#  Counterexamples to $L^{p}$ boundedness of wave operators for classical and
higher order Schrödinger operators

M. Burak Erdoğan, Michael Goldberg and William R. Green  Department of
Mathematics
University of Illinois
Urbana, IL 61801, U.S.A<EMAIL_ADDRESS>Department of Mathematics
University of Cincinnati
Cincinnati, OH 45221 U.S.A<EMAIL_ADDRESS>Department of Mathematics
Rose-Hulman Institute of Technology
Terre Haute, IN 47803, U.S.A<EMAIL_ADDRESS>
###### Abstract.

We consider the higher order Schrödinger operator $H=(-\Delta)^{m}+V(x)$ in
$n$ dimensions with real-valued potential $V$ when $n>4m-1$, $m\in\mathbb{N}$.
We show that for any $\frac{2n}{n-4m+1}<p\leq\infty$ and
$0\leq\alpha<\frac{n+1}{2}-2m-\frac{n}{p}$, there exists a real-valued,
compactly supported potential $V\in C^{\alpha}(\mathbb{R}^{n})$ for which the
wave operators $W^{\pm}$ are not bounded on $L^{p}(\mathbb{R}^{n})$. As a
consequence of our analysis we show that the wave operators for the usual
second order Schrödinger operator $-\Delta+V$ are unbounded on
$L^{p}(\mathbb{R}^{n})$ for $n>3$ and $\frac{2n}{n-3}<p\leq\infty$ for
insufficiently differentiable potentials $V$, and show a failure of
$L^{p^{\prime}}\to L^{p}$ dispersive estimates that may be of independent
interest.

The first author was partially supported by the NSF grant DMS-2154031 and
Simons Foundation Grant 634269. The second author is partially supported by
Simons Foundation Grant 635369. The third author is partially supported by
Simons Foundation Grant 511825.

##  1\. Introduction

We consider the higher order Schrödinger equation

$\displaystyle i\psi_{t}=(-\Delta)^{m}\psi+V\psi,\qquad
x\in\mathbb{R}^{n},\quad m\in\mathbb{N},$

with a real-valued and decaying potential $V$. We denote the free higher order
Schrödinger operator by $H_{0}=(-\Delta)^{m}$ and the perturbed operator by
$H=(-\Delta)^{m}+V(x)$. We study the wave operators, which are defined by

$W_{\pm}=s\text{\ --}\lim_{t\to\pm\infty}e^{itH}e^{-itH_{0}}.$

Recent work by the first and third authors, [3], showed that for $m>1$ and
$n>2m$ the wave operators extend to bounded operators on
$L^{p}(\mathbb{R}^{n})$ for $1\leq p\leq\infty$ for sufficiently smooth small
potentials. The case $m=1$ was established by Yajima in [16]. Here we show
that when $n>4m-1$ and $\frac{2n}{n-4m+1}<p\leq\infty$ the $L^{p}$ boundedness
of the wave operators may fail even for compactly supported continuous
potentials if the potential is not sufficiently smooth. Our main result is the
following.

######  Theorem 1.1.

Fix $m\in\mathbb{N}$, let $n>4m-1$ and $\frac{2n}{n-4m+1}<p\leq\infty$, for
all $0\leq\alpha<\frac{n+1}{2}-2m-\frac{n}{p}$ there exists a real-valued
compactly supported potential of class $C^{\alpha}(\mathbb{R}^{n})$ for which
the wave operators $W_{\pm}$ are not bounded on $L^{p}(\mathbb{R}^{n})$.

For the convenience of the reader, in Section 3 we prove Theorem 1.1 when
$p=\infty$, then adapt the argument in Section 4 to the remaining cases of
$\frac{2n}{n-4m+1}<p<\infty$. By considering frequency-localized dispersive
estimates, we provide a direct argument for arbitrary integer order $m$ that
simplifies the $m=1$ argument for the dispersive bounds in [6] and extends to
$m\geq 1$.

For comparison, the results on boundedness in [3] require some smoothness on
the potential when $n\geq 4m-1$. Writing $\langle x\rangle$ to denote
$(1+|x|^{2})^{\frac{1}{2}}$, $\mathcal{F}(f)$ or $\widehat{f}$ to denote the
Fourier transform of $f$ and defining the norm
$\|f\|_{H^{\delta}}=\|\langle\cdot\rangle^{\delta}\widehat{f}(\cdot)\|_{2}$,
we recall the relevant statements below.

######  Theorem 1.2 (Theorem 1.1 in [3]).

Let $n>2m$. Assume that the $V$ is a real-valued potential on $\mathbb{R}^{n}$
and fix $0<\delta\ll 1$. Then $\exists C=C(\delta,n,m)>0$ so that the wave
operators extend to bounded operators on $L^{p}(\mathbb{R}^{n})$ for all
$1\leq p\leq\infty$, provided that

  1. i)

$\big{\|}\langle\cdot\rangle^{\frac{4m+1-n}{2}+\delta}V(\cdot)\big{\|}_{2}<C$
when $2m<n<4m-1$,

  2. ii)

$\big{\|}\langle\cdot\rangle^{1+\delta}V(\cdot)\big{\|}_{H^{\delta}}<C$ when
$n=4m-1$,

  3. iii)

$\big{\|}\mathcal{F}(\langle\cdot\rangle^{\sigma}V(\cdot))\big{\|}_{L^{\frac{n-1-\delta}{n-2m-\delta}}}<C$
for some $\sigma>\frac{2n-4m}{n-1-\delta}+\delta$ when $n>4m-1$.

See [3] for other statements that remove the smallness requriement on the
potential. The arbitrarily small differentiability assumption when $n=4m-1$
appears to be an artifact of the method in [3]. We note that the norm used
when $n>4m-1$ is finite when $\langle x\rangle^{\sigma}V(x)$ has more than
$\frac{n}{n-1}(\frac{n+1}{2}-2m)$ derivatives in $L^{2}(\mathbb{R}^{n})$. We
believe this requirement on smoothness of the potential is not sharp, in light
of the counterexample constructed here. When $m=1$ the counterexample in [6]
and the positive result of the first and third authors in [2] show that
$\alpha=\frac{n-3}{2}$ is sharp for $L^{1}\to L^{\infty}$ dispersive
estimates, at least when $n=5,7$. In effect, our result shows that one cannot
expect decay of a class of potentials alone suffice to ensure the boundedness
of the wave operators as is the case in lower dimensions, when $2m<n<4m-1$
above. Instead, one also expects a degree of differentiability on the
potential is needed to ensure boundedness in high dimensions.

The potentials considered here also suffice to imply, see for example [13, 1,
14], the existence, $L^{2}$-boundedness, asymptotic completeness, and
intertwining identity for the wave operators. In particular, we have the
identity

(1) $\displaystyle f(H)P_{ac}(H)=W_{\pm}f((-\Delta)^{m})W_{\pm}^{*}.$

Here $P_{ac}(H)$ is the projection onto the absolutely continuous spectral
subspace of $H$, and $f$ is any Borel function. One use of (1) is to obtain
$L^{p}$-based mapping properties for the perturbed operator $f(H)P_{ac}(H)$
from the simpler free operator $f((-\Delta)^{m})$. The boundedness of the wave
operators on $L^{p}(\mathbb{R}^{n})$ for any choice of $p\geq 2$ with the
function $f(\cdot)=e^{-it(\cdot)}$ yield dispersive estimates of the form

(2) $\displaystyle\|e^{-itH}P_{ac}(H)\|_{L^{p^{\prime}}\to
L^{p}}\lesssim|t|^{-\frac{n}{2m}+\frac{n}{pm}},\qquad\frac{1}{p}+\frac{1}{p^{\prime}}=1.$

To establish our results, we appeal to the intertwining identity and show that
certain dispersive estimates fail for a specific choice of $f(\cdot)$ for
small time when localized to large frequencies.

We note that our results include the case of $m=1$, the usual second order
Schrödinger operator in dimensions $n>3$. The results of Theorem 1.1 are, to
the best of the authors’ knowledge, new even in this case. In particular
showing that the wave operators need not be bounded on
$\frac{2n}{n-3}<p\leq\infty$ for insufficiently smooth potentials. We also
show that the $L^{p}\to L^{p^{\prime}}$ dispersive estimates

$\bigg{\|}e^{it(-\Delta+V)}P_{ac}(-\Delta+V)\bigg{\|}_{p\to
p^{\prime}}\lesssim|t|^{-\frac{n}{2}(\frac{2}{p}-1)},\quad\text{when
}n>3\text{ and }1\leq p<\frac{2n}{n+3}$

fail in this case, see Corollary 4.1 below.

Our work is inspired by recent work by the first and third authors on the
$L^{p}$-continuity of the higher order wave operators in [3] and the
counterexample to dispersive estimates in the $m=1$ case of the second author
and Visan in [6]. The recent work on the $L^{p}$-boundedness of the wave
operators for higher order Schrödinger operators was sparked by the work of
Feng, Soffer, Wu and Yao on weighted $L^{2}$-based “local dispersive
estimates” for higher order Schrödinger operators considered in [4], which
extends the $m=1$ result of Jensen [8]. In addition, the recent work on the
$L^{p}(\mathbb{R}^{3})$ boundedness of the wave operators for the fourth order
($m=2$) Schrödinger operators by the second and third authors [5], and the
extensive works in the case of $m=1$ most notably that of Yajima, [16, 17,
18]. The $L^{p}(\mathbb{R})$ boundedness has recently been investigated by
Mizutani, Wan and Yao in [12]. The $L^{2}$ existence and other properties of
the higher order wave operators have been studied by many authors, for example
by Agmon [1], Kuroda [10, 11], Hörmander [7], and Schechter, [13, 14].

Similar to the usual second order Schrödinger operator, there is a Weyl
criterion and $\sigma_{ac}(H)=\sigma_{ac}(H_{0})=[0,\infty)$ for sufficiently
decaying potentials. In contrast, decay of the potential is not sufficient to
ensure the lack of eigenvalues embedded in the continuous spectrum for the
higher order operators, [4]. Even perturbing with compactly supported, smooth
potentials may induce embedded eigenvalues. For the potentials we consider, we
show that the eigenvalues cannot be too large, which allows the use of a
limiting absorption principle for the perturbed operator on the interval
$[C_{V},\infty)$ for a sufficiently large $C_{V}>0$, see Lemma 3.2 below.

To prove Theorem 1.1 we show the failure of a dispersive estimate based on
detailed analysis of oscillatory integrals involving the resolvent operators.
The splitting identity for $z\in\mathbb{C}\setminus[0,\infty)$, (c.f. [4])
allows us to study the resolvent $((-\Delta)^{m}-\lambda)^{-1}$ in terms of
$R_{0}(z)=(-\Delta-z)^{-1}$, the usual ($2^{nd}$ order) Schrödinger resolvent.

(3)
$\mathcal{R}_{0}(z)(x,y):=((-\Delta)^{m}-z)^{-1}(x,y)=\frac{1}{mz^{1-\frac{1}{m}}}\sum_{\ell=0}^{m-1}\omega_{\ell}R_{0}(\omega_{\ell}z^{\frac{1}{m}})(x,y)$

here $\omega_{\ell}=\exp(i2\pi\ell/m)$ are the $m^{th}$ roots of unity. It is
convenient to use the change of variables $z=\lambda^{2m}$ with $\lambda$
restricted to the sector in the complex plane with $0<\arg(\lambda)<\pi/m$,

(4)
$\mathcal{R}_{0}(\lambda^{2m})(x,y):=((-\Delta)^{m}-\lambda^{2m})^{-1}(x,y)=\frac{1}{m\lambda^{2m-2}}\sum_{\ell=0}^{m-1}\omega_{\ell}R_{0}(\omega_{\ell}\lambda^{2})(x,y).$

The kernal of higher order Schrödinger resolvents
$\mathcal{R}_{0}(\lambda^{2m})$ grow like $\lambda^{\frac{n+1}{2}-2m}$ in the
spectral variable when $n>4m-1$, this growth necessitates a control over
derivatives of the potential which was controlled in terms of
$\mathcal{F}L^{r}$ norms in [3], similar to the conditions for the second
order Schrödinger established by Yajima, [16]. When $m=1$ this growth may be
exploited to cause a failure of $L^{1}\to L^{\infty}$ dispersive estimates in
dimensions greater than three without sufficient smoothness of the potential;
a counterexample was constructed by the second author and Visan, [6]. We adapt
and simplify this argument by considering $L^{1}\to L^{\infty}$ estimates of
the operators of the form
$H^{\frac{n(m-1)}{2m}}e^{itH}\psi(H/L^{2m})P_{ac}(H)$, with $\psi(s)$ is a
cut-off to frequencies of size $s\approx 1$ and $L$ is a sufficiently large
constant. Treating this operator as an element of the functional calculus, the
modified Stone’s formula is

$H^{\frac{n(m-1)}{2m}}e^{itH}\psi(H/L^{2m})P_{ac}(H)f(x)=\frac{1}{2\pi
i}\int_{0}^{\infty}\lambda^{\frac{n(m-1)}{2m}}e^{it\lambda}\psi(\lambda/L^{2m})\,dE_{ac}(\lambda)f(x).$

Here the difference of the limiting resolvent operators,
$\mathcal{R}_{V}^{\pm}(\lambda)=\lim_{\epsilon\to
0^{+}}((-\Delta)^{m}+V-\lambda\mp i\epsilon)^{-1}$, provides the spectral
measure
$dE_{ac}(\lambda)=[\mathcal{R}_{V}^{+}(\lambda)-\mathcal{R}_{V}^{-}(\lambda)]\,d\lambda$.
These operators are well-defined between weighted spaces by Agmon’s limiting
absorption principle, and are well studied in [3]. By appropriately relating
the frequency and time, we show a dispersive estimate that the free operator
(when $V=0$) satisfies cannot hold for the perturbed operator as $t\to 0$.
Using the intertwining identity, (1), we show the wave operators are not
bounded on $L^{\infty}$. By appropriately rescaling both the powers of $H$ and
the time decay, we extend the argument to $p<\infty$.

The presence of an operator $H^{\frac{n(m-1)}{2m}}$ in the dispersive estimate
is natural in the following sense: For $m>1$, the free fundamental solution
$e^{itH_{0}}(x,y)$ has a central peak where $|x-y|\lesssim|t|^{\frac{1}{2m}}$,
then experiences a combination of polynomial decay and oscillation (similar to
an Airy function) for larger distances $|x-y|$. Differentiating $n(m-1)$ times
yields an operator whose kernel is approximately the same size for all values
of $|x-y|$, so the $L^{1}\to L^{\infty}$ bound can be achieved at every length
scale or when localized to any frequency band.

The paper is organized as follows. In Section 2 we show that the free operator
satisfies a family of dispersive estimates. In Section 3 we show that the
perturbed operator cannot satisfy the same $L^{1}\to L^{\infty}$ dispersive
estimate as the free operator for a specifically constructed potential. As a
consequence, the wave operator is not bounded on $L^{\infty}$. Finally, in
Section 4 we show how the argument may be adapted to construct a class of
potentials for which the wave operators are unbounded on the larger range of
$\frac{2n}{n-4m+1}<p\leq\infty$.

##  2\. The free estimate

In this section we establish a family of dispersive estimates that the free
operator $H_{0}=(-\Delta)^{m}$ satisfies. In Section 3 we construct a
compactly supported potential for which the perturbed operator cannot satisfy
the analogous bound.

######  Proposition 2.1.

$\big{\|}H_{0}^{\frac{n(m-1)}{2m}}e^{itH_{0}}\big{\|}_{1\to\infty}\lesssim|t|^{-\frac{n}{2}},$

and for all $\sigma\in\mathbb{R}$,

$\big{\|}H_{0}^{\frac{n(m-1)+i\sigma}{2m}}e^{itH_{0}}\big{\|}_{1\to\infty}\lesssim(1+|\sigma|)^{\frac{n+2}{2}}|t|^{-\frac{n}{2}}.$

###### Proof.

Using the splitting identity (4), along with the fact that
$R_{0}^{+}(\omega_{\ell}\lambda^{2})=R_{0}^{-}(\omega_{\ell}\lambda^{2})$ for
$\omega_{\ell}\notin\mathbb{R}$, that is when $\omega_{\ell}\neq 1$, we have

(5)
$\displaystyle[\mathcal{R}_{0}^{+}-\mathcal{R}_{0}^{-}](\lambda^{2m})(x,y)=\frac{1}{m\lambda^{2m-2}}[R_{0}^{+}-R_{0}^{-}](\lambda^{2})(x,y).$

Hence, one need only understand the usual second order free resolvent to
understand the estimates in Propostion 2.1. As usual, we estimate the
evolution by using the functional calculus, in this case, we’ll estimate
(ignoring constants)

(6)
$\displaystyle\sup_{x,y\in\mathbb{R}^{n}}\bigg{|}\int_{0}^{\infty}e^{it\lambda^{2m}}\lambda^{n(m-1)+1+i\sigma}[R_{0}^{+}-R_{0}^{-}](\lambda^{2})(x,y)\,d\lambda\bigg{|}.$

We utilize the “symbol class” representation for the Schrödinger operators
used by the second author and Visan in [6] to write

$[R_{0}^{+}-R_{0}^{-}](\lambda^{2})(x,y)=\lambda^{n-2}\big{(}e^{i\lambda|x-y|}A_{1}(\lambda|x-y|)-e^{-i\lambda|x-y|}A_{2}(\lambda|x-y|)\big{)},$

where $A_{1}(s),A_{2}(s)=\widetilde{O}(\langle s\rangle^{\frac{1-n}{2}})$.
Here $f(s)=\widetilde{O}(\langle s\rangle^{j})$ means
$|\partial_{s}^{k}f(s)|\lesssim\langle s\rangle^{j-k}$ for each
$k=0,1,2,\dots$

We only consider the case $t>0$ and when there is a ‘$-$’ sign on the phase;
the other cases are similar or easier. Therefore, we only prove that

$\sup_{x,y\in\mathbb{R}^{n},L>0,t>0}t^{n/2}\bigg{|}\int_{0}^{\infty}e^{it\lambda^{2m}-i\lambda|x-y|}\lambda^{nm-1+i\sigma}A_{2}(\lambda|x-y|)\chi(\lambda/L)\,d\lambda\bigg{|}\lesssim
1.$

Only the function $A_{2}(\lambda|x-y|)$ is present, so we omit the subscript
in the rest of the calculation. Changing the variable
$\lambda|x-y|\mapsto\lambda$ and appropriately renaming the variables
$t\mapsto t|x-y|^{2m}$, $L|x-y|\mapsto L$, it suffices to prove that

(7)
$\displaystyle\sup_{t>0,L>0}t^{n/2}\bigg{|}\int_{0}^{\infty}e^{it\lambda^{2m}-i\lambda}\lambda^{nm-1+i\sigma}A(\lambda)\chi(\lambda/L)\,d\lambda\bigg{|}\lesssim
1.$

Note that the phase $\phi(\lambda)=t\lambda^{2m}-\lambda$ has a critical point
at $\lambda_{0}=(2mt)^{-\frac{1}{2m-1}}$. Let $\psi_{1}$ be a smooth cutoff
for the set $|\lambda|\approx\lambda_{0}$, $\psi_{2}$ for the set
$|\lambda|\ll\lambda_{0}$ and $\psi_{3}$ for the set $|\lambda|\gg\lambda_{0}$
so that $\psi_{1}+\psi_{2}+\psi_{3}\equiv 1$. Let

$I_{j}:=\int_{0}^{\infty}e^{it\lambda^{2m}-i\lambda}\lambda^{nm-1+i\sigma}A(\lambda)\psi_{j}(\lambda)\chi(\lambda/L)\,d\lambda,\,\,\,\,j=1,2,3.$

Note that in the support of $\psi_{1}(\lambda)$, we have
$\phi^{\prime\prime}(\lambda)\approx t\lambda_{0}^{2m-2}$. Therefore by the
corollary on page 334 of [15], we have

$|I_{1}|\lesssim(t\lambda_{0}^{2m-2})^{-1/2}\int\big{|}\partial_{\lambda}\big{(}\lambda^{nm-1+i\sigma}A(\lambda)\psi_{1}(\lambda)\chi(\lambda/L)\big{)}\big{|}d\lambda\\\
\lesssim
t^{-1/2}\langle\sigma\rangle\lambda_{0}^{1-m+nm-1}\langle\lambda_{0}\rangle^{\frac{1-n}{2}}\lesssim
t^{-1/2}\langle\sigma\rangle\lambda_{0}^{(2m-1)(n-1)/2}\approx\langle\sigma\rangle
t^{-n/2}.$

Note that on the support of $\psi_{2}$, where $|\lambda|\ll\lambda_{0}$, we
have $|\phi^{\prime}|\gtrsim 1$. Therefore we may integrate by parts $N\leq
nm-1$ times to obtain the bound

$|I_{2}|\lesssim\int_{0}^{\infty}\Big{|}\Big{[}\partial_{\lambda}\frac{1}{\phi^{\prime}(\lambda)}\Big{]}^{N}\big{(}\lambda^{nm-1+i\sigma}A(\lambda)\psi_{2}(\lambda)\chi(\lambda/L)\big{)}\Big{|}d\lambda.$

Noting that the effect of a derivative on each term can be bounded by
multiplication by $\langle\sigma\rangle\lambda^{-1}$, we have

$|I_{2}|\lesssim\langle\sigma\rangle^{N}\int_{0}^{\lambda_{0}}\lambda^{nm-1-N}d\lambda\lesssim\langle\sigma\rangle^{N}\lambda_{0}^{nm-N}\lesssim\langle\sigma\rangle^{N}t^{-\frac{nm-N}{2m-1}},\,\,\,0\leq
N\leq nm-1.$

Picking $N=0$ when $t>1$ and $\frac{n}{2}\leq N\leq nm-1$ for $0<t<1$ yields
stronger bounds than $t^{-n/2}$.

It remains to consider $I_{3}$. On the support of $\psi_{3}$, we have
$|\phi^{\prime}|\gtrsim t\lambda^{2m-1}\gtrsim 1$. Therefore by integration by
parts $N$ times we obtain

$|I_{3}|\lesssim\langle\sigma\rangle^{N}\int_{\lambda_{0}}^{\infty}\lambda^{nm-1-N}\langle\lambda\rangle^{\frac{1-n}{2}}[t\lambda^{2m-1}]^{-N}d\lambda=\langle\sigma\rangle^{N}t^{-N}\int_{\lambda_{0}}^{\infty}\lambda^{nm-1-2mN}\langle\lambda\rangle^{\frac{1-n}{2}}d\lambda.$

When $t<1$, $\lambda_{0}\gtrsim 1$ and taking
$N>\frac{nm+\frac{1}{2}-\frac{n}{2}}{2m}$ implies the stronger bound
$\langle\sigma\rangle^{N}t^{-\frac{n}{2}}t^{\frac{N-\frac{1}{2}}{2m-1}}$. For
$t>1$, take $N=0$ on $[\lambda_{0},t^{-\frac{1}{2m}}]$, and take $N>n/2$ on
$[t^{-\frac{1}{2m}},\infty)$ to obtain

$\lesssim\int_{0}^{t^{-\frac{1}{2m}}}\lambda^{nm-1}d\lambda+\langle\sigma\rangle^{N}t^{-N}\int_{t^{-\frac{1}{2m}}}^{\infty}\lambda^{nm-1-2mN}d\lambda\lesssim
t^{-n/2}+\langle\sigma\rangle^{N}t^{-N}t^{-\frac{1}{2m}(nm-2mN)}\lesssim\langle\sigma\rangle^{N}t^{-n/2}.$

At no point was integration by parts needed more than $\frac{n}{2}+1$ times,
so the sum $I_{1}+I_{2}+I_{3}$ is bounded by
$\langle\sigma\rangle^{\frac{n+2}{2}}t^{-n/2}$. ∎

The above bound is all we need in Section 3 to establish the result of Theorem
1.1 when $p=\infty$. For the full range of $p$ considered in Section 4, we
need the following corollary.

######  Corollary 2.2.

Let $\psi$ be a smooth cutoff for the set
$\\{\lambda\in\mathbb{R}:\lambda\approx 1\\}$. For all $p\in[1,2]$ and $L>0$
we have

$\|H_{0}^{\frac{n(m-1)}{2m}(\frac{2}{p}-1)}e^{itH_{0}}\psi(H_{0}/L)\|_{p\to
p^{\prime}}\lesssim|t|^{-\frac{n}{p}+\frac{n}{2}},$

where the implict constant is independent of $p$ and $L$.

###### Proof.

The claim without $\psi$ follows from complex interpolation of the bound in
Proposition 2.1 with the $L^{2}$ conservation law. The claim with $\psi$
follows from this and the fact that $\widehat{\psi}\in L^{1}$. ∎

##  3\. Failure of the dispersive estimate

In this section we show that the perturbed evolution cannot satisfy the same
dispersive bound as the free evolution. For clarity, we concentrate in this
section on the failure of the $L^{1}\to L^{\infty}$ dispersive estimate and
consequently the unboundedness of the wave operator on $L^{\infty}$. In
Section 4, we show how to adapt this argument to a larger range of $p$.

Let $\psi$ be a smooth cutoff for frequencies $\lambda\approx 1$. Note that
the $L^{\infty}$ boundedness of wave operators together with asymptotic
completeness, the intertwining identity (1) and the bound in Corollary 2.2
imply that the bound

(8)
$\big{\|}H^{\frac{n(m-1)}{2m}}e^{itH}\psi(H/L^{2m})P_{ac}(H)\big{\|}_{L^{1}\to
L^{\infty}}\lesssim|t|^{-n/2}$

holds for all $L>0$ uniformly in $L$.

Let $C^{\alpha}(B(0,2))$ be the Banach space of real-valued $C^{\alpha}$
functions supported in $B(0,2)$. We prove that the inequality

(9)
$\sup_{t>0,L>0}t^{n/2}\big{\|}H^{\frac{n(m-1)}{2m}}e^{itH}\psi(H/L^{2m})P_{ac}(H)\big{\|}_{L^{1}\to
L^{\infty}}\leq C_{V}$

cannot hold for all $V\in C^{\alpha}(B(0,2))$ if
$0\leq\alpha<\tfrac{n+1}{2}-2m$. By showing that this dispersive bound fails
for insufficiently smooth potentials $V$ for small times $t\to 0$, we show
that the wave operators are not bounded on $L^{\infty}(\mathbb{R}^{n})$.

To do so we iterate the resolvent identity to expand the perturbed resolvent
into a Born series

(10)
$\displaystyle\mathcal{R}_{V}(z)=\sum_{j=0}^{2M-1}\big{[}\mathcal{R}_{0}(z)(-V\mathcal{R}_{0}(z))^{j}\big{]}-(\mathcal{R}_{0}(z)V)^{M}\mathcal{R}_{V}(z)(V\mathcal{R}_{0}(z))^{M}.$

We note that the $j=0$ term corresponds to the free evolution. In the
following subsections we first show that the first term of the Born series,
when $j=1$, for large frequencies and small times does not satisfy the
dispersive bound. In the following subsection, we show that the final term in
the identity (the “tail”) obeys the dispersive bound. Finally, we use these
facts to show that the full evolution can’t satisfy the dispersive bound.

###  3.1. The first term of the Born series

First we consider the first term of the Born series of the operator in (9).
Ignoring the constants, we write it’s kernel, $K_{L,t}$, as a difference of
kernels

$\displaystyle
K^{\pm}_{L,t}(x,y)=\int_{0}^{\infty}\int_{\mathbb{R}^{n}}e^{it\lambda^{2m}}\lambda^{2m-1}\lambda^{n(m-1)}\mathcal{R}_{0}^{\pm}(\lambda^{2m})(x,z)V(z)\mathcal{R}_{0}^{\pm}(\lambda^{2m})(z,y)\psi(\lambda^{2m}/L^{2m})dzd\lambda$

Let $f$ be the $L^{1}$ normalized characteristic function of $B(0,1)$, and let
$f_{L}(x)=(CL)^{n}f(x(CL))$. Below $C$ will be chosen to be large to guarantee
that the relevant values of $|x|$ and $|y|$ are $\ll 1$. In what follows we
prove that the dispersive estimate fails by letting $t=L^{-(2m-1)}$ and taking
$L\to\infty$. With $t^{\frac{n}{2}}=L^{\frac{n}{2}-nm}$, we define

$a_{1,L}(V)=L^{\frac{n}{2}-nm}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}K_{L,L^{-(2m-1)}}(x,y)f_{L}(x)f_{L}(y)dxdy.$

Note that $a_{1,L}$ is a linear operator on $C^{\alpha}(B(0,2))$. By uniform
boundedness principle, if the bound in (9) holds for the first term of the
Born series and for all $V\in C^{\alpha}(B(0,2))$, then we have

(11) $\forall V\in C^{\alpha}(B(0,2)),\,\,\,\,\sup_{L>0}|a_{1,L}(V)|\leq
C_{\alpha}\|V\|_{C^{\alpha}(B(0,2))}.$

Therefore, it suffices to find a sequence $\\{V_{L}\\}_{L>1}\subset
C^{\alpha}(B(0,2))$ so that

$\lim_{L\to\infty}\frac{|a_{1,L}(V_{L})|}{\|V_{L}\|_{C^{\alpha}(B(0,2))}}=\infty.$

Note that for $\lambda|x-z|>1$, we have

(12) $R_{0}^{\pm}(\lambda^{2})(x,z)=e^{\pm
i\lambda|x-z|}\lambda^{n-2}\omega^{\pm}(\lambda|x-z|),$

where
$\omega^{\pm}(s)=c_{\pm}s^{\frac{1-n}{2}}+\widetilde{O}(s^{-\frac{1+n}{2}}),$
$s>1$. In our argument below, we have $\lambda\approx L$ large and
$|x-z|\approx 1$, which allows us to avoid the logarithmic behavior of even
dimensional resolvents when $\lambda|x-z|\ll 1$. Using this in the splitting
formula for $\mathcal{R}_{0}^{\pm}(\lambda^{2m})$, (4), and noting the
exponential decay of the remaining terms in the splitting identity allows them
to be absorbed into $\omega^{\pm}(\lambda|x-z|)$, we have for $\lambda|x-z|>1$

$\mathcal{R}_{0}^{\pm}(\lambda^{2m})(x,z)=e^{\pm
i\lambda|x-z|}\lambda^{n-2m}\omega^{\pm}(\lambda|x-z|).$

Here, by a slight abuse of notation, $\omega^{\pm}$ satisfies the same bounds
as $\omega^{\pm}$ in (12). One can also see Lemmas 3.2 and 6.2 in [3] for more
detailed representations of the resolvent. Using this in the formula for
$K^{\pm}_{L,t}(x,y)$, we have

$K^{\pm}_{L}(x,y):=K^{\pm}_{L,L^{-(2m-1)}}\\\
=\int_{0}^{\infty}\int_{\mathbb{R}^{n}}e^{i\frac{\lambda^{2m}}{L^{2m-1}}\pm
i\lambda R}\lambda^{mn+n-2m-1}\omega^{\pm}(\lambda r)\omega^{\pm}(\lambda
s)V(z)\psi(\lambda^{2m}/L^{2m})dzd\lambda$

where $R:=r+s:=|x-z|+|z-y|$. Letting
$\varphi(\lambda)=\lambda^{mn+n-2m-1}\psi(\lambda^{2m})$ be a modified cutoff
to the interval $\lambda\approx 1$, we rewrite the kernel above as

$L^{mn+n-2m-1}\int_{0}^{\infty}\int_{\mathbb{R}^{n}}e^{i\frac{\lambda^{2m}}{L^{2m-1}}\pm
i\lambda R}\omega^{\pm}(\lambda r)\omega^{\pm}(\lambda
s)V(z)\varphi(\lambda/L)dzd\lambda.$

In what follows we have $r,s=1+o(1)$ and $R=2+o(1)$, this is accomplished by
taking $|x|,|y|$ small and $V(z)$ supported on a sufficiently small
neighborhood of $|z|=1$. Changing the variable $\lambda\to L\lambda$ we have

$K^{\pm}_{L}(x,y)=L^{mn+n-2m}\int_{\mathbb{R}^{n}}\int_{0}^{\infty}e^{iL(\lambda^{2m}\pm\lambda
R)}\omega^{\pm}(\lambda Lr)\omega^{\pm}(\lambda
Ls)V(z)\varphi(\lambda)dzd\lambda,$

Note that the contribution of $+$ sign above is $O(L^{-N}\|V\|_{L^{1}})$ by
nonstationary phase since $\varphi$ is smooth and supported on the set
$\lambda\approx 1$ and $|\partial_{\lambda}^{k}\omega^{\pm}(\lambda Lr)|\leq
C_{k}$ for all $k$. For the $-$ sign we have a critical point at
$\lambda_{0}=(R/2m)^{\frac{1}{2m-1}}\approx 1$, and hence by stationary phase
we have

$K^{-}_{L}(x,y)=C_{m}L^{mn+n-2m-\frac{1}{2}}\int_{\mathbb{R}^{n}}\omega^{-}(\lambda_{0}Lr)\omega^{-}(\lambda_{0}Ls)\varphi(\lambda_{0})e^{ic_{m}L(\frac{R}{2})^{\frac{2m}{2m-1}}}V(z)dz\\\
+O(L^{mn-2m-\frac{1}{2}}\|V\|_{L^{1}}),$

where $c_{m}=(\tfrac{1}{m})^{\frac{2m}{2m-1}}(1-2m)$. For the error term, we
used the bound
$|\omega^{\pm}(\lambda)|\approx\langle\lambda\rangle^{\frac{1-n}{2}}.$ Using
the asymptotic expansion above for $\omega^{\pm}$ and slightly modifying
$\varphi$, we can rewrite this as

$K^{-}_{L}(x,y)=C_{m}L^{mn-2m+\frac{1}{2}}\int_{\mathbb{R}^{n}}(rs)^{\frac{1-n}{2}}\varphi(\lambda_{0})e^{ic_{m}L(\frac{R}{2})^{\frac{2m}{2m-1}}}V(z)dz+O(L^{mn-2m-\frac{1}{2}}\|V\|_{L^{1}}),$

Let

$V_{L}(z)=\cos\big{(}c_{m}L|z|^{\frac{2m}{2m-1}}\big{)}\rho_{\delta}(z),$

where $\delta\ll 1$, and $\rho_{\delta}$ is a smooth cutoff for the set
$|z|\in(1-\delta,1+\delta)$. We determine $\delta$ later. However, $\delta\ll
1$ is fixed and we later let $L\to\infty$. Therefore, for sufficiently large
$L$,

$\|V_{L}\|_{C^{\alpha}(B(0,2))}\approx\delta^{-\alpha}+L^{\alpha}\approx
L^{\alpha}$

For $x,y\in$supp$(f_{L})$, i.e., $|x|,|y|\leq\frac{1}{CL}$, we consider

$K_{L}(x,y)=K^{+}_{L}(x,y)-K^{-}_{L}(x,y)=O(L^{mn-2m-\frac{1}{2}}\delta)+\\\
\frac{C_{m}}{2i}L^{mn-2m+\frac{1}{2}}\int_{\mathbb{R}^{n}}(rs)^{\frac{1-n}{2}}\varphi(\lambda_{0})e^{ic_{m}L[(\frac{R}{2})^{\frac{2m}{2m-1}}-|z|^{\frac{2m}{2m-1}}]}\rho_{\delta}(z)dz\\\
+\frac{C_{m}}{2i}L^{mn-2m+\frac{1}{2}}\int_{\mathbb{R}^{n}}(rs)^{\frac{1-n}{2}}\varphi(\lambda_{0})e^{ic_{m}L[(\frac{R}{2})^{\frac{2m}{2m-1}}+|z|^{\frac{2m}{2m-1}}]}\rho_{\delta}(z)dz.$

Note that by nonstationary phase, for all $N\in\mathbb{N}$, the last integral
is $O(L^{mn-2m+\frac{1}{2}-N}\delta^{1-N})$ since
$|\nabla^{k}\rho_{\delta}|\lesssim\delta^{-k}$. Using this with $N=1$, the
error terms are combined to $O(L^{mn-2m-\frac{1}{2}})$. On the other hand,
since in the support of $f_{L}$

$\big{|}\frac{R}{2}-|z|\big{|}\leq\frac{1}{2}(|x|+|y|)\leq\frac{1}{CL},$

by choosing $C$ in the definition of $f_{L}$ sufficiently large depending only
on $m$, we see that the phase in the first integral is $o(1)$. Therefore, we
have the following lower bound for the first integral:

$L^{mn-2m+\frac{1}{2}}\delta.$

We can conclude that there is a constant $c\in\mathbb{C}$:

$\Re\big{[}cK_{L}^{+}(x,y)-cK_{L}^{-}(x,y)\big{]}\geq
L^{mn-2m+\frac{1}{2}}\delta,$

for $\delta\ll 1$ fixed, $L$ sufficiently large, and
$|x|,|y|\leq\frac{1}{CL}$. Therefore, we have

$\frac{|a_{1,L}(V_{L})|}{\|V_{L}\|_{C^{\alpha}(B(0,2))}}\gtrsim
L^{\frac{n+1}{2}-2m-\alpha}\delta\to\infty$

as $L\to\infty$ unless $\alpha\geq\frac{n+1}{2}-2m$.

###  3.2. The Tail of the Born Series

Here we establish that, upon sufficient iteration of the Born series, that the
tail of the Born series is bounded as $t\to 0$. Specifically, we show the
following.

######  Proposition 3.1.

Assume $V\in L^{\infty}(\mathbb{R}^{n})$ is supported on $B(0,2)$, given any
choice of $L>C_{V,n,m}$, where $0<C_{V,n,m}<\infty$ is a constant depending on
the size of the potential, spatial dimension and order of the operator, there
exists an $M$ so that the tail of the Born series satisfies the following
bound

$\displaystyle\sup_{x,y\in\mathbb{R}^{n}}\bigg{|}\int_{0}^{\infty}e^{it\lambda}\psi(\lambda/L^{2m})\lambda^{\frac{n(m-1)}{2m}}(\mathcal{R}_{0}^{\pm}(\lambda)V)^{M}\mathcal{R}_{V}^{\pm}(\lambda)(V\mathcal{R}_{0}^{\pm}(\lambda))^{M}(x,y)\,d\lambda\bigg{|}\lesssim
1.$

We utilize the limiting absorption principle established in [4]. For fixed
$m,n$, there are no positive eigenvalues of $H$ on the support of the cut-off
provided we take $L$ sufficiently large, depending only on the size of $V$.

######  Lemma 3.2.

Fix $\gamma>\frac{1}{2}$. Assume that $|V(x)|\leq M\langle
x\rangle^{-2\gamma}$, $x\in\mathbb{R}^{n}$. Then, there exists a constant
$C=C(n,m,\gamma,M)>0$ such that any $\lambda>C$ cannot be an eigenvalue of
$H$.

###### Proof.

Assume that $\lambda>0$ is an eigenvalue of $H$, then there exists
$0\neq\Psi\in L^{2}$ such that

$((-\Delta)^{m}+V)\Psi=\lambda\Psi,\quad\Rightarrow\quad((-\Delta)^{m}-\lambda-i\epsilon)\Psi=-i\epsilon\Psi-V\Psi.$

Here $\epsilon>0$, so that $\lambda+i\epsilon$ is in the resolvent set of
$H_{0}$. Acting the resolvent operator $\mathcal{R}_{0}(\lambda+i\epsilon)$ on
both sides of the above expression yields

$\Psi=\mathcal{R}_{0}(\lambda+i\epsilon)[-i\epsilon\Psi-V\Psi].$

Hence,

$\|\Psi\|_{L^{2,-\gamma}}\leq\|\mathcal{R}_{0}(\lambda+i\epsilon)\epsilon\Psi\|_{L^{2}}+\|\mathcal{R}_{0}(\lambda+i\epsilon)V\Psi\|_{L^{2,-\gamma}}\lesssim\bigg{\|}\frac{\epsilon}{|\xi|^{2m}-\lambda-i\epsilon}\hat{\Psi}\bigg{\|}_{L^{2}_{\xi}}+\lambda^{\frac{1-2m}{2m}}\|V\Psi\|_{L^{2,\gamma}},$

where we invoked the limiting absorption principle for the free operator on
the second summand. Taking $\epsilon\to 0^{+}$, the dominated convergence
theorem suffices to conclude the first summand vanishes. Using the bound on
$V$, we have $\|V\Psi\|_{L^{2,\gamma}}\leq M\|\Psi\|_{L^{2,-\gamma}}$, hence

$\displaystyle\|\Psi\|_{L^{2,-\gamma}}\lesssim\lambda^{\frac{1-2m}{2m}}M\|\Psi\|_{L^{2,-\gamma}}\lesssim\lambda^{\frac{1-2m}{2m}}M\|\Psi\|_{L^{2,-\gamma}}.$

From here, we conclude for large enough $\lambda$ that
$\|\Psi\|_{L^{2,-\gamma}}=0$, and have that $\Psi=0$ a.e. Thus $\lambda$
cannot be an eigenvalue of $H$.

∎

The above suffices to allow us to use the limiting absorption principle for
the class of potentials we consider. In the statement below $B(s,-s^{\prime})$
is the space of bounded linear operators mapping $L^{2,s}\to
L^{2,-s^{\prime}}$, and assumes the lack of embedded eigenvalues in the
continuous spectrum of $H$.

######  Theorem 3.3 (Theorem 3.9 in [4]).

For $k=0,1,2,3\dots,$ let $|V(x)|\lesssim\langle x\rangle^{-\beta}$ for some
$\beta>2+2k$, then for $s,s^{\prime}>k+\frac{1}{2}$,
$\mathcal{R}_{V}^{(k)}(z)\in B(s,-s^{\prime})$ is continuous for $z>0$.
Furthermore, we have

$\big{\|}\mathcal{R}_{V}^{(k)}(z)\big{\|}_{L^{2,s}\to
L^{2,-s^{\prime}}}\lesssim|z|^{\frac{1-2m}{2m}(1+k)}.$

Note that, in particular, these bounds hold for the free resolvent, for which
there are no embedded eigenvalues, as well as for the perturbed operator when
$|z|$ is sufficiently large by Lemma 3.2.

###### Proof of Proposition 3.1.

The proof here mirrors closely the high energy argument in [3]. Since we need
only show boundedness, the argument is straight-forward. We write
$M=\ell_{1}+\ell_{2}$ where $\ell_{1}=\lfloor\frac{n}{4m}\rfloor+1$ is the
number of iterations of the resolvent required to ensure
$(\mathcal{R}_{0}^{\pm}V)^{\ell_{1}}$ is locally $L^{2}$, and $\ell_{2}$ is
selected large enough to ensure that there is sufficient decay in $\lambda$
using the limiting absorption principle in Theorem 3.3. This suffices to
ensure the desired integral is bounded.

Following the argument in Propositions 5.3 and 6.5 [3], we have the bound
$\sigma>\frac{1}{2}$ and $\ell_{1}=\lfloor\frac{n}{4m}\rfloor+1$ we have

(13)
$\displaystyle\|(V\mathcal{R}_{0}^{+})^{\ell_{1}-1}V\mathcal{R}_{0}^{\pm}(\lambda^{2m})(\cdot,y)\|_{L^{2,\sigma}}\lesssim\frac{\lambda^{\ell_{1}(\frac{n+1}{2}-2m)}}{\langle
y\rangle^{\frac{n-1}{2}}},$

and similarly,

(14)
$\displaystyle\|\mathcal{(}\mathcal{R}_{0}^{+}V)^{\ell_{1}}(\lambda^{2m})(x,\cdot)\|_{L^{2,\sigma}}\lesssim\frac{\lambda^{\ell_{1}(\frac{n+1}{2}-2m)}}{\langle
x\rangle^{\frac{n-1}{2}}}.$

Selecting $L$ large enough so there are no eigenvalues on the support of the
cut-off allow for iterated use of Theorem 3.3 which yields

(15)
$\displaystyle\|(\mathcal{R}_{0}^{+}(\lambda^{2m})V)^{\ell_{2}}\mathcal{R}_{V}^{+}(\lambda^{2m})(V\mathcal{R}_{0}^{+}(\lambda^{2m}))^{\ell_{2}}\|_{L^{2,\sigma}\to
L^{2,-\sigma}}\lesssim\lambda^{(2\ell_{2}+1)(1-2m)}.$

Combing these, we arrive at the bound

(16)
$\bigg{|}\int_{0}^{\infty}e^{it\lambda^{2m}}\psi(\lambda^{2m}/L^{2m})\lambda^{(n+2)m-n-1}(\mathcal{R}_{0}^{\pm}(\lambda^{2m})V)^{M}\mathcal{R}_{V}^{\pm}(\lambda^{2m})(V\mathcal{R}_{0}^{\pm}(\lambda^{2m}))^{M}\,d\lambda\bigg{|}\\\
\lesssim\frac{1}{(\langle x\rangle\langle
y\rangle)^{\frac{n-1}{2}}}\int_{0}^{\infty}\psi(\lambda^{2m}/L^{2m})\lambda^{((n+2)m-n-1+2\ell_{1}(n+1-4m)+(2\ell_{2}+1)(1-2m))}\,d\lambda.$

Having selected $\ell_{1}=\lfloor\frac{n}{4m}\rfloor+1$ we selected
$\ell_{2}\in\mathbb{N}$ so that

$\ell_{2}>\frac{1+n(m-1)+2\ell_{1}(n+1-4m)}{4m-2}.$

Thus, we may bound

$\displaystyle|\eqref{eqn:hi tail int}|\lesssim\frac{1}{(\langle
x\rangle\langle
y\rangle)^{\frac{n-1}{2}}}\int_{0}^{\infty}\psi(\lambda^{2m}/L^{2m})\lambda^{-2}\,d\lambda\lesssim\int_{\lambda\approx
L}\lambda^{-2}\,d\lambda\lesssim L^{-1}.$

This is uniformly bounded in $x,y\in\mathbb{R}^{n}$ and $L>1$.

∎

We note that these bounds can be shown to hold for a much larger class of
potentials $V$, and one can show that the tail decays in $t$ by utilizing the
oscillation in the integral. Such bounds are interesting, but are not needed
for our purpose here.

###  3.3. Failure of the dispersive bound for the full evolution

We now prove that the full evolution cannot satisfy the dispersive bound, and
consequently that the wave operators are not bounded on $L^{\infty}$. The
proof follows that in [6] for the case of $m=1$, though the frequency
localization allows us to avoid many technical issues such as regularity of
the threshold energies or considering larger classes of potentials. Namely we
show the following.

######  Proposition 3.4.

Suppose that $n>3$ and $n>4m-1$, $\psi$ is a smooth cut-off to $\lambda\approx
1$ and $0\leq\alpha<\frac{n+1}{2}-2m$. There cannot exist a bound of the form

$\sup_{t>0,L>0}t^{\frac{n}{2}}\|H^{\frac{n(m-1)}{2m}}e^{itH}\psi(H/L^{2m})P_{ac}(H)f\|_{\infty}\leq
C(V)\|f\|_{1}$

with $C(V)<\infty$ for all $V\in C^{\alpha}(B(0,2))$.

###### Proof.

Assume such a bound can hold. Write $V=\theta W$ with $\theta\in[0,1]$ and
$W\in C^{\alpha}(B(0,2))$. The assumed dispersive bound implies the following
holds uniformly in $L>C_{V}$ where $C_{V}$ is the constant in the proof of
Proposition 3.1 chosen to ensure there are no embedded eigenvalues on the
support of the cut-off. Taking $L>C_{V}$ suffices to ensure the argument for
the tail holds for any choice of $0\leq\theta\leq 1$.

$\sup_{0<t<1,L>C_{V}}t^{\frac{n}{2}}|\langle
H^{\frac{n(m-1)}{2m}}e^{itH}\psi(H/L^{2m})P_{ac}(H)f,g\rangle|\\\
=\sup_{0<t<1,L>C_{V}}\frac{m\,t^{\frac{n}{2}}}{\pi}\bigg{|}\int_{0}^{\infty}e^{it\lambda^{2m}}\lambda^{2m-1+n(m-1)}\psi(\lambda^{2m}/L^{2m})\langle[\mathcal{R}_{\theta
W}^{+}(\lambda^{2m})-\mathcal{R}_{\theta
W}^{-}(\lambda^{2m})]f,g\rangle\,d\lambda\bigg{|}\\\ \leq
C(\theta,W)\|f\|_{1}\|g\|_{1}.$

We take both $f$ and $g$ to be the function $f_{L}$ defined above (11).
Expanding the perturbed resolvent into a Born series as in (10) allows us to
express the evolution as a sum a polynomial in $\theta$ of degree $2M$ and the
tail of the Born series. The coefficients of this polynomial depend on $t$,
$W$, and $L$, are defined by

(17) $a_{k}(W,L)=\frac{m\,t^{\frac{n}{2}}}{\pi
i}\int_{0}^{\infty}e^{it\lambda^{2m}}\lambda^{2m-1+n(m-1)}\psi(\lambda^{2m}/L^{2m})\\\
\langle[\mathcal{R}_{0}^{+}(\lambda^{2m})[W\mathcal{R}_{0}^{+}(\lambda^{2m})]^{k}-\mathcal{R}_{0}^{-}(\lambda^{2m})[W\mathcal{R}_{0}^{-}(\lambda^{2m})]^{k}]f_{L},f_{L}\rangle\,d\lambda$

Proposition 3.1 shows that the tail obeys the desired bound provided $L$ is
sufficiently large. It follows that the Born series terms, the polynomial in
$\theta$, must also obey the bound as well. Writing

$P_{L,t}(\theta)=\sum_{k=0}^{2M}a_{k}(W,L)\theta^{k}$

By assumption, $\sup_{0<t<1,L>C_{V}}|P_{L,t}(\theta)|$ is finite for each
$0\leq\theta\leq 1$. Hence the maximum $0\leq j\leq 2M$ of
$\sup_{0<t<1,L>C_{V}}|P_{L,t}(\frac{j}{2M})|$ is bounded. The value of the
polynomial at $2M+1$ points suffices to solve for the values of the
coefficients, and allows us to conclude that each of the coefficients is
bounded. Hence, we conclude that

$\sup_{0<t<1,L>C_{V}}t^{\frac{n}{2}}|a_{1}(W,L)|\leq C(W)<\infty.$

This, however, is false in light of the estimate in subsection 3.1. Hence, the
assumption is false and such a bound cannot hold.

∎

The intertwining identity (1) quickly establishes the following corollary,
which is the $p=\infty$ statement in Theorem 1.1.

######  Corollary 3.5.

Suppose that $n>3$ and $n>2m$. Then for any $\alpha<\frac{n+1}{2}-2m$, there
exists a real-valued compactly supported potential $V$ in
$C^{\alpha}(\mathbb{R}^{n})$ for which the wave operators $W^{\pm}$ do not
extend to bounded operators on $L^{\infty}(\mathbb{R}^{n})$.

##  4\. Extension to $L^{p}$

In this section we complete the proof of Theorem 1.1 to consider the full
range of $p$ on which our counterexample to $L^{p}$ boundedness applies. The
argument presented in the previous section can be adapted to show the failure
of dispersive bounds from $L^{p}$ to $L^{p^{\prime}}$ for $1\leq p\leq 2$,
which implies unboundedness of the wave operators on $L^{p^{\prime}}$. With
$\psi$ the same cut-off to frequencies $\lambda\approx 1$, by Corollary 2.2
the free evolution satisfies the bound

$\left\|H_{0}^{\frac{n(m-1)}{2m}(\frac{2}{p}-1)}e^{itH_{0}}\psi(H_{0}/L^{2m})\right\|_{p\to
p^{\prime}}\lesssim|t|^{\frac{n}{2}(1-\frac{2}{p})}.$

Here we show that the dispersive bound for the perturbed operator fails by
showing that one cannot have a bound of the form

(18)
$\displaystyle\sup_{L>0,t>0}t^{\frac{n}{2}(1-\frac{2}{p})}\bigg{|}\left\langle
H^{\frac{n(m-1)}{2m}(\frac{2}{p}-1)}e^{itH}\psi(H/L^{2m})f,g\right\rangle\bigg{|}\leq
C(V)$

for $f,g$ unit vectors in $L^{p}$. As in the previous section, we’ll select
$f=g$ to be functions that concentrate at zero as $L\to\infty$. Namely, we’ll
use $f_{L,p}(x)=(CL)^{\frac{n}{p}}f(x(CL))$ where $f$ is the
$L^{p}$-normalized characteristice function of $B(0,1)$. The argument proceeds
analogously, we show that the first term of the Born series grows too fast as
$t\to 0^{+}$ and the tail remains bounded.

For the first term of the Born series, we note that we now consider the
difference of kernels

$K^{\pm}_{L,p,t}(x,y)\\\
=\int_{0}^{\infty}\int_{\mathbb{R}^{n}}e^{it\lambda^{2m}}\lambda^{2m-1}\lambda^{n(m-1)(\frac{2}{p}-1)}\mathcal{R}_{0}^{\pm}(\lambda^{2m})(x,z)V(z)\mathcal{R}_{0}^{\pm}(\lambda^{2m})(z,y)\psi(\lambda^{2m}/L^{2m})dz\,d\lambda.$

The oscillatory integral argument applies ver batim, there are three changes.
Due to the different power of $\lambda$, the $\lambda\mapsto\lambda L$
rescaling produces a power of $n(m-1)(\frac{2}{p}-1)$ on $L$. The time factor
in front of the integral $t^{\frac{n}{2}(1-\frac{2}{p})}$ becomes
$L^{nm-\frac{n}{2}+\frac{n}{p}-\frac{2mn}{p}}$ using the scaling
$t=L^{-(2m-1)}$. The final change comes from the fact that the positive
functions $f_{L,p}$ are integrated directly, we have
$\|f_{L,p}\|_{1}=(CL)^{n/p-n}\|f\|_{1}\approx L^{n/p-n}\|f\|_{p}$ since $f$ is
a normalized characteristic function of $B(0,1)$.

Hence we consider the linear operator

$a_{1,p,L}(V)=L^{nm-\frac{n}{2}+\frac{n}{p}-\frac{2mn}{p}}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}K_{L,L^{-(2m-1)}}(x,y)f_{L,p}(x)f_{L,p}(y)dxdy.$

With the rescaling accounted for, the stationary phase arguments apply
directly to show that there is a $C_{p}\in\mathbb{C}$ so that

$\Re\big{[}C_{p}K_{L,p}^{+}(x,y)-C_{p}K_{L,p}^{-}(x,y)\big{]}\gtrsim
L^{\frac{n}{p}+\frac{1-n}{2}-2m}\delta,$

for $\delta\ll 1$ fixed, and for $L$ sufficiently large and
$|x|,|y|\ll\frac{1}{L}$. Therefore, we have

$\frac{|a_{1,p,L}(V_{L})|}{\|V_{L}\|_{C^{\alpha}(B(0,2))}}\gtrsim
L^{\frac{n}{p}+\frac{1-n}{2}-2m-\alpha}\delta\to\infty$

as $L\to\infty$ unless $\alpha\geq\frac{n}{p}+\frac{1-n}{2}-2m$, provided
$p<\frac{2n}{4m+n-1}$ to ensure the right side of the inequality is positive.

For the tail of the Born series, the extension is straight-forward. By
selecting $M$ appropriately large depending on the parameters $j,k$ below, the
proof of Proposition 3.1 can be adapted to show

$\displaystyle\sup_{x,y\in\mathbb{R}^{n}}\bigg{|}\int_{0}^{\infty}e^{it\lambda}\psi(\lambda/L^{2m})\lambda^{k}(\mathcal{R}_{0}^{\pm}(\lambda)V)^{M}\mathcal{R}_{V}^{\pm}(\lambda)(V\mathcal{R}_{0}^{\pm}(\lambda))^{M}(x,y)\,d\lambda\bigg{|}\lesssim
L^{-j},$

for any choice of $j,k\in\mathbb{N}$. Again using the $L^{p}$-normalized
function $f_{L,p}$ we have $\|f_{L,p}\|_{1}\approx L^{n/p-n}\|f\|_{p}$. Hence,
we have

$\bigg{|}\int_{0}^{\infty}e^{it\lambda}\psi(\lambda/L^{2m})\lambda^{k}\langle(\mathcal{R}_{0}^{\pm}(\lambda)V)^{M}\mathcal{R}_{V}^{\pm}(\lambda)(V\mathcal{R}_{0}^{\pm}(\lambda))^{M}(x,y)f_{L,p},f_{L,p}\rangle\,d\lambda\bigg{|}\\\
\lesssim\sup_{x,y\in\mathbb{R}^{n}}\bigg{|}\int_{0}^{\infty}e^{it\lambda}\psi(\lambda/L^{2m})\lambda^{k}(\mathcal{R}_{0}^{\pm}(\lambda)V)^{M}\mathcal{R}_{V}^{\pm}(\lambda)(V\mathcal{R}_{0}^{\pm}(\lambda))^{M}(x,y)\,d\lambda\bigg{|}\|f_{L,p}\|_{1}^{2}\lesssim
L^{-1}.$

uniformly in $L>C_{n,m,V}$, where $C_{n,m,V}$ is the constant chosen so that
there are no embedded eigenvalues on the support of the cut-off.

The polynomial argument now applies with coefficients defined by

(19) $a_{k}(W,L,p)=\frac{m\,t^{\frac{n}{2}(\frac{2}{p}-1)}}{\pi
i}\int_{0}^{\infty}e^{it\lambda^{2m}}\lambda^{2m-1+n(m-1)(\frac{2}{p}-1)}\psi(\lambda^{2m}/L^{2m})\\\
\langle[\mathcal{R}_{0}^{+}(\lambda^{2m})[-W\mathcal{R}_{0}^{+}(\lambda^{2m})]^{k}-\mathcal{R}_{0}^{-}(\lambda^{2m})[-W\mathcal{R}_{0}^{-}(\lambda^{2m})]^{k}]f_{L,p},f_{L,p}\rangle\,d\lambda.$

Hence, we are able to conclude that the $L^{p}\to L^{p^{\prime}}$ dispersive
bound in (18) fails to hold for all $V\in C^{\alpha}(B(0,2))$ if
$0\leq\alpha<\frac{n}{p}+\frac{1-n}{2}-2m$. By appealing to the intertwining
identity (1), this shows that the wave operators are unbounded on $L^{q}$ for
$\frac{2n}{n-4m+1}<q\leq\infty$.

A consequence of these bounds are results that are, to the best of the
authors’ knowledge, new for the classical $m=1$ Schrödinger evolution. Namely,
with $p^{\prime}$ the Hölder conjugate of $p$ we have the following.

######  Corollary 4.1.

Suppose that $n>3$ and $1\leq p<\frac{2n}{n+3}$. Then for any
$0\leq\alpha<\frac{n}{p}-\frac{n+3}{2}$ there exists a real-valued compactly
supported potential $V$ in $C^{\alpha}(\mathbb{R}^{n})$ for which the
dispersive bound

$\bigg{\|}e^{it(-\Delta+V)}P_{ac}(-\Delta+V)\bigg{\|}_{p\to
p^{\prime}}\lesssim|t|^{-\frac{n}{2}(\frac{2}{p}-1)}$

fails.

In particular, the wave operators $W^{\pm}$ do not extend to bounded operators
on $L^{p^{\prime}}(\mathbb{R}^{n})$. Consequently, we conclude that for all
$\frac{2n}{n-3}<q\leq\infty$ there exist a compactly supported continuous
potential for which the wave operators are unbounded on
$L^{q}(\mathbb{R}^{n})$.

###### Proof.

Assuming that the $L^{p}\to L^{p\prime}$ bound in the corollary holds and
using Theorem 2.1 in [9] for the uniform $L^{p}$ boundedness of $\psi(H/L)$ we
conclude that the bound (18) should hold, which is a contradiction. ∎

## References

  * [1] S. Agmon, Spectral properties of Schrödinger operators and scattering theory. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151–218.
  * [2] M. B. Erdoğan, and W. R. Green, _Dispersive estimates for the Schrödinger equation for $C^{\frac{n-3}{2}}$ potentials in odd dimensions,_ Int. Math. Res. Notices 2010:13, 2532–2565.
  * [3] M. B. Erdoğan, and W. R. Green, _The $L^{p}$-continuity of wave operators for higher order Schrödinger operators,_ Adv. Math. 404 (2022), Paper No. 108450.
  * [4] H. Feng, A. Soffer, Z. Wu, and X. Yao, Decay estimates for higher order elliptic operators, Trans. Amer. Math. Soc. 373 (2020), no. 4, 2805–-2859.
  * [5] M. Goldberg, and W. Green, _On the $L^{p}$ boundedness of the Wave Operators for fourth order Schrödinger operators,_. Trans. Amer. Math. Soc. 374 (2021), 4075–4092.
  * [6] M. Goldberg, and M. Visan. _A Counterexample to Dispersive Estimates_. Comm. Math. Phys. 266 (2006), no. 1, 211-238.
  * [7] L. Hörmander, _The existence of wave operators in scattering theory._ Math. Z. 146 (1976), no. 1, 69–91.
  * [8] A. Jensen, _Spectral properties of Schrödinger operators and time-decay of the wave functions results in $L^{2}(R^{m})$, $m\geq 5$_. Duke Math. J. 47 (1980), no. 1, 57–80.
  * [9] A. Jensen, and S. Nakamura, _Mapping properties of functions of Schrödinger operators between $L^{p}$-spaces and Besov spaces_. Spectral and scattering theory and applications, 187–209, Adv. Stud. Pure Math., 23, Math. Soc. Japan, Tokyo, 1994.
  * [10] S. Kuroda, _Scattering theory for differential operators. I._ J. Math. Soc. Japan 25 (1973), 75–104.
  * [11] S. Kuroda, _Scattering theory for differential operators. II. Self-adjoint elliptic operators._ J. Math. Soc. Japan 25 (1973), 222–234.
  * [12] H. Mizutani, Z. Wan, and X. Yao, _$L^{p}$ -boundedness of wave operators for fourth-order Schrödinger operators on the line_, preprint, 2022. arXiv:2201.04758
  * [13] M. Schechter, _Scattering theory for pseudodifferential operators,_ Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 111–121.
  * [14] M. Schechter, _Scattering theory for elliptic operators of arbitrary order._ Comment. Math. Helv. 49 (1974), 84–113.
  * [15] Stein, E. _Harmonic analysis real-variable methods, orthogonality, and oscillatory integrals_. Princeton Univ. Press, Princeton, NJ, 1993.
  * [16] K. Yajima, _The $W^{k,p}$-continuity of wave operators for Schrödinger operators._ J. Math. Soc. Japan 47 (1995), no. 3, 551–581.
  * [17] K. Yajima, _The $W^{k,p}$-continuity of wave operators for Schrödinger operators. II. Positive potentials in even dimensions $m\geq 4$._ Spectral and scattering theory (Sanda, 1992), 287–300, Lecture Notes in Pure and Appl. Math., 161, Dekker, New York, 1994.
  * [18] K. Yajima, _The $W^{k,p}$-continuity of wave operators for Schrödinger operators. III. Even-dimensional cases $m\geq 4$._ J. Math. Sci. Univ. Tokyo 2 (1995), no. 2, 311–346.