\documentclass[reqno]{amsart}
\begin{document}
{\noindent\small {\em Electronic Journal of Differential Equations},
Vol.~2000(2000), No.~55, pp.~18.\newline
ISSN: 10726691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)}
\thanks{\copyright 2000 Southwest Texas State University and
University of North Texas.}
\vspace{1cm}
\title[\hfilneg EJDE2000/55\hfil A generalization of Gordon's theorem]
{ A generalization of Gordon's theorem and applications to quasiperiodic
Schr\"odinger operators }
\author[ D. Damanik \& G. Stolz \hfil EJDE2000/55\hfilneg]
{ David Damanik \& G\"unter Stolz }
\address{David Damanik \hfill\break
Department of Mathematics 25337, California Institute of Technology \hfill\break
Pasadena, CA 91125, USA \hfill\break
and Fachbereich Mathematik, Johann Wolfgang GoetheUniversit\"at \hfill\break
60054 Frankfurt, Germany}
\email{damanik@its.caltech.edu}
\address{G\"unter Stolz \hfill\break
Department of Mathematics, University of Alabama at Birmingham \hfill\break
Birmingham, AL 35294, USA}
\email{stolz@math.uab.edu}
\date{}
\thanks{Submitted May 12, 2000. Published July 18, 2000.}
\thanks{(D. D.) Supported by the German Academic Exchange Service
through \hfill\break\indent
Hochschulsonderprogramm III (Postdoktoranden). \hfill\break\indent
(G. S.) Partially supported by NSF Grant DMS 9706076.}
\subjclass{34L05, 34L40, 81Q10}
\keywords{ Schr\"odinger operators, eigenvalue problem, quasiperiodic potentials}
\begin{abstract}
We present a criterion for absence of eigenvalues for onedimensional
Schr\"odinger operators. This criterion can be regarded as an $L^1$version of
Gordon's theorem and it has a broader range of application.
Absence of eigenvalues is then established for quasiperiodic potentials
generated by Liouville frequencies and various types of functions such as
step functions, H\"older continuous functions and functions with powertype
singularities. The proof is based on Gronwalltype a priori estimates for
solutions of Schr\"odinger equations.
\end{abstract}
\maketitle
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{coro}[theorem]{Corollary}
\newcommand{\tr}{\operatorname{\rm tr}}
\newcommand{\osc}{\operatorname{\rm osc}}
\section{Introduction}
In this paper we study onedimensional Schr\"odinger operators of the form
\begin{equation}\label{operator}
H = \frac{d^2}{dx^2} + V ( x ),
\end{equation}
acting on $L^2({\mathbb R})$, with some realvalued $L^1_{\rm loc}$potential $V$.
We will be particularly interested in potentials of the form
\begin{equation}\label{potential}
V ( x ) = V_1 ( x ) + V_2 ( x \alpha + \theta ),
\end{equation}
where we assume that $V_1$ and $V_2$ are $1$periodic and locally integrable,
and $\alpha , \theta \in [0,1)$. If $\alpha = \frac{p}{q}$ is rational,
then the potential $V$ is $q$periodic
and $H$ has purely absolutely continuous spectrum. If $\alpha$ is irrational,
then the potential is quasiperiodic and the spectral theory of $H$ is far from
trivial; compare \cite{e,fk1,fk2,fsw,k,ss}.
We want to study the eigenvalue problem for $H$. More precisely, we are
interested in methods that allow one to exclude the presence of eigenvalues.
A notion that has proved to be useful in
this context is the following. A bounded potential $V$ on $(\infty, \infty)$
is called a \textit{Gordon potential} if there exist $T_m$periodic potentials
$V^{(m)}$ such that $T_m\rightarrow \infty$ and for every $m$,
$$ \sup_{2T_m \le x \le 2T_m} V(x)  V^{(m)}(x) \le C m^{T_m} $$
for some suitable constant $C$. It has been shown by Gordon \cite{g}
(see also Simon \cite{s1}) that $H$ has no eigenvalues if $V$ is a Gordon
potential. For discrete Schr\"odinger operators, certain variants of this
result have been established by Delyon and Petritis
\cite{dp} and by S\"ut\H{o} \cite{s2}; see \cite{d} for a survey of the
applications of criteria in this spirit. The applications in the discrete
case include in particular results for models that are generated by
discontinuous functions, for example, step functions. The interest in
such models stems from the theory of onedimensional quasicrystals;
compare \cite{d}. It is clear that in the continuum case, these functions
are outside the scope of Gordon's result. This motivates our attempt to find
a more general criterion for absence of eigenvalues.
Let us call $V$ a \textit{generalized Gordon potential} if
$V \in L^1_{{\rm loc,unif}}({\mathbb R})$, that is,
$$
\V\_{1,{\rm unif}} = \sup_{x\in{\mathbb R}} \int_x^{x+1} V(x) dx < \infty
$$
and there exist $T_m$periodic potentials $V^{(m)}$ such that
$T_m \rightarrow \infty$ and for
every $C < \infty$, we have
\begin{equation}\label{ggpcond}
\lim_{m \rightarrow \infty} \exp(C T_m) \cdot \int_{T_m}^{2T_m}
V(x)  V^{(m)}(x) dx = 0.
\end{equation}
Clearly, every Gordon potential is a generalized Gordon potential.
Our main result is the following:
\begin{theorem}\label{main}
Suppose $V$ is a generalized Gordon potential. Then the operator $H$ in
\eqref{operator} has empty point spectrum.
\end{theorem}
As in the classical case \cite{g,s1}, the proof gives the stronger
result that for every energy $E$, the solutions of
\begin{equation}\label{ode}
u'' (x) + V(x) u(x) = E u(x)
\end{equation}
do not tend to zero as $x \rightarrow \infty$, that is, $u(x_n)^2
+ u'(x_n)^2 \ge D$ for some constant $D > 0$ and a sequence
$(x_n)_{n \in {\mathbb N}}$ which obeys $x_n \rightarrow
\infty$ as $n \rightarrow \infty$. Thus there are no $L^2$solutions since
$u\in L^2({\mathbb R})$ would imply $u(x)^2+u'(x)^2 \rightarrow 0$ as
$x \rightarrow \infty$ by Harnack's inequality (see \cite{semi}).
Note that this uses $V\in L^1_{{\rm loc,unif}}({\mathbb R})$, which also guarantees
that the operator $H$ can be defined by form methods or via SturmLiouville
theory.
Let us now discuss the application of Theorem \ref{main} to quasiperiodic $V$
given by \eqref{potential}. Given some irrational $\alpha \in [0,1)$,
we consider its continued fraction expansion
$$
\alpha = \cfrac{1}{a_1+ \cfrac{1}{a_2+ \cfrac{1}{a_3 + \cdots}}}
$$
with uniquely determined $a_m \in {\mathbb N}$ and the continued fraction approximants
$\alpha_m = p_m/q_m$ defined by
\begin{alignat*}{3}
p_0 &= 0, &\quad p_1 &= 1, &\quad p_m &= a_m p_{m1} + p_{m2},\\
q_0 &= 1, & q_1 &= a_1, & q_m &= a_m q_{m1} + q_{m2};
\end{alignat*}
compare \cite{khin,lang}. Recall that $\alpha$ is called a
\textit{Liouville number} if
\begin{equation}\label{liouville}
 \alpha  \alpha_m  \le B m^{q_m}
\end{equation}
for some suitable $B$, and that the set of Liouville numbers is a dense
$G_\delta$set of zero Lebesgue measure. Given $V$ as in \eqref{potential},
we consider the $q_m$periodic approximants $V^{(m)}$ defined by
\begin{equation}\label{approximants}
V^{(m)} ( x ) = V_1 ( x ) + V_2 ( x \alpha_m + \theta ).
\end{equation}
We immediately obtain the following corollary to Theorem \ref{main}.
\begin{coro}
Suppose that for every $C$, we have
\begin{equation}\label{condition}
\lim_{m \rightarrow \infty} \exp(C q_m) \int_{q_m}^{2q_m} V_2( x
\alpha + \theta )  V_2 ( x \alpha_m + \theta) dx = 0 .
\end{equation}
Then $V$ {\rm (}as given by \eqref{potential}{\rm )} is a generalized Gordon
potential and $H$ {\rm (}as given by \eqref{operator}{\rm )} has empty point
spectrum.
\end{coro}
Note that for $\alpha,\theta$ fixed, the class of functions $V_2$ obeying
\eqref{condition} is a linear space, that is, it is closed under taking
finite sums and under multiplication by constants. Moreover, we shall show
that condition \eqref{condition} is satisfied, for example, if $V_2$ is a
H\"older continuous function, a step function, or a function with powertype
singularities, and $\alpha$ is Liouville and $\theta$ arbitrary.
The organization of this paper is as follows. In Section 2 we establish
estimates on solutions of \eqref{ode} which will imply Theorem \ref{main}.
The examples for condition (\ref{condition}) are
discussed in Section 3.
\section{GronwallType Solution Estimates and Proof of Theorem \ref{main}}
In this section we study the solutions to the eigenvalue equations associated
to two potentials. These two potentials will later be given by a generalized
Gordon potential and one of its approximants. We assume that the solutions
have the same initial conditions at $0$. By an a priori estimate for the
equivalent first order systems, found by a standard application of Gronwall's
lemma (e.g., \cite{Walter}), we can bound the distance of the two solutions
by an integral expression involving the distance of the potentials.
It is this estimate which allows us to use $L^1$ rather than
$L^{\infty}$bounds in (\ref{ggpcond}). Theorem \ref{main} follows from this
bound combined with some useful properties of solutions to periodic
eigenvalue equations.
Fix two potentials $W_1 \in L^1_{{\rm loc,unif}}({\mathbb R})$,
$W_2 \in L^1_{{\rm loc}}({\mathbb R})$ and some energy $E$ and consider the solutions
$u_1,u_2$ of
$$
u_1''(x) + W_1 (x) u_1 (x) = E u_1 (x), \; u_2''(x) + W_2 (x) u_2 (x)
= E u_2 (x),
$$
subject to
$$
u_1(0) = u_2(0), \; u_1'(0) = u_2'(0), \; u_1(0)^2 + u_1'(0)^2
= u_2(0)^2 + u_2'(0)^2 = 1.
$$
\begin{lemma}\label{estimate}
There exists $C = C(\W_1E\_{1,{\rm unif}})$ such that for every $x$,
we have
\begin{equation} \label{u1u2est}
\left \ \left( \begin{array}{c} u_1(x)\\u_1'(x) \end{array} \right)
 \left( \begin{array}{c} u_2(x)\\u_2'(x) \end{array} \right) \right\
\le C \exp(C x ) \int_{\min(0,x)}^{\max(0,x)}
W_1(t)  W_2(t) \cdot u_2(t) dt.
\end{equation}
\end{lemma}
\noindent\textit{Proof.} We consider the case $x \ge 0$ (the modifications for
$x < 0$ are obvious). We have
\begin{align*}
\left( \begin{array}{c} u_1(x)  u_2(x) \\ u_1'(x)  u_2'(x) \end{array}
\right) = & \int_0^x \left( \begin{array}{c} u_1'(t)  u_2'(t) \\
(W_1(t)  E)u_1(t)  (W_2(t)  E)u_2(t) \end{array}
\right) dt\\
= & \int_0^x \left( \begin{array}{c} 0 \\ (W_1(t)  W_2(t)) u_2(t) \end{array}
\right) dt \, + \\
& + \int_0^x \left( \begin{array}{c} u_1'(t)  u_2'(t) \\ (W_1(t)  E) (u_1(t)

u_2(t)) \end{array} \right) dt\\
= & \int_0^x \left( \begin{array}{c} 0 \\ (W_1(t)  W_2(t)) u_2(t) \end{array}
\right) dt \, +\\
& + \int_0^x \left( \begin{array}{cc} 0 & 1 \\ W_1(t)  E & 0 \end{array}
\right) \cdot \left( \begin{array}{c} u_1(t)  u_2(t) \\ u_1'(t)  u_2'(t)
\end{array} \right) dt.
\end{align*}
Hence
\begin{align*}
\left\ \left( \begin{array}{c} u_1(x)  u_2(x) \\ u_1'(x)  u_2'(x)
\end{array} \right) \right\ \le & \int_0^x  (W_1(t)  W_2(t)) \cdot u_2(t)
dt \, +\\
& + \int_0^x \left\ \left( \begin{array}{cc} 0 & 1 \\ W_1(t)  E & 0
\end{array} \right) \right\ \cdot \left\ \left( \begin{array}{c} u_1(t)
 u_2(t) \\ u_1'(t)  u_2'(t) \end{array} \right) \right\ dt.
\end{align*}
By Gronwall's lemma \cite{Walter} we therefore get
\begin{align*}
\left \ \left( \begin{array}{c} u_1(x)\\u_1'(x) \end{array} \right)
 \left( \begin{array}{c} u_2(x)\\u_2'(x) \end{array} \right) \right\
\le & \int_0^x  (W_1(t)  W_2(t)) \cdot
u_2(t) dt \, \times\\
& \times \exp \left( \int_0^x \left\ \left( \begin{array}{cc} 0 & 1 \\
W_1(t)  E & 0 \end{array} \right) \right\ dt \right) .
\end{align*}
Choosing $C$ suitably, the assertion of the lemma follows. \hfill $\Box$
\medskip
We see that we can control the difference of the solutions in terms of an
integral condition involving the difference of the potentials. The other key
ingredient in the proof of Theorem \ref{main} is the fact that for periodic
potentials, we have some knowledge about the norm of the solution vector
$(u(x),u'(x))^T$ at certain points $x$. This is made explicit in the
following lemma which is essentially well known (particularly in the discrete
case \cite{d,dp}).
\begin{lemma}\label{perestimate}
Suppose $W$ is $p$periodic and $E$ is some arbitrary energy. Then every
solution of
\begin{equation}\label{help}
u''(x) + W(x) u(x) = E u(x),
\end{equation}
normalized in the sense that
\begin{equation}\label{normal}
u(0)^2 + u'(0)^2 = 1,
\end{equation}
obeys the estimate
$$
\max \left( \; \left\ \left( \begin{array}{c} u(p)\\u'(p) \end{array}
\right) \right\ , \left\ \left( \begin{array}{c} u(p)\\u'(p) \end{array}
\right) \right\ , \left\ \left( \begin{array}{c} u(2p)\\u'(2p) \end{array}
\right) \right\ \; \right) \ge \frac{1}{2}.
$$
\end{lemma}
\noindent\textit{Proof.} This follows by the same reasoning as in the discrete
case; compare \cite{d,dp}. For the reader's convenience, we sketch the
argument briefly. Consider the solutions $u$ of \eqref{help}.
For $x,y \in {\mathbb R}$, $x < y$, the mapping
\begin{equation}\label{transfer}
M(x,y) : \left( \begin{array}{c} u(x)\\u'(x) \end{array} \right)
\mapsto \left( \begin{array}{c} u(y)\\u'(y) \end{array} \right)
\end{equation}
is clearly linear and depends only on the energy $E$ and the potential on the
interval $(x,y)$. Thus, since $W$ is $p$periodic, we have
\begin{equation}\label{repeat}
M(p,0) = M(0,p) = M(p,2p) =: M.
\end{equation}
Moreover, by the CayleyHamilton theorem, we have
\begin{equation}\label{cht}
M^2  \tr (M) M + I = 0.
\end{equation}
If $\tr (M) \le 1$, we apply this equation to $(u(0),u'(0))^T$ obeying
\eqref{normal} and obtain, using \eqref{repeat},
$$
\max \left( \; \left\ \left( \begin{array}{c} u(p)\\u'(p) \end{array} \right)
\right\ , \left\ \left( \begin{array}{c} u(2p)\\u'(2p) \end{array} \right)
\right\ \; \right) \ge \frac{1}{2},
$$
since $(u(0),u'(0))^T$ has norm one. If $\tr (M) > 1$, we apply \eqref{cht}
along with \eqref{repeat} to $(u(p),u'(p))^T$ and obtain
$$
\max \left( \; \left\ \left( \begin{array}{c} u(p)\\u'(p) \end{array} \right)
\right\ , \left\ \left( \begin{array}{c} u(p)\\u'(p) \end{array} \right)
\right\ \;
\right) \ge \frac{1}{2},
$$
again since the vector $(u(0),u'(0))^T$ has norm one. Put together, we obtain
the claimed result. \hfill $\Box$
We are now in a position to prove the main result.
\medskip
\noindent\textit{Proof of Theorem \ref{main}.} Let $V$ be a generalized Gordon
potential and let $V^{(m)}$ be the $T_m$periodic approximants obeying
\eqref{ggpcond}. Fix some $m$
and apply Lemma \ref{estimate} with $W_1 = V$ and $W_2 = V^{(m)}$. We obtain
\begin{equation} \label{uumest}
\left \ \left( \begin{array}{c} u(x)\\u'(x) \end{array} \right)
 \left( \begin{array}{c} u_m (x)\\u_m'(x) \end{array} \right) \right\
\le C_1 \exp(C_1 x ) \int_{\min(0,x)}^{\max(0,x)} V(t)  V^{(m)}(t)
u_m(t) dt,
\end{equation}
where $u$ solves $u''(x) + V(x) u(x) = E u(x)$, $u_m$ solves
$u_m''(x) + V^{(m)}(x) u_m(x) = E u_m(x)$), and $u,u_m$ are both normalized
at the origin and obey the same boundary
condition there. We conclude from \eqref{ggpcond} that
$\V^{(m)}\_{1,{\rm unif}}$ is bounded in $m$. Thus a second application of
Lemma \ref{estimate} with $W_1 = V^{(m)}$ and $W_2 = 0$, noting that the
constant in \eqref{u1u2est} only depends on the $L^1_{{\rm loc,unif}}$norm of
$W_1E$, leads to
$$
\left \ \left( \begin{array}{c} u_m(x)\\u_m'(x) \end{array} \right)  \left(
\begin{array}{c} u_0 (x)\\u_0'(x) \end{array} \right) \right\ \le C_2
\exp(C_2 x) \int_{\min(0,x)}^{\max(0,x)} V^{(m)}(t) u_0(t) dt,
$$
where $C_2$ does not depend on $m$ and $u_0$ is a normalized solution of
$u_0''=Eu_0$. Noting that $u_0$ is exponentially bounded, this gives
$$
u_m(x) \le C_3 \exp(C_3 x)
$$
with $C_3$ independent of $m$. This and \eqref{uumest} yield
$$
\left \ \left( \begin{array}{c} u(x)\\u'(x) \end{array} \right)
 \left( \begin{array}{c} u_m (x)\\u_m'(x) \end{array} \right) \right\
\le C \exp(Cx) \int_{\min(0,x)}^{\max(0,x)}
V(t)  V^{(m)}(t) dt.
$$
By \eqref{ggpcond} we find some $m_0$ such that for $m \ge m_0$, we have
$$
\left \ \left( \begin{array}{c} u(x)\\u'(x) \end{array} \right)  \left(
\begin{array}{c} u_m (x)\\u_m' (x) \end{array} \right) \right\
\le \frac{1}{4}
$$
for every $x$ with $T_m \le x \le 2T_m$. Combining this with Lemma
\ref{perestimate}, we can conclude the proof. \hfill $\Box$
\section{Examples of Generalized Gordon Potentials}
In this section we give examples of functions $V_2$ that obey condition
\eqref{condition} for Liouville frequencies $\alpha$ and hence induce
quasiperiodic functions $V$ by \eqref{potential} which are generalized Gordon
potentials. These will include H\"older continuous functions, step functions,
functions with local singularities, and linear combinations thereof.
Let us observe the following:
\begin{prop}
For fixed $\alpha,\theta$, the class of functions $V_2$ obeying
\eqref{condition} is a linear space, that is, it is closed under taking finite
sums and under multiplication by constants.
\end{prop}
\noindent\textit{Proof.} This is obvious. \hfill $\Box$
\medskip
Define for some $1$periodic function $f$,
$$
\osc_{f,\varepsilon}(x) = \sup_{y,z \in (x  \varepsilon, x
+ \varepsilon)}  f(y)  f(z) .
$$
Then we have the following proposition.
\begin{prop}\label{boundex}
{\rm (a)} If there are $0 < \delta , D < \infty$ such that
\begin{equation} \label{osc}
\int_0^1 \osc_{V_2,\varepsilon}(x) dx \le D \varepsilon^\delta
\end{equation}
for all sufficiently small $\varepsilon > 0$, then for every Liouville number
$\alpha \in [0,1)$ and every $\theta \in [0,1)$, condition \eqref{condition}
is satisfied.\\[1mm]
{\rm (b)} Condition \eqref{osc} holds for all H\"older continuous functions
and for all step functions.
\end{prop}
\noindent\textit{Proof.} (a) Fix some $C$. Then by \eqref{liouville} and
\eqref{osc}, we have
\begin{align*}
\limsup_{m \rightarrow \infty} \exp(C q_m) \int_{q_m}^{2 q_m} 
& V_2(x \alpha + \theta)  V_2(x \alpha_m + \theta) dx \le \\
& \le \limsup_{m \rightarrow \infty} \exp(C q_m)
\frac{3 q_m \alpha + 1}{\alpha} \int_0^1 \osc_{V_2,2 q_m \alpha  \alpha_m}
(x) dx\\
& \le \limsup_{m \rightarrow \infty} \exp(C q_m) \frac{3 q_m \alpha + 1}{\alpha}
D (2 q_m \alpha  \alpha_m)^\delta\\
& \le \limsup_{m \rightarrow \infty} \exp(C q_m) \frac{3 q_m \alpha + 1}{\alpha}
D 2^\delta q_m^\delta B^\delta m^{\delta q_m}\\
& = 0.
\end{align*}
(b) This is straightforward. \hfill $\Box$
\medskip
The class for which \eqref{condition} was established in Proposition
\ref{boundex} contains only bounded potentials. We finally provide an example
which shows that the use of generalized Gordon potentials allows one to exclude
eigenvalues for some unbounded quasiperiodic potentials. We will exhibit some
$V_2$ that has an integrable powerlike singularity and which satisfies
\eqref{condition}, and therefore $H$ defined by \eqref{operator} and
\eqref{potential} has empty point spectrum. Note that by linearity this also
gives examples with negative singularities and multiple singularities with
different values for $\gamma$.
\begin{prop}
Let $0<\gamma<1$ and $V_2(x)$ be the $1$periodic potential which for
$1/2 \le x \le 1/2$ is given by $V_2(x) = x^{\gamma}$. Then for every
Liouville number $\alpha \in [0,1)$ and every $\theta \in [0,1)$, condition
\eqref{condition} is satisfied.
\end{prop}
\noindent\textit{Proof.} For simplicity, we will only establish
\eqref{condition} for $\theta = 0$. The calculations for general $\theta$ are
similar but slightly more tedious. Start by writing
\begin{equation} \label{split}
\int_{q_m}^{2q_m} V_2(\alpha x)  V_2(\alpha_m x) dx = \frac{q_m}{p_m}
\sum_{n=p_m}^{2p_m  1} \int_n^{n+1} \leftV_2 \left( \frac{\alpha q_m}{p_m}
y \right)  V_2(y) \right dy
\end{equation}
and
\begin{equation} \label{shift}
\int_n^{n+1} \left V_2 \left( \frac{\alpha q_m}{p_m} y \right)  V_2(y)
\right dy = \int_0^1 \left V_2 \left( y+ \left( \frac{\alpha q_m}{p_m}
1 \right) (y+n) \right)  V_2(y) \right dy.
\end{equation}
We have $\frac{\alpha q_m}{p_m}1 y+n \le 2p_m \frac{\alpha q_m}{p_m} 1
=: \delta < 1/4$ for $m$ sufficiently large and can estimate
\begin{eqnarray} \label{halfest}
& & \int_0^{1/2} \left V_2 \left(y+ \left( \frac{\alpha q_m}{p_m} 1 \right)
(y+n) \right)  V_2(y) \right dy\\
& & \le C\delta + C \delta^{1\gamma} + \left \int_0^{1/2} \left( V_2
\left( y+ \left( \frac{\alpha q_m}{p_m} 1 \right)(y+n) \right)  V_2(y) \right)
dy \right, \nonumber
\end{eqnarray}
where the $\delta^{1\gamma}$ term arises from the singularity of $V_2$ at $0$,
and the monotonicity of $V_2$ in $[0,1/2]$ was used to take the absolute value
outside the integral. The integral on the right can be calculated explicitly,
which eventually leads to an estimate
$C(p_m \frac{\alpha q_m}{p_m}1)^{1\gamma}$ for its absolute value and thus
also for \eqref{halfest}.
In a similar way we get the same estimate for the integral from $1/2$ to $1$
on the right hand side of \eqref{shift}. Inserting into \eqref{split}
we finally find
$$
\int_{q_m}^{2q_m} \left V_2(\alpha x)  V_2(\alpha_m x) \right dx
\le C p_m^{2\gamma} \left \frac{\alpha q_m}{p_m} 1 \right^{1\gamma}.
$$
In view of \eqref{liouville} this suffices to imply \eqref{condition}.
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