|_{s,\gamma}\left\|g\right\|_{s,\gamma},\qquad\forall\lambda\in(-r,r)\setminus\\{0\\},\,\forall
f,g\in\dot{\mathcal{H}}^{s,\gamma}_{0},$

where $C>0$ does not depend on $f$, $g$ or $\lambda$.

###### Remark 4.8.

1\. In the case $\alpha\leq 1$ the constraint (4.6) is satisfied for all
$s>1/2$ with $\gamma=0$. This means that for $\alpha\leq 1$, the subspace
$\dot{\mathcal{H}}^{s,0}_{0}$ is sufficient to get the estimate of the DoS.

2\. In the case $\alpha>1$ however, the constraint (4.6) is stronger and
working in the subspace $\dot{\mathcal{H}}^{s,0}_{0}$ is not sufficient to
obtain estimates of the DoS. This is mainly due to the fact that the
eigenvalues $\lambda_{m,k}=km^{\alpha}$ concentrate at $0$ faster as $m\to 0$
and $k\to\infty$. To balance that effect, more regularity in the $\theta$
variable is required, as the equation $\lambda=\lambda_{m,k}$ allows to trade
control of high frequencies in smallness in $m$.

$\theta$$m$$\alpha<1$ $\theta$$m$$\alpha>1$

###### Proof of Theorem 4.7.

As in the case of the toy model presented in Section 3, we write

$\displaystyle(E(\lambda)f,g)_{L^{2}(\mathcal{A})}$
$\displaystyle=\int_{[0,1]}\left(E_{m}(\lambda)f_{m},g_{m}\right)_{L^{2}(\mathbb{S})}\,\mathrm{d}m$
$\displaystyle=\int_{[0,1]}\sum_{\begin{subarray}{c}k\in\mathbb{Z}\\\
\lambda_{m,k}\leq\lambda\end{subarray}}\left(P_{m}(\lambda_{m,k})f_{m},g_{m}\right)_{L^{2}(\mathbb{S})}\,\mathrm{d}m$

where $P_{m}(\lambda_{m,k})$ is the projection on the Fourier coefficient
$\widehat{f_{m}}(k)$ of $f_{m}$.

As before, we compute the DoS starting from the definition of a derivative,
i.e. the limit of
$\frac{1}{h}\left\\{(E(\lambda+h)f,g)_{L^{2}(\mathcal{A})}-(E(\lambda)f,g)_{L^{2}(\mathcal{A})}\right\\}$
as $h\to 0$. We define the energy band

$\mathcal{B}(\lambda,m,h)=\left\\{k\in\mathbb{Z}\,:\,\lambda<km^{\alpha}\leq\lambda+h\right\\}$

and have that

$\displaystyle\frac{1}{h}\left\\{(E(\lambda+h)f,g)_{L^{2}(\mathcal{A})}-(E(\lambda)f,g)_{L^{2}(\mathcal{A})}\right\\}$
$\displaystyle\quad=\frac{1}{h}\int_{[0,1]}\sum_{k\in\mathcal{B}(\lambda,m,h)}\left(P_{m}(\lambda_{m,k})f_{m},g_{m}\right)_{L^{2}(\mathbb{S})}\,\mathrm{d}m$
$\displaystyle\quad=\frac{1}{h}\int_{[0,1]}\sum_{k\in\mathcal{B}(\lambda,m,h)}\,\widehat{f_{m}}(k)\overline{\widehat{g_{m}}(k)}\,\mathrm{d}m.$

We would now like to commute the integration and summation, to obtain the sum
of integrals over small subintervals of $[0,1]$, much like in the computations
in (3.2), in order to then take the limit $h\to 0$. However, for fixed $h$ the
sets $\mathcal{B}(\lambda,m,h)$ will contain arbitrarily many elements as
$m\to 0$. Moreover, as $m\to 0$ the preimages of
$\lambda<km^{\alpha}\leq\lambda+h$ in $m\in[0,1]$ are no longer disjoint,
resulting in many redundant integrations. Indeed, intervals
$(\lambda/m^{\alpha},(\lambda+h)/m^{\alpha}]$ are of size larger than one if
$m<h^{1/\alpha}$, and then may contain more than one integer. Our strategy is
to split the previous integral into two parts

$\frac{1}{h}\int_{[0,1]}\sum_{k\in\mathcal{B}(\lambda,m,h)}\,\widehat{f_{m}}(k)\overline{\widehat{g_{m}}(k)}\,\mathrm{d}m=\mathcal{I}(h)+\mathcal{R}(h)$

where

$\mathcal{R}(h)=\frac{1}{h}\int_{[0,h^{1/\alpha})}\sum_{k\in\mathcal{B}(\lambda,m,h)}\,\widehat{f_{m}}(k)\overline{\widehat{g_{m}}(k)}\,\mathrm{d}m.$

and

$\mathcal{I}(h)=\frac{1}{h}\int_{[h^{1/\alpha},1]}\sum_{k\in\mathcal{B}(\lambda,m,h)}\,\widehat{f_{m}}(k)\overline{\widehat{g_{m}}(k)}\,\mathrm{d}m\\\
$

_1\. The term $\mathcal{R}(h)$_. We show that $\lim_{h\to 0}\mathcal{R}(h)=0$.
Using the inequalities

$\left|\sum_{k\in\mathcal{B}(\lambda,m,h)}\,\widehat{f_{m}}(k)\overline{\widehat{g_{m}}(k)}\right|\leq\sum_{k\in\mathcal{B}(\lambda,m,h)}\left|\widehat{f_{m}}(k)\overline{\widehat{g_{m}}(k)}\right|\leq\sum_{k>\lambda/m^{\alpha}}\,\left|\widehat{f_{m}}(k)\overline{\widehat{g_{m}}(k)}\right|$

and taking $f,g\in\dot{\mathcal{H}}^{s,\gamma}_{0}$ we have

$\displaystyle\left|\mathcal{R}(h)\right|$
$\displaystyle\leq\frac{1}{h}\int_{[0,h^{1/\alpha})}\sum_{k>\lambda/m^{\alpha}}\,|k|^{-2\gamma}\left(|k|^{\gamma}\left|\widehat{f_{m}}(k)\right|\right)\,\left(|k|^{\gamma}\left|\overline{\widehat{g_{m}}(k)}\right|\right)\,\mathrm{d}m$
$\displaystyle\leq\frac{1}{h}\int_{[0,h^{1/\alpha})}\sup_{k>\lambda/m^{\alpha}}\left(|k|^{-2\gamma}\right)\,\left\|f_{m}\right\|_{\dot{H}^{\gamma}}\left\|g_{m}\right\|_{\dot{H}^{\gamma}}\,\mathrm{d}m,$

using the Hölder inequality for the last line. We use now inequality (4.4)
that gives decay as $m\to 0$ for the norm
$\left\|f_{m}\right\|_{\dot{H}^{\gamma}}$ and compute then

$\displaystyle\left|\mathcal{R}(h)\right|$
$\displaystyle\leq\frac{1}{h}\int_{[0,h^{1/\alpha})}\left(\frac{\lambda}{m^{\alpha}}\right)^{-2\gamma}\left\|f_{m}\right\|_{\dot{H}^{\gamma}}\left\|g_{m}\right\|_{\dot{H}^{\gamma}}\,\mathrm{d}m$
$\displaystyle\leq
C\,\|f\|_{s,\gamma}\,\|g\|_{s,\gamma}\,\frac{\lambda^{-2\gamma}}{h}\int_{[0,h^{1/\alpha})}m^{2\alpha\gamma+2(s-1/2)}\,\mathrm{d}m$
$\displaystyle=C\|f\|_{s,\gamma}\,\|g\|_{s,\gamma}\,{\lambda^{-2\gamma}}h^{2\gamma+2s/\alpha-1}.$

As soon as the constraint (4.6) on $s$ and $\gamma$ is satisfied there holds

$\lim_{h\to 0}\mathcal{R}(h)=0.$

_2\. The term $\mathcal{I}(h)$_. Whenever $m\geq h^{\/\alpha}$, the energy
band $\mathcal{B}(\lambda,m,h)$ contains at most one integer. For
$\mathcal{B}(\lambda,m,h)$ to contain one integer, by definition $m$ has to be
in an interval of the form
$(\left(\lambda/k^{\prime}\right)^{1/\alpha},\left((\lambda+h)/k^{\prime}\right)^{1/\alpha}]$
for some $k^{\prime}\in\mathbb{N}$. As $m\in[h^{1/\alpha},1]$, this implies
for the bounds of such an interval that

$h^{1/\alpha}\leq\left(\lambda/k^{\prime}\right)^{1/\alpha}\quad\text{and}\quad\left((\lambda+h)/k^{\prime}\right)^{1/\alpha}\leq
1.$

The integer $k^{\prime}$ has then to satisfy the bounds

$N(\lambda,h):=\lfloor\lambda/h\rfloor\geq k^{\prime}\geq
k_{0}:=\max(1,\lfloor\lambda\rfloor),$

reminding that $k^{\prime}\geq 1$. We write then

$\displaystyle\mathcal{I}(h)$
$\displaystyle=\frac{1}{h}\int_{[h^{1/\alpha},1]}\sum_{k\in\mathcal{B}(\lambda,m,h)}\,\widehat{f_{m}}(k)\overline{\widehat{g_{m}}(k)}\,\mathrm{d}m$
$\displaystyle=\frac{1}{h}\sum_{N(\lambda,h)\geq k^{\prime}\geq
k_{0}}\int_{(\left(\lambda/k^{\prime}\right)^{1/\alpha},\left((\lambda+h)/k^{\prime}\right)^{1/\alpha}]}\sum_{k\in\mathcal{B}(\lambda,m,h)}\,\widehat{f_{m}}(k)\overline{\widehat{g_{m}}(k)}\,\mathrm{d}m$
$\displaystyle=\frac{1}{h}\sum_{N(\lambda,h)\geq k^{\prime}\geq
k_{0}}\int_{(\left(\lambda/k^{\prime}\right)^{1/\alpha},\left((\lambda+h)/k^{\prime}\right)^{1/\alpha}]}\,\widehat{f_{m}}(k^{\prime})\overline{\widehat{g_{m}}(k^{\prime})}\,\mathrm{d}m.$

Next, we use the same kind of computation as in the proof of Theorem 3.3,
where the prefactor $\frac{1}{\mathcal{E}^{\prime}(m)}$ appears. In the
current computation, the function $\mathcal{E}(m)$ becomes
$k\varphi(m)=km^{\alpha}$ so that $\mathcal{E}^{\prime}(m)$ becomes $k\alpha
m^{\alpha-1}$. Substituting $\left(\frac{\lambda}{k}\right)^{1/\alpha}$ for
$m$ we find that the prefactor should be $\frac{1}{\alpha
k}\left(\frac{\lambda}{k}\right)^{1/\alpha-1}$. Furthermore, we use
Proposition 4.6 which ensures some uniform regularity for the Fourier
coefficients and allows us to make pointwise (in $m$) evaluations. This leads
to

$\lim_{h\to
0}\frac{1}{h}\int_{(\left(\lambda/k\right)^{1/\alpha},\left((\lambda+h)/k\right)^{1/\alpha}]}\,\widehat{f_{m}}(k)\overline{\widehat{g_{m}}(k)}\,\mathrm{d}m=\frac{1}{\alpha
k}\left(\frac{\lambda}{k}\right)^{1/\alpha-1}\,\widehat{f}_{(\lambda/k)^{1/\alpha}}(k)\overline{\widehat{g}_{(\lambda/k)^{1/\alpha}}(k)}$

which gives

$\lim_{h\to 0}\mathcal{I}(h)=\sum_{k\geq k_{0}}\frac{1}{\alpha
k}\left(\frac{\lambda}{k}\right)^{1/\alpha-1}\,\widehat{f}_{(\lambda/k)^{1/\alpha}}(k)\,\overline{\widehat{g}_{(\lambda/k)^{1/\alpha}}(k)}.$

by uniform boundedness. To conclude we use inequality (4.5) and the fact that
$f_{0}$ is trivial in $\dot{H}^{\gamma}$ to obtain

$\left|\widehat{f}_{(\lambda/k)^{1/\alpha}}(k)\right|\leq
C\|f\|_{s,\gamma}|k|^{-\gamma}\left|(\lambda/k)^{1/\alpha}\right|^{s-1/2}$

with the constant $C>0$ independent of $k$ and $\lambda$. There holds now for
$\lim_{h\to 0}\mathcal{I}(h)$ the bound

$\displaystyle\left|\lim_{h\to 0}\mathcal{I}(h)\right|$
$\displaystyle=\frac{1}{\alpha}\left|\sum_{k\geq
k_{0}}\frac{1}{k}\left(\frac{\lambda}{k}\right)^{1/\alpha-1}\,\widehat{f}_{(\lambda/k)^{1/\alpha}}(k)\overline{\widehat{g}_{(\lambda/k)^{1/\alpha}}(k)}\right|$
$\displaystyle\leq
C\,\|f\|_{s,\gamma}\,\|g\|_{s,\gamma}\,\frac{\lambda^{1/\alpha-1}}{\alpha}\,\sum_{k\geq
k_{0}}k^{{}-1/\alpha-2\gamma}\left(\frac{\lambda}{k}\right)^{2(s-1/2)/\alpha}$
$\displaystyle=C\,\|f\|_{s,\gamma}\,\|g\|_{s,\gamma}\,\frac{\lambda^{2s/\alpha-1}}{\alpha}\,\sum_{k\geq
k_{0}}k^{{}-(1/\alpha+2\gamma+2(s-1/2)/\alpha)}$ $\displaystyle\leq
C\,\|f\|_{s,\gamma}\,\|g\|_{s,\gamma}\,\frac{\lambda^{2s/\alpha-1}}{\alpha}\,\sum_{k\geq
1}k^{{}-(2\gamma+2s/\alpha)}$

where we use $k_{0}\geq 1$ in the last inequality. The sum over $k$ is then
finite if the constraint (4.6) on $s$ and $\gamma$ is satisfied. This
completes the proof of Theorem 4.7. ∎

###  4.4. Uniform ergodic theorem: proof of Theorem 1.1

Finally we are ready to prove Theorem 1.1. In [BAM19] we prove the following
result:

###### Theorem 4.9 (Corollary 1.5 in [BAM19]).

Let $A:D(A)\subset\mathcal{H}\to\mathcal{H}$ be self-adjoint and assume that
there exist a Banach subspace $\mathcal{X}\subset\mathcal{H}$ which is dense
in $\mathcal{H}$ in the topology of $\mathcal{H}$ that is continuously
embedded in $\mathcal{H}$ (and therefore the norm $\|\cdot\|_{\mathcal{X}}$ is
stronger than the norm $\|\cdot\|_{\mathcal{H}}$), and positive numbers
$C,p,r>0$ such that:

$\left|\frac{\mathrm{d}}{\mathrm{d}\lambda}\left(E(\lambda)f,g\right)_{\mathcal{H}}\right|\leq
C|\lambda|^{p-1}\|f\|_{\mathcal{X}}\|g\|_{\mathcal{X}},\qquad\forall
f,g\in\mathcal{X},\,\forall\lambda\in(-r,r)\setminus\\{0\\}.$

Then the operators $P^{T}f:=\frac{1}{2T}\int_{-T}^{T}e^{itA}f\,\mathrm{d}t$
and $P:=\text{orthogonal projection of $\mathcal{H}$ onto }\ker A$ satisfy

$\|P^{T}-P\|_{\mathcal{X}\to\mathcal{H}}\leq
C(p)T^{-\frac{p}{2+p}},\qquad\forall T>1.$

This result readily allows us to prove our main theorem:

###### Proof of Theorem 1.1.

We can invoke Theorem 4.9. From Theorem 4.7 we know that the bound on the
density of states is satisfied with

  * •

$\mathcal{H}=L^{2}(\mathcal{A})$,

  * •

$\mathcal{X}=\dot{\mathcal{H}}^{s,\gamma}_{0}$,

  * •

$p=\frac{2s}{\alpha}$.

Observing that
$\frac{p}{2+p}=\frac{\frac{2s}{\alpha}}{\frac{2s}{\alpha}+2}=\frac{s}{s+\alpha}$
we deduce that

$\|P^{T}-P\|_{\dot{\mathcal{H}}^{s,\gamma}_{0}\to L^{2}(\mathcal{A})}\leq
CT^{-\frac{s}{s+\alpha}},\qquad\forall T>1,$

where $s>1/2$ and $\gamma\geq 0$ satisfy $\gamma+s/\alpha>1/2$, and where
$C>0$ is a constant that does not depend on $T$. This completes the proof. ∎

We finish with a few remarks:

###### Remark 4.10.

1\. Note that the rate does not depend on $\gamma$. Note also that, even if
there is a threshold regarding $\alpha$ less or greater than $1$ in the
regularity constraint (4.6) (see Remark 4.8), there is no such thing regarding
the exponent $-\frac{s}{s+\alpha}$.

2\. As $\alpha\downarrow 0$ the exponent tends to $1$ which is the rate of
convergence in the case of a spectral gap. Intuitively, smaller $\alpha$
brings us closer to the case of a spectral gap in the sense that there are
“fewer” slow trajectories.

3\. We also observe that $\frac{s}{s+\alpha}\to 1$ as $s\to+\infty$.
Informally, this expresses the fact that “infinite regularity” in the $m$
variable gives the rate of convergence of the spectral gap case.

## References

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