# Regularity of the Scattering Matrix for Nonlinear Helmholtz Eigenfunctions

Jesse Gell-Redman  School of Mathematics and Statistics, University of
Melbourne, Melbourne, Victoria, Australia<EMAIL_ADDRESS>,  Andrew
Hassell  Mathematical Sciences Institute, Australian National University,
Acton, ACT, Australia<EMAIL_ADDRESS>and  Jacob Shapiro
Department of Mathematics, University of Dayton, Dayton, OH 45469-2316
<EMAIL_ADDRESS>
###### Abstract.

We study the nonlinear Helmholtz equation
$(\Delta-\lambda^{2})u=\pm|u|^{p-1}u$ on $\mathbb{R}^{n}$, $\lambda>0$,
$p\in\mathbb{N}$ odd, and more generally $(\Delta_{g}+V-\lambda^{2})u=N[u]$,
where $\Delta_{g}$ is the (positive) Laplace-Beltrami operator on an
asymptotically Euclidean or conic manifold, $V$ is a short range potential,
and $N[u]$ is a more general polynomial nonlinearity. Under the conditions
$(p-1)(n-1)>4$ and $k>(n-1)/2$, for every $f\in
H^{k}(\mathbb{S}^{n-1}_{\omega})$ of sufficiently small norm, we show there is
a nonlinear Helmholtz eigenfunction taking the form

$u(r,\omega)=r^{-(n-1)/2}\Big{(}e^{-i\lambda r}f(\omega)+e^{+i\lambda
r}b(\omega)+O(r^{-\epsilon})\Big{)},\qquad\text{as }r\to\infty,$

for some $b\in H^{k}(\mathbb{S}_{\omega}^{n-1})$ and $\epsilon>0$. That is,
the scattering matrix $f\mapsto b$ preserves Sobolev regularity, which is an
improvement over the authors’ previous work [GHSZ20] with Zhang, that proved a
similar result with a loss of four derivatives.

The authors were supported in part by the Australian Research Council through
grant DP180100589

##  1\. Introduction and statement of results

We consider a Hamiltonian $H=\Delta_{g}+V$ defined on $\mathbb{R}^{n}$, where
$g$ is an asymptotically Euclidean Riemannian metric in the sense defined
below (an example is any smooth, compactly supported perturbation of the flat
metric), and $V\in C^{\infty}(\mathbb{R}^{n})$ is a real valued potential
function which is short range and satisfies symbolic estimates in the sense
that

$|D_{z}^{\alpha}V(z)|\leq C\langle z\rangle^{-\gamma-|\alpha|}$

for some $\gamma>1$. We study the scattering problem for the nonlinear
Helmholtz equation

$(H-\lambda^{2})u=N[u],$ (1.1)

for certain polynomial nonlinearities $N$. The admissible nonlinearities are
defined below, but for now we note the examples include

$N[u]=\big{(}c_{1}|u|^{p-1}+c_{2}|\nabla u|^{p-1}\big{)}u,$

where $p\geq 3$ is an odd integer. For prescribed, sufficiently small data
$f\in H^{k}(\mathbb{S}^{n-1})$, we seek $u$ solving equation (1.1), such that

$u(r,y)\sim r^{-(n-1)/2}\Big{(}e^{-i\lambda r}f(y)+e^{+i\lambda
r}b(y)\Big{)},\qquad b\in H^{k^{\prime}}(\mathbb{S}^{n-1}).$ (1.2)

Here $f$ is the “incoming” data and $b$ is the “outgoing” data. We refer to
the association $f\mapsto b$ as the “nonlinear scattering matrix.” The results
from [GHSZ20] include the following.

######  Theorem.

Assume that $(p-1)(n-1)>4$ and suppose that $k-4>(n-1)/2$, $k\in\mathbb{N}$.
Then there is $\epsilon>0$ sufficiently small, such that for all $f\in
H^{k}(\mathbb{S}^{n-1})$ with $\|f\|_{H^{k}}<\epsilon$, there is a solution
$u$ to (1.1) satisfying (1.2) with $k^{\prime}=k-4$.

The purpose of this article is to prove, as is expected from the inherent
symmetry in the determination of $b$ from $f$ and vice-versa, that the value
of $k^{\prime}$ can be taken equal to $k$, i.e. that the nonlinear scattering
matrix preserves Sobolev regularity.

######  Theorem 1.

Assume that $(p-1)(n-1)>4$ and suppose that $k>(n-1)/2$, $k\in\mathbb{N}$.
There is an $\epsilon>0$ such that for all $f\in H^{k}(\mathbb{S}^{n-1})$ with
$\|f\|_{H^{k}}<\epsilon$, there is a solution $u$ to (1.1) satisfying (1.2)
with $b\in H^{k}(\mathbb{S}^{n-1})$.

Moreover, $u$ is unique in the sense of the main theorem of [GHSZ20],
described in detail in Section 4, and the error term in the asymptotic
expansion (1.2),

$E_{r}:=u-r^{-(n-1)/2}\Big{(}e^{-i\lambda r}f+e^{+i\lambda r}b\Big{)}\text{
satisfies
}\|E_{r}\|_{H^{k-2}(\mathbb{S}^{n-1})}=O(r^{-(n-1)/2-\epsilon^{\prime}}).$

To elaborate on the uniqueness statement, we show that, for an appropriate
microlocalizing pseudodifferential operator $A_{-}$ (see (2.12)) to the
incoming radial set, with $u_{-}=A_{-}u_{0}$ and $u_{0}=P(\lambda)f$ the
linear generalized eigenfunction with incoming data $f$, then $u-u_{-}$ is
uniquely determined in a small ball around the origin in the Hilbert space
$H_{+}^{s,-1/2-\delta;1,k-1}$ defined in Section 2.2.

Note that the topology of convergence is lower regularity than the limit. This
reflects the well known phenomenon from the linear setting whereby an
asymptotic expansion for an incoming (or outgoing) approximate eigenfunction
is produced by computing successive terms in an asymptotic expansion. This
process in general produces a “distributional” expansion, in which
coefficients of higher order terms have decreasing regularity. For example, in
flat Euclidean space, an incoming approximate eigenfunction
$(\Delta_{0}-\lambda^{2})u_{-}\in\mathscr{S}(\mathbb{R}^{n})$ with incoming
data $f\in C^{\infty}(M)$ admits an asymptotic expansion

$\begin{gathered}u\sim r^{-(n-1)/2}e^{i\lambda
r}\sum_{j=0}^{\infty}r^{-j}v_{j}(y),\\\ v_{0}=f,\quad
v_{j+1}=\frac{1}{2i(j+1)\lambda}\left(\Delta_{\mathbb{S}^{n-1}}+\frac{(n-1)(n-3)}{4}-j(j+1)\right)v_{j}.\end{gathered}$

From this one sees immediately that with $f\in H^{k}$ one can obtain a partial
expansion of an approximate eigenfunction in which each subsequent term has a
coefficient two orders rougher than the previous one. It therefore seems very
natural that the convergence in our theorem takes place in
$H^{k-2}(\mathbb{S}^{n-1})$.

Our methods extend to prove a generalization of this result in the setting of
asymptotically conic manifolds. These are Riemannian manifolds $(M^{\circ},g)$
where $M^{\circ}$ is the interior of a compact manifold $M$ with boundary
$\partial M$ and $g$ is a so-called scattering metric, meaning it takes the
form

$g=\frac{dx^{2}}{x^{4}}+\frac{h(x,y,dy)}{x^{2}},$ (1.3)

in a neighborhood of $\partial M$ where $x$ is a boundary defining function,
i.e. $\partial M=\\{x=0\\}$ and $x\geq 0$ has that $dx$ is nonvanishing over
$\partial M$, and $y$ are coordinates on $\partial M$. Here $h$ is a smooth
$(0,2)$-tensor that restricts to a metric on $\partial M$. Flat Euclidean
space is an example of an asymptotically conic space; write
$M^{\circ}=\mathbb{R}^{n}$ and include
$\mathbb{R}^{n}\hookrightarrow\overline{\mathbb{R}^{n}}=\\{w\in\mathbb{R}^{n}:|w|\leq
1\\}=:M$ where the inclusion can be realized by the map $z\mapsto z/(1+\langle
z\rangle)$ and note that the metric form is realized by writing the flat
metric in polar coordinates and setting $x=1/r$.

In the Euclidean case, $\partial M$ is the sphere $\mathbb{S}^{n-1}$ at
infinity with its standard metric, and $h$ is independent of $x$. In general,
if $(\partial M,h(0))$ is the sphere with its standard metric, then we call
$(M^{\circ},g)$ an asymptotically Euclidean metric.

On a general asymptotically conic manifold, writing $r=1/x$ one obtains an
analogue of the radial variable in this more general context, and the metric
then takes the form near infinity

$g=dr^{2}+r^{2}h(\frac{1}{r},y,dy).$ (1.4)

The admissible nonlinearities are those $N[u]:=N(u,\overline{u},\nabla
u,\nabla\overline{u},\nabla^{(2)}u,\nabla^{(2)}\overline{u})$ which are a sum
of monomial terms, each of which has degree not less than $p$, where $p$ is an
_odd_ integer

$\frac{(p-1)(n-1)}{2}>2,$ (1.5)

in $u$ and $\overline{u}$ and their derivatives up to order two, with
coefficients smooth on $M$.

######  Theorem 2 (Main Theorem, asymptotically conic case).

Let $(M^{\circ},g)$ be an asymptotically conic manifold of dimension $n$, and
let $V$ be a conormal short range potential, that is, a smooth potential on
$M^{\circ}$ satisfying estimates near infinity of the form

$\Big{|}(rD_{r})^{k}D_{y}^{\alpha}V(r,y)\Big{|}\leq C\langle
r\rangle^{-\gamma}\text{ for all }k\geq 0,\ \alpha\in\mathbb{N}^{n-1}$ (1.6)

for some $\gamma>1$. Let $H=\Delta_{g}+V$ where $\Delta_{g}$ is the Laplace-
Beltrami operator on $(M^{\circ},g)$. Let $N[u]$ be an admissible
nonlinearity, and let $k\in\mathbb{N}$, $k>\min(1,(n-1)/2)$. There exists
$\epsilon>0$ sufficiently small, such that for every $f\in H^{k}(\partial M)$
with $\|f\|_{H^{k}(\partial M)}<\epsilon$, there is a solution $u$ to

$(H-\lambda^{2})u=N[u]$

on $M^{\circ}$ satisfying

$u(r,y)=r^{-(n-1)/2}\Big{(}e^{-i\lambda r}f(y)+e^{+i\lambda
r}b(y)+O_{H^{k-2}}(r^{-\epsilon^{\prime}})\Big{)}$ (1.7)

for some $b\in H^{k}(\partial M)$ and some $\epsilon^{\prime}>0$.

For ease of exposition, we return to the Euclidean case in the remainder of
this introduction. Given $f\in L^{2}(\mathbb{S}^{n-1})$, the linear solution
$u_{0}$ to $(H-\lambda^{2})u_{0}=0$ with incoming data $f$ is the image of $f$
under the _incoming Poisson operator_ , $P(\lambda)$. This $u_{0}$ can be
written (non-uniquely) as a sum of incoming and outgoing terms
$u_{0}=u_{-}+u_{+}$, where, roughly speaking, $u_{\pm}\sim r^{-(n-1)/2}e^{\pm
ir\lambda}f_{\pm}$ as $r$ goes to infinity. For adequate decompositions
$u_{\pm}$, solutions $u$ to (1.1) satisfying (1.2) can be constructed by a
contraction mapping argument in which one writes $u=u_{-}+w=u_{0}+(w-u_{+})$
where $w$ will be outgoing (in a sense to be made precise below) and a fixed
point of the mapping

$\Phi(w)=u_{+}+R(\lambda+i0)N[u_{-}+w],$

where $R(\lambda+i0)$ is the outgoing resolvent, that is,
$(H-(\lambda+i0)^{2})^{-1}$. The true nonlinear solution therefore satisfies

$u=u_{0}+R(\lambda+i0)N[u].$ (1.8)

At issue in this paper is the regularity of the outgoing data of $w$, and how
this can be understood in terms of the mapping properties of the Poisson
operator, the decomposition $u_{\pm}$, and the resolvent.

The construction of the nonlinear eigenfunction $u$ requires a decomposition
$u_{0}=u_{-}+u_{+}$ for the free solution, and we improve on the reults in
[GHSZ20] by using the Schwartz kernel of the Poisson operator $P(\lambda)$
(see Section 3) to obtain a decomposition with optimal regularity. Indeed, the
Poisson operator is given by the action of a well understood oscillatory
integral kernel, due to [MeZw96] in the asymptotically conic case, with many
antecedents in the Euclidean case, e.g. [Ik60, Is82]. Using this we prove a
relationship between the regularity of the incoming data $f\in
H^{k}(\mathbb{S}^{n-1})$ for $k\geq 0$ and the corresponding linear
generalized eigenfunction $u_{0}$. This is best understood using the theory of
scattering pseudodifferential operators [Me94]. It is well-known that $u_{0}$
lies in weighted Sobolev spaces $H^{s,-1/2-\epsilon}(\mathbb{R}^{n})$, where
$s\in\mathbb{R}$ and $\epsilon>0$ are arbitrary. This means that

$\langle z\rangle^{-1/2-\epsilon}u_{0}\in H^{s}(\mathbb{R}^{n}),$

where $H^{s}(\mathbb{R}^{n})$ are the standard $L^{2}$-based Sobolev spaces on
$\mathbb{R}^{n}$. (In particular, $u_{0}$ is a smooth function in the
interior, a consequence of elliptic regularity.) What we prove below is that
$u_{0}$ can be decomposed into $u_{0}=u_{-}+u_{+}$ where each $u_{\pm}$ has
$k$ additional order of ‘module regularity’, specifically each remains in
$H^{s,-1/2-\epsilon}$ after application of $k-$fold combinations of angular
derivatives and the radial annihilators $r(D_{r}\mp\lambda)$ of the
oscillatory factors $e^{\pm i\lambda r}$. As we describe below, these
operators are determined directly by the microlocal structure of the problem;
they comprise the modules $\mathscr{M}_{\pm}$ of scattering pseudodifferential
operators in $\Psi_{\mathrm{sc}}^{1,1}$ which are characteristic on the
incoming/outgoing ($-$/$+$) radial sets $\mathscr{R}_{\pm}$ of the operators
$H-\lambda^{2}$, thought of as a _non-elliptic_ scattering operator with non-
degenerate characteristic set over spatial infinity.

The paper is organized as follows. In Section 2 we recall the basic
definitions and structures that will be used in the paper, including
scattering pseudodifferential operators and the weighted Sobolev spaces
between which they act. We also recall there the definitions of the module
regularity spaces used in [GHSZ20] and the crucial mapping properties of the
resolvent between such spaces. In Section 3 we discuss the mapping properties
of the Poisson operator. These properties are very closely related to mapping
properties of the incoming and outgoing resolvents, due to formula (3.5). The
key result is Proposition 3.4 which shows that the resolvent applied to
certain functions admit asymptotic expansions corresponding to one of the
terms in (1.7). The PDE-type argument, however, does not give the optimal
regularity for the leading coefficient. The optimal regularity is obtained in
Propositions 3.5 and 3.6, which relates the map taking a function $F$ to the
leading expansion of $R(\lambda\pm i0)F$ to the _adjoint_ of the Poisson
operator, $P(\mp\lambda)^{*}$, applied to $F$. In Section 4 we put these
results together to prove the main theorem.

Our analysis of the Poisson operator is based on the description of its
Schwartz kernel as an oscillatory integral, which is due to Melrose-Zworski
[MeZw96]. The powerful tools from microlocal scattering theory we employ were
developed in [Me94] (radial point estimates), [HMV04] (test modules), [Va13]
(Fredholm theory for real principal type operators on anisotropic Sobolev
spaces) and [Va18] (radial point estimates and Fredholm theory in the
scattering calculus), and we refer the reader to [GHSZ20] for the detailed
discussion of related literature.

##  2\. Microlocal analysis of the resolvents $R(\lambda\pm i0)$

We review the relevant objects here only briefly as they are discussed in
detail in other work. A detailed introduction to scattering differential
operators on $\mathbb{R}^{n}$ can be found in Vasy’s minicourse notes [Va18],
while the more general development on scattering manifolds is due to Melrose
[Me94]. See also Sections 2 and 3 of [GHSZ20].

###  2.1. Weighted Sobolev spaces and scattering pseudodifferential operators

We confine most of our introductory discussion to the case of Euclidean space.
Letting $\mathscr{S}(\mathbb{R}^{n})$ denote the space of Schwartz functions
and $\mathscr{S}^{\prime}$ the tempered distributions, each
$u\in\mathscr{S}^{\prime}$ lies in some weighted $L^{2}$-based Sobolev space

$H^{m,l}(\mathbb{R}^{n})=\langle z\rangle^{-l}H^{m}(\mathbb{R}^{n}),$ (2.1)

for $m,l\in\mathbb{R}$.

Recall the scattering symbols and scattering pseudodifferential operators,
defined for $m,l\in\mathbb{R}$ by

$\begin{gathered}S^{m,l}(\mathbb{R}^{n})=\\{a(z,\zeta)\in
C^{\infty}(\mathbb{R}^{n}_{z}\times\mathbb{R}^{n}_{\zeta}):\|a\|_{m,l;N}<\infty\mbox{
for all }N\in\mathbb{N}_{0}\\}\\\
\Psi_{\mathrm{sc}}^{m,l}(\mathbb{R}^{n})=\\{\text{Op}(a)+B:a\in
S^{m,l}(\mathbb{R}^{n}),\quad\kappa_{B}(z,z^{\prime})\in\mathscr{S}(\mathbb{R}^{n}_{z}\times\mathbb{R}^{n}_{z^{\prime}})\\},\end{gathered}$

where

$\|a\|_{m,l;N}=\sum_{|\alpha|+|\beta|\leq N}\sup_{z,\zeta}|\langle
z\rangle^{-l+|\alpha|}\langle\zeta\rangle^{-m+|\beta|}D_{z}^{\alpha}D_{\zeta}^{\beta}a(z,\zeta)|,$

Here $\text{Op}(a)$ denotes the operator with integral kernel $\int
e^{-i(z-z^{\prime})}a(z,\zeta)d\zeta$ and, for any operator $B$, $\kappa_{B}$
denotes its Schwartz kernel. Thus up to smoothing operators the scattering
pseudodifferential operators are by definition (here) the left quantizations
of scattering symbols.

There is a notion of principal symbol attached to scattering PsiDO’s which
includes their leading order behavior at spatial infinity. The (filtered)
algebra of scattering PsiDO’s admits a natural mapping to the graded algebra
of scattering symbols,

$\Psi_{\mathrm{sc}}^{m,l}\longrightarrow S^{[m,l]}:=S^{m,l}/S^{m-1,l-1}.$

Given $A\in\Psi_{\mathrm{sc}}^{m,l}$ its principal symbol will be denoted by
$\sigma_{m,l}(A)$.

We work here exclusively with classical scattering symbols, which are
functions $a(z,\zeta)$ with joint asymptotic expansions in $z,\zeta$ as
$|z|,|\zeta|\to\infty$. We will need to construct explicit symbols whose
corresponding operators will function as microlocal cutoffs. In general,
classical symbols can be written

$a\equiv\langle z\rangle^{l}\langle\zeta\rangle^{m}a_{0}(z,\zeta),$

where $a_{0}(z,\zeta)$ is bounded and is smooth in $r^{-1}=|z|^{-1}$ and
$|\zeta|^{-1}$. This characterization of the asymptotic behavior of $a_{0}$ is
equivalent to the statement that $a_{0}$ extends to an element in
$C^{\infty}(\overline{\mathbb{R}^{n}}\times\overline{\mathbb{R}^{n}})$ where
$\overline{\mathbb{R}^{n}}$ is the radial compactification of $\mathbb{R}^{n}$
in which the map $z\mapsto z/(1+\langle z\rangle)$ realizes $\mathbb{R}^{n}$
as the interior of the unit ball $\mathbb{B}^{n}$, and the overline notation
denotes this entire structure (the map from Euclidean space into the ball as
opposed to just the ball.)

For $[a]\in S^{[m,l]}$, i.e. $[a]$ the principal symbol of a scattering PsiDO,
the value of $a_{0}$ is determined here only to the addition of elements in
$S^{m-1,l-1}$, so in particular, in sets of bounded frequency, $|\zeta|<C$,
with $\mathop{\rm
supp}(\chi\colon\mathbb{R}_{t}\longrightarrow\mathbb{R})\subset\\{|t|\leq
C\\}$,

$\chi(\zeta)(a_{0}(z,\zeta)-a(\hat{z},\zeta))\equiv 0\qquad(\mbox{mod
}S^{m-1,l-1}),$

with a similar expression in the $|z|<C$ regions. An
$A\in\Psi_{\mathrm{sc}}^{m,l}$ is by definition _scattering elliptic_ if its
principal symbol is invertible in the graded algebra $\bigcup_{m,l}S^{[m,l]}$.
The microlocal notion of scattering ellipticity will be used as follows. On
regions of large frequency $|\zeta|>C$, scattering ellipticity is implied by
the uniform estimate

$\sigma_{m,l}(A)(z,\zeta)\geq C\langle
z\rangle^{l}\langle\zeta\rangle^{m}\mbox{ for }|\zeta|>C.$

Below, the operators of interest are (scattering) elliptic on such large
frequency regions but are not globally scattering elliptic, hence the more
detailed estimates coming from propagation phenomena will arise from analysis
on sets of bounded frequency, whence we will typically need only to discuss
the “spatial” principal symbol of $A\in\Psi_{\mathrm{sc}}^{m,l}$, defined for
classical scattering symbols by

$\sigma_{\text{base},\,l}(A)(\hat{z},\zeta)=\lim_{r\to\infty}\langle
z\rangle^{-l}a(z,\zeta).$

Note that the behavior in $\zeta$ of this function is in general symbolic but
not homogeneous. In this region, (scattering) ellipticity is the same familiar
ellipticity but in the $z$ directions; $(\hat{z}_{0},\zeta_{0})$ is in the
elliptic set if $\sigma_{m,l}(A)(z,\zeta)\geq C\langle z\rangle^{l}$ in a
region $|z|>C,|\hat{z}-\hat{z}_{0}|,|\zeta-\zeta_{0}|<\epsilon$. It is
straight forward to check that this is equivalent to
$\sigma_{\text{base},\,l}(A)(\hat{z}_{0},\zeta_{0})\neq 0$.

The basic boundedness property of scattering pseudodifferential operators is
that for $A\in\Psi_{\mathrm{sc}}^{m,l}$,

$A\colon H^{s,k}\longrightarrow H^{s-m,k-l}\mbox{ is bounded.}$ (2.2)

The residual operators are

$\Psi^{-\infty,-\infty}=\bigcap_{m,l}\Psi_{\mathrm{sc}}^{m,l},$

which have Schwartz kernels in
$\mathscr{S}(\mathbb{R}^{n}\times\mathbb{R}^{n})$, and in particular
$A\in\Psi_{\mathrm{sc}}^{-\infty,-\infty}$ is bounded between _any_ two
weighted $L^{2}$-based Sobolev spaces; in particular since $H^{s,l}\subset
H^{s^{\prime},l^{\prime}}$ for any $s>s^{\prime},l>l^{\prime}$, residual $A$’s
define compact operators on $L^{2}$. If $A$ is globally scattering elliptic
then the map in (2.2) is Fredholm, since in that case there is an approximate
inverse $B\in\Psi_{\mathrm{sc}}^{-m,-l}$ such that
$\operatorname{Id}-AB,\operatorname{Id}-BA\in\Psi_{\mathrm{sc}}^{-\infty,-\infty}$.

All of the foregoing material extends directly to the general asymptotically
conical case; see [GHSZ20, Section 2.2] for further details. In the general
case, one has the fiberwise radial compactification of the scattering
cotangent bundle ${}^{\mathrm{sc}}\overline{T}^{*}M$ which is a manifold with
corners of codimension $2$; there are boundary defining fuctions $x$ for
spatial infinity and $\rho$ for fiber infinity (which in the Euclidean case
can be taken to be $\langle z\rangle^{-1}$ and $\langle\zeta\rangle^{-1}$,
respectively) and the symbol estimates are essentially the same as the
Euclidean space symbol estimates written in terms of $x$ and $\rho$. The
scattering pseudodifferential operators $\Psi_{\mathrm{sc}}^{s,k}(M)$ are the
quantizations of these symbols, and the scattering Sobolev spaces, denoted
$H^{s,k}_{\mathrm{sc}}(M)$, consist of distributions $u$ with $Au\in L^{2}(M)$
for all $A\in\Psi_{\mathrm{sc}}^{s,k}(M)$.

###  2.2. Mapping properties of $H-\lambda^{2}$ and the resolvent

Analysis of $H-\lambda^{2}$ as a scattering differential operator was first
carried out by Melrose in [Me94]. We review the relevant material again on
$\mathbb{R}^{n}$. We have $H-\lambda^{2}\in\Psi_{\mathrm{sc}}^{2,0}$, with

$\sigma_{2,0}(H-\lambda^{2})=|\zeta|^{2}_{g}+V-\lambda^{2}=\langle\zeta\rangle^{2}\frac{|\zeta|^{2}_{g}+V-\lambda^{2}}{\langle\zeta\rangle^{2}}=:\langle
z\rangle^{2}a_{0}(\zeta),$

so $H-\lambda^{2}$ is elliptic in the region $|\zeta|>(1+\epsilon)\lambda$ for
any $\epsilon>0$. In the general asymptotically Euclidean case and adding a
short range potential we get that the spatial principal symbol is rather
simple:

$\sigma_{\text{base},\,2}(H-\lambda^{2})(\hat{z},\zeta)=|\zeta|^{2}-\lambda^{2}.$
(2.3)

In particular $H-\lambda^{2}$ is not globally scattering elliptic. Its
characteristic set $\Sigma$, which is a subset of
$\mathbb{S}^{n-1}_{\hat{z}}\times\mathbb{R}^{n}_{\zeta}$, is the vanishing
locus of the fiber principal symbol:

$\Sigma=\\{(\hat{z},\zeta):|\zeta|^{2}-\lambda^{2}=0\\}.$ (2.4)

To understand the action of the flow of the Hamilton vector field on $\Sigma$,
we can work in polar coordinates $(r,y)$, where $y$ are arbitrary coordinates
in a coordinate patch on the sphere $\mathbb{S}^{n-1}$; these automatically
give rise to dual coordinates $(\nu,\eta)$, and we rescale the angular dual
variable $\eta$ by writing $r\mu=\eta$. Via this localization and rescaling we
effectively pass to the case of an arbitrary asymptotially conic manifold.

Noting that in these variables we have

$|(x=0,y,\nu,\mu)|^{2}=\nu^{2}+\|\mu\|^{2}_{h_{y}^{-1}},$

where $h_{y}$ is the Riemannian metric on the round sphere (or more generally
on $\partial M$) at the point $y$. Apart from giving the obvious rewriting of
the characteristic set as
$\Sigma=\\{\nu^{2}+\|\mu\|^{2}_{h_{y}^{-1}}=\lambda^{2}\\}$, it clarifies the
behavior of the Hamilton vector field, which, recall, is defined with respect
to arbitrary coordinates and their canonical dual coordinates to be

$H_{p}:=\frac{\partial p}{\partial\nu}\frac{\partial}{\partial
r}-\frac{\partial a}{\partial
r}\frac{\partial}{\partial\nu}+\sum_{j=1}^{n-1}\Big{(}\frac{\partial
a}{\partial\tilde{\mu}_{j}}\frac{\partial}{\partial y_{j}}-\frac{\partial
a}{\partial y_{j}}\frac{\partial}{\partial\tilde{\mu}_{j}}\Big{)}.$

Scattering calculus propagation results are phrased in terms of the natural
conformal rescaling (or reweighting) of this vector field, namely
$\mathsf{H}_{p}:=\langle z\rangle H_{p},$ and with
$p=\sigma_{2,0}(H-\lambda^{2})$, we obtain

$\mathsf{H}_{p}=-2\nu(x\partial_{x}+R_{\mu})+2|\mu|^{2}_{y}\partial_{\nu}+H_{\partial
M,h},$ (2.5)

where $R_{\mu}$ is the radial vector field in $\mu$ and $H_{\partial M,h}$ is
the Hamilton vector field on $\partial M$ for the metric $h$. (Note that the
apparent discrepancy in sign between (2.5) and [Me94, Eq. 8.19] is due to our
usage of $\nu$, the dual variable to $r$, in contrast with Melrose’s use of
$\tau=-\nu$.)

From this one deduces that the two submanifolds

$\mathscr{R}_{\pm}:=\\{\mu=0=x,\ \nu=\pm\lambda\\}$ (2.6)

are sinks ($+$) or sources ($-$) for the (rescaled) Hamilton flow, i.e. the
flow of $\mathsf{H}_{p}$, on the characteristic set $\Sigma$. The well-known
formula for the principal symbol of a commutator of pseudodifferential
operators also extends to the scattering setting, namely if
$A\in\Psi_{\mathrm{sc}}^{s,l},B\in\Psi_{\mathrm{sc}}^{s^{\prime},l^{\prime}}$
with $a=\sigma_{s,l}(A),b=\sigma_{s^{\prime},l^{\prime}}(B)$, then
$[A,B]\in\Psi_{\mathrm{sc}}^{s+s^{\prime}-1,l+l^{\prime}-1}$ and

$\sigma_{s+s^{\prime}-1,l+l^{\prime}-1}(i[A,B])=\\{a,b\\}=H_{a}b,$

where here $\\{\cdot,\cdot\\}$ denote the Poisson bracket.

The most important example of the use of commutators for us will be those of
the form $[H,Q]$ where $H$ is the Hamiltonian and
$Q\in\Psi_{\mathrm{sc}}^{0,0}$, in particular we will take $Q$ to have various
microsupport properties (discussed below) which will allow them to act as
microlocal cutoffs to various portions of scattering phase space. Particularly
simple examples include symbols of the form

$q(x,y,\nu,\mu)=\chi(x)\chi((|\mu|^{2}+\nu^{2})/2\lambda^{2})\phi(\nu)$

where $\chi$ is a smooth function with $\chi(t)=1$ for $t\leq 1$ and
$\chi(t)=0$ for $t\geq 2$. (This $q$ is then a function whose support lies in
a neighborhood of $\Sigma$ and whose restriction to $\Sigma$ depends only on
$\nu$.) Then in fact $Q\in\Psi_{\mathrm{sc}}^{-\infty,0}$ and
$[H,Q]\in\Psi_{\mathrm{sc}}^{-\infty,-1}$, whence

$\langle
z\rangle\sigma_{-\infty,-1}(i[H,Q])=\mathsf{H}_{p}q=2|\mu|^{2}_{y}\phi^{\prime}(\nu)\mbox{
in the region }|x|<1,\ |\mu|^{2}+\nu^{2}<2\lambda^{2},$ (2.7)

and $\sigma_{-\infty,-1}(i[H,Q])=0$ for $|\mu|^{2}+\nu^{2}\geq 2$.

We can now define the modules of pseudodifferential operators
$\mathscr{M}_{\pm}$ used to measure the regularity of distributions in the
domain and range of our formulation of the resolvent mapping. Specifically

$\mathscr{M}_{\pm}:=\\{A\in\Psi_{\mathrm{sc}}^{1,1}:\mathscr{R}_{\pm}\subset\Sigma_{1,1}(A)\\},$
(2.8)

or, in words, $\mathscr{M}_{+}$ is the vector space of scattering
pseudodifferential operators of order $(1,1)$ which are characteristic on the
“outgoing” radial set $\mathscr{R}_{+}$ and $\mathscr{M}_{-}$ is the same for
$\mathscr{R}_{-}$. Locally an element $A\in\Psi_{\mathrm{sc}}^{1,1}$ lies in
$\mathscr{M}_{\pm}$ if and only if it can be written

$A=B_{0}r(D_{r}\mp\lambda)+\sum_{i=1}^{n-1}B_{i}D_{y_{i}}+B^{\prime}$

with $B_{0},B_{i},B^{\prime}\in\Psi_{\mathrm{sc}}^{0,0}$. Note that
$\mathscr{M}_{+}$ and $\mathscr{M}_{-}$ both contain the identity operator.
Also, $\mathscr{M}_{\pm}$ contain a elliptic element of order $(1,1)$, namely
the radial operator $r(D_{r}\mp\lambda)$, at the opposite radial set
$\mathscr{R}_{\mp}$.

We also need the ‘small’ module of angular derivatives

$\mathscr{N}=\\{\sum_{i=1}^{n-1}B_{i}D_{y_{i}}+B^{\prime}:B_{i},B^{\prime}\in\Psi_{\mathrm{sc}}^{0,0}\\}=\mathscr{M}_{+}\cap\mathscr{M}_{-}.$

Theorem 2.7 of [GHSZ20] then gives the following mapping property for the
resolvent. Let $s,l\in\mathbb{R},k,k^{\prime}\in\mathbb{N}_{0}$, and define

$\begin{gathered}\mathscr{Y}^{s,l;k,k^{\prime}}_{\pm}:=H^{s,l;k,k^{\prime}}_{\pm}=\\{u\in
H^{s,l}:\mathscr{N}^{k^{\prime}}\mathscr{M}_{\pm}^{k}u\subset H^{s,l}\\}\\\
\mathscr{X}^{s,l;k,k^{\prime}}_{\pm}:=\\{u\in
H_{\pm}^{s,l;k,k^{\prime}}:(H-\lambda^{2})u\in
H_{\pm}^{s-2,l+1;k,k^{\prime}}\\}.\end{gathered}$ (2.9)

Then

$R(\lambda\pm
i0)\colon\mathscr{Y}_{\pm}^{s,l+1;k,k^{\prime}}\longrightarrow\mathscr{X}_{\pm}^{s+2,l;k,k^{\prime}}\mbox{
provided }k\geq 1,\ l\in(-\frac{1}{2}-k,-\frac{1}{2}).$ (2.10)

Here, the value $l=-1/2$ is referred to as the “threshold” value; it is the
critical spatial order for which we need different “radial propagation
estimates” for $l$ greater than vs. less than $-1/2$. See [GHSZ20, Section 3].
The condition in (2.10) is that $l$ is below threshold, but that $l+k$ is
above threshold. That means that module regularity for $\mathscr{M}_{\pm}$ of
order $k$ gives above threshold regularity at the opposite radial set
$\mathscr{R}_{\mp}$, which is the key to obtaining estimates such as (2.10).
This statement can be rephrased as follows: for such $k\in\mathbb{N}$ and
$l\in\mathbb{R}$, the operator $H-\lambda^{2}$ is an isomorphism from
$\mathscr{X}_{\pm}^{s,l;k,k^{\prime}}$ to
$\mathscr{Y}_{\pm}^{s-2,l+1;k,k^{\prime}}$, and its inverse is the resolvent
$R(\lambda\pm i0)$.

The reason for introducing the small module $\mathscr{N}$ to treat the
nonlinear problem is explained in detail in the introduction of [GHSZ20]. We
remark here only that for multiplication of distributions in
$H_{+}^{s,l;k,k^{\prime}}$, large module regularity produces loss in decay –
see e.g. [GHSZ20, Cor. 2.10] from which one has $H_{+}^{s,l;k,k^{\prime}}\cdot
H_{+}^{s,l;k,k^{\prime}}\subset H_{+}^{s,2l-k;k,k^{\prime}}$. Small module
regularity is used to minimize this loss.

###  2.3. Microlocalization

It will be important to microlocalize distributions both near to and away from
the radial set. To this end we recall some features of the operator wavefront
set $\operatorname{WF}^{\prime}(A)$ of $A\in\Psi_{\mathrm{sc}}^{m,l}(M)$. We
will work mostly with those $A$ which are compactly microlocalized in
frequency. Concretely, this means that

$A=\text{Op}(a(z,\zeta))+E,\quad\mathop{\rm
supp}a(z,\zeta)\subset\\{|\zeta|<C\\},\
\kappa_{E}\in\mathscr{S}(\mathbb{R}^{n}\times\mathbb{R}^{n}),$

for some $C>0$. (Recall $\kappa_{E}$ is the Schwartz kernel of $E$.) The
condition on $\kappa_{E}$ is equivalent to $E$ being a _residual_ operator,
meaning

$E\in\Psi_{\mathrm{sc}}^{-\infty,-\infty}:=\bigcap_{m,l\in\mathbb{R}\times\mathbb{R}}\Psi_{\mathrm{sc}}^{m,l}.$

For such $A$,
$\operatorname{WF}^{\prime}(A)\subset\mathbb{S}^{n-1}_{\hat{z}}\times\mathbb{R}^{n}_{\zeta}$
is by definition the complement of the set $(\hat{z},\zeta)$ such that $a$ is
Schwartz in $\zeta$ in the $\hat{z}$ direction.

Clearly $\operatorname{WF}^{\prime}(A)=\varnothing\implies
A\in\Psi_{\mathrm{sc}}^{-\infty,-\infty}$, and thus for any
$M,N\in\mathbb{R}$,

$\operatorname{WF}^{\prime}(A)=\varnothing\implies A\colon
H^{M,N}(\mathbb{R}^{n})\longrightarrow\mathscr{S}(\mathbb{R}^{n}).$

Also, wavefront sets have the expected algebraic property of supports, namely,
for
$A\in\Psi_{\mathrm{sc}}^{m,l},B\in\Psi_{\mathrm{sc}}^{m^{\prime},l^{\prime}}$,

$\operatorname{WF}^{\prime}(AB)\subset\operatorname{WF}^{\prime}(A)\cap\operatorname{WF}^{\prime}(B).$

Two particularly useful examples of such operators are those of the form
$Q_{\pm}\in\Psi_{\mathrm{sc}}^{0,0}$ which are microlocalized near the two
components of the wavefront set $\mathscr{R}_{\pm}$. This can be done by
defining $Q_{\pm}$ explicitly in a way similar to the definition of $Q$ above
(2.7), specifically one can take

$Q_{\pm}=\text{Op}(q_{\pm}),\quad
q_{\pm}(x,y,\nu,\mu)=\chi(x)\chi((|\mu|^{2}+\nu^{2})/2\lambda^{2})\chi((\nu\mp\lambda)/\epsilon),$
(2.11)

where again $\chi(s)=1$ for $s\leq 1$ and $\chi(s)=0$ for $s\geq 2$.

Alternatively, one can simply break up phase space into a region where $\nu$
is positive and its complement, excluding one radial set from each. We choose
an operator $A_{+}\in\Psi^{0,0}_{\text{cl}}(M)$ such that $A_{+}$ is
microlocally equal to the identity in a neighbourhood of $\mathscr{R}_{+}$,
and microlocally equal to $0$ in a neighbourhood of $\mathscr{R}_{-}$. We also
let $A_{-}=\operatorname{Id}-A_{+}$; thus, $A_{-}$ is microlocally equal to
the identity in a neighbourhood of $\mathscr{R}_{-}$, and microlocally equal
to $0$ in a neighbourhood of $\mathscr{R}_{+}$. It is convenient to choose
$A_{+}$ such that its principal symbol $a$ is a function only of $\nu$ in a
neighbourhood of the characteristic variety of $H$, and is monotone. Indeed,
we can take

$A_{+}=\text{Op}(a),\quad
a(x,\nu)=\chi(x)\chi((|\mu|^{2}+\nu^{2})/2\lambda^{2})\tilde{\chi}(\nu),$
(2.12)

where $\chi$ is as in the definition of $Q_{\pm}$ and $\tilde{\chi}(\nu)\equiv
1$ in $\nu>\lambda/4$ and $\tilde{\chi}(\nu)\equiv 0$ for $\nu<-\lambda/4$,
and is monotone in between.

$\mathscr{R}_{-}$ at 134 5 $\mathscr{R}_{+}$ at 133 182 $\frac{\lambda}{4}$ at
-7 115 $-\frac{\lambda}{4}$ at -11 70 $a\equiv 1$ at 0 145 $a\equiv 0$ at 0 40
$\lambda$ at 83 177 $-\lambda$ at 78 12 $\mu$ at 183 85 $\nu$ at 84 193

Figure 1. The bicharacteristic flow in the characteristic set
$\Sigma(P)=\\{x=0,\,\nu^{2}+|\mu|^{2}_{y}=\lambda^{2}\\}$ with
$P=\Delta-\lambda^{2}$ and the principal symbol of the microlocalizer $A_{+}$.

While microlocalization gives us a concrete mechanism for analyzing frequency-
localized spatial decay, it will be useful to note that this is also possible
using anisotropic Sobolev spaces, meaning Sobolev spaces $H^{s,\mathsf{l}}$ in
which the parameters $s,\mathsf{l}$ are themselves functions on phase space.
Since the operators under consideration here are elliptic outside a compact
set in frequency, we will always take $s$ constant, but we shall employ
variable spatial weights $\mathsf{l}_{\pm}\in S^{0,0}$ satisfying

$\begin{gathered}\text{$\mathsf{l}_{+}$ is equal to $-1/2$ outside small
neighbourhoods of $\mathscr{R}_{+}$ and $\mathscr{R}_{-}$,}\\\
\text{$\mathsf{l}_{+}$ equals $-1/2\mp\delta$ in smaller neighbourhoods of
$\mathscr{R}_{\pm}$, }\\\ \text{$\mathsf{l}_{+}$ is nonincreasing along the
Hamilton flow of $P$ within $\Sigma(P)$,}\\\ \text{and
$\mathsf{l}_{-}=-1-\mathsf{l}_{+}$ has corresponding properties with
$\mathscr{R}_{-}$ and $\mathscr{R}_{+}$ switched.}\end{gathered}$ (2.13)

For definiteness, we suppose that for some sufficiently small
$\delta\in[0,0.01]$, we have $\mathsf{l}_{+}$ equal to $-1/2$ outside the sets
$\\{x^{2}+|\mu|^{2}+(\nu\mp\lambda)^{2}\leq 4\delta\lambda^{2}\\}$, and is
equal to $-1/2\mp\delta$ within the sets
$\\{x^{2}+|\mu|^{2}+(\nu\mp\lambda)^{2}\leq\delta\lambda^{2}\\}$. Thus,
distributions in $H^{s,\mathsf{l}_{+}}$ lie in $H^{s,-1/2-\epsilon}$ and are
microlocally $H^{s,-1/2+\epsilon}$ near $\mathscr{R}_{-}$; that is, they are
below threshold regularity at $\mathscr{R}_{+}$ and below threshold at
$\mathscr{R}_{-}$, with a corresponding statement true with reversed signs for
$H^{s,\mathsf{l}_{-}}$. Then by Theorem 3.2 of [GHSZ20] we have isomorphisms

$\begin{gathered}H-\lambda^{2}\colon\mathscr{X}_{\pm}^{s,\mathsf{l}_{\pm}}\longrightarrow\mathscr{Y}_{\pm}^{s-2,\mathsf{l}_{\pm}+1},\\\
\mathscr{Y}_{\pm}^{s-2,\mathsf{l}_{\pm}+1}=H^{s-2,\mathsf{l}_{\pm}+1},\quad\mathscr{X}_{\pm}^{s,\mathsf{l}_{\pm}}=\\{u\in
H^{s,\mathsf{l}_{\pm}}:(H-\lambda^{2})u\in\mathscr{Y}_{\pm}^{s-2,\mathsf{l}_{\pm}+1}\\}.\end{gathered}$
(2.14)

The inverse maps are the incoming/outgoing resolvent $R(\lambda\pm i0)$. As
with (2.10), we notice the sharp difference of one in the spatial regularity,
and the avoidance of the threshold value at the radial sets.

Moreover, we can combine variable order spaces with module regularity,
obtaining isomorphisms

$\begin{gathered}H-\lambda^{2}\colon\mathscr{X}_{\pm}^{s,\mathsf{l}_{\pm};k,k^{\prime}}\longrightarrow\mathscr{Y}_{\pm}^{s-2,\mathsf{l}_{\pm}+1;k,k^{\prime}},\\\
\mathscr{Y}_{\pm}^{s^{\prime},\mathsf{l}^{\prime};k,k^{\prime}}:=\\{u\in\mathscr{Y}_{\pm}^{s^{\prime},\mathsf{l}^{\prime}}\mid\mathscr{N}^{k^{\prime}}\mathscr{M}_{\pm}^{k}\,u\subset\mathscr{Y}_{\pm}^{s^{\prime},\mathsf{l}^{\prime}}\\}\\\
\mathscr{X}_{\pm}^{s,\mathsf{l}_{\pm};k,k^{\prime}}:=\\{u\in\mathscr{Y}_{\pm}^{s,\mathsf{l}_{\pm};k,k^{\prime}}:(H-\lambda^{2})u\in\mathscr{Y}_{\pm}^{s-2,\mathsf{l}_{\pm}+1;k,k^{\prime}}\\}.\end{gathered}$
(2.15)

We emphasize that in (2.15), we can allow $k=0$, that is, only consider small
module regularity, in contrast to (2.10).

These variable order module regularity results do not appear explicitly in our
previous work [GHSZ20], but such results follow readily from the propagation
estimates in Section 3 of that paper. (The reason for restricting the module
regularity spaces to constant orders in [GHSZ20] is that it is more convenient
to analyze multiplicative properties of such spaces when the orders are
constant.) We shall not review variable order spaces in detail here, referring
the reader to [Va18]. Here we only mention one elementary property of such
spaces. Namely, we can conclude containment in these types of variable order
spaces using microlocalization. For example, it is straightforward to show
that, given a tempered distribution $u$, if there exists
$Q_{+}\in\Psi_{\mathrm{sc}}^{0,0}$ as in (2.11) such that $Q_{+}u\in
H^{s,-1/2-\epsilon}$ and $(\operatorname{Id}-Q_{+})u\in H^{s,-1/2+\epsilon}$,
then $u\in H^{s,\mathsf{l}_{+}}$ (provided $\delta>0$ in the definition of
$\mathsf{l}_{+}$ is sufficiently small.)

We note that in all these considerations, the value of $\mathsf{l}_{\pm}$ off
$\Sigma$ is essentially irrelevant, since off $\Sigma$ one has elliptic
estimates for $H-\lambda^{2}$.

##  3\. Mapping properties of the linear Poisson operator

The Poisson operator furnishes a distorted or generalized Fourier transform
for the operator $H$. Constructions in the Euclidean case go back a long way
in scattering theory, see e.g. [Ik60], [Is82]. We use the results of Melrose
and Zworski [MeZw96], who constructed the Poisson operator $P(\lambda)$,
$\lambda>0$, for the Helmholtz operator $H=\Delta-\lambda^{2}$ on
asymptotically conic manifolds (scattering metrics). Their results extend
readily to the case of a Schrödinger operator with potential in the conormal
space $\mathscr{A}^{\gamma}(M)$, for $\gamma>1$, which is by definition the
space of smooth functions $V$ on $M^{\circ}$ satisfying estimates near
$\partial M$ of the form

$\Big{|}(xD_{x})^{j}D_{y}^{\alpha}V(x,y)\Big{|}\leq Cx^{\gamma},\quad j\geq
0,\ \alpha\in\mathbb{N}^{n-1}.$ (3.1)

These estimates on $V$ are equivalent to (1.6) in the case of Euclidean space.

The Poisson operator has the property that it maps a smooth function $\phi\in
C^{\infty}(\partial M)$ to a function $u_{0}$ satisfying $Hu_{0}=0$, with the
asymptotics

$u_{0}=u_{-}+u_{+},\quad u_{\pm}=r^{-(n-1)/2}e^{\pm i\lambda r}f_{\pm},\quad
f_{\pm}\in C(M),\quad f_{-}\big{|}_{\partial M}=\phi.$ (3.2)

In fact the $f_{\pm}$ can be taken to lie in
$C^{\infty}(M)+\mathscr{A}^{\gamma-1}(M)$ (which are continuous up to the
boundary, as $\gamma-1>0$); moreover, if $V$ is equal to $r^{-2}$ times a
smooth function on $M$, then the $f_{\pm}$ can be taken in $C^{\infty}(M)$. We
shall also define $P(-\lambda)$, again for $\lambda>0$, to be the operator
mapping $\phi\in C^{\infty}(\partial M)$ to a function $u_{0}$ satisfying
$Hu_{0}=0$, with the asymptotics

$u_{0}=u_{-}+u_{+},\quad u_{\pm}=r^{-(n-1)/2}e^{\pm i\lambda r}f_{\pm},\quad
f_{\pm}\in C^{\infty}(M)+\mathscr{A}^{\gamma-1}(M),\quad
f_{+}\big{|}_{\partial M}=\phi.$

Recall that $C^{\infty}$ functions on $M$, as opposed to $C^{\infty}$
functions on $M^{\circ}$, are (by definition) smooth functions of $x=1/r$ and
$y$ near $\partial M$.

The adjoint operator $P(\lambda)^{*}$ will also play a role in this article.
To describe it, let $v$ be a Schwartz function111By a Schwartz function on $M$
we mean a smooth function that vanishes, with all its derivatives, at the
boundary. When $M$ is the radial compactification of $\mathbb{R}^{n}$ this
corresponds to the usual meaning of Schwartz function. on $M$, and let
$u_{\pm}=\pm R(\lambda\pm i0)v$. Then each $u_{\pm}$ has a similar expansion
as in (3.2), but with only the outgoing ($+$)/incoming($-$) oscillation:

$u_{\pm}=r^{-(n-1)/2}e^{\pm i\lambda r}f_{\pm},\quad f_{\pm}\in
C^{\infty}(M)+\mathscr{A}^{\gamma-1}(M).$ (3.3)

We claim that $P(\lambda)^{*}v=-2i\lambda f_{-}|_{\partial M}$. To see that
this is true, we use a pairing identity that follows from Green’s formula.
Suppose that $u_{1}$ and $u_{2}$ are two functions of the form

$u_{i}=r^{-(n-1)/2}\Big{(}e^{-i\lambda r}f_{i,-}+e^{+i\lambda
r}f_{i,+}\Big{)},\quad f_{i,\pm}\in C^{\infty}(M)+\mathscr{A}^{\gamma-1}(M).$

Suppose further that $(H-\lambda^{2})u_{i}=O(r^{-(n+1)/2-\epsilon})$ for some
$\epsilon>0$. Then the following identity holds [Me94, Equation (13.1)] (see
also [HV99, Section 5]):

$\int_{M}\Big{(}u_{1}\overline{(H-\lambda^{2})u_{2}}-((H-\lambda^{2})u_{1})\overline{u_{2}}\Big{)}d\mathrm{vol}_{g}=2i\lambda\int_{\partial
M}\Big{(}f_{1,+}\overline{f_{2,+}}-f_{1,-}\overline{f_{2,-}}\Big{)}d\mathrm{vol}_{h},$

where we take the restrictions to $\partial M$ in the integral over the
boundary. To prove this one notes that the LHS is absolutely integrable, so it
can be obtained as the limit, as $R\to\infty$, of the integral restricted to
$\\{r\leq R\\}$. Then one applies Green’s formula and uses the asymptotic form
of the functions $u_{i}$ to see that the limit as $R\to\infty$ is the RHS.

We apply this with $u_{1}=P(\lambda)a$, $a\in C^{\infty}(\partial M)$ and
$u_{2}=u_{-}$ above. Then $(H-\lambda^{2})u_{1}=0$, and $f_{2,+}=0$, while
$f_{1,-}=a$. We obtain

$-\int_{M}P(\lambda)a\overline{v}=2i\lambda\int_{\partial
M}a\overline{f_{-}}.$ (3.4)

This immediately yields $P(\lambda)^{*}v=-2i\lambda f_{-}|_{\partial M}$. A
similar argument with $u_{+}$ shows that $P(-\lambda)v=-2i\lambda
f_{+}|_{\partial M}$.

Moreover, applying $P(\lambda)$ to $f_{-}$, we obtain $u_{+}+u_{-}$, since
this is the unique eigenfunction with incoming data equal to $f_{-}$. It
follows that we have

$P(\lambda)P(\lambda)^{*}=\frac{2\lambda}{i}\Big{(}R(\lambda+i0)-R(\lambda-i0)\Big{)},$
(3.5)

acting on Schwartz functions. A similar argument shows that
$P(-\lambda)P(-\lambda)^{*}$ is equal to the same quantity. Notice that the
quantity in (3.5) is equal to the spectral measure, up to a factor of
$d\lambda/2\pi$.

Our first purpose in this section to analyze the range of $P(\lambda)$ on the
$L^{2}$-based Sobolev space $H^{m}(\partial M)$. We begin with a basic mapping
property for the Poisson operator on $L^{2}(\partial M)$. For any $\delta>0$,
identity (3.5) implies that we have a continuous mapping

$P(\lambda)\colon L^{2}(\partial M)\longrightarrow H^{s,-1/2-\delta}(M)$ (3.6)

Indeed, first note that both $R(\lambda\pm i0)$ map $H^{-1,1/2+\delta}(M)$ to
$H^{1,-1/2-\delta}(M)$. This result is well known but follows in particular
from (2.14) taking $s=1$ since for either choice of $\pm$ and any $s$,

$H^{s-2,1/2+\delta}\subset\mathscr{Y}_{\pm}^{s-2,\mathsf{l}_{\pm}+1},\qquad\mathscr{X}_{\pm}^{s,\mathsf{l}_{\pm}}\subset
H^{s,-1/2-\delta}.$

Formula (3.5) then shows that $P(\lambda)$ extends from $C^{\infty}(\partial
M)$ to a bounded map from $L^{2}(\partial M)$ to $H^{1,-1/2-\delta}(M)$, as
this holds if and only if $P(\lambda)P(\lambda)^{*}$ maps
$H^{-1,1/2+\delta}(M)$ to $H^{1,-1/2-\delta}(M)$ (since $H^{-1,1/2+\delta}(M)$
is the dual of $H^{1,-1/2-\delta}(M)$), whence (3.6) holds for $s=1$. Finally,
since the image of $P(\lambda)$ is contained in solutions to
$(H-\lambda^{2})u=0$, the differential index $1$ can be replaced by any $s$
using elliptic regularity.

Now we note that (3.6) can be improved easily using the microlocalization
discussed above. If we define

$\mathsf{l}_{\min}=\min(\mathsf{l}_{+},\mathsf{l}_{-}),\quad\mathsf{l}_{\max}=\max(\mathsf{l}_{+},\mathsf{l}_{-}),$
(3.7)

then these are smooth functions on phase space for $\delta^{\prime}>0$
sufficiently small, and (again for arbitrary $s$),

$(H^{s,\mathsf{l}_{\min}})^{*}=H^{-s,-\mathsf{l}_{\min}}=H^{-s,\mathsf{l}_{\max}+1}$

by the final property in (2.13). Repeating our argument to deduce mapping
properties for $P(\lambda)$ from (3.5), we use

$H^{s-2,\mathsf{l}_{\max}+1}\subset\mathscr{Y}_{\pm}^{s-2,\mathsf{l}_{\pm}+1},\qquad\mathscr{X}_{\pm}^{s,\mathsf{l}_{\pm}}\subset
H^{s,\mathsf{l}_{\min}}$

to conclude that for any $s\in\mathbb{R}$,

$\begin{gathered}P(\lambda)\colon L^{2}(\partial M)\longrightarrow
H^{s,\mathsf{l}_{\min}}(M),\\\ P(\lambda)^{*}\colon
H^{s,\mathsf{l}_{\max}+1}(M)\longrightarrow L^{2}(\partial M),\end{gathered}$
(3.8)

the latter following from adjunction of the former. This improves (3.6) since
$\mathsf{l}_{min}=-1/2>-1/2-\delta$ away from the radial sets.

Again we can phrase this in terms of microlocalization, giving improved decay
for $P(\lambda)f$, $f\in L^{2}(\partial M)$ away from the radial sets. That is
to say, if $Q\in\Psi_{\mathrm{sc}}^{-\infty,0}$ has
$\operatorname{WF}^{\prime}(Q)\cap\mathscr{R}_{+}\cup\mathscr{R}_{-}=\varnothing$,
then

$QP(\lambda)\colon L^{2}(\partial M)\longrightarrow H^{s,-1/2}(M),$ (3.9)

as follows immediately from (3.8) and the fact that for $\delta^{\prime}$
taken sufficiently small in the definition of $\mathsf{l}_{\pm}$, we have
$Q\colon H^{s,\mathsf{l}_{\min}}\longrightarrow H^{s,-1/2}$. Proposition 3.1,
a crucial step in proving our main theorem, improves on this; it asserts that
increased regularity of the boundary distribution $f$ corresponds to increased
decay of $P(\lambda)f$ away from the radial sets.

The Poisson operator extends to a map on distributions, or equivalently on
negative order Sobolev spaces $H^{m}(\partial M)$ for all $m\in\mathbb{R}$.
The representation of the Poisson kernel as a distribution associated to a
intersecting pair of Legendre submanifolds with conic points in [MeZw96] shows
that there is a mapping property of the form

$P(\pm\lambda):H^{m}(\partial M)\to H^{s,m+c}(M)\text{ for all }m<0$ (3.10)

for some $c\in\mathbb{R}$; this is obtained by passing derivatives from the
distribution onto the Poisson kernel, thus reducing to the case $m\geq 0$. The
next proposition, plus positive commutator estimates as in [GHSZ20, Section
3], show that in fact $c=-1/2$ is sharp for (3.10), although we do not need
this fact in the present article. Notice that this has the following simple
consequence for the adjoint:

$P(\pm\lambda)^{*}$ maps Schwartz functions on $M$ to smooth functions on
$\partial M$.  (3.11)

######  Proposition 3.1.

Let $Q\in\Psi^{-\infty,0}$ satisfy
$\operatorname{WF}^{\prime}(Q)\cap(\mathscr{R}_{+}\cup\mathscr{R}_{-})=\varnothing$.
Then for all $s\in\mathbb{R}$, $m\in\mathbb{R}$,

$QP(\lambda)\colon H^{m}(\partial M)\longrightarrow H^{s,m-1/2}(M).$ (3.12)

Before we prove the proposition we note the straightforward corollary.

######  Corollary 3.2.

Let $A_{+}$ be chosen as in (2.12). Then for $f\in H^{m}(\partial M)$,

$[H,A_{\pm}]P(\lambda)f\in H^{s,m+1/2}(M)$

and is microlocally trivial in a neighborhood of
$\mathscr{R}_{+}\cup\mathscr{R}_{-}$.

The statement about microlocal triviality here means that there is a
neighborhood $U$ of $\mathscr{R}_{-}\cup\mathscr{R}_{+}$ so that for any
$Q\in\Psi_{\mathrm{sc}}^{0,0}$ with $\operatorname{WF}^{\prime}Q\subset U$,
$Q[H,A_{\pm}]P(\lambda)f\in H^{M,N}$ for any $M,N$, i.e. is rapidly decaying.

We remark that when $H=H_{0}:=\Delta_{\mathbb{R}^{n}}$ is the flat Lapacian,
Proposition 3.1 follows without use of the structure of the Poisson kernel
used in the more general proof below. Let $P_{0}(\lambda)$ denote the Poisson
operator in the flat case. Assuming without loss of generality that
$\mathop{\rm supp}Q\subset\\{r\geq 1\\}$, one can make use of the generators
of rotation on $\mathbb{S}^{n-1},$ which form a family $\mathrm{Rot}:=\\{V\\}$
of vector fields commuting with $\Delta_{\mathbb{S}^{n-1}}$ and generating the
cotangent space at every point, in particular $f\in
H^{k}(\mathbb{S}^{n-1})\iff\mathrm{Rot}^{j}f\subset L^{2}(\mathbb{S}^{n-1})$
for all $j\leq k$. Let $V$ denote an arbitrary such vector field. Using
$[H_{0},V]=0$ in $r>0$, we claim that for $f\in H^{k}$,

$P_{0}(\lambda)Vf=VP_{0}(\lambda)f.$ (3.13)

This follows from the explicit formula for the Poisson operator and
integration by parts, but in using the results presented here one can argue as
follows. For $f\in C^{\infty}(\mathbb{S}^{n-1})$, the expansion for
$P_{0}(\lambda)f$ gives that $P_{0}(\lambda)Vf-VP_{0}(\lambda)f$ lies in
$\mathscr{X}^{s,\mathsf{l}_{+}}_{+}$, but we have
$(H_{0}-\lambda^{2})(P_{0}(\lambda)Vf-VP_{0}(\lambda)f)=0$, so since
$H_{0}-\lambda^{2}$ is injective on this domain, (3.13) follows. The condition
on $\operatorname{WF}^{\prime}(Q)$, means that
$p:=(\hat{z},\nu,\mu)\in\operatorname{WF}^{\prime}(Q)$ then $\mu\neq 0$, and
thus we can find $V$ with $\sigma_{1,1}(V)(p)\neq 0$. Since
$QV\in\Psi^{-\infty,1}$ is elliptic at $p$, if $f\in H^{k}$, since
$QP_{0}(\lambda)Vf\in H^{s,-1/2}$ we have by scattering elliptic regularity
that $QP_{0}(\lambda)\in H^{s,-1/2+k}$. We will use the structure of the
Schwartz kernel of the Poisson operator to deduce this same result for more
general Hamiltonians.

###### Proof of Proposition 3.1.

This will follow by using the well understood stucture of the integral kernel
of $P(\lambda)$, due in this generality to Melrose and Zworski [MeZw96]. As we
show below, using propagation of singularities, it will suffice to restrict
the microsupport of $Q$ to punctured neighborhoods of $\mathscr{R}_{-}$ over
small balls on the boundary, as follows. First pick a small coordinate patch
$V$ on $\partial M$ with coordinates $y$, and then a small neighborhood
$V^{\prime}$ of $\mathscr{R}_{-}\cap{}^{\mathrm{sc}}T^{*}_{V}M$; our $Q$ will
be supported in the punctured neighborhood
$U=V^{\prime}\setminus\mathscr{R}_{-}$. Concretely, this can be parametrized
as
$\\{(x,y,\nu,\mu):x,\nu^{2}+|\mu|^{2}_{h}-\lambda^{2}<\epsilon,0<|\nu+\lambda|<\epsilon\\}$
with $y$ in the coordinate patch.

For such $Q$, the operator $QP(\lambda)$ takes the form

$e^{-i\lambda\cos
d_{h}(y,y^{\prime})/x}\tilde{a}(x,y,y^{\prime})+e(x,y,y^{\prime})$

where $d_{h}$ is the distance function on $(\partial M,h)$, $a$ is smooth and
is supported where $y$ is in a deleted neighborhood of $y^{\prime}$, with
$y^{\prime}$ varying in the same chosen coordinate patch as $y$. Here $e$ is a
smooth function which is rapidly decaying as $x\to 0$ which we subsequently
ignore.

Since $\mathscr{F}(H^{s,m}(\mathbb{R}^{n}))=H^{m,s}(\mathbb{R}^{n})$, were we
working in Euclidean space, it would now suffice to prove that
$\mathscr{F}\circ QP(\lambda)\colon L^{2}\longrightarrow H^{-1/2+m,s}$. To use
this approach, we identify the open set $(0,1)_{x}\times V\subset M^{\circ}$
with a subset of Euclidean space by first choosing any local diffeomorphism
$\hat{z}=\hat{z}(y)$ to a coordinate patch on $\mathbb{S}^{n-1}$ and then
writing $r=1/x$, $r\hat{z}(y)=z$. Then $\mathscr{F}\circ QP(\lambda)$ has
Schwartz kernel

$\tilde{K}(\zeta,y^{\prime}):=\int e^{-iz\cdot\zeta}e^{-i\lambda|z|\cos
d_{h}(\hat{z},y^{\prime})}a(z,y^{\prime})\,dz.$

where $a(z,y)=\tilde{a}(x,y,y^{\prime})$ is a classical symbol in the
$z$-variable of order $0$, smooth in $y$. We claim that for $\chi\in
C^{\infty}(\mathbb{R}^{n})$ with $\chi(\zeta)=1$ on $|\zeta|<2\lambda$ and
$\mathop{\rm supp}\chi\subset\\{|\zeta|<4\\}$, then

$K(\zeta,y^{\prime}):=\int e^{-iz\cdot\zeta}e^{-i\lambda|z|\cos
d_{h}(\hat{z},y^{\prime})}a(z,y^{\prime})\chi(\zeta)\,dz,\quad a\in S^{0}$

satisfies

$\tilde{K}(\bullet,y^{\prime})-K(\bullet,y^{\prime})\in\mathscr{S}(\mathbb{R}^{n}_{\zeta}),$
(3.14)

smoothly in $y^{\prime}$, i.e. it is rapidly decreasing as $\zeta\to\infty$,
as follows from non-stationary phase.

We now view $K$ as a Fourier integral operator mapping from the sphere to the
dual $\mathbb{R}^{n}$, parametrized by $\zeta$, with $z$ playing the role of a
homogeneous phase variable. We use the mapping properties of FIOs found in
Chapter 25 of [Hvol4], in particular Theorem 25.3.8. Recall throughout that
$a$ is supported where $y$ is close to $y^{\prime}$ but not equal to it. Here
$K$ is an FIO of order $m=0+((n-1)+n-2n)/4=1/4$ with phase function
$\phi(\zeta,z,y)=-iz\cdot\zeta-i\lambda|z|\cos d_{h}(\hat{z},y^{\prime})$
where $z$ is the auxiliary variable. We compute that $\phi$ defines the
canonical relation

$C:=\\{(\zeta,z;y^{\prime},\mu^{\prime})\mid\zeta=\lambda\hat{z}\cos
d_{h}(\hat{z},y^{\prime})+\lambda\theta_{2}\sin
d_{h}(\hat{z},y^{\prime}),\quad\mu^{\prime}=\lambda|z|\theta_{1}\sin
d_{h}(\hat{z},y^{\prime})\\}$

where $\theta_{1}$ and $\theta_{2}$ are the initial, resp. final, fibre
coordinates of the unit length geodesic between $y^{\prime}$ and $\hat{z}$
with respect to the Riemannian metric $h$ on $\partial M$, interpreted as
being in $T^{*}_{y^{\prime}}\partial M$ in the first case and
$T^{*}_{y(\hat{z})}\partial M$ in the second. The dimension of the canonical
relation is $2n-1$,and it is parametrized by $z,y^{\prime}$ since for
$y,y^{\prime}$ sufficiently close (recall that $y$ parametrizes $\hat{z}$),
the $\theta_{i}$ are determined by $y,y^{\prime}$ and therefore by $\hat{z}$
and $y$ and the other variables $\mu^{\prime},\zeta$ are given in terms of
these. On the other hand the projection of $C$ to the ‘right’ factor
$T^{*}\partial M_{y^{\prime}}$ is a surjective submersion since
$\partial\mu^{\prime}/\partial\hat{z}$ is full rank for $y,y^{\prime}$
sufficiently close but not equal. Thus the pull back of the symplective form
on $T^{*}\partial M_{y^{\prime}}$ to $C$ is rank $2(n-1)$ or corank $1$.
Finally, at no point are either of the radial vector fields
$\mu^{\prime}D_{\mu^{\prime}}$ in $T^{*}\partial M\setminus\\{0\\}$ nor
$zD_{z}$ in $T^{*}\mathbb{R}^{n}_{\zeta}\setminus\\{0\\}$ tangential to $C$.

Therefore [Hvol4, Theorem 25.3.8] together with the standard argument (see
e.g. [Hvol4, Corollary 25.3.2]) in which one pre/post composes an FIO by
invertible elliptic pseudodifferential operators, give that $K$ maps
$H^{m}(\partial M)$ to $H^{m-1/2}(\mathbb{R}^{n}_{\zeta})$ continuously. (Note
that the theorem concludes a mapping to
$H^{m-1/2}_{loc}(\mathbb{R}^{n}_{\zeta})$ but $K$ is compactly supported in
$\zeta$.) Compact support of $\tilde{K}$ in $\zeta$ and (3.14) then give that
$\langle\zeta\rangle^{s}\tilde{K}$ maps $H^{m}(\partial M)$ to
$H^{m-1/2,s}(\mathbb{R}^{n}_{\zeta})$ for any $s$, whence composing with the
inverse Fourier transform gives (3.9) for $Q$ microlocalized near the radial
sets and locally over the boundary.

To treat general $Q$ with
$\operatorname{WF}^{\prime}(Q)\cap(\mathscr{R}_{+}\cup\mathscr{R}_{-})=\varnothing$,
we recall that the assertion of continuity in (3.12) can be microlocalized in
the sense that it will follow if, for all $q\in\operatorname{WF}^{\prime}(Q)$
there is $\tilde{Q}\in\Psi_{\mathrm{sc}}^{0,0}$ such that $q\in\mathop{\rm
Ell}_{0,0}(\tilde{Q})$ and for some $C>0$

$\|\tilde{Q}P(\lambda)f\|_{H^{s,m-1/2}(M)}\leq C\|f\|_{H^{m}(\partial M)}.$

for all $f\in H^{m}(\partial M)$. For $q\in\mathop{\rm
Ell}_{2,0}(H-\lambda^{2})$ this follows from the microlocal elliptic estimate
(see e.g. [GHSZ20, Prop. 3.3]

$\begin{gathered}\|\tilde{Q}P(\lambda)f\|_{H^{s,m-1/2}(M)}\lesssim\|(H-\lambda^{2})P(\lambda)f\|_{H^{s-2,m-1/2}(M)}+\|P(\lambda)f\|_{H^{-N,-M}})\\\
=\|P(\lambda)f\|_{H^{-N,-M}(M)}\lesssim\|f\|_{H^{m}(\partial
M)}\end{gathered}$

where $N,M$ are chosen sufficiently large and the final bound comes from
(3.10). (Here $\lesssim$ means there exists a $C>0$ such that the LHS is
bounded by $C$ times the RHS.) For
$q\in\Sigma_{2,0}(H-\lambda^{2})\setminus(\mathscr{R}_{+}\cup\mathscr{R}_{-})$,
then there is a bicharacteristic ray $\gamma$ in $\Sigma$ such that
$\gamma(0)=q$ and $\lim_{\sigma\to-\infty}\gamma(\sigma)\in\mathscr{R}_{-}$.
The first part of our proof implies the existence of a $\sigma_{0}<<0$ and
$Q\in\Psi_{\mathrm{sc}}^{0,0}$ such that $\gamma(\sigma_{0})\in\mathop{\rm
Ell}(Q)$ and $Q$ satisfies (3.12). On the other hand, by the standard
propagation of singularities estimate (see e.g. [GHSZ20, Prop. 3.4]), there
exists $\tilde{Q}\in\Psi_{\mathrm{sc}}^{0,0}$ such that

$\begin{gathered}\|\tilde{Q}P(\lambda)f\|_{H^{s,m-1/2}(M)}\phantom{\|\tilde{Q}P(\lambda)f\|_{H^{s,m-1/2}(M)}\|\tilde{Q}P(\lambda)f\|_{H^{s,m-1/2}(M)}\|\tilde{Q}P(\lambda)f\|}\\\
\lesssim\|QP(\lambda)f\|_{H^{s,m-1/2}(M)}+\|(H-\lambda^{2})P(\lambda)f\|_{H^{s-2,m-1/2}(M)}+\|P(\lambda)f\|_{H^{-N,-M}})\\\
=\|QP(\lambda)f\|_{H^{s,m-1/2}(M)}+\|P(\lambda)f\|_{H^{-N,-M}}\\\
\lesssim\|QP(\lambda)f\|_{H^{s,m-1/2}(M)}+\|f\|_{H^{m}}\end{gathered}$

where we choose $M,N$ sufficiently large and use (3.10) to estimate
$P(\lambda)f$ by $f$. Then, using
$\|QP(\lambda)f\|_{H^{s,m-1/2}(M)}\lesssim\|f\|_{H^{m}}$ gives the result for
$q$ on the characteristic set away from the radial sets, completing the proof.
∎

The next proposition is known to experts, see e.g. [Va98] and [HMV04, Section
3] in a slightly different context. It is implicit in older works, e.g.
[IK84]; no doubt many other references could be given. Notice that in (3.15),
$R(\lambda+i0)-R(\lambda-i0)$ is, up to a constant, the ‘spectral projection’
$dE(\lambda^{2})$ for the operator $H$. This operator is not an actual
projection, however; in fact, it is not bounded on any natural Hilbert space.
The proposition shows that a modification of this operator leads to a genuine
projection, that acts as the identity on the generalized eigenfunctions.

######  Proposition 3.3.

Let $u=P(\lambda)f$, for $f\in H^{k}(\partial M)$, k = 0, 1, 2, …. Then we can
express

$u=\Big{(}R(\lambda+i0)-R(\lambda-i0)\Big{)}[H,A_{+}]u,$ (3.15)

where, by Corollary 3.2, $[H,A_{+}]u$ is in $H^{s,k+1/2}(M)$, and microlocally
trivial near $\mathscr{R}_{+}\cup\mathscr{R}_{-}$.

Conversely, let $w\in H^{0,k+1/2}(M)$ be such that $w$ is microlocally trivial
in a neighbourhood of $\mathscr{R}_{-}\cup\mathscr{R}_{+}$. Then, defining $u$
by

$u:=\Big{(}R(\lambda+i0)-R(\lambda-i0)\Big{)}w,$

$u$ is the image under $P(\lambda)$ of a function $f\in
H^{k}(\mathbb{S}^{n-1})$, namely the function $f=(i/2\lambda)P(\lambda)^{*}w$.

###### Proof.

First we show (3.15). Using the identity $(H-\lambda^{2})u=0$ we evaluate the
right hand side.

$\displaystyle\Big{(}R(\lambda+i0)-R(\lambda-i0)\Big{)}[H,A_{+}]u$
$\displaystyle=R(\lambda+i0)[H-\lambda^{2},A_{+}]u+R(\lambda-i0)[H-\lambda^{2},A_{-}]u$
$\displaystyle=R(\lambda+i0)(H-\lambda^{2})A_{+}u+R(\lambda-i0)(H-\lambda^{2})A_{-}u.$

We next claim that $A_{+}u$ is in the space $\mathscr{X}^{s,\mathsf{l}_{+}}$
(for any $s$). To see this, we use (3.6), which shows that $P(\lambda)f$ is in
$H^{s,-1/2-\epsilon}$ globally, and Proposition 3.1, which shows that it is in
$H^{s,-1/2}$ microlocally away from $\mathscr{R}_{+}$, together with the
definition of the operator $A_{+}$ which microlocalizes away from
$\mathscr{R}_{-}$. Similarly, $A_{-}u$ is in the space
$\mathscr{X}^{s,\mathsf{l}_{-}}$. On these spaces, $R(\lambda+i0)$,
respectively $R(\lambda-i0)$ is a left inverse to $H-\lambda^{2}$. It follows
that the RHS is equal to $A_{+}u+A_{-}u=u$, proving the identity (3.15).

We show the converse first for $k=0$. This follows immediately from (3.5) and
(3.8) since the assumption on $w$ implies that, for $\delta,\delta^{\prime}$
in the definition of $\mathsf{l}_{\pm}$ sufficiently small, $w\in
H^{s,\mathsf{l}_{max}+1}$. For any positive integer $k$, we write

$P(\lambda)^{*}f=P(\lambda)^{*}Qf+P(\lambda)^{*}(\operatorname{Id}-Q)f,$

where $Q$ is microlocally equal to the identity on the microsupport of $w$,
and microlocally trivial near the radial sets. Then $(\operatorname{Id}-Q)f$
is Schwartz, so using (3.11), $P(\lambda)^{*}(\operatorname{Id}-Q)f$ is
$C^{\infty}$. For the other term, the adjoint of Proposition 3.1 shows that
$P(\lambda)^{*}Qf$ is in $H^{k}(\partial M)$. ∎

We now begin our analysis of the incoming/outgoing data of distributions in
the image of $R(\lambda\pm i0)$. The main improvement in the regularity of
this incoming/outgoing data comes from a combination of the mapping properties
of the Poisson operator and its adjoint, together with the reproducing formula
(3.15).

######  Proposition 3.4.

Let $s\geq 0$, and assume $F$ be in
$\mathscr{Y}_{+}^{s,\mathsf{l}_{\max}+1;1,k-1}$, $k\geq 2$, where
$\mathsf{l}_{\max}$ is defined in (3.7). Assume that $\epsilon$ satisfies

$0<2\epsilon<\min(1,\gamma-1),$ (3.16)

where $\gamma$ is as in (1.6). Then $u_{+}:=R(\lambda+i0)F$ is such that, in a
collar neighbourhood of the boundary $\partial M\times(R,\infty)_{r}$, the
limit

$\mathscr{L}(-\lambda)u_{+}:=\lim_{r\to\infty}r^{(n-1)/2}e^{-i\lambda
r}u_{+}(r,\cdot)$

exists in $H^{k-2}(\partial M)$, with estimates

$\begin{gathered}\Big{\|}\mathscr{L}(-\lambda)u_{+}\Big{\|}_{H^{k-2}(\partial
M)}\leq C\|F\|_{\mathscr{Y}_{+}^{s,\mathsf{l}_{\max}+1;1,k-1}},\\\
\big{\|}r^{(n-1)/2}e^{-i\lambda
r}u_{+}(r,\cdot)-\mathscr{L}(-\lambda)u_{+}\big{\|}_{H^{k-2}(\partial
M)}=O(r^{-\epsilon^{\prime}})\end{gathered}$ (3.17)

for some $\epsilon^{\prime}>0$. Similarly, if
$F\in\mathscr{Y}_{-}^{s,\mathsf{l}_{\max}+1;1,k-1}$, $k\geq 2$, then
$u_{-}:=R(\lambda-i0)F$ is such that the limit

$\mathscr{L}(\lambda)u_{-}:=\lim_{r\to\infty}r^{(n-1)/2}e^{+i\lambda
r}u_{-}(r,\cdot)$

exists in $H^{k-2}(\partial M)$, with estimates

$\begin{gathered}\Big{\|}\mathscr{L}(\lambda)u_{-}\Big{\|}_{H^{k-2}(\partial
M)}\leq C\|F\|_{\mathscr{Y}_{-}^{s,\mathsf{l}_{\max}+1;1,k-1}},\\\
\big{\|}r^{(n-1)/2}e^{+i\lambda
r}u_{-}(r,\cdot)-\mathscr{L}(\lambda)u_{-}\big{\|}_{H^{k-2}(\partial
M)}=O(r^{-\epsilon^{\prime}})\end{gathered}$ (3.18)

for some $\epsilon^{\prime}>0$.

###### Proof.

We follow the argument of Section 4 of [GHSZ20], which in turn is based on
[Me94]. We do this only for $u_{+}$, as the argument is similar for $u_{-}$.
From now on we denote $u_{+}$ by $u$.

We use the microlocalizing operators $Q_{\pm}$ from Section 2.3, and complete
these to a partition of unity, $\operatorname{Id}=Q_{+}+Q_{-}+Q_{3}$. Thus,
$Q_{+}$ is microsupported close to $\mathscr{R}_{+}$, $Q_{-}$ is
microsupported close to $\mathscr{R}_{-}$, and $Q_{3}$ is microsupported away
from $\mathscr{R}_{+}\cup\mathscr{R}_{-}$. Assuming that the microsupports of
$Q_{\pm}$ are sufficiently close to $\mathscr{R}_{\pm}$, the assumption on $F$
implies that $Q_{\pm}F$ are in the space
$\mathscr{Y}_{+}^{s,1/2+\epsilon;1,k}$, while $Q_{3}F$ is in
$\mathscr{Y}_{+}^{s,1/2;1,k}$.

We notice that all of the commutators $[H,Q_{\bullet}]$ have boundary order
$1$ and microlocally vanish near $\mathscr{R}_{+}\cup\mathscr{R}_{-}$, so
$r^{2}[H,Q_{\bullet}]$ belongs in both modules, $\mathscr{M}_{+}$ and
$\mathscr{M}_{-}$. We write $u_{1}=Q_{+}u$, $u_{2}=Q_{-}u$, and
$u_{3}=Q_{3}u$.

Write $\tilde{u}_{1}=\chi(r)r^{(n-1)/2}e^{-i\lambda r}u_{1}$, where $\chi$ is
supported in $r>R$ and identically $1$ near $r\geq 2R$. Our first goal is to
show that $\tilde{u}_{1}(r,y)$ has a limit $b(y)$ in $H^{k-2}(\partial M)$,
and that $\tilde{u}_{1}(r,y)-b(y)=O_{H^{k-2}(\partial
M)}(r^{-\epsilon^{\prime}})$ as $r\to\infty$. To do this, we write the
operator $H-\lambda^{2}$ in the form

$H-\lambda^{2}=D_{r}^{2}-i(n-1)r^{-1}D_{r}+r^{-2}L+r^{-2}\tilde{L}+V-\lambda^{2},$
(3.19)

where $L$ is a second order differential operator involving only tangential
$D_{y_{j}}$ derivatives, and $\tilde{L}$ is a scattering differential operator
of order $(1,0)$. Since $(H-\lambda^{2})u_{+}=F$, we obtain

$\Big{(}D_{r}+\lambda\Big{)}\Big{(}D_{r}-\lambda-\frac{i(n-1)}{2r}\Big{)}u_{1}\\\
=Q_{+}F+[H,Q_{+}]u_{1}+\frac{i(n-1)}{2r^{2}}(r(D_{r}-\lambda))u_{1}+\Big{(}\frac{n-1}{2r^{2}}-r^{-2}L-r^{-2}\tilde{L}-V\Big{)}u_{1}.$
(3.20)

According to (2.15), $u_{+}$ is in the space
$\mathscr{X}_{+}^{s+2,\mathsf{l}_{+};1,k-1}$. Hence
$u_{1}\in\mathscr{Y}_{+}^{s+2,-1/2-\epsilon;1,k-1}$. We claim that each term
on the right side of (3.20) is in $\mathscr{Y}_{+}^{0,1/2+\epsilon;0,k-2}$.
For the $Q_{+}F$ term this has already been noted. For the second term, we use
the fact that $[H,Q_{+}]$ is $r^{-2}$ times a small module element, so this
term is in $\mathscr{Y}_{+}^{s+2,3/2-\epsilon;0,k-2}$, which is contained in
$\mathscr{Y}_{+}^{s+2,1/2+\epsilon;0,k-2}$ since $\epsilon<1/2$ according to
(3.16). For the third, resp. fifth, term on the RHS it follows similarly from
the fact that $u_{+}$ has large module regularity of order $1$, resp. small
module regularity of order two. For the fourth term this is obvious due to the
$r^{-2}$ prefactor, while for the sixth term we only need small module
regularity of order 1. Finally, for the $V$ term this follows from the fact
that $V$ satisfies the conormal estimates (1.6), with $\gamma$ satisfying
(3.16).

Now observe the operator $D_{r}+\lambda$ is elliptic everywhere on the
microsupport of $u_{1}$, since $Q_{+}$ is microsupported near
$\mathscr{R}_{+}$. Thus we may invert this operator microlocally, obtaining

$\begin{gathered}\Big{(}D_{r}-\lambda-\frac{i(n-1)}{2r}\Big{)}u_{1}\in\mathscr{Y}^{0,1/2+\epsilon;0,k-2},\\\
\Big{\|}\Big{(}D_{r}-\lambda-\frac{i(n-1)}{2r}\Big{)}u_{1}\Big{\|}_{\mathscr{Y}^{0,1/2+\epsilon;0,k-2}}\leq
C\|F\|_{\mathscr{Y}_{+}^{s,\mathsf{l}_{\max}+1;1,k-1}}.\end{gathered}$ (3.21)

Now, observing that

$D_{r}\tilde{u}_{1}=\chi(r)r^{(n-1)/2}e^{-i\lambda
r}\Big{(}D_{r}-\lambda-\frac{i(n-1)}{2r}\Big{)}u_{1}+(D_{r}\chi)r^{(n-1)/2}e^{-i\lambda
r}u_{1},$

we find that $D_{r}\tilde{u}_{1}\in H^{0,1/2+\epsilon-(n-1)/2;0,k-2}$, or
equivalently

$D_{r}\tilde{u}_{1}\in
r^{n/2-1-\epsilon}L^{2}\big{(}([R,\infty),r^{n-1}dr);H^{k-2}(\partial
M)\big{)},$

with a corresponding norm estimate (where we used the support property of
$D_{r}\chi$ for the inclusion in $H^{0,1/2+\epsilon-(n-1)/2;0,k}$ of the
second term). We can express this with respect to the measure $dr$ as follows:

$\begin{gathered}D_{r}\tilde{u}_{1}\in
r^{-1/2-\epsilon}L^{2}\big{(}([R,\infty),dr);H^{k-2}(\partial
M)\big{)}\subseteq
r^{-\epsilon^{\prime}}L^{1}\big{(}([R,\infty),dr);H^{k-2}(\partial
M)\big{)},\\\
\Big{\|}r^{\epsilon^{\prime}}D_{r}\tilde{u}_{1}\Big{\|}_{L^{1}\big{(}([R,\infty),dr);H^{k-2}(\partial
M)\big{)}}\leq
C\|F\|_{\mathscr{Y}_{+}^{s,\mathsf{l}_{\max}+1;1,k-1}}.\end{gathered}$ (3.22)

We note that, since $\tilde{u}_{1}$ is locally $H^{1}$ in $r$ with values in
$H^{k-2}(\partial M)$, it is in fact continuous in $r$ with values
$H^{k-2}(\partial M)$. By (3.22), we can integrate to infinity to find

$b(y)=\int_{R}^{\infty}\partial_{r^{\prime}}\tilde{u}_{1}(r^{\prime},y)\,dr^{\prime},\quad\|b\|_{H^{k-2}(\partial
M)}\leq C\|F\|_{\mathscr{Y}_{+}^{s,\mathsf{l}_{\max}+1;1,k-1}}$

is well-defined as an element of $H^{k-2}(\partial M)$. Moreover,

$\begin{gathered}\tilde{u}_{1}(r,y)-b(y)=-\int_{r}^{\infty}\partial_{r^{\prime}}\tilde{u}_{1}(r^{\prime},y)\,dr^{\prime},\\\
\Big{\|}\tilde{u}_{1}(r,y)-b(y)\Big{\|}_{H^{k-2}(\partial M)}\leq
Cr^{-\epsilon^{\prime}}\Big{\|}r^{\epsilon^{\prime}}D_{r}\tilde{u}_{1}\Big{\|}_{L^{1}\big{(}([R,\infty),dr);H^{k-2}(\partial
M)\big{)}}\\\ \leq
Cr^{-\epsilon^{\prime}}\|F\|_{\mathscr{Y}_{+}^{s,\mathsf{l}_{\max}+1;1,k-1}}.\end{gathered}$
(3.23)

To treat $u_{2}$, we apply a similar argument with the role of
$\mathscr{R}_{+}$ and $\mathscr{R}_{-}$ interchanged. In this case we define
$\tilde{u}_{2}=\chi(r)r^{(n-1)/2}e^{+i\lambda r}u_{2}$ and write (3.20) in the
form

$\Big{(}D_{r}-\lambda\Big{)}\Big{(}D_{r}+\lambda-\frac{i(n-1)}{2r}\Big{)}u_{2}\\\
=Q_{-}F+[H,Q_{-}]u_{2}+\frac{i(n-1)}{2r^{2}}(r(D_{r}+\lambda))u_{+}+\Big{(}\frac{n-1}{2r^{2}}-r^{-2}L-r^{-2}\tilde{L}-V\Big{)}u_{+}.$
(3.24)

Folllowing the same reasoning as above, we find that $\tilde{u}_{2}$ has a
limit as $r\to\infty$, with the same $O(r^{-\epsilon^{\prime}})$ of
convergence. But in case (ii), the mapping property of the _outgoing_
resolvent near the _incoming_ radial set (the inverse mapping to (2.15), see
[GHSZ20, Theorem 3.1]) shows that $u_{2}$ is actually in
$H^{s,-1/2+\epsilon}$, i.e. the spatial order is above threshold, since
$\mathsf{l}_{+}=-1/2+\epsilon$ on the microsupport of $Q_{-}$. Therefore,
since we know that $\tilde{u}_{2}$ has a limit as $r\to\infty$, it must be
zero (were the limit nonzero, then $u$ would fail to lie in the space
$H^{s,-1/2+\epsilon}$ for $\epsilon>0$). It follows that the limit of
$\chi(r)r^{(n-1)/2}e^{-i\lambda r}u_{2}=e^{-2i\lambda r}\tilde{u}_{2}$ also
has a zero limit.

It remains to discuss $u_{3}$. We claim that $u_{3}$ is an element of
$\mathscr{Y}_{+}^{s+2,1/2-\epsilon;1,k-2}$. We argue separately in the
microlocal regions (i) near the characteristic variety $\Sigma$, and (ii) away
from $\Sigma$. The mapping property (2.15) shows that $u$ is in
$\mathscr{Y}_{+}^{s+2,-1/2-\epsilon;1,k-1}$. In region (i), since $u_{3}$ is
microsupported away from the radial sets, the small module $\mathscr{N}$ is
elliptic there, so we can trade one order of small module regularity for a
gain of one spatial order. In region (ii), we already have
$u\in\mathscr{Y}_{+}^{s+2,+1/2-\epsilon;1,k-1}$ since $H-\lambda^{2}$ is
elliptic there and we incur no loss in the spatial regularity from applying
the resolvent in this region.

Now using the $\mathscr{M}_{+}$ module regularity, and replacing the
differential order with zero, we see that

$r\big{(}D_{r}-\lambda-i\frac{n-1}{2r}\big{)}u_{3}\in
H^{0,1/2-\epsilon;0,k-2},$

that is,

$\big{(}D_{r}-\lambda-i\frac{n-1}{2r}\big{)}u_{3}\in
H^{0,3/2-\epsilon;0,k-2},$

which is stronger than (3.21) as $\epsilon<1/2$. We can thus apply the same
reasoning as for $u_{1}$ to obtain a limit for $\chi(r)r^{(n-1)/2}e^{-i\lambda
r}u_{3}$ (which is necessarily zero, for the same reason as for $u_{2}$), with
the same rate of convergence.

The estimates in (3.17) are obtained by adding the contributions from $u_{1}$,
$u_{2}$ and $u_{3}$. The estimates (3.18) are obtained similarly.

∎

######  Remark 3.1.

From the proof above we see the following: if $Q\in\Psi_{\mathrm{sc}}^{0,0}$
is microsupported away from $\mathscr{R}_{+}$, then
$\mathscr{L}(-\lambda)(Qu_{+})=0$. Similarly, if
$Q^{\prime}\in\Psi_{\mathrm{sc}}^{0,0}$ is microsupported away from
$\mathscr{R}_{-}$, then $\mathscr{L}(\lambda)(Q^{\prime}u_{-})=0$.

######  Proposition 3.5.

Let $f\in H^{k}(\partial M)$, with $k\in\mathbb{N}\geq 2$, and let
$u=P(\lambda)f$. Then $A_{-}u$ is such that the limit

$\mathscr{L}(\lambda)(A_{-}u)=\lim_{r\to\infty}r^{(n-1)/2}e^{i\lambda
r}(A_{-}u)(r,\cdot)$ (3.25)

exists in $H^{k-2}(\partial M)$. Moreover, this limit is $f$. Similarly, the
limit

$\mathscr{L}(-\lambda)(A_{+}u)=\lim_{r\to\infty}r^{(n-1)/2}e^{-i\lambda
r}(A_{+}u)(r,\cdot)$ (3.26)

exists in $H^{k-2}(\partial M)$ as $r\to\infty$. Moreover, this limit is in
$H^{k}(\partial M)$.

Both limits are achieved with an $O(r^{-\epsilon^{\prime}})$ convergence rate,
as in Proposition 3.4.

###### Proof.

As we have already seen, in the proof of Proposition 3.3, $A_{-}u$ is the
incoming resolvent applied to $[H,A_{-}]u$, and $A_{+}u$ is the outgoing
resolvent applied to $[H,A_{+}]u=-[H,A_{-}]u$. We have also seen that for
$f\in H^{k}(\partial M)$, $[H,A_{\pm}]u$ is in the space $H^{s,k+1/2}(M)$ and
microlocally trivial near $\mathscr{R}_{-}\cup\mathscr{R}_{+}$ (Corollary
3.2), and therefore in the module regularity space
$\mathscr{Y}_{\pm}^{s,\mathsf{l}_{\max}+1;k,0}$. Therefore, the existence of
the limits (3.25) and (3.26) in $H^{k-2}(\partial M)$ follows from Proposition
3.4.

To see that the limit for $A_{-}u$ is $f$, we first note that this is true for
all $f\in C^{\infty}(\partial M)$; this is a defining property of the Poisson
operator, and is easily obtained from the Schwartz kernel of $P(\lambda)$
using the method of stationary phase. Now for arbitrary $f\in H^{k}(\partial
M)$, we choose a sequence of smooth $f_{j}$ converging to $f$ in
$H^{k}(\partial M)$. Let $u_{j}=P(\lambda)f_{j}$; then $[H,A_{-}]u_{j}$
converges to $[H,A_{-}]u$ in $H^{s,k+1/2}(M)$ (Corollary 3.2 again), and
therefore in $\mathscr{Y}_{-}^{s,\mathsf{l}_{\max}+1;1,k-1}$. As above, we
have $A_{-}u_{j}=R(\lambda-i0)[H,A_{-}]u_{j}$. Using (3.17) in Proposition
3.4, $\lim_{j\to\infty}\mathscr{L}(\lambda)A_{-}u_{j}$ exists in
$H^{k-2}(\partial M)$ and is $\mathscr{L}(\lambda)A_{-}u$. On the other hand,
since the $f_{j}$ are smooth, $\mathscr{L}(\lambda)A_{-}u_{j}$ is precisely
$f_{j}$, which converge to $f$ in $H^{k}$, and so _a fortiori_ in $H^{k-2}$,
showing that $\mathscr{L}(\lambda)A_{-}u=f$.

To obtain the result for $A_{+}u$, we could just appeal to the main result of
[MeZw96] that says that the limit is the scattering matrix $S(\lambda)$
applied to $f$, and since $S(\lambda)$ is an FIO of order zero, then
$S(\lambda)f$ is in $H^{k}$. However, we prefer a direct argument. Using the
formula (3.5), we see that $u$ can be expressed as

$u=\frac{1}{2\pi}P(-\lambda)P(-\lambda)^{*}i[H,A_{+}]u.$

That is, $u$ is equal to $P(-\lambda)f^{\prime}$, where
$f^{\prime}=i(2\pi)^{-1}P(-\lambda)^{*}[H,A_{+}]u$. Now, arguing as in the
proof of the converse to Proposition 3.3, but for $P(-\lambda)$ instead of
$P(\lambda)$, we see that $f^{\prime}$ is in $H^{k}(\partial M)$. Now applying
the argument in the first half of this proof, with signs switched, we conclude
that the limit (3.26) exists in $H^{k-2}(\partial M)$ and is equal to
$f^{\prime}\in H^{k}(\partial M)$.

The statement about the convergence rate is shown by applying Proposition 3.4
to $F=[H,A_{\pm}]u$. ∎

######  Remark 3.2.

The previous proof shows that the operators $\mathscr{L}(\pm\lambda)\circ
R(\lambda\mp i0)$ in Proposition 3.4 coincide with $P(\pm\lambda)^{*}$.

We combine the previous two propositions to obtain

######  Proposition 3.6.

The limits in Proposition 3.4 lie in $H^{k}(\partial M)$.

###### Proof.

In the notation of Proposition 3.4, we have
$F\in\mathscr{Y}^{s,1/2+\epsilon;0,k}$, and therefore it lies in both of the
variable order module regularity spaces
$\mathscr{Y}^{s,\mathsf{l}_{\pm}+1;0,k}$. It follows that
$u:=u_{+}-u_{-}=(R(\lambda+i0)-R(\lambda-i0))F$ lies in
$\mathscr{X}^{s+2,\mathsf{l}_{\pm};0,k}$. In particular, $u$ is in
$H^{s+2,k-1/2}$ microlocally away from small neighbourhoods of
$\mathscr{R}_{+}$ and $\mathscr{R}_{-}$. (Notice that small module regularity
is as good as full module regularity away from the radial sets.) Applying
Proposition 3.3, with $k=0$, we have
$u=(R(\lambda+i0)-R(\lambda-i0))[H,A_{+}]u$. But then, since $[H,A_{+}]u$ is
in $H^{s+2,k-1/2}$, and microlocalized away from $\mathscr{R}_{+}$ and
$\mathscr{R}_{-}$, Proposition 3.3 shows that $u=P(\lambda)f$ with $f\in
H^{k}(\partial M)$.

Consider the limit $\mathscr{L}(-\lambda)u_{+}$. We claim that is the same as
$\mathscr{L}(-\lambda)A_{+}u$. In fact, the difference is

$\begin{gathered}\mathscr{L}(-\lambda)(u_{+}-A_{+}u)\\\
=\mathscr{L}(-\lambda)(u_{+}-A_{+}u_{+}-A_{+}u_{-})\\\
=\mathscr{L}(-\lambda)(A_{-}u_{-}-A_{+}u_{-}).\end{gathered}$

Using Remark 3.1, we see that $\mathscr{L}(-\lambda)A_{-}u_{+}=0$ and
$\mathscr{L}(\lambda)A_{+}u_{-}=0$, which implies that
$\mathscr{L}(-\lambda)A_{+}u_{-}=0$. Thus
$\mathscr{L}(-\lambda)u_{+}=\mathscr{L}(-\lambda)A_{+}u$. Similarly,
$\mathscr{L}(\lambda)u_{-}=\mathscr{L}(\lambda)A_{-}u$. The conclusion then
follows from applying Proposition 3.5 to $u$. ∎

##  4\. Proof of the main theorem

We now elaborate on the construction and regularity of nonlinear Helmholtz
eigenfunction $u$, whose asymptotic behavior is the subject of the main
theorem. We begin by discussing (linear) generalized eigenfunctions.

Given $f\in H^{k}(\partial M)$, we let $u_{0}=P(\lambda)f$ and decompose into

$u_{0}=u_{-}+u_{+},\qquad u_{\pm}=A_{\pm}u_{0},$

where $A_{\pm}$ are as in (2.12). According to Proposition 3.5, we have

$u_{0}=u_{-}+u_{+}=r^{-(n-1)/2}\Big{(}e^{-i\lambda r}f(\omega)+e^{+i\lambda
r}b_{0}(\omega)+O_{H^{k-2}}(r^{-\epsilon^{\prime}})\Big{)},$ (4.1)

where $b_{0}$ is in $H^{k}(\partial M)$.

To address the nonlinear problem, following [GHSZ20], we obtain a nonlinear
Helmholtz eigenfunction $u$ satisfying

$\begin{gathered}u=u_{-}+w,\quad u\mbox{ solves \eqref{eq:equation}},\\\
u_{-}\in\mathscr{X}_{-}^{s,-1/2-\epsilon;1,k-1}\quad
w\in\mathscr{X}_{+}^{s,-1/2-\epsilon;1,k-1}\\\
w=u_{+}+R(\lambda+i0)N[u_{-}+w].\end{gathered}$

Moreover, if the nonlinearity $N$ involves products of degree $p$, then, as
described in detail in Section 4.2 of [GHSZ20],

$N[u_{-}+w]\in H_{+}^{s,\ell^{\prime};1,k-1}$

provided $\ell^{\prime}\leq p(-1/2-\delta)+\frac{(p-1)n}{2}-1$, whence the
bound for $p$ in (1.5) allows for $\ell^{\prime}=1/2+\epsilon$ for $\epsilon$
sufficiently small. Thus $F=N[u_{-}+w]$ satisfies the assumptions of
Proposition 3.4. We conclude that

$w-u_{+}=r^{-(n-1)/2}e^{i\lambda
r}\Big{(}b_{1}(y)+O_{H^{k-2}}(r^{-\epsilon})\Big{)},\quad b_{1}\in
H^{k}(\partial M),$ (4.2)

using Proposition 3.6 for the regularity of $b_{1}$. Combining
$u=u_{0}+(w-u_{+})$ using (4.1) and (4.2) proves the asymptotic behavior
stated in the main theorem, with $b=b_{0}+b_{1}\in H^{k}(\partial M)$.

Uniqueness follows from the same considerations as in [GHSZ20]. Namely, given
$u_{-}\in\mathscr{X}_{-}^{s,-1/2-\epsilon;k,0}$, as the function $w$ above is
produced using a contraction mapping, $w=u-u_{-}$ is uniquely determined in a
small ball in $\mathscr{X}_{+}^{s,-1/2-\epsilon;k,0}$.

## References

  * [GHSZ20] J. Gell-Redman, A. Hassell, J. Shapiro, and J. Zhang. Existence and asymptotics of Nonlinear Helmholtz eigenfunctions, to appear, SIAM J. Math. Anal.
  * [HV99] A. Hassell, A. Vasy, The spectral projections and the resolvent for scattering metrics. J. Anal. Math. 79 (1999), 241–298.
  * [HMV04] A. Hassell, R. B. Melrose, A. Vasy, Spectral and scattering theory for symbolic potentials of order zero, Adv. Math. 181 (2004), no. 1, 1–87.
  * [Hvol4] L. Hörmander, The analysis of linear partial differential operators IV, Springer, 1985.
  * [Ik60] T. Ikebe, Eigenfunction expansions associated with the Schrödinger operators and their applications to scattering theory. Arch. Rational Mech. Anal. 5 (1960), 1–34 (1960).
  * [Is82] H. Isozaki, On the generalized Fourier transforms associated with Schrödinger operators with long-range perturbations. J. Reine Angew. Math. 337 (1982), 18–67.
  * [IK84] H. Isozaki, H. Kitada, Microlocal resolvent estimates for 2-body Schrödinger operators. J. Funct. Anal. 57 (1984), no. 3, 270–300.
  * [Me94] R. Melrose. Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces. In Spectral and scattering theory (Sanda, 1992), volume 161 of Lecture notes in Pure and Appl. Math., 85–120, Dekker, New York, 1994.
  * [MeZw96] R. Melrose and M. Zworski. Scattering metrics and geodesic flow at infinity. Invent. Math. 123(3), (1996), 389–436.
  * [Va98] A. Vasy, Geometric scattering theory for long-range potentials and metrics. Internat. Math. Res. Notices 1998, no. 6, 285–315.
  * [Va13] A. Vasy, Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov). Invent. Math. 194 (2013), no. 2, 381–513.
  * [Va18] A. Vasy.  A minicourse on microlocal analysis for wave propagation.  In Asymptotic analysis in general relativity, volume 443 of London Math. Soc. Lecture Note Ser., pages 219–374. Cambridge Univ. Press, Cambridge, 2018.