# Dispersive Estimates for Maxwell’s equations in the exterior of a sphere

Yan-long Fang , Alden Waters Department of Mathematics, University College
London, WC1H 0AY, United Kingdom<EMAIL_ADDRESS>for
Analysis, Department of Mathematics, Leibniz University Hannover, Welfengarten
1, 30167 Hannover, Germany<EMAIL_ADDRESS>
###### Abstract

The goal of this article is to establish general principles for high frequency
dispersive estimates for the $p$-form Laplacian with relative boundary
conditions on co-closed forms. In dimension $3$ for the case $p=1$, we show
that the propagator corresponding to Maxwell’s equations on compactly
supported co-closed forms satisfies the same dispersive estimates as in
$\mathbb{R}^{3}$ for the corresponding wave equation in the exterior of a
ball- but only for certain polarizations. In particular we show that some, but
not all, polarizations of electromagnetic waves scatter at the same rate as
the usual wave equation and this rate is not expected to hold in general. The
Dirichlet Laplacian wave equation $L^{1}-L^{\infty}$ scattering rate does not
hold true for the $1-$form Laplacian with relative boundary conditions by
itself in the exterior of a sphere, or any smooth obstacle for that matter. We
also do not expect it to hold in general for Maxwell’s equations because of
the presence of $L^{2}$ harmonic $1$ forms.

keywords: Maxwell’s equations, dispersive estimates

MSC classes: 35R01, 35R30, 35L20, 58J45, 35A22

##  1 Introduction

Determination of decay rates for Maxwell’s equations with time independent
metrics is otherwise known as the scattering problem. Maxwell’s equations are
vector-valued and in the presence of an obstacle mixed boundary conditions
make it so they are not reducible to standard evolution equations even in the
constant coefficient case. The Maxwell scattering problem is therefore more
difficult than the wave or Helmholtz problem as the presence of a nontrivial
topology can amplify or spread the initial pulse of waves. This type of
geometric analysis is important because Maxwell’s equations are fundamental in
electromagnetism, astrophysics, and nanotechnology. Here we are interested in
the specific question of the rate of decay of these waves in time around
obstacles and using geometric scattering theory to determine this rate.

To see some of the difficulties involved, we consider a simple example.
Imagine you hold a flashlight in front of a ball in a darkly lit room. The
light from the shadow of the ball creates a pattern on the wall. At the center
of this pattern is a bright spot known as the Poisson-Arago spot. The property
of a wave to spread out as time goes by while keeping its energy unchanged is
called dispersion. In three dimensions it is claimed that light bends
effectively at the same rate as the free space [22] (Euclidean, no obstacle
case). Therefore there is no change in what is known as dispersive estimates.
In higher dimensions [22], the analogue of the Poisson-Arago spot changes the
rate at which the initial pulse decays. There is no longer a direct comparison
to the free-space dispersion, that is, the rate at which waves scatter from
their initial state without the ball in the room. The authors of [22] use this
example and the wave equation to state their claims about light scattering
when they consider a single convex obstacle. The wave equation more
appropriately describes sound waves. But this important work [22] is the first
to consider the case of dispersion with an obstacle present, in this case a
sphere, and prove a non-local estimate.

Light is not governed by the wave equation, but by Maxwell’s equations.
However surprisingly for electromagnetic radiation the global exterior
dispersive decay rate remains unknown for in the presence of obstacles. This
decay rate is unknown even just for a sphere. In our case, we show using Mie
scattering that a slightly weaker $L^{1}\textendash L^{\infty}$ dispersive
estimate still holds uniformly in the exterior of a sphere for Maxwell’s
propagator. If the obstacle, any obstacle in fact, has a charge, corresponding
to the electric field no longer being divergence free, then we provide counter
examples to show this rate no longer holds. We also show using spectral theory
for the Maxwell and 1-form Laplacian that the bottom of the spectrum, near
$\lambda=0$, creates counter-examples to the usual dispersive estimates. These
phenomena have their roots in the Hodge-decomposition theorem.

Figure 1: Three obstacles in $\mathbb{R}^{3}$ with complement $M$, each object
scatters at a different rate, but the ball and the cuboid scatter the same to
leading order, while the torus does not

The study of dispersive estimates has a long history in the function case.
Exterior dispersion was studied for the Schrödinger and wave case in [22, 23].
While energy estimates have been done for Maxwell in a variety of mixed spaced
settings, genuine $L^{1}-L^{\infty}$ estimates have not been examined. In the
noncompact case in general geometric settings, $L^{r}-L^{q}$ for $r>1$
estimates for $p$-forms has been examined in the work of [13]. Resolvent
estimates which are related for $p$-forms can be found in [14, 15] and the
often used here [41]. In particular for the counter examples we use the
presence of certain low energy resolvent terms first found in [14] and given
precision in [41]. The technique of constructing a generalized eigenfunction
is similar to those asymptotic expansions used to establish general scattering
rates for functions in [16]. Related literature in the case of convex bodies
for the Dirichlet Laplacian includes [44, 5, 2, 18, 21] and Mie scattering in
dimension 2 [12]. The estimates here could be useful for establishing
existence and uniqueness for semi-linear Maxwell’s equations as done in the
Dirichlet function case of the sphere and convex bodies [7] and [39],
respectively. We mention that recently Strichartz estimates have been
established for Lipschitz domains for the interior Maxwell problem in [4] also
using the language of $p$-forms. We hope that the construction of the kernel
using spectral theory could be useful in the related examination of Price’s
law [19, 31] in the future.

##  2 Precise setup and notations

We consider the space $\mathbb{R}^{d}$ and a single smooth obstacle
$\mathcal{O}$, or the union of any smooth non-overlapping obstacles, which is
removed from the free space. The set $\mathbb{R}^{d}\setminus\mathcal{O}=M$.
We denote the differentials $\mathrm{d}:C^{\infty}_{0}(M;\Lambda T^{*}M)\to
C^{\infty}_{0}(M;\Lambda T^{*}M)$ be the differential on smooth forms and
$\delta:C^{\infty}_{0}(M;\Lambda T^{*}M)\to C^{\infty}_{0}(M;\Lambda T^{*}M)$
its formal adjoint. The Laplace-Beltrami operator $\Delta$ on differential
forms is defined as $\Delta=\mathrm{d}\delta+\delta\mathrm{d}$. It has a
restriction to forms of degree $p$ which we denote by $\Delta_{p}$. Boundary
conditions on $\Delta_{p}$ which make this into an essentially self-adjoint
operator are called relative and absolute boundary conditions. For a
differential form $\omega\in C^{\infty}_{0}(M;\Lambda^{p}T^{*}M)$ its
restriction to $\partial\mathcal{O}$ we define by
$\omega|_{\partial\mathcal{O}}$. Then whenever $\iota:\partial\mathcal{O}\to
M$ is the natural inclusion map the restriction
$\omega|_{\partial\mathcal{O}}$ is therefore a section in the pull back bundle
$\iota^{*}(\Lambda^{p}T^{*}M)$. This bundle is canonically isomorphic to
$\Lambda^{p}T^{*}(\partial\mathcal{O})\oplus\Lambda^{p-1}T^{*}(\partial\mathcal{O})$,
with the induced splitting being the split of $\omega_{\partial\mathcal{O}}$
into tangential and normal components
$\omega|_{\partial\mathcal{O}}=\omega|_{\partial\mathcal{O},\mathrm{tan}}+\omega|_{\partial\mathcal{O},\mathrm{nor}}$.
The tangential component is the same as the pull-back $\iota^{*}\omega$ of the
differential form $\omega$ to $\partial\mathcal{O}$. There are several
distinguished boundary conditions for the Laplace operator that lead to self-
adjoint extensions of the Laplace operator on compactly supported smooth
forms. These boundary conditions are Relative boundary conditions. For the
Laplace operator these are defined as

$\displaystyle\omega|_{\partial\mathcal{O},\mathrm{tan}}=0,\quad(\delta\omega)|_{\partial\mathcal{O},\mathrm{tan}}=0.$

We also have Absolute boundary conditions which are defined as

$\displaystyle(\omega)|_{\partial\mathcal{O},\mathrm{nor}}=0,\quad(\mathrm{d}\omega)|_{\partial\mathcal{O},\mathrm{nor}}=0.$

Whenever $\omega$ satisfies relative boundary conditions, then $*\omega$
satisfies absolute boundary conditions. Here $*$ is the Hodge star operator.

We will denote by $\Delta_{p,\mathrm{rel}}$ and $\Delta_{p,\mathrm{abs}}$ the
self-adjoint extensions of unbounded operators in $L^{2}(M,\Lambda^{p}T^{*}M)$
of $\Delta_{p}$ resulting from the respective boundary conditions. Since
$*\Delta_{p,\mathrm{rel}}=\Delta_{n-p,\mathrm{abs}}*$ the Hodge star operator
allows us to pass from relative to absolute boundary conditions. The relative
Laplacian $\Delta_{\mathrm{rel}}$ acting on differential forms can be written
as the square of a self-adjoint operator
$Q_{\mathrm{rel}}=\overline{\delta}+\overline{d}_{0}$ (see for example [11, 6,
3]). Here $\overline{\delta}$ is the closure of the operator
$\delta:C^{\infty}_{0}(M,\Lambda T^{*}M)\to C^{\infty}_{0}(M,\Lambda T^{*}M)$,
and $\overline{d}_{0}$ is the closure of the restriction of
$d|_{C^{\infty}_{0}(M_{\mathrm{int}},\Lambda
T^{*}M_{\mathrm{int}})}:C^{\infty}_{0}(M_{\mathrm{int}},\Lambda
T^{*}M_{\mathrm{int}})\to C^{\infty}_{0}(M_{\mathrm{int}},\Lambda
T^{*}M_{\mathrm{int}})$. Here
$M_{\mathrm{int}}=M\backslash\partial\mathcal{O}$ is the interior of $M$. If
$p=0$ then relative boundary conditions correspond to Dirichlet boundary
conditions imposed on $\partial\mathcal{O}$, and absolute boundary conditions
correspond to Neumann boundary conditions.

The Hilbert space $L^{2}(M,\Lambda^{p}T^{*}M)$ decomposes orthogonally into
three invariant subspaces for $\Delta_{p,\mathrm{rel}}$ as follows (see [6]
and also [11])

$L^{2}(M;\Lambda^{p}T^{*}M)=\overline{\delta
C^{\infty}_{0}(M;\Lambda^{p+1}T^{*}M)}\oplus\overline{\mathrm{d}C^{\infty}_{0}(M_{\mathrm{int}};\Lambda^{p-1}T^{*}M_{\mathrm{int}})}\oplus\mathcal{H}^{p}_{\mathrm{rel}}(M),$
(2.1)

where $\mathcal{H}^{p}_{\mathrm{rel}}=\ker{\Delta_{p,\mathrm{rel}}}$ is the
space of relative $L^{2}$-harmonic $p$-forms, i.e. the space of square
integrable forms that are closed, co-closed and satisfy relative boundary
conditions.

##  3 Statement of the main theorems

We start with the following definition

######  Definition 1.

We say a cut-off function is:

  * •

A high frequency cut-off function if $\varphi\in
C^{\infty}_{0}(\mathbb{R}^{+})$ is a non-negative function localising in a
neighbourhood of $\lambda=a$ for some positive $a$ and it satisfies
$\varphi(\lambda)=1$ in a smaller neighbourhood of $\lambda=a$. Without loss
of generality, we can assume
$\operatorname{supp}{\varphi}\subset(\frac{1}{2}a,\frac{3}{2}a)$.

  * •

A low frequency cut-off function if $\varphi\in
C^{\infty}(\overline{\mathbb{R}^{+}})$ is a non-negative function and
satisfies $\varphi(0)=1$ and $\varphi$ has compact support.

In our estimates we only consider high frequency cutoffs. When there is no
obstacle in the problem, the spectral representation of the wave operator for
the $p$-form Laplacian can be calculated explicitly in terms of Bessel
functions. For the free case, we use the subscript free to emphasise this fact
for various quantities. The Theorems below are specialized to the case
$X=\mathbb{R}^{d}$, however the propositions and corollaries for spectral
theory of the Maxwell operator are done for generic $M$ in Section 5. Our
second main theorem 2 is for $M=\mathbb{R}^{3}\setminus\mathcal{O}$ where
$\mathcal{O}$ is an open ball of fixed radius $\rho>0$. We consider the domain
$W^{2,1}(M,\Lambda^{1}T^{*}M)$, as the underlying domain for the operators in
question so that the estimates make sense if $Q$, $\delta$ or $d$ are applied
to the forms in question. The constant $C$ in all of the theorems depends on
at most the dimension, $a$ and $\rho$. The interpretation of our Maxwell
propagator in 3 dimensions with relative boundary conditions applied to co-
closed forms is the electric field. Let $D_{t}=-i\partial_{t}$. We then have
the following theorems about dispersion of the Maxwell propagator in the free
space and in the exterior of a ball of radius $\rho$, respectively.

######  Theorem 1.

Let $\Delta_{\textrm{free},p}$ be the free Laplace on $p$-forms, $d\geq 3$,
$d$ odd and $\varphi$ be a high frequency cut-off function and
$M=\mathbb{R}^{3}$. We let $Q=\delta d$ or $d\delta$. Then for any
$h\in(0,1)$, the free space Maxwell equation propagator satisfies an estimate
of the form:

$\left\|\varphi(hD_{t})e^{\mathrm{i}t\sqrt{\Delta_{\textrm{free},p}}}Q\right\|_{L^{1}(M,\Lambda^{1}T^{*}M)\rightarrow
L^{\infty}(M,\Lambda^{1}T^{*}M)}\leq
Ch^{-(d+2)}\mathrm{min}\left\\{1,\left(\frac{h}{|t|}\right)^{\frac{d-1}{2}}\right\\}.$
(3.1)

If $H_{1}$ and $H_{2}$ denote the subspace of co-closed and closed $p$-forms
respectively, then for $j=1,2$

$\left\|\varphi(hD_{t})e^{\mathrm{i}t\sqrt{\Delta_{\textrm{free},p}}}\right\|_{L^{1}(M,\Lambda^{1}T^{*}M)\cap
H_{j}\rightarrow L^{\infty}(M,\Lambda^{1}T^{*}M)}\leq
Ch^{-d}\mathrm{min}\left\\{1,\left(\frac{h}{|t|}\right)^{\frac{d-1}{2}}\right\\}.$
(3.2)

######  Example 1.

The free space dispersive estimate for the wave equation in dimension $3$ does
not hold in the exterior of _any_ obstacle for the operator

$\varphi(hD_{t})e^{\mathrm{i}t\sqrt{\Delta_{rel,\mathcal{O},1}}}.$

The operator $e^{\mathrm{i}t\sqrt{\Delta_{rel,\mathcal{O},1}}}$ with domain
restricted to $H$ also does not obey the free space estimate unless the domain
of the operator is further restricted to exclude $\mathcal{H}^{1}_{rel}(M)$.

We examine the mechanism behind this failure in the examples section. It
demonstrates that the scattering rate for obstacles with
$e^{\mathrm{i}t\sqrt{\Delta_{\mathrm{rel},\mathcal{O},1}}}\Delta_{\mathrm{rel},\mathcal{O},1}$
and $e^{\mathrm{i}t\sqrt{\Delta_{\mathrm{rel},\mathcal{O},1}}}Q$ with
$Q=\delta d$ are genuinely different from that of the function case. As such
we have the following theorem whose proof suggests the dispersion is a
manifestation of the good behaviour of the spectral representation of the
operator away from $\mathcal{H}_{rel}^{1}(M)$, and correspondingly,
$\lambda=0$. The ideal Dirichlet Laplacian case dispersive estimate we cannot
obtain here for Maxwell’s equations but we have a theorem with a loss of $h$.
This has to do with the fact that the scattering matrix in special coordinates
contains TWO terms only one of which behaves like the function case. It is not
likely this near field estimate can be improved in terms of powers of
$h^{-1}$.

######  Theorem 2.

Let $d=3$ and $\mathcal{O}$ be a sphere of radius $\rho$ centered at the
origin. Let $R$ be a constant and $M\cap B_{R}(0)=M_{R}$ where $B_{R}(0)$ is a
ball of radius $R$. Furthermore let $\varphi$ be a high frequency cutoff,
$\rho\geq\frac{2}{a}$, $a>h,\rho\geq 1$. Let $C$ be a nonzero constant
depending on $\rho$ and $a$ only. The following dispersive estimate holds with
$\Delta_{\mathrm{rel},\mathcal{O},1}$ for any $h\in(0,\frac{1}{4})$:

$\displaystyle\left\|\varphi(hD_{t})e^{\mathrm{i}t\sqrt{\Delta_{\mathrm{rel},\mathcal{O},1}}}\right\|_{L^{1}(M_{R},\Lambda^{1}T^{*}M_{R})\cap
H_{1}\rightarrow L^{\infty}(M_{3R},\Lambda^{1}T^{*}M_{3R})}\leq
Ch^{-5}\mathrm{min}\left\\{1,\left(\frac{R}{|t|}\right)\right\\}$ (3.3)

Under the same conditions, we also have the following corollary

######  Corollary 1.

We have that

$\displaystyle\left\|\varphi(hD_{t})e^{\mathrm{i}t\sqrt{\Delta_{\mathrm{rel},\mathcal{O},1}}}\right\|_{L^{1}(M,\Lambda^{1}T^{*}M)\cap
H_{1}\rightarrow L^{\infty}(M,\Lambda^{1}T^{*}M)}\leq
Ch^{-5}\mathrm{min}\left\\{1,\left(\frac{1}{|t|}\right)\right\\}$ (3.4)

where $C$ depends only on $\rho$ and $a$.

The goal of this paper is to analyze the mechanism behind the Maxwell
scattering versus wave scattering - the divergence free condition encoded in
the operators $\delta d$ and shape of the obstacles. The outline of this paper
is as follows, in Section 4 we re-visit the question of generalized
eigenfunctions for $p$-form scattering in the presence of obstacles. This
section sets up the background from [41] needed for the paper. In Section 5 we
show how using [42] Maxwell’s equations in the presence of obstacles can be
related to the formalism developed in [41]. This Section 5 also establishes
the spectral theory representations for the kernels
$e^{\mathrm{i}t\sqrt{\Delta_{\mathcal{O},1,\mathrm{rel}}}}Q$ for general
manifolds and isotropic Maxwell dielectric and permittivity coefficients. The
free space dispersive estimates are proved in Section 6 using spectral theory.
The basic theory of Mie scattering to cope with relative boundary conditions
are in Section 7. Here we also develop generalized eigenfunctions for
Maxwell’s equations which have a specific form. The development of these
generalized eigenfunctions is a major contribution of the paper. Some examples
are computed in Section 8. The later section, Section 9 proves Theorem 2, and
Section 10 improves the result to a global one by presenting the proof of
Corollary 1, which is built on the works of [27, 28, 29] and [24].

##  4 Review

###  4.1 Generalized Eigenfunctions

Since $M$ is Euclidean at infinity there is a compact set $K$ with $M\setminus
K$ which is isometric to $\mathbb{R}^{d}\setminus B_{R}(0)$. On the set
$M\setminus K$ we have a natural coordinate system. We use both Cartesian
coordinates $x\in\mathbb{R}^{d}$ and spherical coordinates
$(r,\omega)\in(\rho,\infty)\times\mathbb{S}^{d-1}$ where $r=|x|$ and
$\omega=\frac{x}{|x|}$ where it is understood. We pick a smooth function
$\chi\in C^{\infty}(M)$ which is supported in $M\setminus K$ such that
$1-\chi$ is compactly supported. Using Cartesian coordinates and the
orthonormal frame, $(\mathrm{d}x^{1},...,\mathrm{d}x^{d})$, we trivialise the
bundle $T^{*}(M\setminus K)$ and therefore identify forms in
$C^{\infty}(M\setminus K;\Lambda^{p}T^{*}M)$ with vector-valued functions in
$C^{\infty}(M\setminus K;\Lambda^{p}\mathbb{R}^{d})$.

###  4.2 Incoming and outgoing forms

Assume that $f\in C^{\infty}(M;\Lambda^{p}T^{*}M)$, then if
$\lambda\in\mathbb{R}$ is not zero, and $g=(\Delta_{p}-\lambda^{2})f$ is
compactly supported then $f$ is called outgoing for $\lambda$ whenever
$f=R_{\lambda}h$ and $h$ is compactly supported. If $f\chi\in
C_{0}^{\infty}(M;\Lambda^{p}T^{*}M)$ then $f$ is outgoing if and only if
$(1-\chi f)$ is outgoing. This definition depends only on the behavior of $f$
at infinity. The notion is independent of the precise structure of the
resolvent and is independent of the compact part $M\setminus K$. Therefore $f$
outgoing on $M$ is equivalent to $f|_{M\setminus K}$ being outgoing on
$\mathbb{R}^{d}$. This fact can be used to see that an outgoing $f$ has an
asymptotic expansion

$\displaystyle f=\frac{e^{\mathrm{i}\lambda
r}}{r^{\frac{d-1}{2}}}\Phi+\mathcal{O}\left(\frac{1}{r^{\frac{d+1}{2}}}\right)\quad
r\rightarrow\infty$ (4.1)

with $\Phi\in C^{\infty}(\mathbb{S}^{d-1};\Lambda^{p}\mathbb{R}^{d})$ the
restriction of an entire function on $\mathbb{C}^{d}$ to the sphere. The
expansion can be differentiated in $r$.

###  4.3 Generalized plane waves

Assume $d$ is odd. Given $\omega\in\mathbb{S}^{d-1}$ and
$v\in\Lambda^{p}\mathbb{R}^{d}$ we define the distorted plane wave

$\displaystyle e_{\lambda}(\omega,v)\in C^{\infty}(M;\Lambda^{p}T^{*}M)$ (4.2)

by

$\displaystyle
e_{\lambda}(\omega,v)(x)=\chi(x)ve^{-\mathrm{i}\lambda\omega\dot{x}}-R_{\lambda}(\Delta_{p}-\lambda^{2})\chi(x)ve^{-\mathrm{i}\lambda\omega\cdot
x}.$ (4.3)

By construction this is a meromorphic function on $\mathbb{C}$ with

$\displaystyle(\Delta_{p}-\lambda^{2})e_{\lambda}(\omega,v)=0$ (4.4)

satisfying relative boundary conditions but is not necessarily in
$L^{2}(M;\Lambda^{p}T^{*}M)$.

###  4.4 Generalized eigenfunctions and the scattering matrix

Given $\Phi\in L^{2}(\mathbb{S}^{d-1},\Lambda^{p}\mathbb{R}^{d})$ we define
the distorted spherical waves $E_{\lambda}(\Phi)$ by

$\displaystyle
E_{\lambda}(\Phi)(x)=\left(\frac{\lambda}{2\pi}\right)^{\frac{d-1}{2}}\int\limits_{\mathbb{S}^{d-1}}e_{\lambda}(\omega,\Phi(\omega))\,d\omega.$
(4.5)

The distorted spherical waves are generalized eigenfunctions and we can
express them in spherical Bessel and Hankel functions directly. For high
frequencies similar expansions are in [36]. To do so, we need to introduce
some notation. The sphere $\mathbb{S}^{d-1}$ gives an orthornormal basis
$(\phi_{\nu})_{\nu\in I}$ in $L^{2}(\mathbb{S}^{d-1})$ which consists of
eigenfunctions on the Laplacian with eigenvalues $\ell_{\nu}(\ell_{\nu}+d-2)$.
The index set $I$, enumerates the spherical harmonics. These spherical
harmonics are obtained by restricting homogeneous harmonic polynomials to the
sphere. Whenever we are given $g\in L^{2}(\mathbb{S}^{d-1})$ we can write
$g=\sum\limits_{\nu}a_{\nu}\phi_{\nu}$, where $a_{\nu}(g)=\langle
g,\phi_{\nu}\rangle_{L^{2}(\mathbb{S}^{d-1})}$ and convergence is in
$L^{2}(\mathbb{S}^{d-1})$. Denoting $\mathcal{H}_{\ell}(\mathbb{S}^{d-1})$ the
spherical harmonics of degree $\ell$, we have the Hilbert space direct sum

$\displaystyle
L^{2}(\mathbb{S}^{d-1})=\bigoplus\limits_{\ell=0}^{\infty}\mathcal{H}_{\ell}(\mathbb{S}^{d-1}).$
(4.6)

The vector valued case the Hilbert space direct sum

$\displaystyle
L^{2}(\mathbb{S}^{d-1};\Lambda^{p}\mathbb{R}^{d})=\bigoplus\limits_{\ell=0}^{\infty}\mathcal{H}^{p}_{\ell}(\mathbb{S}^{d-1}),$
(4.7)

where $\mathcal{H}_{\ell}^{p}(\mathbb{S}^{d-1})\subset
C^{\infty}(\Lambda^{p}\mathbb{R}^{d})$. As our generalised eigenfunctions are
closely related to Bessel functions $J_{\ell}$ and Hankel functions
$H^{(1)}_{\ell}$, $H^{(2)}_{\ell}$ (or simply $H^{(1,2)}_{\ell}$), it will be
useful to introduce their spherical counterparts at first.

######  Definition 2.

For $\Phi\in L^{2}(\mathbb{S}^{d-1},\Lambda^{\bullet}\mathbb{C}^{d})$, we
define $\tilde{j}_{\lambda}(\Phi)$, $\tilde{h}_{\lambda}^{(1)}(\Phi)$ and
$\tilde{h}_{\lambda}^{(2)}(\Phi)$ as follows

$\tilde{j}_{\lambda}(\Phi)(r,\theta)=2\lambda^{\frac{d-1}{2}}\sum_{\nu}a_{\nu}(\Phi)\Phi_{\nu}(\theta)j_{d,\ell_{\nu}}(\lambda
r)(-\mathrm{i})^{\ell_{\nu}},$ (4.8)

where
$j_{d,\ell}(x)=\sqrt{\frac{\pi}{2}}x^{\frac{2-d}{2}}J_{\ell+\frac{d-2}{2}}(x)$
and $a_{\nu}(\Phi)=\langle\Phi,\Phi_{\nu}\rangle$. Similarly,

$\tilde{h}_{\lambda}^{(1,2)}(\Phi)(r,\theta)=\lambda^{\frac{d-1}{2}}\sum_{\nu}a_{\nu}(\Phi)\Phi_{\nu}(\theta)h_{d,\ell_{\nu}}^{(1,2)}(\lambda
r)(-\mathrm{i})^{\ell_{\nu}},$ (4.9)

where
$h_{d,\ell}^{(1,2)}(x)=\sqrt{\frac{\pi}{2}}x^{\frac{2-d}{2}}H^{(1,2)}_{\ell+\frac{d-2}{2}}(x)$.

The generalised eigenfunctions $E_{\lambda}(\Phi)$ can also be considered as
distributions in $\lambda$ with values in the Schwartz space of functions
$\mathcal{S}(M;\Lambda^{p}T^{*}M)$. Whenever $g\in
C_{0}^{\infty}(\mathbb{R}^{+})$ then

$\int\limits_{\mathbb{R}}g(\lambda)E_{\lambda}(\Phi)\,d\lambda$

is square integrable. The inner product

$\displaystyle\langle
E_{\lambda}(\Phi),E_{\mu}(\Psi)\rangle_{L^{2}(M)}=(4\pi\lambda)\delta(\lambda^{2}-\mu^{2})\langle\Phi,\Psi\rangle_{L^{2}(\mathbb{S}^{d-1})}$
(4.10)

as a bidistribution in
$\mathcal{D}^{\prime}(\mathbb{R}_{+}\times\mathbb{R}_{+})$. The way this is
computed is by taking the limit

$\displaystyle\lim\limits_{\mathbb{R}\rightarrow\infty}(\langle\Delta_{p}E_{\lambda}(\Phi),\chi_{R}E_{\mu}(\Psi)\rangle_{L^{2}(M)}-(\langle
E_{\lambda}(\Phi),\chi_{R}\Delta_{p}E_{\mu}(\Psi)\rangle_{L^{2}(M)}.$ (4.11)

We have the following formula which is from [41].

######  Proposition 1 (Prop 2.2 in [41]).

We have that

$E_{\lambda}(\Phi)=\chi\tilde{j}_{\lambda}(\Phi)-R_{\lambda}(\Delta_{p}-\lambda^{2})(\chi\tilde{j}_{\lambda}(\Phi))=\chi\tilde{j}_{\lambda}(\Phi)-R_{\lambda}[\Delta_{p},\chi](\tilde{j}_{\lambda}(\Phi)).$

We also have that

######  Proposition 2 (Prop 2.4 in [41]).

For every $\lambda\in\mathbb{R}\setminus\\{0\\}$ and $\Phi\in
L^{2}(\mathbb{S}^{d-1},\Lambda^{p}\mathbb{R}^{d})$ there exists a unique
$A_{\lambda}(\Phi)\in C^{\infty}(\mathbb{S}^{d-1},\Lambda^{p}\mathbb{R}^{d})$
such that

$\displaystyle E_{\lambda}(\Phi)|_{M\setminus
K}=\tilde{j}_{\lambda}(\Phi)+\tilde{h}_{\lambda}^{(1)}(A_{\lambda}\Phi).$
(4.12)

These formulas will be the basis for our spectral representations later in the
paper.

##  5 Maxwell’s equations

We would like to recast electromagnetic scattering theory in dimension $d=3$
into the language of forms, and we follow the treatment in [42]. We let
$D,E\in C^{\infty}(\mathbb{R}_{t}\times M,T^{*}M)$ denote the electric field
and the magentic field as $\underline{H},\underline{B}\in
C^{\infty}(\mathbb{R}_{t}\times M,T^{*}M)$. Maxwell’s equations in linear
matter can be written as

$\displaystyle\mathrm{curl}E=-\underline{\dot{B}},\quad\mathrm{div}D=0,$
$\displaystyle\mathrm{div}\underline{B}=0,\quad\mathrm{curl}\,\underline{H}=\dot{D},$
$\displaystyle D=\epsilon E,\quad\underline{B}=\mu\underline{H},$

the dot for the $t$ derivative is given to us by $\dot{E}=\partial_{t}E$. The
permittivity $\epsilon$ and the permeability $\mu$ are incorporating the
linear effects of matter. These are both positive definite matrix-valued
functions, which for simplicity we assume are smooth. Perfectly conducting
boundary conditions for smooth obstacles correspond to the tangential
component of $E$ and the normal component of $\underline{B}$ vanishing at the
boundary. In order to understand this in terms of differential forms, we
denote the co-vector field $\underline{B}$ corresponding to the two-form $B$
as defined by $\underline{B}=*B$, where $*$ is the Hodge-star operator on $M$.

We then think of $\epsilon$ as a smooth
$\mathrm{End}(\Lambda^{1}T^{*}M)$-valued function on $M$ which is pointwise
positive definite. Similarly, $\mu$ can be thought of as a smooth
$\mathrm{End}(\Lambda^{2}T^{*}M)$-valued function on $M$ via Hodge-star
operator. One can now define a metric on $\Lambda^{p}T_{x}^{*}M$ by

$(v,w)=\begin{cases}vw,\;&p=0,\\\ (v,\epsilon\,w)_{g},&p=1,\\\
(v,\mu^{-1}w)_{g},&p=2,\\\ (v,(\epsilon\mu^{2})^{-1}w)_{g},&p=3,\end{cases}$
(5.1)

where $v,w\in\Lambda^{p}T_{x}^{*}M$ and $(\cdot,\cdot)_{g}$ is the metric
induced inner product on $\Lambda^{p}T_{x}^{*}M$. This defines a bundle metric
on the direct sum $\Lambda^{\bullet}T_{x}^{*}M$. Let $\delta$ be the formal
adjoint of exterior derivative $\mathrm{d}$ with respect to the inner product
defined in (5.1), one computes

$\displaystyle\mathrm{div}(\epsilon E)=-\delta
E,\quad*\mathrm{d}E=\mathrm{curl}E,\quad\delta(\mu
H)=\epsilon^{-1}\mathrm{curl}\underline{H}.$

We therefore obtain for Maxwell’s equations the differential form version

$\displaystyle\mathrm{d}E=-\dot{B},\quad\mathrm{d}B=0,$ $\displaystyle\delta
E=0,\quad\delta H=\dot{E}.$

Matter effects in the form of $\epsilon$ and $\mu$ have then been absorbed
into the bundle metric, we have that $E$ and $B$ are time-dependent
differential forms. Since $E$ and $B$ must have vanishing tangential
components, then $\underline{B}$ has zero normal component. This absorption of
the parameters into the bundle metric for the isotropic which was detailed in
[42] we use in the later sections to show that our scattering estimates can
include a nontrivial geometry and $\epsilon$ and $\mu$ whenever these are
compactly supported. Our spectral analysis is then done in the flat region.
For $E$ one obtains the equation

$\displaystyle\epsilon\ddot{E}+\mathrm{curl}(\mu^{-1}\mathrm{curl}E)=0,\quad\mathrm{div}(\epsilon
E)=0,$ (5.2)

this then simplifies to

$\ddot{E}+\Delta E=0,\quad\delta E=0.$

Therefore spectral theory of the operator $\Delta$ on divergence-free vector
fields (co-closed forms) determines the operator $E$. The effect of matter is
in encoded in the operator $\delta$. In each case $\epsilon=\mu=1$, then
accordingly one has
$\Delta=\mathrm{curl}\,\mathrm{curl}-\mathrm{grad}\;\mathrm{div}$.

Unlike the study of dispersive estimates for scalars, we would need to analyse
the behaviour of wave propagators on closed and co-closed forms for Maxwell’s
system. In order to establish a functional calculus on closed or co-closed
forms, we need to decompose the scattering matrix into normal and tangential
parts. Let $\mathrm{d}r$ be the unit conormal on the sphere. Define
$\theta=\mathrm{d}r\wedge$ as the operator of exterior multiplication by
$\mathrm{d}r$ and then $\theta^{*}=\iota_{\mathrm{d}r}$ is the operator of
inner multiplication by $\mathrm{d}r$. If $Q_{1}=\delta\mathrm{d}$ and
$Q_{2}=\mathrm{d}\delta$, we can define the projection operator associated
with $Q_{1,2}$ on $L^{2}(\mathbb{S}^{d-1},\Lambda^{\bullet}\mathbb{C}^{d})$ as
follows

$\left\\{\begin{aligned} P_{Q_{1}}=\theta^{*}\theta,\\\
P_{Q_{2}}=\theta\theta^{*}.\end{aligned}\right.$

Note that $\theta^{*}\theta$ is the projection onto the tangential part
$L_{t}^{2}(\mathbb{S}^{d-1},\Lambda^{\bullet}\mathbb{C}^{d})$ and
$\theta\theta^{*}=1-\theta^{*}\theta$ is the projection onto the normal part
$L_{n}^{2}(\mathbb{S}^{d-1},\Lambda^{\bullet}\mathbb{C}^{d})$. Before giving
an analogue of Stone’s formula for closed or co-closed differential forms, we
first state some useful properties of the generalised eigenfunctions.

######  Proposition 3.

For $\lambda\in\mathbb{R}\backslash\\{0\\}$, the generalised eigenfunction,
$E_{\lambda}$, as a map from
$L^{2}(\mathbb{S}^{d-1},\Lambda^{\bullet}\mathbb{C}^{d})$ to
$C^{\infty}(M\backslash K,\Lambda_{\mathbb{C}}^{\bullet}T^{*}(M\backslash K))$
is given by

$E_{\lambda}(\Phi)|_{M\backslash
K}=\tilde{j}_{\lambda}(\Phi)+\tilde{h}_{\lambda}^{(1)}(A_{\lambda}\Phi),$
(5.3)

where
$A_{\lambda}:L^{2}(\mathbb{S}^{d-1},\Lambda^{\bullet}\mathbb{C}^{d})\mapsto
C^{\infty}(\mathbb{S}^{d-1},\Lambda^{\bullet}\mathbb{C}^{d})$ is uniquely
defined. Moreover, $E_{\lambda}$ satisfies the following identities

$\displaystyle\mathrm{d}E_{\lambda}(\Phi)=-\mathrm{i}\lambda
E_{\lambda}(\mathrm{d}r\wedge\Phi),$ $\displaystyle\delta
E_{\lambda}(\Phi)=\mathrm{i}\lambda E_{\lambda}(\iota_{\mathrm{d}r}\Phi).$
(5.4)

###### Proof.

This is an application of Rellich’s uniqueness theorem and a detailed proof
can be found in [41, Proposition 2.4 & 2.6]. ∎

A version of Stone’s formula for differential forms is the following.

######  Theorem (c.f. [41], a variant of Stone’s theorem).

For any $\lambda>0$ and $f\in C^{\infty}_{0}(M,\Lambda^{p}T^{*}M)$, we have

$(R_{\lambda}-R_{-\lambda})f=\frac{\mathrm{i}}{2\lambda}\sum\limits_{\nu}E_{\lambda}(\Phi_{\nu})\langle
f,E_{\lambda}(\Phi_{\nu})\rangle,$

where the equality is understood in the sense of
$C^{\infty}(M,\Lambda^{p}T^{*}M)$ and $R_{\lambda}$ is the outgoing resolvent,
i.e. $(\Delta-\lambda^{2})^{-1}$. Therefore, for any bounded Borel function
$k:\mathbb{R}\to\mathbb{C}$ one obtains

$\displaystyle\langle k(\Delta_{p,\mathrm{rel}})f,g\rangle$ (5.5)
$\displaystyle=k(0)\sum_{j=1}^{N}\langle f,u_{j}\rangle\langle
u_{j},g\rangle+\frac{1}{2\pi}\sum\limits_{\nu}\int_{0}^{\infty}k(\lambda^{2})\langle
f,E_{\lambda}(\Phi_{\nu})\rangle\langle
E_{\lambda}(\Phi_{\nu}),g\rangle\,\mathrm{d}\lambda,$

for any $g\in C^{\infty}_{0}(M,\Lambda^{p}T^{*}M)$. Here $(u_{j})_{j=1}^{N}$
are the $L^{2}$-normalised-eigenfunctions in the kernel of
$\Delta_{p,\mathrm{rel}}$, $\mathcal{H}_{rel}^{1}(M)$.

We have the following Theorem from [41]

######  Proposition 4.

If $k$ is a Borel function with $k=O((1+\lambda^{2})^{-q})$ for all
$q\in\mathbb{N}$ we have that $k(\Delta_{p,\mathrm{rel}})$ has smooth integral
kernel $\kappa_{k}\in C_{0}^{\infty}(M\times
M;\Lambda^{p}T^{*}M\boxtimes(\Lambda^{p}T^{*}M)^{*})$ and

$\displaystyle\kappa_{k}(x,y)=k(0)\sum_{j=1}^{N}u_{j}(x)\otimes(u_{j}(y))^{*}+\frac{1}{2\pi}\sum\limits_{\nu}\int_{0}^{\infty}k(\lambda^{2})E_{\lambda}(\Phi_{\nu})(x)\otimes
E_{\lambda}(\Phi_{\nu})^{*}(y)\,\mathrm{d}\lambda,$ (5.6)

where the sum converges in $C^{\infty}_{0}(M\times
M;\Lambda^{p}T^{*}M\boxtimes(\Lambda^{p}T^{*}M)^{*})$.

From the above Proposition 4, we obtain the following corollary.

######  Corollary 2.

For $\lambda\in\mathbb{R}\backslash\\{0\\}$ and $Q=Q_{1,2}$,
$\sum\limits_{\nu}E_{\lambda}((1-P_{Q})\Phi_{\nu})\langle\cdot,E_{\overline{\lambda}}(P_{Q}\Phi_{\nu})\rangle$
equals zero as an operator from $C^{\infty}_{0}(M,\Lambda^{p}T^{*}M)$ to
$C^{\infty}(M,\Lambda^{p}T^{*}M)$.

###### Proof.

Let $f\in C^{\infty}_{0}(M,\Lambda^{p}T^{*}M)$, we have from Stone’s Theorem
above that

$\displaystyle(\Delta-Q)Q(R_{\lambda}-R_{-\lambda})f=$
$\displaystyle(\Delta-Q)(R_{\lambda}-R_{-\lambda})Qf$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}}{2\lambda}\sum\limits_{\nu}(\Delta-Q)E_{\lambda}(\Phi_{\nu})\langle
Qf,E_{\lambda}(\Phi_{\nu})\rangle.$

Recall that the Green’s formula [8] for complex differential forms is given by

$\langle\mathrm{d}f_{1},f_{2}\rangle=\langle f_{1},\delta
f_{2}\rangle+\int_{\partial M}f_{1}\wedge\ast\overline{f_{2}}.$

Using the compactness of $f$, we could get rid of the boundary term in the
Green’s formula and thereby obtain

$\displaystyle(\Delta-Q)Q(R_{\lambda}-R_{-\lambda})f=$
$\displaystyle\frac{\mathrm{i}}{2\lambda}\sum\limits_{\nu}(\Delta-Q)E_{\lambda}(\Phi_{\nu})\langle
f,QE_{\lambda}(\Phi_{\nu})\rangle$ $\displaystyle=$
$\displaystyle\frac{\mathrm{i}\lambda^{3}}{2}\sum\limits_{\nu}E_{\lambda}((1-P_{Q})\Phi_{\nu})\langle
f,E_{\lambda}(P_{Q}\Phi_{\nu})\rangle,$

where we have used properties (5.4) of generalised eigenfunctions in the last
step. As $\lambda\neq 0$ and $(\Delta-Q)Q=0$, we conclude the proof of the
corollary. ∎

We are now ready to state a modification of Stone’s formula for the closed and
co-closed forms. This is a Corollary to Stone’s Theorem above

######  Corollary 3.

Let $f,g\in C^{\infty}_{0}(M,\Lambda^{p}T^{*}M)$ and $\lambda>0$. For any
bounded Borel function $k:\mathbb{R}\to\mathbb{C}$ we have

$\langle
k(\Delta_{p,\mathrm{rel}})Qf,g\rangle=\frac{1}{2\pi}\sum\limits_{\nu}\int_{0}^{\infty}\lambda^{2}k(\lambda^{2})\langle
f,E_{\lambda}(P_{Q}\Phi_{\nu})\rangle\langle
E_{\lambda}(P_{Q}\Phi_{\nu}),g\rangle\,\mathrm{d}\lambda.$ (5.7)

If $\delta f=0$, then the Stone’s formula for co-closed forms is given by

$\displaystyle(R_{\lambda}-R_{-\lambda})f=\frac{\mathrm{i}}{2\lambda}\sum\limits_{\nu}E_{\lambda}(P_{Q_{1}}\Phi_{\nu})\langle
f,E_{\lambda}(P_{Q_{1}}\Phi_{\nu})\rangle$ (5.8)

in $C^{\infty}(M,\Lambda^{p}T^{*}M)$. This implies

$\langle
P_{ac}k(\Delta_{p,\mathrm{rel}})f,g\rangle=\frac{1}{2\pi}\sum\limits_{\nu}\int_{0}^{\infty}k(\lambda^{2})\langle
f,E_{\lambda}(P_{Q_{1}}\Phi_{\nu})\rangle\langle
E_{\lambda}(P_{Q_{1}}\Phi_{\nu}),g\rangle\,\mathrm{d}\lambda,$ (5.9)

where $P_{ac}$ is the projection onto the absolutely continuous spectrum.
Similarly, if $\mathrm{d}f=0$, then Stone’s formula for closed forms becomes

$\displaystyle(R_{\lambda}-R_{-\lambda})f=\frac{\mathrm{i}}{2\lambda}\sum\limits_{\nu}E_{\lambda}(P_{Q_{2}}\Phi_{\nu})\langle
f,E_{\lambda}(P_{Q_{2}}\Phi_{\nu})\rangle,$ (5.10)

which implies

$\langle
P_{ac}k(\Delta_{p,\mathrm{rel}})f,g\rangle=\frac{1}{2\pi}\sum\limits_{\nu}\int_{0}^{\infty}k(\lambda^{2})\langle
f,E_{\lambda}(P_{Q_{2}}\Phi_{\nu})\rangle\langle
E_{\lambda}(P_{Q_{2}}\Phi_{\nu}),g\rangle\,\mathrm{d}\lambda.$ (5.11)

###### Proof.

Using Proposition 3 and Corollary 2, one obtains

$(R_{\lambda}-R_{-\lambda})Qf=\frac{\mathrm{i}\lambda}{2}\sum\limits_{\nu}E_{\lambda}(P_{Q}\Phi_{\nu})\langle
f,E_{\lambda}(P_{Q}\Phi_{\nu})\rangle.$ (5.12)

Combining with the standard functional calculus for sectorial operators and
equation (5.12), one then has

$\displaystyle\langle
k(\Delta_{p,\mathrm{rel}})Qf,g\rangle=k(0)\sum_{j=1}^{N}\langle
Qf,u_{j}\rangle\langle u_{j},g\rangle+\langle
P_{ac}k(\Delta_{p,\mathrm{rel}})Qf,g\rangle$
$\displaystyle=\frac{1}{2\pi}\sum\limits_{\nu}\int_{0}^{\infty}\lambda^{2}k(\lambda^{2})\langle
f,E_{\lambda}(P_{Q}\Phi_{\nu})\rangle\langle
E_{\lambda}(P_{Q}\Phi_{\nu}),g\rangle\,\mathrm{d}\lambda,$

where $k$ is any continuous function vanishing at infinity. Note that the
contribution from the point spectrum is anhilated by $Qf$ using integration by
parts. Now, by Riesz representation theorem, one could further extend $k$ to
be any bounded Borel function. This proves equation (5.7).

For the other parts, we know that $P_{Q_{1}}+P_{Q_{2}}=1$ on
$L^{2}(\mathbb{S}^{d-1},\Lambda^{\bullet}\mathbb{C}^{d})$ and Stone’s Theorem,

$(R_{\lambda}-R_{-\lambda})f=\frac{\mathrm{i}}{2\lambda}\sum\limits_{\nu}E_{\lambda}(\Phi_{\nu})\langle
f,E_{\lambda}((P_{Q_{1}}+P_{Q_{2}})\Phi_{\nu})\rangle,$

for $f\in C^{\infty}_{0}(M,\Lambda^{p}T^{*}M)$. If $\delta f=0$, then we have,
by equation (5.4) and integration by parts,

$\displaystyle(R_{\lambda}-R_{-\lambda})f=\frac{\mathrm{i}}{2\lambda}\sum\limits_{\nu}E_{\lambda}(\Phi_{\nu})\langle
f,E_{\lambda}(P_{Q_{1}}\Phi_{\nu})\rangle.$ (5.13)

Now using Corollary 2, we can easily obtain equation (7.17) and then equation
(7.18) follows immediately. Similar arguments apply to the case
$\mathrm{d}f=0$ and therefore we also have equations (5.10) and (5.11) proved.
∎

##  6 Dispersive estimates: the free p-form case

Dispersive estimates for the free scalar wave equations is well known and
derivations can be found in various places (see e.g. [1, 37]). In this
section, we give a proof of dispersive estimates for the free $p$-form Laplace
via a spectral decomposition. More importantly, we will see that the well
known dispersive estimates are controlled by asymptotic behaviours of
generalised eigenfunctions at infinity. We will also see that projections onto
co-closed (divergence-free) and closed (curl-free) does not change the
behaviour, in the free case. We use spectral theory here rather than the
Fourier transform as we also setup the notation for the other sections.

###### Proof of Theorem 1.

Applying formula (5.7) to
$k(\lambda^{2})=\varphi(h\lambda)e^{\mathrm{i}\lambda t}$, we know that the
kernel of $k(\Delta_{\textrm{free},p})Q$ is given by

$\displaystyle K(x,x^{\prime};h)=$
$\displaystyle\frac{1}{2\pi}\sum\limits_{\nu}\int_{0}^{\infty}\lambda^{2}\varphi(h\lambda)e^{\mathrm{i}\lambda
t}E_{\textrm{free},\lambda}(P_{Q}\Phi_{\nu})(x)(E_{\textrm{free},\lambda}(\Phi_{\nu})(x^{\prime}))^{*}\mathrm{d}\lambda=$
$\displaystyle\frac{1}{2\pi
h^{3}}\sum\limits_{\nu}\int_{0}^{\infty}\lambda^{2}\varphi(\lambda)e^{\mathrm{i}\frac{\lambda}{h}t}E_{\textrm{free},\frac{\lambda}{h}}(P_{Q}\Phi_{\nu})(x)(E_{\textrm{free},\frac{\lambda}{h}}(\Phi_{\nu})(x^{\prime}))^{*}\mathrm{d}\lambda.$

In the above expression, we consider
$(E_{\textrm{free},\lambda}(\Phi_{\nu})(x^{\prime}))^{*}$ instead of
$(E_{\textrm{free},\lambda}(P_{Q}\Phi_{\nu})(x^{\prime}))^{*}$ as in Theorem
3, which will be sufficient for our purpose. Note that
$E_{\textrm{free},\lambda}(\Phi_{\nu})=\tilde{j}_{\lambda}(\Phi_{\nu})$ and it
is given by

$\displaystyle
E_{\textrm{free},\lambda}(\Phi_{\nu})(r,\theta)=\sqrt{2\pi}(-\mathrm{i})^{\ell_{\nu}}\Phi_{\nu}(\theta)\lambda^{\frac{1}{2}}r^{\frac{2-d}{2}}J_{\ell_{\nu}+\frac{d-2}{2}}(\lambda
r),$ $\displaystyle
E_{\textrm{free},\lambda}(P_{Q}\Phi_{\nu})(r,\theta)=2\lambda^{\frac{d-1}{2}}\sum_{\mu}a_{\mu}(P_{Q}\Phi_{\nu})\Phi_{\mu}(\theta)j_{d,\ell_{\mu}}(\lambda
r)(-\mathrm{i})^{\ell_{\mu}},$

where we use $x=(r,\theta)$ and $x^{\prime}=(r^{\prime},\theta^{\prime})$ to
denote polar coordinates. This means

$E_{\textrm{free},\lambda}(P_{Q}\Phi_{\nu})(r,\theta)=\sum_{\mu}a_{\mu}(P_{Q}\Phi_{\nu})E_{\textrm{free},\lambda}(\Phi_{\mu})(r,\theta).$

From [34], Section 10.53 and 10.57, we know that

$\displaystyle|E_{\textrm{free},\lambda}(\Phi_{\mu})(r,\theta)|\leq\frac{\sqrt{2\pi}\|\Phi_{\mu}\|_{L^{\infty}}}{\Gamma(\ell_{\mu}+\frac{d}{2})}|\lambda|^{\ell_{\mu}+\frac{d-1}{2}}r^{\ell_{\mu}}e^{|\operatorname{Im}\lambda
r|}\leq
C\frac{\sigma_{\ell_{\mu}}^{\frac{d-2}{2}}}{\Gamma(\ell_{\mu}+\frac{d}{2})}|\lambda|^{\ell_{\mu}+\frac{d-1}{2}}r^{\ell_{\mu}}e^{|\operatorname{Im}\lambda
r|},$ (6.1)

for $\lambda\in\mathbb{C}$, where $\sigma_{\ell_{\mu}}^{2}$ is the eigenvalue
of $1$-form Laplace corresponding to $\Phi_{\mu}$. We have used the fact that
$\|\Phi_{\mu}\|_{L^{\infty}}\leq
C\sigma_{\ell_{\mu}}^{\frac{d-2}{2}}\|\Phi_{\mu}\|_{L^{2}}$ from [38]. Also,
from [34], we have

$|E_{\textrm{free},\lambda}(\Phi_{\mu})(r,\theta)|\leq\sqrt{2\pi}\|\Phi_{\mu}\|_{L^{\infty}}|\lambda|^{\frac{1}{2}}r^{\frac{2-d}{2}}\leq
C\sigma_{\ell_{\mu}}^{\frac{d-2}{2}}r^{\frac{2-d}{2}},$ (6.2)

for $\lambda\in\mathbb{R}\backslash\\{0\\}$. Hence we have that

$|E_{\textrm{free},\lambda}(\Phi_{\mu})(r,\theta)|\leq
C\sigma_{\ell_{\mu}}^{\frac{d-2}{2}}\min\left\\{\frac{|\lambda|^{\ell_{\mu}+\frac{d-1}{2}}r^{\ell_{\mu}}e^{|\operatorname{Im}\lambda
r|}}{\Gamma(\ell_{\mu}+\frac{d}{2})},r^{\frac{2-d}{2}}\right\\}.$ (6.3)

Since $P_{Q}$ is a projection operator that maps

$H_{\ell}(\mathbb{S}^{d-1},\Lambda^{\bullet}\mathbb{C}^{d})\to
H_{\ell-2}(\mathbb{S}^{d-1},\Lambda^{\bullet}\mathbb{C}^{d})\oplus
H_{\ell}(\mathbb{S}^{d-1},\Lambda^{\bullet}\mathbb{C}^{d})\oplus
H_{\ell+2}(\mathbb{S}^{d-1},\Lambda^{\bullet}\mathbb{C}^{d})$

and $\langle\Phi_{\nu},\Phi_{\mu}\rangle=0$ for $\mu\neq\nu$, one has

$a_{\mu}(P_{Q}\Phi_{\nu})=\langle P_{Q}\Phi_{\nu},\Phi_{\mu}\rangle=0$

for $\ell_{\mu}\neq\ell_{\nu},\ell_{\nu}\pm 2$. This simplifies
$E_{\textrm{free},\lambda}(P_{Q}\Phi_{\nu})$ to

$E_{\textrm{free},\lambda}(P_{Q}\Phi_{\nu})=\sum_{\ell_{\mu}=\ell_{\nu},\ell_{\nu}\pm
2}a_{\mu}(P_{Q}\Phi_{\nu})E_{\textrm{free},\lambda}(\Phi_{\mu}).$

By Cauchy-Schwartz inequality, we obtain

$|a_{\mu}(P_{Q}\Phi_{\nu})|=|\langle
P_{Q}\Phi_{\nu},\Phi_{\mu}\rangle|\leq\|P_{Q}\Phi_{\nu}\|_{L^{2}}\|\Phi_{\mu}\|_{L^{2}}\leq
1.$

Therefore, we have that

$|E_{\textrm{free},\lambda}(P_{Q}\Phi_{\nu})|\leq\sum_{\ell_{\mu}=\ell_{\nu},\ell_{\nu}\pm
2}|E_{\textrm{free},\lambda}(\Phi_{\mu})|.$

Taking into account of $h$, we have

$|E_{\textrm{free},\frac{\lambda}{h}}(\Phi_{\mu})(r,\theta)|\leq
C\sigma_{\ell_{\mu}}^{\frac{d-2}{2}}\min\left\\{\frac{|\frac{\lambda}{h}|^{\ell_{\mu}+\frac{d-1}{2}}r^{\ell_{\mu}}e^{|\operatorname{Im}\frac{\lambda}{h}r|}}{\Gamma(\ell_{\mu}+\frac{d}{2})},r^{\frac{2-d}{2}}\right\\}$
(6.4)

and

$\displaystyle|E_{\textrm{free},\frac{\lambda}{h}}(P_{Q}\Phi_{\nu})(r,\theta)|\leq
C\sum_{\ell_{\mu}=\ell_{\nu},\ell_{\nu}\pm
2}\sigma_{\ell_{\mu}}^{\frac{d-2}{2}}\min\left\\{\frac{|\frac{\lambda}{h}|^{\ell_{\mu}+\frac{d-1}{2}}r^{\ell_{\mu}}e^{|\operatorname{Im}\frac{\lambda}{h}r|}}{\Gamma(\ell_{\mu}+\frac{d}{2})},r^{\frac{2-d}{2}}\right\\}.$
(6.5)

We now break the estimates on the kernel into three cases.

  1. 1.

Case 1: $\left|\frac{\lambda r}{h}\right|<1$ and $t<h$.
Without loss of generality we take $r^{\prime}\leq r$. Using estimates (6.4)
and (6.5), we obtain the following:

$\displaystyle|K(r,\theta,r^{\prime},\theta^{\prime};h)|$ (6.6)
$\displaystyle=\frac{1}{2\pi
h^{3}}\left|\sum\limits_{\nu}\int_{0}^{\infty}\lambda^{2}\varphi(\lambda)e^{\mathrm{i}\frac{\lambda}{h}t}E_{\textrm{free},\frac{\lambda}{h}}(P_{Q}\Phi_{\nu})(r,\theta)E^{*}_{\textrm{free},\frac{\lambda}{h}}(\Phi_{\nu})(r^{\prime},\theta^{\prime})\mathrm{d}\lambda\right|$
$\displaystyle\leq\frac{1}{2\pi
h^{3}}\sum\limits_{\nu}\int_{0}^{\infty}\lambda^{2}\varphi(\lambda)\left|E_{\textrm{free},\frac{\lambda}{h}}(P_{Q}\Phi_{\nu})(r,\theta)E^{*}_{\textrm{free},\frac{\lambda}{h}}(\Phi_{\nu})(r^{\prime},\theta^{\prime})\right|\mathrm{d}\lambda$
$\displaystyle\leq\frac{1}{2\pi
h^{3}}\sum\limits_{\nu}\int_{0}^{\infty}\lambda^{2}\varphi(\lambda)\sum_{\ell_{\mu}=\ell_{\nu},\ell_{\nu}\pm
2}|E_{\textrm{free},\frac{\lambda}{h}}(\Phi_{\mu})(r,\theta)||E_{\textrm{free},\frac{\lambda}{h}}^{*}(\Phi_{\nu})(r^{\prime},\theta^{\prime})|\mathrm{d}\lambda$
$\displaystyle\leq\frac{C}{2\pi
h^{3}}\sum\limits_{\nu}\int_{0}^{\infty}\lambda^{2}\varphi(\lambda)\sum_{\ell_{\mu}=\ell_{\nu},\ell_{\nu}\pm
2}\sigma_{\ell_{\mu}}^{\frac{d-2}{2}}\frac{|\frac{\lambda}{h}|^{\ell_{\mu}+\frac{d-1}{2}}r^{\ell_{\mu}}}{\Gamma(\ell_{\mu}+\frac{d}{2})}\sigma_{\ell_{\nu}}^{\frac{d-2}{2}}\frac{|\frac{\lambda}{h}|^{\ell_{\nu}+\frac{d-1}{2}}(r^{\prime})^{\ell_{\nu}}}{\Gamma(\ell_{\nu}+\frac{d}{2})}\mathrm{d}\lambda\leq$
$\displaystyle\frac{C}{2\pi
h^{3}}\sum\limits_{\nu}\sum_{\ell_{\mu}=\ell_{\nu},\ell_{\nu}\pm
2}\int_{0}^{\infty}\lambda^{2}\varphi(\lambda)\frac{\sigma_{\ell_{\mu}}^{\frac{d-2}{2}}\sigma_{\ell_{\nu}}^{\frac{d-2}{2}}}{\Gamma(\ell_{\mu}+\frac{d}{2})\Gamma(\ell_{\nu}+\frac{d}{2})}\left(\frac{\lambda}{h}\right)^{\ell_{\mu}+\ell_{\nu}+d-1}r^{\ell_{\mu}}(r^{\prime})^{\ell_{\nu}}\mathrm{d}\lambda\leq$
$\displaystyle Ch^{-(d+2)}.$

Here we have used that it is due to the Weyl’s Law that $\sigma_{\ell}$ is
polynomially bounded in $\ell$. Therefore, this last estimate shows that the
kernel $K(x,x^{\prime};h)$ converges in
$L^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d})$ when $|\lambda r|\leq h$.

  2. 2.

Case 2 when $|\frac{\lambda r}{h}|<1$ and $t\geq h$
We could have also integrated by parts with $e^{\mathrm{i}\frac{t\lambda}{h}}$
to arrive at the bound when $t>h$ using essentially the same estimate as (6.6)
that is:

$\displaystyle|K(r,\theta,r^{\prime},\theta^{\prime};h)|$ (6.7)
$\displaystyle=\frac{1}{2\pi
h^{2}t}\left|\sum\limits_{\nu}\int_{0}^{\infty}e^{\mathrm{i}\frac{\lambda}{h}t}\frac{\mathrm{d}}{\mathrm{d}\lambda}\left(\lambda^{2}\varphi(\lambda)(E_{\textrm{free},\frac{\lambda}{h}}(P_{Q}\Phi_{\nu})(r,\theta)E^{*}_{\textrm{free},\frac{\lambda}{h}}(\Phi_{\nu})(r^{\prime},\theta^{\prime}))\right)\mathrm{d}\lambda\right|\leq$
$\displaystyle\frac{C}{2\pi
h^{2}t}\sum\limits_{\nu}\sum_{\ell_{\mu}=\ell_{\nu},\ell_{\nu}\pm
2}\int_{0}^{\infty}(\lambda\varphi(\lambda)+\lambda^{2}\varphi^{\prime}(\lambda))(\ell_{\nu}+2)\times$
$\displaystyle\frac{\sigma_{\ell_{\mu}}^{\frac{d-2}{2}}\sigma_{\ell_{\nu}}^{\frac{d-2}{2}}}{\Gamma(\ell_{\mu}+\frac{d}{2})\Gamma(\ell_{\nu}+\frac{d}{2})}\left(\frac{\lambda}{h}\right)^{\ell_{\mu}+\ell_{\nu}+d-1}r^{\ell_{\mu}}(r^{\prime})^{\ell_{\nu}}\mathrm{d}\lambda\leq
Ch^{-(d+2)}.$

Integration by parts is only possible once since $\lambda^{d-2-k}$ is not a
polynomial if $k$ is larger than $d-2$, and then the factors of $\frac{\lambda
r}{h},\frac{\lambda r^{\prime}}{h}$ will not be necessarily be bounded.

  3. 3.

Case $|\frac{\lambda r}{h}|>1$
To get a bound on the kernel in the case where $|\lambda r|>h$, we will use a
stationary phase argument. As the free Laplace commutes with the translation
operator, we can, without loss of generality, fix $x^{\prime}=0$. This
translation simplification comes from the dispersive estimates argument in [1]
for the free space wave operator applied to functions. From estimate (6.3), we
know that $E_{\textrm{free},\lambda}(\Phi_{\nu})(0)=0$ for $\ell_{\nu}>0$. Now
the kernel simplifies to

$\displaystyle K(r,\theta,0,\theta^{\prime};h)=$ (6.8)
$\displaystyle\frac{1}{2\pi
h^{3}}\sum\limits_{\ell_{\nu}=0}\int_{0}^{\infty}\lambda^{2}\varphi(\lambda)e^{\mathrm{i}\frac{\lambda}{h}t}E_{\textrm{free},\frac{\lambda}{h}}(P_{Q}\Phi_{\nu})(r,\theta)(E_{\textrm{free},\frac{\lambda}{h}}(\Phi_{\nu})(0,\theta^{\prime}))^{*}\mathrm{d}\lambda=$
$\displaystyle\frac{1}{\sqrt{2\pi}2^{\frac{d-2}{2}}\Gamma(\frac{d}{2})h^{\frac{d+5}{2}}}\sum\limits_{\ell_{\nu}=0}\int_{0}^{\infty}\lambda^{2}\varphi(\lambda)\lambda^{\frac{d-1}{2}}e^{\mathrm{i}\frac{\lambda}{h}t}E_{\textrm{free},\frac{\lambda}{h}}(P_{Q}\Phi_{\nu})(r,\theta)\Phi_{\nu}^{*}(\theta^{\prime})\mathrm{d}\lambda=$
$\displaystyle\frac{1}{\sqrt{2\pi}2^{\frac{d-2}{2}}\Gamma(\frac{d}{2})h^{\frac{d+5}{2}}}$
$\displaystyle\times\sum\limits_{\ell_{\nu}=0}\int_{0}^{\infty}\varphi(\lambda)\lambda^{\frac{d+3}{2}}e^{\mathrm{i}\frac{\lambda}{h}t}\sum_{\ell_{\mu}=0,2}a_{\mu}(P_{Q}\Phi_{\nu})E_{\textrm{free},\frac{\lambda}{h}}(\Phi_{\mu})(r,\theta)\Phi_{\nu}^{*}(\theta^{\prime})\mathrm{d}\lambda=$
$\displaystyle\frac{1}{\sqrt{2\pi}2^{\frac{d-2}{2}}\Gamma(\frac{d}{2})h^{d+2}}$
$\displaystyle\times\sum\limits_{\ell_{\nu}=0}\int_{0}^{\infty}\varphi(\lambda)\lambda^{d+1}e^{\mathrm{i}\frac{\lambda}{h}t}\sum_{\ell_{\mu}=0,2}a_{\mu}(P_{Q}\Phi_{\nu})(-\mathrm{i})^{\ell_{\mu}}j_{d,\ell_{\mu}}\left(\frac{\lambda
r}{h}\right)(\Phi_{\mu})(\theta)\Phi_{\nu}^{*}(\theta^{\prime})\mathrm{d}\lambda$

There is then an exact formula for the free space spherical Bessel functions
in this case using (12.1) and the fact that $2j=h^{(1)}+h^{(2)}$. This reduces
the integral to the following estimate when $\ell_{\mu}=0$

$\displaystyle\left|\int\limits_{0}^{\infty}\frac{e^{\mathrm{i}\frac{\lambda}{h}(t-r)}}{r^{\frac{d-1}{2}}}\varphi(\lambda)\,d\lambda\right|\leq\frac{C}{r^{\frac{d-1}{2}}}\leq
C\left(\frac{1}{t}\right)^{\frac{d-1}{2}}$ (6.9)
$\displaystyle|t-r|<\sqrt{h},\quad t\geq 1,\quad h<1,$

and we also have that

$\displaystyle\left|\int\limits_{0}^{\infty}\frac{e^{\mathrm{i}\frac{\lambda}{h}(t-r)}}{r^{\frac{d-1}{2}}}\varphi(\lambda)\,d\lambda\right|\leq
C_{N}h^{\frac{N}{2}}\qquad|t-r|\geq{\sqrt{h}}.$ (6.10)

Here the constant $C_{N}$ depends on the $N$-th derivative of $\varphi$. In
the first estimate we have used the fact that $|\frac{1}{1-x}|\leq 2$ for
$|x|<1$. We have similar estimates for the spherical Bessel functions
$j_{d,2}(\lambda r)$ when $\ell_{\mu}=2$ because they have an explicit form.
Combining the last two inequalities and the one for $\ell_{\mu}=2$ yields the
desired result. We do not have to worry about the norms of the
$\Phi_{\mu}(\theta)$ growing polynomially as we are summing over finitely many
small indices. When we lose translation invariance because we have introduced
an object into the computations, control over the sum becomes much more
difficult. The second set of estimates in proof of Theorem 3.1 follow
trivially by removing the $\lambda^{2}$ from the previous integrals using
Corollary 3.

∎

##  7 Scattering estimates in the case of the ball

###  7.1 Generalized Eigenfunctions for Maxwell’s equations

For Maxwell’s equations we are only using the case $d=3$. For these equations
there is a special orthonormal basis of spherical harmonics. These allow us to
calculate scattering matrices more easily as in this basis the scattering
matrix is diagonal in the exterior of a sphere. Let $\ell\in\mathbb{N}$ and
$|m|\leq\ell$. We start by defining the ordinary scalar harmonics

$\displaystyle Y_{\ell
m}(\theta,\phi)=\sqrt{\frac{\epsilon_{m}}{2\pi}}\sqrt{\frac{2\ell+1}{2}\frac{(\ell-|m|)!}{(\ell+|m|)!}}P_{\ell}^{|m|}(\cos\theta)\begin{cases}\cos
m\phi\\\ \sin m\phi\end{cases}$ (7.1)

where the Neumann factor $\epsilon_{m}$ is defined as

$\displaystyle\epsilon_{m}=2-\delta_{m0}\quad\epsilon_{0}=1\quad\epsilon_{m}=2\quad
m>0.$ (7.2)

We could replace the orthonormal basis $\Phi_{\mu}$ in each of these
representations in Definition (2) with an orthonormal basis of vector valued
spherical harmonics $Y_{\mu}$ which would consist of a three dimensional
vector with the ordinary spherical harmonics. This change of basis would then
be immediate because there is an explicit change of basis formula between
spherical harmonics and the harmonic polynomials on the sphere. But this still
leads to a complicated scattering matrix, therefore it makes sense to change
basis to what is known as vector spherical harmonics, or VSH.

For our calculations and representation theorems to hold with vector spherical
harmonics we will have to construct new generalized eigenfunctions for
Maxwell’s equations. However we have to make some changes to definitions of
the generalized eigenfunctions because the vector spherical harmonics do not
solve the same equations that the harmonic polynomials on the sphere do. There
is also a corresponding new scattering matrix $S_{\lambda,Y}^{t}$.

We start with the definition of the vector spherical harmonics, sometimes
abbreviated VSH in the literature. Let $x\in\mathbb{R}^{3}$ so that
$x=r\hat{r}$ where
$\hat{r}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ and $r>0$ is a
scalar. Now we can define the vector spherical harmonics on the surface
$\\{x\in\mathbb{R}^{3}:|x|=r\\}$ as follows

$\displaystyle\Psi_{1,\ell,m}=\frac{1}{\sqrt{\ell(\ell+1)}}\nabla Y_{\ell
m}\times x\qquad\Psi_{2,\ell,m}=\frac{r}{\sqrt{\ell(\ell+1)}}\nabla Y_{\ell
m}\qquad\Psi_{3,\ell,m}=\hat{r}Y_{\ell m}$ (7.3)

We re-label them simply as $Y_{\mu}$ when we restrict to the sphere $r=1$,
this is elaborated on in the Appendix subsection 12.5. The index $\mu$ is then
$(j,\ell,m)$ where $j=1,2,3$, $\ell\in\mathbb{N}$ and $|m|\leq\ell$. These VSH
form an orthonormal basis for the space
$L^{2}(\mathbb{S}^{2},\Lambda^{1}\mathbb{C}^{3})$. The $\Psi_{1,\ell,m}$ are
divergence free on the sphere, the $\Psi_{2,\ell,m}$ are curl free on the
sphere and $\Psi_{3,\ell,m}\in\mathcal{H}^{1}_{rel}(\mathbb{S}^{2})$,
corresponding to the Hodge decomposition theorem. Notice that
$\Psi_{1,\ell,m}$ is harmonic of degree $\ell$, whereas $\Psi_{2,\ell,m}$ and
$\Psi_{3,\ell,m}$ are linear combinations of spherical harmonics of degree
$\ell-1$ and $\ell+1$. This will become important for later computations. We
supress the subscript $d$ where it is understood as when we refer to the VSH
$d=3$.

We can then define the free space and general Hankel functions corresponding
to $\tilde{j}_{\lambda}^{(1)}(\Phi)$, $\tilde{h}_{\lambda}^{(1)}(\Phi)$, and
$\tilde{h}_{\lambda}^{(2)}(\Phi)$ respectively in this basis. Because we are
only in dimension 3, it makes more sense to use the language of vectors rather
than forms for this section and the related ones. We define the space
$L^{2}_{t}(\mathbb{S}^{2},\mathbb{C}^{3})$ as

$\displaystyle L^{2}_{t}(\mathbb{S}^{2},\mathbb{C}^{3})=\\{\nu\cdot
f=0\,\,\,\mathrm{on}\,\,\,\mathbb{S}^{2}\\}.$ (7.4)

Then we further have $\Psi_{j,\ell,m}$ with $j=1,2$ form an orthonormal basis
for the subspace $L^{2}_{t}(\mathbb{S}^{2},\mathbb{C}^{3})$.

A main observation of this paper is to establish the relation between Mie
scattering and spectral representations for operators governing the evolution
of Maxwell equations. When we consider perfectly conducting boundary
conditions, we will be able to reconstruct a replacement of Proposition 7 in
this coordinate system. Mie scattering has a long history of successful use in
modelling electromagnetic fields outside of obstacles starting with the work
of Gustav Mie [30].

In order to introduce the analogue of Proposition 7, we start with the
description of the time-harmonic Maxwell equations. We denote by $E$, $H$ the
electric and magnetic fields, respectively. The time-harmonic Maxwell-system
for a generic $M$ is

$\displaystyle\mathrm{curl}\,E-\mathrm{i}\lambda H=0$
$\displaystyle\mathrm{div}E=0$
$\displaystyle\mathrm{curl}\,H+\mathrm{i}\lambda E=0$ (7.5)
$\displaystyle\mathrm{div}\,H=0$ $\displaystyle\nu\times
E=F\quad\mathrm{on}\quad\partial\cal{O}$
$\displaystyle\langle\nu,H\rangle=F_{H}\quad\mathrm{on}\quad\partial\cal{O}.$

Here $\nu$ is the almost everywhere defined outward pointing unit normal
vector field on $\partial\cal{O}$. An important observation which will show up
often here is the direct relationship between solutions of the Helmholtz
equation and Maxwell’s equations. Recall that we take the definition of the
Laplacian which has positive spectrum. We see that

######  Lemma 1 (Lemma 2.42 in [17]).

Let $\lambda\neq 0$. If $u(x)$ is a scalar solution to the Helmholtz equation
in $\mathbb{R}^{3}$, that is $\Delta u-\lambda^{2}u=0$, then the pairs
$(E_{1},H_{1})$ and $(E_{2},H_{2})$ where

$\displaystyle E_{1}(x)=\mathrm{curl}(xu(x))\quad\mathrm{and}\quad
H_{1}(x)=\frac{1}{\mathrm{i}\lambda}\mathrm{curl}E_{1}(x)$

and

$\displaystyle
E_{2}(x)=\mathrm{curl}\mathrm{curl}(xu(x))\quad\mathrm{and}\quad
H_{2}(x)=\frac{1}{\mathrm{i}\lambda}\mathrm{curl}E_{2}(x)$

are solutions to the time-harmonic Maxwell system (7.1) whenever
$\mathcal{O}=\emptyset$.

######  Remark 1.

The main advantage of Lemma 1 is that electric field solutions, denoted by,
$E_{\lambda}$, to the system (7.1) can be expressed in forms of $E_{1}$ and
$E_{2}$ in the Lemma. This is because, by the Hodge decomposition (2.1) of
$\Delta_{\mathrm{rel}}$, we know that when $\lambda\neq 0$,
$E_{\lambda}=\delta A_{\lambda}+\mathrm{d}\phi_{\lambda}$ with $A_{\lambda}$
being a $2$-form and $\phi_{\lambda}$ being a scalar. Since we want $\delta
E_{\lambda}=0$, this gives $\Delta\phi_{\lambda}=0$, which implies
$\phi_{\lambda}=0$. Therefore, we can seek $E_{\lambda}$ in the form of
$\mathrm{curl}\underline{A_{\lambda}}$, where
$\underline{A_{\lambda}}=*A_{\lambda}$. In other words, $E_{\lambda}$ is in
the form of $E_{1}$ and $E_{2}$ of the Lemma. Similar arguments hold for
magnetic field $H_{\lambda}$.

Indeed, when there is no obstacle, Bessel functions can be used in this way to
construct a solution to the time-harmonic Maxwell system (7.1). This is taken
from [17].

######  Theorem 3 (Theorem 2.43 in [17]).

Let $x=r\hat{r}$. The vector field
$E_{1}:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ defined by

$\displaystyle E_{1}(x)=\sqrt{\ell(\ell+1)}\mathrm{curl}(xj_{\ell}(\lambda
r)Y_{\ell m})=j_{\ell}(\lambda r)\Psi_{1,\ell,m}$

and the corresponding vector field

$\displaystyle
H_{1}(x)=\frac{\mathrm{curl}E(x)}{\mathrm{i}\lambda}=\frac{\ell(\ell+1)}{\mathrm{i}\lambda
r}j_{\ell}(\lambda
r)\Psi_{3,\ell,m}+\frac{\sqrt{\ell(\ell+1)}}{\mathrm{i}\lambda
r}(xj_{\ell}(x))^{\prime}|_{x=\lambda r}\Psi_{2,\ell,m}$

solve the corresponding Maxwell system (7.1).
Analogously the vector fields

$\displaystyle E_{2}(x)=\mathrm{curl}\mathrm{curl}(xj_{\ell}(\lambda r)Y_{\ell
m})=$ $\displaystyle\ell(\ell+1)\frac{j_{\ell}(\lambda
r)}{r}\Psi_{3,\ell,m}+\sqrt{\ell(\ell+1)}\frac{(xj_{\ell}(x))|_{x=\lambda
r}^{\prime}}{r}\Psi_{2,\ell,m}$

and

$H_{2}(x)=\frac{\mathrm{curl}E(x)}{\mathrm{i}\lambda}$

also solve the Maxwell system (7.1).

###### Proof.

We sketch the proof of the first part directly from [17] for only for
completeness. Using the fact that $\mathrm{curl}x=0$ and

$\mathrm{curl}(\lambda F)=(\nabla\lambda)\times F+\lambda\mathrm{curl}F$

for $F$ and $\lambda$ scalar functions, we have that

$\displaystyle\mathrm{curl}(xj_{\ell}(\lambda r)Y_{\ell m})$
$\displaystyle=\nabla(j_{\ell}(\lambda r)Y_{\ell m})\times(r\hat{r})$
$\displaystyle=\left(\frac{1}{r}j_{\ell}(\lambda
r)\mathrm{Grad}_{\mathbb{S}^{2}}Y_{\ell m}+\frac{\partial}{\partial
r}\left(j_{\ell}(\lambda r)\right)Y_{\ell m}\hat{r}\right)\times x.$

This gives the first part. For the second part we have that
$\mathrm{curl}\mathrm{curl}=\Delta+\nabla\mathrm{div}$ and the fact that
$w=j_{\ell}(\lambda r)Y_{\ell m}$ solves the Helmholtz equation. We see that

$\mathrm{curl}E(x)=\Delta(xw(x))+\nabla\mathrm{div}(xw(x))=-2\nabla
w(x)+\lambda^{2}xw(x)+3\nabla w(x)+\nabla(x\cdot\nabla w(x))$

which when all the derivatives are written out can be simplified using
Bessel’s equation (12.14) gives the desired result. ∎

We would like to show the following analogue of Proposition 1.2 in [43] is
applicable in three dimensions with $\Phi$ replaced by $Y$.

######  Proposition 5 (Analogue Proposition 1.2 in [43]).

For fixed $\lambda>0$ the generalised eigenfunctions $E_{\lambda}(Y)$ are
completely determined by the following

  1. 1.

$(\Delta_{1,\mathrm{rel}}-\lambda^{2})E_{\lambda}(Y)=0$,

  2. 2.

$\chi E_{\lambda}\in\mathrm{dom}(\Delta_{1,\mathrm{rel}})$ for any $\chi\in
C^{\infty}_{0}(M)$ with $\nabla\chi=0$ near $\partial\Omega$,

  3. 3.

The asymptotic expansion

$E_{\lambda}(Y)=\frac{\mathrm{e}^{-\mathrm{i}\lambda
r}\mathrm{e}^{\frac{i\pi}{2}}}{r}Y+\frac{\mathrm{e}^{\mathrm{i}\lambda
r}\mathrm{e}^{-\frac{i\pi}{2}}}{r}\Psi_{\lambda}+O\left(\frac{1}{r^{2}}\right),\quad\textrm{for}\,\,\,r\to\infty.$

for some $\Psi_{\lambda}\in C^{\infty}(\mathbb{S}^{2};\mathbb{C}^{3})$.

As a result in three dimensions $\Psi_{\lambda}$ is uniquely determined and
implicitly defines a linear mapping

$\displaystyle S_{\lambda}:C^{\infty}(\mathbb{S}^{2};\mathbb{C}^{3})\to
C^{\infty}(\mathbb{S}^{2};\mathbb{C}^{3}),\quad Y\mapsto\tau\Psi_{\lambda},$

where

$\tau:C^{\infty}(\mathbb{S}^{2};\mathbb{C}^{3})\to
C^{\infty}(\mathbb{S}^{2};\mathbb{C}^{3})$

is the pull-back of the antipodal map. The map
$S_{\lambda}:C^{\infty}(\mathbb{S}^{2},\mathbb{C}^{3})\to
C^{\infty}(\mathbb{S}^{2},\mathbb{C}^{3})$ is called the scattering matrix,
and $A_{\lambda}=S_{\lambda}-\mathrm{id}$ is called the scattering amplitude.
Similar to the Hodge-Helmholtz decomposition, the scattering matrix is then of
the form

$\displaystyle S_{\lambda}=\left(\begin{matrix}S_{n,\lambda}&0\\\
0&S_{t,\lambda}\end{matrix}\right),$ (7.6)

if $C^{\infty}(\mathbb{S}^{2},\mathbb{C}^{3})$ is decomposed as
$C^{\infty}_{n}(\mathbb{S}^{2},\mathbb{C}^{3})\oplus
C^{\infty}_{t}(\mathbb{S}^{2},\mathbb{C}^{3})$ into normal and tangential
parts. If we had this proposition, then this will allow us to define
divergence free generalized eigenfunctions for the electric field in $M$. In
order to try to prove this analogue of Proposition 1.2 from [43] we use
Theorem 3 in [17]. We make the definitions

######  Definition 3.

For $Y\in L^{2}_{t}(\mathbb{S}^{2},\mathbb{C}^{3})$, let $a_{j,\ell,m}=\langle
Y,\Psi_{j,\ell,m}\rangle$ then we define

$\displaystyle\tilde{j}_{\lambda,Y}(Y)=2\sum\limits_{\ell,m}a_{1,\ell,m}\Psi_{1,\ell,m}\lambda
j_{\ell}(\lambda r)(-\mathrm{i})^{\ell}+$ (7.7) $\displaystyle
a_{2,\ell,m}\Psi_{2,\ell,m}\frac{\left(xj_{\ell}(x)\right)^{\prime}}{r}|_{x=\lambda
r}(-\mathrm{i})^{\ell-1}+a_{2,\ell,m}\Psi_{3,\ell,m}\frac{\sqrt{\ell(\ell+1)}j_{\ell}(\lambda
r)}{r}(-\mathrm{i})^{\ell-1}$

and also

$\displaystyle\tilde{h}^{(2)}_{\lambda,Y}(Y)=\sum_{\ell,m}a_{1,\ell,m}\Psi_{1,\ell,m}\lambda
h^{(2)}_{\ell}(\lambda r)(-\mathrm{i})^{\ell}+$ (7.8) $\displaystyle
a_{2,\ell,m}\Psi_{2,\ell,m}\frac{\left(xh^{(2)}_{\ell}(x)\right)^{\prime}}{r}|_{x=\lambda
r}(-\mathrm{i})^{\ell-1}+a_{2,\ell,m}\Psi_{3,\ell,m}\frac{\sqrt{\ell(\ell+1)}h^{(2)}_{\ell}(\lambda
r)}{r}(-\mathrm{i})^{\ell-1}$

and

$\displaystyle\tilde{h}^{(1)}_{\lambda,Y}(Y)=\sum_{\ell,m}a_{1,\ell,m}\Psi_{1,\ell,m}\lambda
h^{(1)}_{\ell}(\lambda r)(-\mathrm{i})^{\ell}+$ (7.9) $\displaystyle
a_{2,\ell,m}\Psi_{2,\ell,m}\frac{\left(xh^{(1)}_{\ell}(x)\right)^{\prime}}{r}|_{x=\lambda
r}(-\mathrm{i})^{\ell-1}+a_{2,\ell,m}\Psi_{3,\ell,m}\frac{\sqrt{\ell(\ell+1)}h^{(1)}_{\ell}(\lambda
r)}{r}(-\mathrm{i})^{\ell-1}$

where these sums are defined when they converge in
$C^{\infty}(M,\mathbb{C}^{3})$.

We can then define a Maxwell generalized eigenfunction using the basis of
vector spherical harmonics, $E_{\lambda}^{\mathrm{curl}}(Y)$, in the exterior
of $\mathcal{O}$ an open ball of radius $\rho$. Notice that in this case

$\mathbb{R}^{3}\setminus\overline{\mathcal{O}}=X\setminus\overline{\mathcal{O}}=M\setminus
K=M^{\circ}.$

We start with the following proposition

######  Proposition 6.

Let $\mathcal{O}$ be an open ball of radius $\rho$. We have that

$\displaystyle\chi\tilde{j}_{\lambda,Y}(Y)-R_{\lambda}(\Delta_{1,\mathrm{rel}}-\lambda^{2})(\chi\tilde{j}_{\lambda,Y}(Y))=E_{\lambda}(Y)$
(7.10)

and this is divergence free. Moreover for $|\lambda|<K$ and
$\mathrm{Im}\lambda\geq 0$ we have

$\displaystyle E_{\lambda}(Y_{\nu})=O_{C^{\infty}(\mathbb{R}^{3}\setminus
0)}\left(\frac{\lambda^{\ell_{\nu}-1}R^{\ell_{\nu}}}{\Gamma\left(\ell_{\nu}+\frac{3}{2}\right)}\right)$
(7.11)

so that $E_{\lambda}(Y)\in C^{\infty}_{0}(M,\mathbb{C}^{3})$.

###### Proof.

We have that $E_{\lambda}(Y)$ solves the desired equation by construction. The
divergence free property holds by (5.4) which also hold in this basis. This
proof follows similarly to Lemma 2.9 in [41] and is included for completeness.
We have for any $K>0$ and $|\lambda|<K$ that

$\displaystyle\tilde{j}_{\lambda,Y}(Y_{\nu})=O_{C^{\infty}(\mathbb{R}^{3}\setminus
0)}\left(\frac{\lambda^{\ell_{\nu}+1}R^{\ell_{\nu}}}{\Gamma\left(\ell_{\nu}+\frac{3}{2}\right)}\right)$
(7.12)

uniformly in $\nu$. This follows from the estimate in Appendix 12.13

$\displaystyle|J_{\mu}(z)|\leq\frac{|\frac{z}{2}|^{\mu}e^{|\operatorname{Im}z|}}{\Gamma(\mu+1)}\quad\mu>-\frac{1}{2}$
(7.13)

and the derivative identity in the Appendix, (12.20). The family
$\lambda^{-\ell_{\nu}}\tilde{j}_{\lambda,Y}(Y_{\nu})$ is bounded in
$L^{2}_{\mathrm{loc}}(\mathbb{R}^{3}\setminus 0)$. From Theorem 3 we have that
$\Delta_{1,\mathrm{rel}}\tilde{j}_{\lambda,Y}(Y_{\nu})=\lambda^{2}\tilde{j}_{\lambda,Y}(Y_{\nu})$.
This gives that the family is bounded in
$H^{s}_{\mathrm{loc}}(\mathbb{R}^{3}\setminus 0)$ for $s\in 2\mathbb{N}$. We
note that $R_{\lambda}=\mathcal{O}(\lambda^{-2})$ if we consider it as a map
$H^{s}_{\mathrm{com}}(M)\rightarrow H_{\mathrm{loc}}^{s+2}(M)$. This gives the
desired result. ∎

We want to relate our divergence free generalized eigenfunction to

$\displaystyle
E_{\lambda}^{\mathrm{curl}}(Y)|_{M}=\tilde{j}_{\lambda,Y}(Y)+\tilde{h}^{(1)}_{\lambda,Y}(A_{\lambda,Y}Y)$
(7.14)

for some matrix $A_{\lambda,Y}$. However before proceeding to establish a
relationship to the functional calculus first we need more information about
the scattering amplitude. In particular without the computation of
$A_{\lambda,Y}$ it is not clear why the representations converge in
$C^{\infty}(M,\mathbb{C}^{3})$. We claim

######  Lemma 2.

If $Y\in L^{2}_{t}(\mathbb{S}^{2},\mathbb{C}^{3})$ then $A_{\lambda,Y}\in
C^{\infty}(\mathbb{S}^{2},\mathbb{C}^{3})$ can be chosen uniquely in (7.14) so
that

  1. 1.

$E_{\lambda}^{\mathrm{curl}}(Y)|_{\partial\cal{O}}=0$

  2. 2.

$\nu\times E_{\lambda}^{\mathrm{curl}}(Y)=0$

  3. 3.

$E_{\lambda}^{\mathrm{curl}}(Y)$ solves the equation
$(\mathrm{curl}\mathrm{curl}-\lambda^{2})u=0$.

  4. 4.

$\varphi
E_{\lambda}^{\mathrm{curl}}(Y)\in\mathrm{dom}(\Delta_{1,\mathrm{rel}})$ for
any $\varphi\in C_{0}^{\infty}(M)$ with $\nabla\varphi=0$ near
$\partial\mathcal{O}$.

###### Proof.

If we define

$g=E_{\lambda}^{\mathrm{curl}}(Y)-\chi\tilde{j}_{\lambda,Y}(Y)$

then $g$ is outgoing for $\lambda\neq 0$ and smooth. It follows from the
Appendix, Lemma 5 that on $M\setminus K=M^{\circ}$ we have that
$g=\tilde{h}_{\lambda}^{(1)}(A_{\lambda,Y}(Y))$ for a unique
$A_{\lambda,Y}(Y)\in C^{\infty}(\mathbb{S}^{2},\mathbb{C}^{3})$. The third and
fourth point follows by the definition of the explicit functions involved.
Uniqueness follows by the asymptotic expansions, as a result of Appendix 12.4.
Indeed we have that if $Y\in L^{2}_{t}(\mathbb{S}^{2},\mathbb{C}^{3})$ is
smooth then we have

$\displaystyle E_{\lambda}^{\mathrm{curl}}(Y)=\frac{e^{-\mathrm{i}\lambda
r}e^{\frac{\mathrm{i}\pi}{2}}}{r}Y+\frac{e^{\mathrm{i}\lambda
r}e^{-\frac{\mathrm{i}\pi}{2}}}{r}(\tau(Y)+\tau(A_{\lambda,Y}Y))+O\left(\frac{1}{r^{2}}\right)\quad
r\rightarrow\infty$ (7.15)

where $\tau:C^{\infty}(\mathbb{S}^{2},\mathbb{R}^{3})\rightarrow
C^{\infty}(\mathbb{S}^{2},\mathbb{R}^{3})$, $f(\theta)\mapsto f(-\theta)$ is
the pull back of the antipodal map. This is the classical way to find the
scatting matrix [26], Section 5.1. Uniqueness of the representation follows
immediately from Rellich’s theorem. In other words, for every $Y\in
C^{\infty}(\mathbb{S}^{2},\mathbb{R}^{3})$ there exists a unique
$\Psi_{\lambda}\in C^{\infty}(\mathbb{S}^{2},\mathbb{R}^{3})$ and a uniqe
solution of $E_{\lambda}^{\mathrm{curl}}(Y)$ of
$(\Delta_{1,\mathrm{rel}}-\lambda^{2})E_{\lambda}(Y)=0$ such that

$\displaystyle E_{\lambda}^{\mathrm{curl}}(Y)=\frac{e^{-\mathrm{i}\lambda
r}e^{\frac{\mathrm{i}\pi}{2}}}{r}Y+\frac{e^{\mathrm{i}\lambda
r}e^{-\frac{\mathrm{i}\pi}{2}}}{r}\Psi_{\lambda}+O\left(\frac{1}{r^{2}}\right)\quad
r\rightarrow\infty$ (7.16)

by comparison with the above we get $\Psi_{\lambda}=\tau(Y+A_{\lambda,Y}Y)$.
Therefore the representation agrees with $E_{\lambda}^{\mathrm{curl}}(Y)$ on
$M^{\circ}$. The boundary conditions on $\mathcal{O}$ match each other so this
representation can be extended to $M$. The explicit form of $A_{\lambda,Y}$ is
found in the next subsection 7.2. ∎

Finding the matrix $A_{\lambda}$ is known as Mie scattering where usually this
is referred to as the T-matrix in the engineering and physics literature. The
problem can also involve mixed boundary conditions for $F$ and $F_{H}$ as in
(7.1) rather than the two boundary conditions in Lemma 2. The verification
that the corresponding time-harmonic solution is in
$C^{\infty}(M,\mathbb{C}^{3})$ is done ad hoc in the physics literature.

Lemma 2 shows that the conditions of Proposition 1.2 in [43] are satisfied in
order, but instead of the relative Laplacian we have the curl operator. We
could, of course, use a lengthy calculation to find $E_{\lambda}(Y)$ in this
basis. However our $E_{\lambda}^{\mathrm{curl}}(Y)$ are already constructed so
that $\mathrm{div}E_{\lambda}^{\mathrm{curl}}(Y)=0$ and $Y\in
L^{2}_{t}(\mathbb{S}^{2},\mathbb{C}^{3})$. The major result of this section is
then the observation that instead of a representation for the time-harmonic
Maxwell equations, Lemma 2 enables us to take this representation one step
further and use the functional calculus to represent evolution operators
associated to perfectly conducting objects.

######  Proposition 7.

Let $f,g\in C^{\infty}_{0}(M,\mathbb{R}^{3})$ and $\lambda>0$. For any bounded
Borel function $k:\mathbb{R}\to\mathbb{C}$ we have If $\mathrm{div}f=0$, and
$F=0$ in Lemma 2 then the Stone’s formula for divergence free vector fields is
given by

$\displaystyle(R_{\lambda}-R_{-\lambda})f=\frac{\mathrm{i}}{2\lambda}\sum\limits_{\nu}E_{\lambda}^{\mathrm{curl}}(Y_{\nu})\langle
f,E_{\lambda}^{\mathrm{curl}}(Y_{\nu})\rangle$ (7.17)

in $C^{\infty}(M,\mathbb{R}^{3})$. This implies

$\langle
P_{ac}k(\Delta_{1,\mathrm{rel}})f,g\rangle=\frac{1}{2\pi}\sum\limits_{\nu}\int_{0}^{\infty}k(\lambda^{2})\langle
f,E_{\lambda}^{\mathrm{curl}}(Y_{\nu})\rangle\langle
E_{\lambda}^{\mathrm{curl}}(Y_{\nu}),g\rangle\,\mathrm{d}\lambda,$ (7.18)

where $P_{ac}$ is the projection onto the absolutely continuous spectrum.

The result of Proposition 1.2 in [43] which follows from Lemma 2 gives that
$E_{\lambda}^{\mathrm{curl}}(Y)=E_{\lambda}(Y)$ is a generalized eigenfunction
of the 1 form Laplacian with relative boundary conditions acting on the
subspace $H_{1}$. Stone’s theorem as well as its corollaries in the previous
section hold by replacing $E_{\lambda}(P_{Q}\Phi_{\nu})$ with
$E_{\lambda,Y}(Y_{\nu})$, using Rellich’s theorem (see Appendix) and the proof
is similar to the proof of Corollary 3. We include it here for completeness
since we have a new eigenbasis.

###### Proof of Proposition 7.

Assume that $f\in C_{0}^{\infty}(M,\mathbb{R}^{3})$ is a divergence free
vector field. Then for fixed $\lambda>0$ we have that

$R_{-\lambda}f$

is incoming. We also know that incoming waves can be expanded as Hankel
functions by Lemma 5 in Appendix 12.5. This has asymptotic behavior

$\displaystyle R_{-\lambda}f=\frac{e^{-\mathrm{i}\lambda
r}e^{\frac{\mathrm{i}\pi}{2}}}{r}\Psi+O\left(\frac{1}{r^{2}}\right)$ (7.19)

as $r\rightarrow\infty$ whenever $\Psi\in
C^{\infty}(\mathbb{S}^{2},\mathbb{C}^{3})$.
Therefore we have that

$\displaystyle(R_{\lambda}-R_{-\lambda})f=\frac{e^{-\mathrm{i}\lambda
r}e^{\frac{\mathrm{i}\pi}{2}}}{r}\Psi+\frac{e^{\mathrm{i}\lambda
r}e^{-\frac{\mathrm{i}\pi}{2}}}{r}\tilde{\Psi}+O\left(\frac{1}{r^{2}}\right)$
(7.20)

as $r\rightarrow\infty$ for some $\tilde{\Psi}\in
C^{\infty}(\mathbb{S}^{2},\mathbb{C}^{3})$.
Because $E_{\lambda}(\Psi)$ and

$\displaystyle(R_{\lambda}-R_{-\lambda})f$ (7.21)

have the same asymptotic term then they necessarily coincide by Rellich’s
uniqueness theorem. Integration by parts results in

$\displaystyle\langle
f,E_{\lambda}(Y_{\nu})\rangle_{L^{2}(M)}=\langle(\Delta-\lambda^{2})R_{-\lambda}f,E_{\lambda}(Y_{\nu})\rangle_{L^{2}(\mathbb{S}^{2})}=-2\mathrm{i}\lambda\langle\Psi,Y_{\nu}\rangle_{L^{2}(\mathbb{S}^{2})}$
(7.22)

Expanding $\Psi$ into the basis of vector spherical harmonics we see that

$\displaystyle\Psi=\sum\limits_{\nu}\langle\Psi,Y_{\nu}\rangle_{L^{2}(M)}Y_{\nu}$
(7.23)

with the sum converging in $C^{\infty}(\mathbb{S}^{2},\mathbb{C}^{3})$ because
$\Psi$ is smooth. Therefore we have that

$\displaystyle\Psi=\frac{i}{2\lambda}\sum\limits_{\nu}\langle
f,E_{\lambda}(Y_{\nu})\rangle_{L^{2}(M)}Y_{\nu}$ (7.24)

It follows that

$\displaystyle(R_{\lambda}-R_{-\lambda})f=E_{\lambda}(\Psi)=\frac{\mathrm{i}}{2\lambda}\sum\limits_{\nu}E_{\lambda}(Y_{\nu})\langle
f,E_{\lambda}(Y_{\nu})\rangle_{L^{2}(M)}.$ (7.25)

From the properties of the resolvent and $E_{\lambda,Y}(Y)$, the map $Y\mapsto
E_{\lambda}(Y)$ for $Y\in L^{2}_{t}(\mathbb{S}^{2},\mathbb{C}^{3})$ is
continuous. The sum converges in $C^{\infty}(M,\mathbb{C}^{3})$ due to the
construction in Lemma 3 and the bounds there. The representation formula
follows. One can compute using the Bessel and Hankel funnction asympototics
that the equalities still hold for $\operatorname{Im}\lambda>0$ uniformly on
compact subsets of the complex plane. Combining these observations, Stone’s
theorem then gives the spectral measure and the complete spectral
decomposition. For a reference see [9], Theorem 4.20 in the black-box setting.
∎

Notice we have circumvented the actual construction of $E_{\lambda}(Y)$ in
this basis which would involve using the operator $\mathrm{div}$ instead of
$\mathrm{curl}\mathrm{curl}$. In further sections we will drop the curl
superscript where it is understood. Proposition 4 also holds in this basis.
Indeed, we have that the analogous statement when
$M=\mathbb{R}^{3}\setminus\mathcal{O}$.

######  Proposition 8.

If $k$ is a Borel function with $k=O((1+\lambda^{2})^{-q})$ for all
$q\in\mathbb{N}$ we have that $k(\Delta_{1,\mathrm{rel}})$ has smooth integral
kernel $\kappa_{k}\in C_{0}^{\infty}(M\times
M;\mathbb{C}^{3}\times\mathbb{C}^{3})$ and

$\displaystyle\kappa_{k}(x,y)=k(0)\sum_{j=1}^{N}u_{j}(x)\otimes(u_{j}(y))^{*}+\frac{1}{2\pi}\sum\limits_{\nu}\int_{0}^{\infty}k(\lambda^{2})E_{\lambda}(Y_{\nu})(x)\otimes
E_{\lambda}(Y_{\nu})^{*}(y)\,\mathrm{d}\lambda,$ (7.26)

where the sum converges in $C^{\infty}_{0}(M\times
M;\mathbb{C}^{3}\times\mathbb{C}^{3})$.

The proof of Proposition 8 is follows basically identically to Proposition 4
after Proposition 7 has been established and therefore will not be repeated
here.

###  7.2 Mie Scattering for the sphere

We have that $E_{\lambda}(Y_{\mu})$ is given by

$E_{Y}(Y_{\mu})|_{\mathbb{R}^{3}\backslash\cal{O}}=\tilde{j}_{\lambda}(Y_{\mu})+\tilde{h}_{\lambda}^{(1)}(A_{\lambda,Y}Y_{\mu})=\tilde{h}_{\lambda}^{(2)}(Y_{\mu})+\tilde{h}_{\lambda}^{(1)}(S_{\lambda,Y}Y_{\mu}).$

We know that $A_{\lambda,Y}=A_{\lambda,Y}^{t}$ on $Y\in
L^{2}_{t}(\mathbb{S}^{2},\mathbb{C}^{3})$. This formula is exact in the
exterior of a sphere in 3 dimensions. In any dimension, representations for
$E_{\lambda}(\Phi)$ using Mie scattering could be done in the basis of
harmonic polynomials $\Phi_{\nu}$, but their corresponding scattering matrix
is not diagonal which is why we chose to switch to this basis. This will allow
us to do calculations with the kernel defined in Proposition 8 explicitly for
the Maxwell propagator. We drop the $Y$ where it is understood. We start with
the following Lemma

######  Lemma 3.

The Maxwell generalized eigenfunction for a perfectly conducting sphere of
radius $\rho$ is given by

$E_{\lambda}(Y_{\mu})|_{\mathbb{R}^{3}\backslash\cal{O}}=\tilde{j}_{\lambda}(Y_{\mu})+\tilde{h}_{\lambda}^{(1)}(A^{t}_{\lambda,Y}Y_{\mu})$
(7.27)

where $A^{t}_{\lambda,Y}$ is a diagonal matrix given by

$\displaystyle\langle
A^{t}_{\lambda,Y}\Psi_{1,\ell,m},\Psi_{1,\ell,m}\rangle=-\frac{2j_{\ell}(\lambda\rho)}{h_{\ell}^{(1)}(\lambda\rho)},$

and

$\displaystyle\langle
A^{t}_{\lambda,Y}\Psi_{2,\ell,m},\Psi_{2,\ell,m}\rangle=-\frac{2(xj_{\ell}(x))^{\prime}|_{x=\lambda\rho}}{(xh_{\ell}^{(1)}(x))^{\prime}|_{x=\lambda\rho}},$

where $\Psi_{j,\ell,m}$ $j=1,2$ form a basis for
$L^{2}_{t}(\mathbb{S}^{2},\mathbb{C}^{3})$.

###### Proof.

Since $E_{\lambda}(Y_{\mu})$ satisfies relative boundary conditions, we have

$\displaystyle
E_{\lambda}(Y_{\mu})|_{\partial\mathcal{O},\mathrm{tan}}=0,\quad\delta
E_{\lambda}(Y_{\mu})|_{\partial\mathcal{O},\mathrm{tan}}=0,$

which is equivalent to

$\left\\{\begin{aligned}
&(\tilde{j}_{\lambda}(Y_{\mu})+\tilde{h}_{\lambda}^{(1)}(A_{\lambda,Y}^{t}Y_{\mu}))|_{\partial\cal{O}}=0,\\\
&\nu\times(\tilde{j}_{\lambda}(Y_{\mu})+\tilde{h}_{\lambda}^{(1)}(A_{\lambda,Y}Y_{\mu}))=0.\end{aligned}\right.$

In the case of the obstacle being a sphere of radius $\rho$,

$\left\\{\begin{aligned} &\text{For $\mu=(1,\ell,m)$},\quad
2\lambda\Psi_{\mu}(\theta)j_{\ell}(\lambda\rho)(-\mathrm{i})^{\ell}+\lambda\sum\limits_{\nu=(j,\ell_{0},m_{0})}(A_{\lambda,Y})_{\mu,\nu}\Psi_{j,\ell_{0},m_{0}}(\theta)h^{(1)}_{\ell_{0}}(\lambda\rho)(-\mathrm{i})^{\ell_{0}}=0.\\\
&\text{For $\mu=(2,\ell,m)$},\quad
2\Psi_{\mu}\frac{\left(xj_{\ell}(x)\right)^{\prime}}{\rho}|_{x=\lambda\rho}(-\mathrm{i})^{\ell-1}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\quad+\sum\limits_{\nu=(j,\ell_{0},m_{0})}(A_{\lambda,Y})_{\mu,\nu}\Psi_{j,\ell_{0},m_{0}}(\theta)\frac{\left(xh^{(1)}_{\ell}(x)\right)^{\prime}}{\rho}|_{x=\lambda\rho}(-\mathrm{i})^{\ell_{0}-1}=0.\end{aligned}\right.$

Since the vector spherical harmonics with $j=1,2$ form a basis for the space
$L^{2}_{t}(\mathbb{S}^{2},\mathbb{C}^{3})$, the tangential part is given by
the diagonal matrix with two different types of diagonal entries. As a
consequence, $(A_{\lambda})_{j,\ell,m}$ is diagonal and its components are
given by

$\displaystyle\langle
A^{t}_{\lambda}\Psi_{1,\ell,m},\Psi_{1,\ell,m}\rangle_{L^{2}(\mathbb{S}^{2})}=-\frac{2j_{\ell}(\lambda\rho)}{h^{(1)}_{\ell}(\lambda\rho)}.$
(7.28)

and also

$\langle
A^{t}_{\lambda}\Psi_{2,\ell,m},\Psi_{2,\ell,m}\rangle_{L^{2}(\mathbb{S}^{2})}=-2\frac{(xj_{\ell}(x))^{\prime}|_{x=\lambda\rho}}{(xh^{(1)}_{\ell}(x))^{\prime}|_{x=\lambda\rho}}.$
(7.29)

This result is popular in the physics literature [10, 30, 40]. This concludes
the result for $A^{t}_{\lambda}$. The result for $S^{t}_{\lambda}$ follows
immediately from $S=A+I$. ∎

Applying Proposition 8 and Proposition 7 to
$k(\lambda^{2})=\varphi(h\lambda)e^{\mathrm{i}\lambda t}$, we have the kernel
of $k(\Delta_{p,\mathrm{rel}})$ on divergence free vector fields is found by
using (5.13). We obtain in this coordinate system

$\displaystyle K^{\mathcal{O},m}(x,x^{\prime},\rho,t,h)=$
$\displaystyle\frac{1}{2\pi
h}\sum\limits_{\mu}\int_{0}^{\infty}\varphi(\lambda)e^{\mathrm{i}\frac{\lambda}{h}t}\left(E_{\mathcal{O},\frac{\lambda}{h}}(Y_{\mu})(r,\theta)\right)\left(E_{\mathcal{O},\frac{\lambda}{h}}(Y_{\mu})(r^{\prime},\theta^{\prime})\right)^{*}\mathrm{d}\lambda.$
(7.30)

From equation (7.27) (and by omitting the arguments in $Y_{\nu}$, we know that
the kernel (7.2) consists of four terms with
$\tilde{h}^{(2)}\left(\tilde{h}^{(2)}\right)^{*}$,
$\tilde{h}^{(2)}\left(\tilde{h}^{(1)}(S^{t})\right)^{*}$,
$\tilde{h}^{(1)}(S^{t})\left(\tilde{h}^{(2)}\right)^{*}$ and
$\tilde{h}^{(1)}(S^{t})\left(\tilde{h}^{(1)}(S^{t})\right)^{*}$ in the
integrands. For instance, the term containing
$\tilde{h}^{(2)}\left(\tilde{h}^{(1)}(S^{t})\right)^{*}$ reads

$\displaystyle\frac{1}{2\pi
h}\sum\limits_{\mu}\int_{0}^{\infty}\varphi(\lambda)e^{\mathrm{i}\frac{\lambda}{h}t}\left(\tilde{h}^{(2)}_{\frac{\lambda}{h}}\left(Y_{\mu}\right)(r,\theta)\right)\left(\tilde{h}^{(1)}_{\frac{\lambda}{h}}\left(S_{\frac{\lambda}{h}}^{t}Y_{\mu}\right)(r^{\prime},\theta^{\prime})\right)^{*}\mathrm{d}\lambda.$
(7.31)

We cannot use the same trick as in the free space case by using rotation
symmetry and moving one of the coordinates to $r=0$ because the Hankel
functions are singular as $r\rightarrow 0$ and also we have the hard boundary
of the sphere at $r=\rho$ with which to contend. We remark here that there are
now TWO terms on the diagonal, whereas in the Dirichlet scattering case of the
3 d sphere you only get a scattering amplitude similar to (7.28), defined on
$\Psi_{1,\ell,m}$.

##  8 Low Energy Examples and Counter-examples

The goal of this section is to show that p form dispersive estimates are quite
different from the functionc ase. We are specifically concentrating on the low
energy case $|\lambda r|<C$, $C$ is a positive constant, in this section,
since we rely on results from [41]. First we recall that in the Dirichlet case
the rate of decay for the sphere was established in [22]. Now we consider the
very different cases for Maxwell’s equation sphere. We need the following
definitions and theorem taken from [41].

Suppose that $f,g:Z\to W$ are functions that take values in a locally convex
topological vector space and $h:Z\to\mathbb{R}$ . As usual we write
$f=g+O_{W}(h)$ if for every continuous semi-norm $p$ on $W$ there is a
constant $C_{p}$ such that $p(f(\lambda)-g(\lambda))\leq C_{p}|h(\lambda)|$
for all $\lambda\in Z$. We recall that

######  Theorem 4 ([41]).

Let $C_{d,\ell}$ be defined by

$C_{d,\ell}=(-\mathrm{i})^{\ell}\sqrt{2\pi}\frac{1}{2^{\ell+\frac{d}{2}-1}}\frac{1}{\Gamma(\ell+\frac{d}{2})},$

and suppose that $\Phi\in
C^{\infty}({\mathbb{S}^{d-1}};\Lambda^{p}\mathbb{R}^{d})$ is a spherical
harmonic of degree $\ell$, then the generalised eigenfunctions have for small
$|\lambda|$ and bounded $|\arg\lambda|$ the following expansion

For $d=3$,

$E_{\mathcal{O},\lambda}(\Phi)=-(d-2+2\ell)C_{d,\ell}\lambda^{\ell+\frac{d-5}{2}}\sum_{j=1}^{N}a_{j}(\Phi)u_{j}\\\
+\mathrm{i}(d-2+2\ell)C_{d,\ell}\lambda^{\ell+\frac{d-3}{2}}\sum_{j,k=1}^{N}a_{kj}^{(1)}a_{j}(\Phi)u_{k}+O_{C^{\infty}(M)}(\lambda^{\ell+\frac{d-1}{2}}),$
(8.1)

where $u_{j}$’s are in the kernel of $\Delta_{\mathrm{rel}}$. If if
$\partial\mathcal{O}=\emptyset$ or $p\not=1$, then $P^{(1)}=0$ and therefore
$a_{kj}^{(1)}=0$ in the previous expansions.

This shows that all $L^{2}$-eigenfunctions appear as expansion coefficients of
generalised eigenfunctions. Let $P_{0}$ denote the orthogonal projection

$P_{0}:L^{2}(M,\Lambda^{\bullet}T^{*}M)\rightarrow\mathcal{H}^{\bullet}(M).$

Below is a summary of Theorem 1.5 in [41] and its more precise statement from
Theorem 3.5 in [42].

######  Theorem 5 ([42, 41]).

Suppose that $d$ is odd and $d\geq 3$. Then for any $s\geq 0$ as a map from
$W^{s}_{\mathrm{comp}}(\overline{M},\Lambda^{\bullet}T^{*}M)\rightarrow
W_{\mathrm{loc}}^{s+2}(\overline{M},\Lambda^{\bullet}T^{*}M)$, the operator
$(\Delta-\lambda^{2})^{-1}$ has an expansion

$\displaystyle(\Delta-\lambda^{2})^{-1}=-\frac{P_{0}}{\lambda^{2}}-\frac{\mathrm{i}B_{-1}}{\lambda}+r(\lambda)$
(8.2)

as $\lambda\rightarrow 0_{+}$ where $r(\lambda)$ is analytic near zero. In the
case that $d>3$ or $\partial\cal{O}=\emptyset$ we have $B_{-1}=0$. In the case
that $\partial\cal{O}\neq\emptyset$ and $d=3$, then
$B_{-1}=a\langle\cdot,\psi\rangle\psi$ where $\psi=\frac{1}{\sqrt{4\pi a}}du$
and $a=\frac{1}{4\pi}C(\cal{O})$.

In particular it follows that in this case $E_{0}(\Phi)\in L^{2}(M)$. If
$\Phi$ is a spherical harmonic of degree higher than $1$ then by the same
reasoning one gets $E_{0}(\Phi)=0$. We recall the following Proposition, which
is Prop. 3.7 in [41].

######  Proposition 9 ([41]).

If $d=3$ then

  * •

$E_{0}(\Phi_{\nu})=0$ if $\ell_{\nu}\not=1$,

  * •

$E_{0}(\Phi_{\nu})=2\mathrm{i}\sum_{j=1}^{N}a_{j}(\Phi_{\nu})u_{j}\in L^{2}$
if $\ell_{\nu}=1$.

Moreover, we have

$B_{-1}=P^{(1)}.$

The proof of this proposition still holds with $P_{Q}\Phi_{\nu}$ replacing
$\Phi_{\nu}$. What is interesting here is that for the case of the sphere,
some of the modes which do not go to $0$ as $\lambda\rightarrow 0$ are
eliminated by the operator $Q$ (correspondingly $P_{Q}$). This result is
encoded in the following corollary.

######  Corollary 4.

When the obstacle is a sphere in $\mathbb{R}^{3}$, we have
$u_{1}=-\frac{\operatorname{cap}(\mathbb{S}^{2})}{4\pi r^{2}}\mathrm{d}r$ with
$\operatorname{cap}(\mathbb{S}^{2})$ being the harmonic capacity of the sphere
and also have $a_{1}(P_{Q_{1}}\Phi)=0$.

###### Proof.

Recall that spherical harmonics on $\mathbb{S}^{2}$ with $\ell=1$ are given by
$Y_{1,-1}(\theta,\varphi)=\frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{-\mathrm{i}\varphi}\sin\theta$,
$Y_{1,0}(\theta,\varphi)=\frac{1}{2}\sqrt{\frac{3}{\pi}}\cos\theta$ and
$Y_{1,1}(\theta,\varphi)=\frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{\mathrm{i}\varphi}\sin\theta$.
Then we have

$u_{1}(r,\theta,\varphi)=\frac{C^{\prime}}{r^{2}}\left(\frac{1}{2}\left(Y_{1,-1}-Y_{1,1}\right)\mathrm{d}x-\frac{1}{2}\left(Y_{1,-1}+Y_{1,1}\right)\mathrm{d}y+\frac{1}{\sqrt{2}}Y_{1,0}\mathrm{d}z\right),$

which implies

$a_{1}(\Phi)=C^{\prime}\left\langle\Phi,\left(\frac{1}{2}\left(Y_{1,-1}-Y_{1,1}\right)\mathrm{d}x-\frac{1}{2}\left(Y_{1,-1}+Y_{1,1}\right)\mathrm{d}y+\frac{1}{\sqrt{2}}Y_{1,0}\mathrm{d}z\right)\right\rangle.$

Since $P_{Q_{1}}\Phi$ is the tangential projection of $\Phi$, we have

$a_{1}(P_{Q_{1}}\Phi)=C^{\prime}\langle P_{Q_{1}}\Phi,\mathrm{d}r\rangle=0.$

∎

Even though we have show that $P_{Q_{1}}\Phi$ is orthogonal to
$\mathcal{H}_{rel}^{1}(M)$ when the degree of the spherical harmonic is 1, it
could be the case that $P_{Q}\Phi$ is not orthogonal to
$\mathcal{H}_{rel}^{1}(M)$ when the degree of the harmonic in the kernel is
$\ell\geq 2$.

###### Demonstration of Counter-example, (Example 1).

Let $d=3$ and let $\Phi$ be a vector valued spherical harmonic of degree 1.
Assume that there is an arbitrary smooth obstacle $\mathcal{O}$ in the set
$K$. In the region where $t<\frac{1}{2a}$, and for any $M\setminus K$, given
$x=(r,\theta)$ and $x^{\prime}=(r^{\prime},\theta)$ we have that from Theorem
4 that

$\left|\int\limits_{0}^{\infty}\cos(\lambda
t)\varphi(h\lambda)\chi_{R}(x)\chi_{R}(x^{\prime})(E_{\lambda}(\Phi)(x))(E_{\lambda}(\Phi)(x^{\prime}))^{*}\,\mathrm{d}\lambda\right|\\\
\geq\frac{1}{2}\left|\int\limits_{0}^{\infty}\varphi(\lambda
h)\frac{\tilde{C}}{r^{2}r^{\prime
2}}\,\mathrm{d}\lambda\right|-\frac{C_{R}}{h^{3}}\geq\frac{C}{h^{5}}$

Let $C$ be a nonzero constant. Then we consider the region where $|\lambda r|$
and $|\lambda r^{\prime}|\leq C$. We have that
$(rr^{\prime})^{-2}>C^{2}\lambda^{4}$. Using change of variables, we obtain
that the leading order terms are bounded below by $Ch^{-5}$ with $C$
independent of the other variables. The rest of the terms are $O(h^{-3})$. The
constant $\tilde{C}$ is nonzero as a result of the proof of Prop. 5.5 in [41]
where it was proved to be nonzero in the presence of obstacles and 0 in the
free space case in dimension $3$. ∎

These terms $\tilde{C}$ may not all appear for the sphere where the structure
of the tangential component of the electromagnetic scattering matrix is
different, which is indicated by the previous corollary. This counter example
is in contrast to the standard wave equation. The counter example is caused by
the modes which lie in the kernel of $\Delta_{rel,\mathcal{O},1}$. This same
setup can be used for any $P_{Q}\Phi$ which is not orthogonal to
$\Delta_{\mathcal{O},1}$ for $\ell\geq 2$, $a\leq h$ to create a counter-
example for Maxwell’s equations as well. An elementary calculation gives the
condition that $a\geq h$, $\rho\geq 1$ eliminates these troublesome elements
of $\mathcal{H}_{\mathrm{rel}}^{1}(M)$. The key point is just because
$P_{Q}\Phi$ eliminates those $\Phi$ in $\mathcal{H}^{1}(\mathbb{S}^{2})$, the
corresponding low energy expansion is not necessarily in
$\mathcal{H}^{1}_{\mathrm{rel}}(M)$. The distinct separation from the bottom
of the spectrum eliminates this issue.

##  9 Proof of Theorem 2

We suppress the dimension $d=3$ for this section where it is understood. We
define

$\displaystyle B_{1,\ell,\lambda}(\rho,r):=-\frac{h^{(1)}_{\ell}(\lambda
r)}{h^{(1)}_{\ell}(\lambda\rho)}$

and

$\displaystyle
B_{2,\ell,\lambda}(\rho,r):=-\frac{\frac{1}{r}(xh^{(1)}_{\ell}(x))^{\prime}|_{x=\lambda
r}}{(xh^{(1)}_{\ell}(x))^{\prime}|_{{}_{x=\lambda\rho}}}$ (9.1)

and lastly

$\displaystyle
B_{3,\ell,\lambda}(\rho,r):=-\frac{\sqrt{\ell(\ell+1)}}{r}\frac{h^{(1)}_{\ell}(\lambda
r)}{(xh^{(1)}_{\ell}(x))^{\prime}|_{x=\lambda\rho}}$ (9.2)

as entries of the corresponding diagonal matrices
$B_{1,\ell,\lambda}(\rho,r)$, $B_{2,\ell,\lambda}(\rho,r)$, and
$B_{3,\ell,\lambda}(\rho,r)$, respectively. We then have the following Lemma

######  Lemma 4.

The following estimates hold for the entries of
$B_{1,\ell,\frac{\lambda}{h}}(\rho,r)$ for $\lambda\in\mathrm{supp}(\varphi)$

$\displaystyle|B_{1,\ell,\frac{\lambda}{h}}(\rho,r)|\leq
1\qquad\left|\frac{\mathrm{d}}{\mathrm{d}\lambda}B_{1,\ell,\frac{\lambda}{h}}(\rho,r)\right|\leq\frac{2}{h}\max\\{r,(\ell+1)h\\}$
(9.3)

and for the entries of $B_{2,\ell,\frac{\lambda}{h}}(\rho,r)$ there is a
constant independent of $r$ and $\ell$ such that for $x=\frac{\lambda\rho}{h}$

$\displaystyle|B_{2,\ell,\frac{\lambda}{h}}(\rho,r)|\leq
C\begin{cases}\frac{1}{h}\quad\ell<xe\\\ \frac{\ell
h}{\lambda}+1\quad\ell>xe\end{cases}\qquad\left|\frac{\mathrm{d}}{\mathrm{d}\lambda}B_{2,\ell,\frac{\lambda}{h}}(\rho,r)\right|\leq
C\begin{cases}\frac{r}{h^{2}}\quad\ell<xe\\\
\frac{\ell(\ell+1)h}{\lambda}+1\quad\ell>xe\end{cases}$ (9.4)

and for the entries of $B_{3,\ell,\frac{\lambda}{h}}(\rho,r)$ there is a
constant independent of $r$ and $\ell$ such that for $x=\frac{\lambda\rho}{h}$

$\displaystyle|B_{3,\ell,\frac{\lambda}{h}}(\rho,r)|\leq
C\begin{cases}\frac{1}{h}\quad\ell<xe\\\ \frac{\ell
h}{\lambda}\quad\ell>xe\end{cases}\qquad\left|\frac{\mathrm{d}}{\mathrm{d}\lambda}B_{3,\ell,\frac{\lambda}{h}}(\rho,r)\right|\leq
C\begin{cases}\frac{r}{h^{2}}\quad\ell<xe\\\
\frac{\ell(\ell+1)h}{\lambda}+1\quad\ell>xe\end{cases}$ (9.5)

###### Proof.

We have from the Appendix (12.13), that the first bound in 9.3 holds. We use
that

$\frac{\mathrm{d}}{\mathrm{d}x}h^{(1)}_{\ell}(x)=h^{(1)}_{\ell-1}(x)-\left(\frac{\ell+1}{x}\right)h_{\ell}^{(1)}(x)=-h^{(1)}_{\ell+1}(x)+\left(\frac{\ell}{x}\right)h_{\ell}^{(1)}(x)$
(9.6)

along with change of variables to get the second bound in 9.3 combined with
(12.13).

For the terms in 9.4, we start by recalling the following derivative
identities for the spherical Hankel functions in dimension 3. Using (9.6) we
obtain

$\frac{\mathrm{d}}{\mathrm{d}x}(xh_{\ell}^{(1)}(x))=xh^{(1)}_{\ell-1}(x)-\ell
h_{\ell}^{(1)}(x)=-xh_{\ell+1}^{(1)}(x)+(\ell+1)h_{\ell}^{(1)}(x)$ (9.7)

and using the equation for the spherical Hankel function itself

$\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}(xh_{\ell}^{(1)}(x))=-xh^{(1)}_{\ell}(x)+\frac{\ell(\ell+1)}{x}h_{\ell}^{(1)}(x).$
(9.8)

The numerators can then be bounded by using (9.7) iteratively. If

$\displaystyle\ell+1<\frac{\lambda\rho}{2h}\quad\lambda\in\mathrm{supp}(\varphi)$
(9.9)

then we get for $x=\frac{\lambda\rho}{h}$

$\displaystyle|xh_{\ell+1}^{(1)}(x)-(\ell+1)h_{\ell}^{(1)}(x)|\geq|xh_{\ell+1}^{(1)}(x)|-(\ell+1)|h_{\ell}^{(1)}(x)|\geq\frac{|xh_{\ell+1}^{(1)}(x)|}{2}.$
(9.10)

We have that for $\ell+1<\frac{\lambda\rho}{2h}$

$|B_{2,\ell,\frac{\lambda}{h}}(\rho,r)|\leq\frac{4}{|\frac{\lambda\rho}{h}h_{\ell+1}^{(1)}(\frac{\lambda\rho}{h})|}\left|\frac{\lambda}{h}h_{\ell+1}^{(1)}\left(\frac{\lambda
r}{h}\right)\right|\leq 4$ (9.11)

Now to derive the first inequality in (9.4) we have to continue to look for a
lower bound for

$\displaystyle|xh^{(1)}_{\ell-1}(x)-\ell h_{\ell}^{(1)}(x)|^{-1}.$ (9.12)

in terms of various orders of $\ell$ with respect to $x$. If $ex<\ell$, then

$|xh^{(1)}_{\ell-1}(x)-\ell
h_{\ell}^{(1)}(x)|\geq\left(\frac{\ell}{x}-1\right)|xh_{\ell}^{(1)}(x)|\geq(e-1)|xh_{\ell}^{(1)}(x)|$
(9.13)

and we get a bound of

$\displaystyle|xh^{(1)}_{\ell-1}(x)-\ell
h_{\ell}^{(1)}(x)|^{-1}\leq\frac{1}{e-1}\left|xh_{\ell}^{(1)}(x)\right|^{-1}.$
(9.14)

This then results in

$\displaystyle|B_{2,\ell,\frac{\lambda}{h}}(\rho,r)|\leq\frac{\ell
h+\lambda}{\lambda\rho(e-1)}$ (9.15)

The difficult region is then when $\frac{\ell}{e}<x<2(\ell+1)$ whenever
$x^{-1}$ is sufficiently small, e.g. $x=\frac{\lambda\rho}{h}$. Using equation
12.13 from the Appendix 12.3 we have that

$2|(xh_{\ell}^{(1)}(x))^{\prime}||xh_{\ell}^{(1)}(x)|\geq\left|\frac{\mathrm{d}}{\mathrm{d}x}\left(\sum\limits_{k=0}^{\ell}\frac{s_{k}(\ell+\frac{1}{2})}{x^{2k}}\right)\right|\geq\sum\limits_{k=0}^{\ell}\frac{2ks_{k}(\ell+\frac{1}{2})}{x^{2k+1}}\geq\frac{|xh_{\ell}^{(1)}(x)|^{2}-1}{x},$

which implies

$\left|(xh_{\ell}^{(1)}(x))^{\prime}\right|^{-1}\leq\frac{2x|xh_{\ell}^{(1)}(x)|}{|xh_{\ell}^{(1)}(x)|^{2}-1}$

with $x=\frac{\lambda\rho}{h}$. We also have for $\ell<\frac{\lambda\rho
e}{h}$ from the identity (9.7)

$\displaystyle\left|\left.\left(xh_{\ell}^{(1)}(x)\right)^{\prime}\right|_{x=\frac{\lambda
r}{h}}\right|\leq
4\left|\frac{\lambda\rho}{h}h_{\ell}^{(1)}\left(\frac{\lambda\rho}{h}\right)\right|.$
(9.16)

This gives

$\displaystyle|B_{2,\ell,\frac{\lambda}{h}}(\rho,r)|\leq\left.\left(\frac{2x|xh_{\ell}^{(1)}(x)|}{r(|xh_{\ell}^{(1)}(x)|^{2}-1)}\right)\right|_{x=\frac{\lambda\rho}{h}}\left|\left.(\tilde{x}h_{\ell}^{(1)}(\tilde{x}))^{\prime}\right|_{\tilde{x}=\frac{\lambda
r}{h}}\right|\leq$
$\displaystyle\left.\left(\frac{8x|xh_{\ell}^{(1)}(x)|^{2}}{|xh_{\ell}^{(1)}(x)|^{2}-1}\right)\right|_{x=\frac{\lambda\rho}{h}}=8x\left.\left(1+\frac{1}{|xh_{\ell}^{(1)}(x)|^{2}-1}\right)\right|_{x=\frac{\lambda\rho}{h}}$

Because we have that

$\displaystyle\left|xh_{\ell}^{(1)}\left(x\right)\right|^{2}\geq
1+\frac{\ell(\ell+1)}{x^{2}}\geq 1+\frac{\ell}{2x}\geq
1+\frac{1}{4}-\frac{1}{2x}.$

it follows that if $h\leq\frac{1}{4}$ we have
$|B_{2,\ell,\frac{\lambda}{h}}(\rho,r)|\leq Ch^{-1}$. This bound is certainly
not tight. The derivative with respect to $\lambda$ of
$B_{2,\ell,\frac{\lambda}{h}}$ follows by using (9.8) and the previously
established lower bounds. The bound for
$|B_{2,\ell,\frac{\lambda}{h}}(\rho,r)|$ then follows.

Similar to (9.11), we have that for $\ell+1<\frac{\lambda\rho}{2h}$

$|B_{3,\ell,\frac{\lambda}{h}}(\rho,r)|\leq\frac{2\ell}{r}\frac{h^{(1)}_{\ell}(\frac{\lambda
r}{h})}{|\frac{\lambda\rho}{h}h_{\ell+1}^{(1)}(\frac{\lambda\rho}{h})|}\leq
2.$

For $ex<\ell$, the same estimate of (9.13) gives

$|B_{3,\ell,\frac{\lambda}{h}}(\rho,r)|\leq\frac{2\ell}{r}\frac{h^{(1)}_{\ell}(\frac{\lambda
r}{h})}{|\frac{\lambda\rho}{h}h_{\ell}^{(1)}(\frac{\lambda\rho}{h})|}\leq\frac{2\ell
h}{\lambda}.$

When $\frac{\ell}{e}<x<2(\ell+1)$, one has

$|B_{3,\ell,\frac{\lambda}{h}}(\rho,r)|\leq\frac{2\ell}{r}\left.\left(\frac{2|xh_{\ell}^{(1)}(x)|^{2}}{r(|xh_{\ell}^{(1)}(x)|^{2}-1)}\right)\right|_{x=\frac{\lambda\rho}{h}}\leq\frac{C}{h}.$

The bounds derivative of $B_{3,\ell,\frac{\lambda}{h}}$ follow similarly. ∎

The impediment to obtaining a better estimate in (9.4) is the lack of precise
asymptotics for the Hankel function derivatives in this regime. In the future
this could perhaps be improved by the use of asymptotics of Airy functions as
in [22] in the Dirichlet function case. However this is significantly more
complicated due to the presence of a derivative in the denominator of the
ratio.

###### Proof of Theorem 2.

We start with the most difficult terms occurring in the kernel to analyze

$\displaystyle\sum\limits_{\nu}\int\limits_{0}^{\infty}\varphi(h\lambda)e^{\mathrm{i}t\lambda}\tilde{h}^{(1)}_{\lambda}(A_{\lambda,Y}^{t}Y_{\nu})(r,\theta)\overline{\tilde{h}^{(1)}_{\lambda}(A_{\lambda,Y}^{t}Y_{\nu})(r^{\prime},\theta^{\prime})}\,\mathrm{d}\lambda.$
(9.17)

By symmetry bounds for the other forms of the terms

$\displaystyle\sum\limits_{\nu}\int\limits_{0}^{\infty}\varphi(h\lambda)e^{\mathrm{i}t\lambda}\tilde{j}_{\lambda}(Y_{\nu})(r,\theta)\overline{\tilde{h}^{(1)}_{\lambda}(A_{\lambda,Y}^{t}Y_{\nu})(r^{\prime},\theta^{\prime})}\,\mathrm{d}\lambda$
(9.18)

and

$\displaystyle\sum\limits_{\nu}\int\limits_{0}^{\infty}\varphi(h\lambda)e^{\mathrm{i}t\lambda}\tilde{j}_{\lambda}(Y_{\nu})(r,\theta)\overline{\tilde{j}_{\lambda}(Y_{\nu})(r^{\prime},\theta^{\prime})}\,\mathrm{d}\lambda$
(9.19)

follow almost immediately in a similar fashion, once we have obtained a bound
for the kernel (9.17).

Using the explicit computation for the scattering matrix coefficients, we see
that

$\displaystyle-\frac{2j_{\ell}(\lambda\rho)}{h^{(1)}_{\ell}(\lambda\rho)}h^{(1)}_{\ell}(\lambda
r)=-2j_{\ell}(\lambda\rho)\frac{h^{(1)}_{\ell}(\lambda
r)}{h^{(1)}_{\ell}(\lambda\rho)}.$ (9.20)

From (9.20) and (9.3) when $Y_{\nu}=\Psi_{1,\ell,m}$ we have that

$\displaystyle\tilde{h}^{(1)}_{\lambda}(A_{\lambda,Y}^{t}Y_{\nu})(r,\theta)=\lambda(A_{\lambda,Y}^{t})_{1,\ell,m}\Psi_{1,\ell,m}(\theta)h_{\ell}^{(1)}(\lambda
r)(-\mathrm{i})^{\ell}$ (9.21)
$\displaystyle=2\lambda\Psi_{1,\ell,m}(\theta)B_{1,\ell,\lambda}(\rho,r)j_{\ell}(\lambda\rho)(-\mathrm{i})^{\ell}=\tilde{j}_{\lambda}(B_{\lambda}Y_{\nu})(\rho,\theta).$

By the Cauchy-Schwarz inequality along with (9.3) and (9.21) we obtain when
$Y_{\nu}=\Psi_{1,\ell,m}$

$\displaystyle\left|\int\limits_{0}^{\infty}\varphi(h\lambda)e^{\mathrm{i}t\lambda}\tilde{h}^{(1)}_{\lambda}(A_{\lambda,Y}^{t}Y_{\nu})(r,\theta)\overline{\tilde{h}^{(1)}_{\lambda}(A_{\lambda,Y}^{t}Y_{\nu})(r^{\prime},\theta^{\prime})}\,\mathrm{d}\lambda\right|\leq$
$\displaystyle\frac{1}{2}\left|\int\limits_{0}^{\infty}\varphi(h\lambda)|\tilde{h}^{(1)}_{\lambda}(A_{\lambda,Y}^{t}Y_{\nu})(r,\theta)|^{2}+|\overline{\tilde{h}^{(1)}_{\lambda}(A_{\lambda,Y}^{t}Y_{\nu})(r^{\prime},\theta^{\prime})}|^{2}\,\mathrm{d}\lambda\right|\leq$
$\displaystyle\frac{1}{2}\int\limits_{0}^{\infty}\varphi(h\lambda)\lambda^{2}\left(|B_{1,\ell,\lambda}(\rho,r)|^{2}+|B_{1,\ell,\lambda}(\rho,r^{\prime})|^{2}\right)\left|Y_{\nu}(\theta)j_{\ell}(\lambda\rho)(-\mathrm{i})^{\ell}\right|^{2}\,\mathrm{d}\lambda\leq$
$\displaystyle\int\limits_{0}^{\infty}\varphi(h\lambda)\lambda^{2}\left|\Psi_{1,\ell,m}(\theta)j_{\ell}(\lambda\rho)(-\mathrm{i})^{\ell}\right|^{2}\,\mathrm{d}\lambda$
(9.22)

We recall the result of Theorem 1, inequality (3.2) with $d=3$ which covers
all cases of compactly supported 1-forms in the Hodge decomposition except
those $\mathcal{H}^{1}_{rel}(M)$. Those are bounded by substituting expansion
in (8.1) at $\lambda=0$, provided $a\geq h,\rho\geq 1$. Summing over
$\nu=(1,\ell,m)$, it follows from (3.2) that (9) is bounded by

$\displaystyle\|K^{\Delta_{1,free}}(\cdot,\cdot,t,h)\|_{L^{\infty}}\leq\left|K^{\Delta_{1,free}}(x,x,0,h)|_{|x|=\rho}\right|\leq\frac{C}{h^{3}}$
(9.23)

where the subscript $1$ denotes only the kernel with $\Psi_{1,\ell,m}$
present. The cross terms are bounded analogously.

The case when $Y_{\nu}=\Psi_{2,\ell,m}$ first coefficient follows by replacing
$B_{1,\ell,\lambda}$ with $B_{2,\ell,\lambda}$ and

$\displaystyle
j_{\ell}\left(\frac{\lambda\rho}{h}\right)\quad\mathrm{with}\quad\left.\frac{(xj_{\ell}(x))^{\prime}}{x}\right|_{x=\frac{\lambda\rho}{h}}.$
(9.24)

Unfortunately because of the different scales of $\ell$ in (9.4) we have to
separate this into two cases, one with $\frac{\lambda\rho e}{h}\leq\ell$ and
the other with $\frac{\lambda\rho e}{h}\geq\ell$. The first case follows from
the previous Cauchy-Schwarz estimate and the first equation in 9.4. We flip
the sums and the integrals where necessary due to Fubini’s theorem and
Proposition 7 combined with Proposition 6. Indeed, we have that writing
$\lambda_{0}=\frac{\lambda}{h}$ for space reasons:

$\displaystyle\frac{1}{h}\left|\int\limits_{0}^{\infty}\sum\limits_{\begin{subarray}{c}\nu,\\\
\ell\leq\lambda_{0}\rho
e\end{subarray}}\chi_{R}(r)\chi_{R}(r^{\prime})\varphi(\lambda)e^{\mathrm{i}t\lambda_{0}}\tilde{h}^{(1)}_{\lambda_{0}}(A_{\lambda_{0},Y}^{t}Y_{\nu})(r,\theta)\overline{\tilde{h}^{(1)}_{\lambda_{0}}(A_{\lambda_{0},Y}^{t}Y_{\nu})(r^{\prime},\theta^{\prime})}\,\mathrm{d}\lambda\right|\leq$
$\displaystyle\frac{1}{2h}\left|\int\limits_{0}^{\infty}\sum\limits_{\begin{subarray}{c}\nu,\\\
\ell\leq\lambda_{0}\rho
e\end{subarray}}\chi_{R}(r)\chi_{R}(r^{\prime})\varphi(\lambda)\left(|\tilde{h}^{(1)}_{\lambda_{0}}(A_{\lambda_{0},Y}^{t}Y_{\nu})(r,\theta)|^{2}+|\overline{\tilde{h}^{(1)}_{\lambda_{0}}(A_{\lambda_{0},Y}^{t}Y_{\nu})(r^{\prime},\theta^{\prime})}|^{2}\right)\,\mathrm{d}\lambda\right|.$
(9.25)

For the first terms in (9) we have that

$\displaystyle\frac{1}{2h}\int\limits_{0}^{\infty}\sum\limits_{\begin{subarray}{c}\ell,m,\\\
\ell\leq\lambda_{0}\rho
e\end{subarray}}\chi_{R}(r)\chi_{R}(r^{\prime})\varphi(\lambda)$ (9.26)
$\displaystyle\times\left(|B_{2,\ell,\lambda_{0}}(\rho,r)|^{2}+|B_{2,\ell,\lambda_{0}}(\rho,r^{\prime})|^{2}\right)\left|\Psi_{2,\ell,m}(\theta)\frac{(xj_{\ell})^{\prime}}{\rho}|_{x=\lambda_{0}\rho}(-\mathrm{i})^{\ell-1}\right|^{2}\,\mathrm{d}\lambda$
$\displaystyle\leq$
$\displaystyle\frac{C}{h^{3}}\int\limits_{0}^{\infty}\sum\limits_{\begin{subarray}{c}\ell,m,\\\
\ell\leq\lambda_{0}\rho
e\end{subarray}}\chi_{R}(r)\chi_{R}(r^{\prime})\varphi(\lambda)\left|\Psi_{2,\ell,m}(\theta)\frac{(xj_{\ell})^{\prime}}{\rho}|_{x=\lambda_{0}\rho}(-\mathrm{i})^{\ell-1}\right|^{2}\,\mathrm{d}\lambda$
$\displaystyle\leq$
$\displaystyle\frac{C}{h^{3}}\int\limits_{0}^{\infty}\sum\limits_{\begin{subarray}{c}\ell,m,\\\
\ell\leq\lambda_{0}\rho
e\end{subarray}}\chi_{R}(r)\chi_{R}(r^{\prime})\varphi(\lambda)\left|\Psi_{2,\ell,m}(\theta)\frac{xj_{\ell-1}(x)-\ell
j_{\ell}(x)}{\rho}|_{x=\lambda_{0}\rho}(-\mathrm{i})^{\ell-1}\right|^{2}\,\mathrm{d}\lambda$
$\displaystyle\leq$
$\displaystyle\frac{|K^{\Delta_{1,free}}(x,x,0,h)|_{|x|=\rho}|}{h^{2}}\leq\frac{C}{h^{5}}.$
(9.27)

For the second terms we obtain

$\displaystyle\frac{1}{2h}\int\limits_{0}^{\infty}\sum\limits_{\begin{subarray}{c}\ell,m,\\\
\ell\leq\lambda_{0}\rho
e\end{subarray}}\chi_{R}(r)\chi_{R}(r^{\prime})\varphi(\lambda)$ (9.28)
$\displaystyle\times\left(|B_{3,\ell,\lambda_{0}}(\rho,r)|^{2}+|B_{3,\ell,\lambda_{0}}(\rho,r^{\prime})|^{2}\right)\left|\Psi_{3,\ell,m}(\theta)(xj_{\ell})^{\prime}|_{x=\lambda_{0}\rho}(-\mathrm{i})^{\ell-1}\right|^{2}\,\mathrm{d}\lambda$
$\displaystyle\leq$
$\displaystyle\frac{C}{h^{3}}\int\limits_{0}^{\infty}\sum\limits_{\begin{subarray}{c}\ell,m,\\\
\ell\leq\lambda_{0}\rho
e\end{subarray}}\chi_{R}(r)\chi_{R}(r^{\prime})\varphi(\lambda)\left|\Psi_{3,\ell,m}(\theta)(xj_{\ell})^{\prime}|_{x=\lambda_{0}\rho}(-\mathrm{i})^{\ell-1}\right|^{2}\,\mathrm{d}\lambda$
$\displaystyle\leq$
$\displaystyle\frac{1}{h^{2}}\|K^{\Delta_{1,free}}(x,x,0,h)|_{|x|=\rho}\|_{L^{\infty}}\leq\frac{C}{h^{5}}.$

For the higher order $\ell$ we have that

$\displaystyle\left|J_{\ell}\left(\frac{\lambda\rho}{h}\right)\right|\leq\frac{1}{\sqrt{2\pi\ell}}\left(\frac{\lambda\rho
e}{2h\ell}\right)^{\ell}$ and
$\displaystyle\left|j_{\ell}\left(\frac{\lambda\rho}{h}\right)\right|\leq
C\ell^{-1}\left(\frac{\lambda\rho e}{2h\ell}\right)^{\ell}.$ (9.29)

Similarly,

$\displaystyle\left|H^{(1)}_{\ell}\left(\frac{\lambda r}{h}\right)\right|\leq
2\sqrt{\frac{2}{\pi\ell}}\left(\frac{\lambda re}{2h\ell}\right)^{-\ell}.$
(9.30)

We then examine the first terms in

$\displaystyle\frac{1}{h}\left|\int\limits_{0}^{\infty}\sum\limits_{\begin{subarray}{c}\nu,\\\
\ell>\lambda_{0}\rho
e\end{subarray}}\varphi(\lambda)e^{\mathrm{i}t\lambda_{0}}\tilde{h}^{(1)}_{\lambda_{0}}(A_{\lambda_{0},Y}^{t}Y_{\nu})(r,\theta)\overline{\tilde{h}^{(1)}_{\lambda_{0}}(A_{\lambda_{0},Y}^{t}Y_{\nu})(r^{\prime},\theta^{\prime})}\,\mathrm{d}\lambda\right|$
(9.31)

We then have from Cauchy-Schwarz and (9.4)

$\displaystyle\frac{1}{h}\int\limits_{0}^{\infty}\sum\limits_{\begin{subarray}{c}(\ell,m),\\\
\ell>\lambda_{0}\rho
e\end{subarray}}\chi_{R}(r)\chi_{R}(r^{\prime})\varphi(\lambda)\left(\frac{\ell
h}{\lambda}+1\right)^{2}\left|\Psi_{2,\ell,m}(\theta)\frac{(xj_{\ell})^{\prime}}{\rho}|_{x=\lambda_{0}\rho}(-\mathrm{i})^{\ell}\right|^{2}\,\mathrm{d}\lambda$
$\displaystyle\leq$ $\displaystyle
Ch\int\limits_{0}^{\infty}\sum\limits_{\begin{subarray}{c}(\ell,m),\\\
\ell>\lambda_{0}\rho
e\end{subarray}}\chi_{R}(r)\chi_{R}(r^{\prime})\varphi(\lambda)\ell^{3}\left|\left.\left(xj_{\ell}(x)-\ell
j_{\ell}(x)\right)\right|_{x=\lambda_{0}\rho}\right|^{2}\,\mathrm{d}\lambda$
$\displaystyle\leq$
$\displaystyle\frac{C}{h}\int\limits_{0}^{\infty}\chi_{R}(r)\chi_{R}(r^{\prime})\varphi(\lambda)\sum\limits_{\ell\geq
1}\frac{\ell^{5}}{4^{\ell}}\,\mathrm{d}\lambda\leq\frac{C}{h}\chi_{R}(r)\chi_{R}(r^{\prime}),$

where the above constants only depend on $\rho$ and $\phi$. In the second
bound we used $\|Y_{\nu}\|_{L^{\infty}}\leq C\sqrt{\ell}$ since as remarked
earlier $Y_{\nu}$ can have degree $\ell-1,\ell$ or $\ell+1$. In the last
bound, we have used (9.29). Using equation (9.7), this allows us to conclude a
bound of $Ch^{-5}$ because of the extra factor of $\ell$ in the first equation
of (9.4). Notice that $\ell$ starts at $1$ for the Maxwell equations. The
bound for the second terms in (9) and (9.31) involving $\Psi_{3,\ell,m}$
follows analogously.

For the case $t>h$ we are looking at (7.31), and the other terms in the
expansion of (7.2) will follow similarly. We separate our investigation into
three cases.

  1. 1.

Case $|\lambda r|<h$, this case is empty by assumption on $a$.

  2. 2.

Case $|\lambda r|>h$, we consider $\ell<\frac{\lambda\rho e}{h}$, where we
have assumed $r\geq r^{\prime}$ without loss of generality. Let the subscript
$-$ denote the restriction to this case. Let $\chi_{R}$ be a $C^{\infty}$
function which is 1 on a ball of radius $2R$ and $0$ outside a ball of radius
$3R$. We examine the difficult terms of the form

$\displaystyle K_{-}(x,x^{\prime},t,h)=$
$\displaystyle\frac{\chi_{R}(r)\chi_{R}(r^{\prime})}{2\pi h}$ (9.32)
$\displaystyle\times\int_{0}^{\infty}\sum\limits_{\nu_{-}}\varphi(\lambda)e^{\mathrm{i}\frac{\lambda}{h}t}\left(\tilde{h}_{\frac{\lambda}{h}}^{(2)}\left(Y_{\nu}\right)(r,\theta)\right)\left(\tilde{h}^{(1)}_{\frac{\lambda}{h}}\left(S_{\frac{\lambda}{h}}^{t}Y_{\nu}\right)(r^{\prime},\theta^{\prime})\right)^{*}\mathrm{d}\lambda.$

The extra decay comes from integrating by parts once. We are interested in the
$L^{\infty}$-bound of $\left|\frac{t}{h}K_{-}\right|$. Note that
$\frac{t}{h}e^{\mathrm{i}\frac{\lambda}{h}t}$ can be replaced by
$-\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}\lambda}e^{\mathrm{i}\frac{\lambda}{h}t}$
in the integrand of (9.32). The derivative of the spherical Bessel function
follows the same relationship as (9.6). We have that

$\frac{\mathrm{d}}{\mathrm{d}\lambda}\left(h^{(1)}_{\ell}\left(\frac{\lambda
r}{h}\right)A_{\frac{\lambda}{h},Y,\ell}\right)=\frac{\mathrm{d}}{\mathrm{d}\lambda}B_{1,\ell,\frac{\lambda}{h}}(\rho,r)\left(j_{\ell}\left(\frac{\lambda\rho}{h}\right)\right)+B_{1,\ell,\frac{\lambda}{h}}(\rho,r)\frac{\mathrm{d}}{\mathrm{d}\lambda}j_{\ell}\left(\frac{\lambda
r}{h}\right)$ (9.33)

When integrating by parts once we can then combine (9.33), and the Cauchy
Schwarz inequality like the proof of (9) where we use the bounds in Lemma 4
and the prefactor from the terms in (9.6) for the second term times $B_{\ell}$
to obtain

$\displaystyle\left|\frac{t}{h}K_{1,-}(x,x^{\prime},t,\rho,h)\right|\leq
CR\left|\frac{t}{h}K^{\Delta_{1,free}}(x,x,t,\rho,h)\right|\leq\frac{CR}{h^{4}}$

whenever $Y_{\nu}=\Psi_{1,\ell,m}$, in the kernel $K_{-}$. Similarly from
Lemma 4, replacing $\frac{\mathrm{d}}{\mathrm{d}\lambda}B_{1,\ell,\lambda}$
with $\frac{\mathrm{d}}{\mathrm{d}\lambda}B_{2,\ell,\lambda}$ or
$\frac{\mathrm{d}}{\mathrm{d}\lambda}B_{3,\ell,\lambda}$, the proof of the
bound for the other polarization directions follow analogously.

  3. 3.

Case $|\lambda r|>h$, without loss of generality assume $r\geq r^{\prime}$ and
$\ell>\frac{\lambda\rho e}{2h}$. Let the subscript $+$ denote the restriction
to this case. We have the large term estimates (9.29) and (9.30) as before.
Then we obtain a bound for the case of $Y_{\nu}=\Psi_{1,\ell,m}$ only using
the bound on $B_{1,\ell,\lambda_{0}}$ in (9.3). That is the sum over
$\ell>\frac{\lambda\rho e}{2h}$ of (9) multiplied by $\frac{t}{h}$ can be
bounded by

$\displaystyle\left|\frac{t}{h}K_{1,+}(x,x^{\prime},t,\rho,h)\right|$
$\displaystyle\leq$
$\displaystyle\frac{C\chi_{R}(r)\chi_{R}(r^{\prime})}{h^{4}}\int\limits_{0}^{\infty}\sum_{\ell>\frac{\lambda\rho
e}{2h}}\varphi(\lambda)\lambda^{2}\max\\{r,(\ell+1)h\\}\left|\Psi_{1,\ell,m}(\theta)j_{\ell}(\lambda_{0}\rho)(-\mathrm{i})^{\ell}\right|^{2}\,\mathrm{d}\lambda$
$\displaystyle\leq$ $\displaystyle\frac{CR}{h^{4}},$

this gives the bound of $K_{1}$ for (9.17) when $Y_{\nu}=\Psi_{1,\ell,m}$. The
rest of the cross terms from (7.2) follow similarly using Cauchy-Schwarz. For
the second case when $Y_{\nu}=\Psi_{2,\ell,m}$, we have that in this case the
first terms in

$\displaystyle\frac{1}{h}\int\limits_{0}^{\infty}\sum\limits_{\begin{subarray}{c}(\ell,m),\\\
\ell>\lambda_{0}\rho
e\end{subarray}}\chi_{R}(r)\chi_{R}(r^{\prime})\varphi(\lambda)\left(\frac{\ell(\ell+1)h}{\lambda}+1\right)^{2}\left|\Psi_{2,\ell,m}(\theta)\frac{(xj_{\ell})^{\prime}}{\rho}|_{x=\lambda_{0}\rho}(-\mathrm{i})^{\ell}\right|^{2}\,\mathrm{d}\lambda$
$\displaystyle\leq\frac{C}{h}\int\limits_{0}^{\infty}\chi_{R}(r)\chi_{R}(r^{\prime})\varphi(\lambda)\sum\limits_{\ell\geq
1}\frac{\ell^{7}}{4^{\ell}}\,\mathrm{d}\lambda\leq\frac{C}{h}\chi_{R}(r)\chi_{R}(r^{\prime}).$

Combining the three cases and summing over all three components of the kernel,
one with $\Psi_{1,\ell,m}$, one with $\Psi_{2,\ell,m}$ and one with
$\Psi_{3,\ell,m}$, with the realization the VSH are orthogonal to each other
on the sphere, there exists a constant $C$ so that for any $f\in
C_{c}^{\infty}(M_{3R},\Lambda^{1}T^{*}M_{3R})$

$\displaystyle\|\chi_{R}(r)\chi(r\leq
r^{\prime})K^{\mathcal{O},m}(x,x^{\prime},\rho,h)f\|_{L^{1}}\leq
C\|f\|_{L^{1}}\left(h^{-5}\min\left\\{1,\frac{R}{t}\right\\}\right).$ (9.34)

The monotone convergence theorem implies the desired result for the dense
class of forms. By density the result holds for $f\in
L^{1}(M_{3R},\Lambda^{1}T^{*}M_{3R})$. ∎

##  10 Finite Speed of Propagation

We show how to improve our result to a more generic one, Corollary 1. Assume
we have generic data $f\in L^{1}_{\mathrm{comp}}(M;\Lambda^{1}T^{*}M)$ where
the subscript compact denotes a compactly supported function. We will show
that initial data of this form evolves according to three tractable scenarios.

  * •

Scenario 1: Let $\mathcal{O}$ be a ball in 3 dimensions of radius $\rho$. Let
$f=f_{1}\in L^{1}(M_{3\rho};\Lambda^{1}T^{*}M_{3\rho})$. As a result of [25],
Theorem 1.6 we only need to prove dispersion for the case $t\leq 6\rho$ (see
Scenario 2), which is covered by Theorem 2 with $3R=9\rho$.

  * •

Scenario 2: We have $f=f_{1}\in L^{1}(M_{3\rho};\Lambda^{1}T^{*}M_{3\rho})$
and $t>6\rho$. Here the free space propagator and the obstacle propagator
differ only by an error which is exponentially decaying in time [22], Lemma
2.6. We review this result here since it needs a small modification to work in
the Maxwell scenario. Let $\Delta$ be the Dirichlet Laplacian in the exterior
of the ball $\mathcal{O}$ of radius $\rho$ in three dimensions. We let
$u(t,x)$ denote the solution to the Cauchy problem for the wave equation in
$M=\mathbb{R}^{3}\setminus\mathcal{O}$:

$\displaystyle\partial_{t}^{2}u-\Delta u=0$ (10.1) $\displaystyle
u(0,x)=U_{0}(x)$ $\displaystyle\partial_{t}u(0,x)=U_{1}(x)$

which is well posed in in $L^{1}([0,T];H^{1}(M))\cap H^{1}([0,T];L^{2}(M))$
for $H^{1}_{0}(M)$ Cauchy data. Since we are using $L^{1}(M)$ spaces, the
$\varphi(hD_{t})$ will play the role of making sure the propagator acting on
$L^{1}(M)$ is again in $L^{1}(M)$. We recall the following definition. Let
$T_{R_{e}}$ be the escape time, that is the time at which there is no geodesic
of length $T_{R_{e}}$ lies entirely within the radius $R_{e}$ of the obstacle.
For smooth Cauchy data we have the following celebrated theorem

######  Theorem 6 (subset of [25], Theorem 1.6).

Consider the Dirichlet laplacian, $\Delta$ in $M$, the exterior of
$\mathcal{O}$ the ball of radius $\rho$ in 3 dimensions. Then there exists a
sequence $\lambda_{j}\in\mathbb{C}$, $\mathrm{Im}(\lambda_{j})<0$ with
$\lim\limits_{j\rightarrow\infty}\operatorname{Im}(\lambda_{j}))=-\infty$ and
associated generalized eigenspaces $V_{j}\subset C^{\infty}(M)$ of dimensions
$m_{j}<\infty$ such that

    1. 1.

$v\in V_{j}\Rightarrow v|_{\partial\mathcal{O}}=0$

    2. 2.

$(\Delta+\lambda_{j}^{2})V_{j}\subset V_{j}$,

    3. 3.

If $(U_{0},U_{1})$ are supported in $\\{x:|x|\leq R\\}$ then there exists
$v_{j,k}\in V_{j}$ such that for all $\epsilon>0$ and $N\in\mathbb{N}$ and a
multi-index $\alpha$ and some constant $C=C(R,N,\epsilon,\alpha)$ the solution
to the Cauchy problem for the wave equation (10.1) satisfies for $t>T_{R_{e}}$

$\displaystyle\sup\limits_{\\{|x|\leq
R_{e}\\}}\left|D_{(t,x)}^{\alpha}\left(u(t,x)-\sum\limits_{j=1}^{N-1}e^{-i\lambda_{j}t}\sum\limits_{k=0}^{m_{j}-1}t^{k}v_{j,k}(x)\right)\right|\leq
Ce^{-(t-T_{R_{e}})(|\mathrm{Im}\lambda_{N}|-\epsilon)}$ (10.2)

The parametrix is for the wave equation in [25]. If we take

$\mathrm{curl}(xu(t,x))=(\nabla u)\times r\hat{r}$

then this solves for the electric field in (7.1) with
$i\lambda\mapsto\partial_{t}$ and $E|_{\partial\cal{O}}=\nu\times E=0$. We
recall here that $\nabla u(t,\rho)=0$ by assumption that $u(t,x)$ solves
(10.1). Taking this solution instead does not change the statement of the
result of [25] Theorem 1.6 very much due to the presence of the derivative in
$\alpha$. By density the result follows for $W^{1,1}_{0}$ data. Notice that
the extra $\varphi(hD_{t})$ gives that $\varphi(hD_{t})e^{it\sqrt{\Delta}}f\in
L^{1}$ whenever $f\in W^{1,1}_{0}$ is.

  * •

Scenario 3: The result of [24] Chapter 8 is that if our input data $f_{2}\in
C_{c}^{\infty}(B_{3\rho}^{c};\Lambda^{1}T^{*}(B_{3\rho}^{c}))$ then the
operators
$\mathrm{cos}(t\Delta_{\textrm{free}})-\mathrm{cos}(t\Delta_{\cal{O},\mathrm{rel}})$
coincide on these forms when $|x-\rho|>t$. This implies that they coincide for
all $t<2\rho$ acting on this particular class of initial data. Their result is
a manifestation of Huygen’s principle. By density the result follows for
$W^{1,1}_{0}$ data. This is scenario 3.

We can then consider $f\in C_{c}^{\infty}(M;\Lambda^{1}T^{*}M)$ as the sum of
data $f=f_{1}+f_{2}$. The missing region is then when the part of the initial
data $f_{2}\in L^{1}(M_{3\rho}^{c};\Lambda^{1}T^{*}(M_{3\rho}^{c}))$ and in
time has evolved beyond $t>2\rho$. We claim that this region is covered by
using data from scenario 1, 2 and 3. Select $R=3\rho$, then since the obstacle
is non-trapping the support of data enters the region described by scenario 1)
when if $t_{0}$ our new starting time for evolution of the data is considered
as $2\rho$ and those contributions of the support of this data which move into
scenario 3) at times $t\geq 2\rho$ already disperse at the correct rate. Thus
our dispersive estimate in Theorem 2 can be extended to Corollary 1.

This is a generalization of an extremely similar argument [22] use to make
their dispersive estimates independent of the source and observer and restrict
their analysis to a very large ball. Better arguments about the asymptotics of
the Bessel and Hankel functions can be used to improve the value of the
constant which depends directly on $R$ in Theorem 2 and therefore the overall
constant’s geometric dependence on $\rho$ but it is not clear that this type
of analysis will result in better powers of $h$ for the Maxwell operator.

##  11 Conclusions

This article presents a novel result for the Maxwell dispersive propagator in
the absence of charge around the sphere. We showed using generalized
eigenfunctions and Mie scattering that in this case the dispersive rate is
what one would anticipate for the free space case for a general class of forms
with compact support, but only for lower order modes. However the ratio in
$B_{2,\ell,\lambda}(r,\rho)$ is not going to be well-behaved around
$\lambda\rho=\ell$. The bottom is nearly zero while the top for large $r$ is
is large. The behavior of this ratio in $\lambda r$ is probably logarithmic
due to (12.15) and (12.16) but this remains to be shown precisely, whence the
higher powers of $h$ for these terms in the dispersive estimates.

While we know that $Q$ does not project out $\mathcal{H}_{rel}^{1}(M)$ which
is responsible for, Example 1. We did show that $P_{Q_{1}}\Phi$ is orthogonal
to the space of $L^{2}$ harmonic 1 forms with spherical degree 1, however
there could be problems with the higher order modes. It is not clear that
$P_{Q_{1}}\Phi$ is orthogonal to $\mathcal{H}_{rel}^{1}(M)$. In this case
examining only the class
$\overline{\mathrm{d}C^{\infty}_{0}(M_{\mathrm{int}};\Lambda^{p-1}T^{*}M_{\mathrm{int}})}$
as the domain of the Maxwell propagator, would require a different technique
that the spectral theory representation we have used here. We leave these open
geometric questions for future study.

We have shown that the question of dispersive estimates for spherical
conductors in the absence of charge is a new and exciting question which can
be examined by geometric scattering theory. Balls with charge, multiple
spheres like in [35], and other obstacles /conductors like the torus will be
analyzed in the future.

##  12 Appendix

###  12.1 Hankel Function Estimates

We recall that in $3d$ we have

$\displaystyle
h_{n}^{(1)}(r)=(-\mathrm{i})^{n+1}\frac{e^{\mathrm{i}r}}{r}\sum\limits_{m=0}^{n}\frac{\mathrm{i}^{m}}{m!(2r)^{m}}\frac{(n+m)!}{(n-m)!}$
(12.1)

is the explicit form for the spherical Hankel function of degree $n$. Let
$a_{0}(\nu)=1$ and define

$\displaystyle
a_{k}(\nu)=\frac{(4\nu^{2}-1)(4\nu^{2}-3^{2})...(4\nu^{2}-(2k-1)^{2})}{k!8^{k}}=\frac{(\frac{1}{2}-\nu)_{k}(\frac{1}{2}+\nu)_{k}}{(-2)^{k}k!}.$
(12.2)

From the Nist digital library [34] eqs. 10.17.13,14, and 15, we have that

$\displaystyle{H^{(1)}_{\nu}}\left(z\right)=\left(\frac{2}{\pi
z}\right)^{\frac{1}{2}}e^{\mathrm{i}\omega}\left(\sum_{k=0}^{\ell-1}(\mathrm{i})^{k}\frac{a_{k}(\nu)}{z^{k}}+R_{\ell}^{\pm}(\nu,z)\right)$
(12.3)

where $\ell\in\mathbb{N}$, $\omega=z-\tfrac{1}{2}\nu\pi-\tfrac{1}{4}\pi$ and

$\left|R_{\ell}^{\pm}(\nu,z)\right|\leq
2|a_{\ell}(\nu)|\mathcal{V}_{z,\pm\mathrm{i}\infty}\left(t^{-\ell}\right)\\*\exp\left(|\nu^{2}-\tfrac{1}{4}|\mathcal{V}_{z,\pm\mathrm{i}\infty}\left(t^{-1}\right)\right),$

where $\mathcal{V}_{z,\mathrm{i}\infty}\left(t^{-\ell}\right)$ may be
estimated in various sectors as follows

$\mathcal{V}_{z,\mathrm{i}\infty}\left(t^{-\ell}\right)\leq\begin{cases}|z|^{-\ell},&0\leq\operatorname{ph}z\leq\pi,\\\
\chi(\ell)|z|^{-\ell},&\parbox[t]{224.037pt}{$-\tfrac{1}{2}\pi\leq\operatorname{ph}z\leq
0$ or $\pi\leq\operatorname{ph}z\leq\tfrac{3}{2}\pi$,}\\\
2\chi(\ell)|\operatorname{Im}z|^{-\ell},&\parbox[t]{224.037pt}{$-\pi<\operatorname{ph}z\leq-\tfrac{1}{2}\pi$
or $\tfrac{3}{2}\pi\leq\operatorname{ph}z<2\pi$.}\end{cases}$

Here, $\chi(\ell)$ is defined by

$\displaystyle\chi(x):=\pi^{1/2}\Gamma\left(\tfrac{1}{2}x+1\right)/\Gamma\left(\tfrac{1}{2}x+\tfrac{1}{2}\right).$

###  12.2 Large order bounds

We have from Exercise 9.1 and equation 9.07 p 255 of [33]

$\displaystyle
J_{\nu}(z)=\frac{(\frac{z}{2})^{\nu}e^{-\mathrm{i}z}}{\Gamma(\nu+1)}M(\nu+\frac{1}{2},2\nu+1,2\mathrm{i}z).$
(12.4)

where

$\displaystyle
M(a,c,z)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int\limits_{0}^{1}t^{a-1}(1-t)^{c-a-1}e^{zt}\,\mathrm{d}t\quad\mathrm{Re}(c)>\mathrm{Re}(a)>0.$
(12.5)

We note that for real $z$ we have

$\displaystyle|M(\nu+\frac{1}{2},2\nu+1,2\mathrm{i}z)|\leq
1\qquad|\frac{d}{dz}M(\nu+\frac{1}{2},2\nu+1,2\mathrm{i}z)|\leq 1$ (12.6)

We also have that

$\displaystyle W\\{J_{n}(z),Y_{n}(z)\\}=\frac{2}{\pi z}.$ (12.7)

We get the large order bounds from the definitions above and Stirling’s
formula. It follows that

$\displaystyle|J_{\nu}(z)|\leq\frac{1}{\sqrt{2\pi\nu}}\left(\frac{ez}{2\nu}\right)^{\nu}$
(12.8)

and

$\displaystyle|Y_{\nu}(z)|\leq\sqrt{\frac{2}{\pi\nu}}\left(\frac{ez}{2\nu}\right)^{-\nu}.$
(12.9)

For $z\to 0$, we have the estimate [34] equations 10.7.7 and 2.

$\displaystyle{H^{(1)}_{\nu}}\left(z\right)$
$\displaystyle\sim\begin{cases}-(\mathrm{i}/\pi)\Gamma\left(\nu\right)(\tfrac{1}{2}z)^{-\nu}&\text{
for }\nu>0\\\ (2\mathrm{i}/\pi)\log z&\text{ for }\nu=0\end{cases}$ (12.10)

Here $\sim$ means that the quotient of left- and right hand side converges to
$1$ as $z\to 0$, and $\log$ is the principal branch of the complex logarithm.

###  12.3 Spherical Bessel functions

The spherical Bessel functions $j_{\ell}$ are usually defined as
$j_{\ell}(x)=\sqrt{\frac{\pi}{2x}}J_{\ell+\frac{1}{2}}(x)$. These functions
appear when separating variables for the three dimensional Helmholtz equation.
Here we will need the higher dimensional analog, which we define as

$j_{d,\ell}(x)=\sqrt{\frac{\pi}{2}}x^{\frac{2-d}{2}}J_{\ell+\frac{d-2}{2}}(x),$

and refer to as the $d$-dimensional spherical Bessel function. Similarly, the
corresponding $d$-dimensional spherical Hankel functions are defined as

$\displaystyle
h^{(1)}_{d,\ell}(x)=\sqrt{\frac{\pi}{2}}x^{\frac{2-d}{2}}H^{(1)}_{\ell+\frac{d-2}{2}}(x),$
(12.11) $\displaystyle
h^{(2)}_{d,\ell}(x)=\sqrt{\frac{\pi}{2}}x^{\frac{2-d}{2}}H^{(2)}_{\ell+\frac{d-2}{2}}(x).$

Recall also that the Wronskian of the spherical Bessel function of the first
kind and the second kind in 3 dimensions is:

$\displaystyle W\\{j_{3,n}(z),y_{3,n}(z)\\}=z^{-2}$ (12.12)

We have from [34] that for real $z$

$\displaystyle
y_{3,n}^{2}(z)+j_{3,n}^{2}(z)=|h_{3,n}^{(1)}(z)|^{2}=\sum\limits_{k=0}^{n}\frac{(2k)!(n+k)!}{2^{2k}(k!)^{2}(n-k)!z^{2k+2}}=\sum\limits_{k=0}^{n}\frac{s_{k}(n+\frac{1}{2})}{z^{2k+2}}$
(12.13)

It follows directly that

$|h^{(1)}_{3,k}(z)|\leq|h^{(1)}_{3,n}(z)|$

for all real $z$ and $k\leq n$. The equation which both the spherical Bessel
and Hankel functions $w=j_{3,\ell}(z)$ or $w=h^{1,2}_{3,\ell}(z)$ satisfy is

$\displaystyle
z^{2}\frac{\mathrm{d}^{2}w}{\mathrm{d}z^{2}}+2z\frac{\mathrm{d}w}{\mathrm{d}z}+(z^{2}-\ell(\ell+1))w=0$
(12.14)

where $\ell$ is a non-negative integer. We also have the following derivatives
with respect to order

$\displaystyle\frac{\partial J_{\pm\nu}\left(z\right)}{\partial\nu}=\pm
J_{\pm\nu}\left(z\right)\ln\left(\tfrac{1}{2}z\right)\mp(\tfrac{1}{2}z)^{\pm\nu}\sum_{k=0}^{\infty}(-1)^{k}\frac{\psi\left(k+1\pm\nu\right)}{\Gamma\left(k+1\pm\nu\right)}\frac{(\tfrac{1}{4}z^{2})^{k}}{k!},$
(12.15)

and

$\displaystyle\frac{\partial
Y_{\nu}\left(z\right)}{\partial\nu}=\cot\left(\nu\pi\right)\left(\frac{\partial
J_{\nu}\left(z\right)}{\partial\nu}-\pi
Y_{\nu}\left(z\right)\right)-\csc\left(\nu\pi\right)\frac{\partial
J_{-\nu}\left(z\right)}{\partial\nu}-\pi J_{\nu}\left(z\right).$ (12.16)

###  12.4 Asymptotic expansions

In this part we give a short proof of the formula for $g\in
L^{2}(\mathbb{S}^{2},\mathbb{C}^{3})$

$\displaystyle\frac{1}{(2\pi)^{\frac{d-1}{2}}}\int\limits_{(\mathbb{S}^{2})^{3}}e^{-\mathrm{i}\lambda
x\cdot\omega}g(\omega)\,\mathrm{d}\omega=2\sum\limits_{\nu}a_{\nu}\phi_{\nu}\left(\frac{x}{r}\right)j_{3,\ell_{\nu}}(\lambda
r)(-\mathrm{i})^{\ell_{\nu}}$ (12.17)

which also holds with $\phi_{\nu}$ being replaced by $\Psi_{1,\ell,m}$ in
dimension 3. We also have in dimension 3

$\displaystyle\frac{1}{(2\pi)}\int\limits_{(\mathbb{S}^{2})^{3}}e^{-\mathrm{i}\lambda
x\cdot\omega}g(\omega)\,\mathrm{d}\omega=2\sum\limits_{\ell,m}a_{2,\ell,m}\Psi_{2,\ell,m}\left(\frac{x}{r}\right)\left.\frac{(yj_{3,\ell}(y))^{\prime}}{y}\right|_{y=\lambda
r}(-\mathrm{i})^{\ell-1}$ (12.18)

We can expand using the identity

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}(xj_{3,\ell}^{(1)}(x))=xj_{3,\ell-1}(x)-\ell
j_{3,\ell}(x).$ (12.19)

We obtain

$\displaystyle\left.\frac{(xj_{3,\ell}(x))^{\prime}}{x}\right|_{x=\lambda
r}=j_{3,\ell-1}(\lambda r)-\frac{\ell}{x}j_{3,\ell}(x).$ (12.20)

Rellich’s uniqueness theorem gives that solutions for real $\lambda>0$ are
determined uniquely by their asymptotic expansion

$\displaystyle f(r\theta)=\frac{e^{\mathrm{i}\lambda
r}}{r^{\frac{d-1}{2}}}g_{+}(\theta)+\frac{e^{-\mathrm{i}\lambda
r}}{r^{\frac{d-1}{2}}}g_{-}(\theta)+\mathcal{O}\left(\frac{1}{r^{\frac{d+1}{2}}}\right)$
(12.21)

Application of the stationary phase lemma gives generically in any dimension:

$\int\limits_{(\mathbb{S}^{d-1})^{3}}e^{-\mathrm{i}\lambda
x\cdot\omega}g(\omega)\,\mathrm{d}\omega=\left(\frac{2\pi}{\lambda
r}\right)^{\frac{d-1}{2}}\left(e^{-\mathrm{i}\lambda
r}e^{\frac{\mathrm{i}(d-1)\pi}{4}}g(\theta)+e^{\mathrm{i}\lambda
r}e^{\frac{-\mathrm{i}(d-1)\pi}{4}}g(-\theta)\right)+\mathcal{O}\left((\lambda
r)^{-\frac{d+1}{2}}\right)$

as $\lambda r\rightarrow\infty$. Here the remainder terms depends on the norm

$\|g(\omega)\|_{H^{2}(\mathbb{S}^{2})^{3}}$

by stationary phase whenever $|\lambda r|$ is large. These expansion terms can
be differentiated term by term in $x$ and then expanded again using stationary
phase to obtain the large $|\lambda r|$ terms. Comparing the spherical Bessel
function terms allows to match the coefficients in the generalized
Eigenfunction expansions. Notice that the terms associated to
$\Psi_{3,\ell,m}$ are genuinely lower order in $r$ as $\lambda
r\rightarrow\infty$.

We recall Proposition C.4 in [41]

######  Proposition 10 (Prop. C.4 in [41]).

Let $\chi\in C^{\infty}(M)$ be supported in $M\setminus K$ be such that
$1-\chi\in C_{0}^{\infty}(M)$. Then $f\in C_{0}^{\infty}(M)$ is outgoing if
and only if $\chi f$ is outgoing for the Laplace operator on $\mathbb{R}^{d}$.

We use this to show the following

######  Lemma 5.

In the exterior of the ball of radius $\rho$, $f$ is outgoing for fixed
$\lambda$ and divergence free then $f$ is of the form

$\displaystyle\tilde{h}^{(1)}_{\lambda,Y}(Y)$ (12.22)

###### Proof.

By repeating the proof of Theorem 3 (Theorem 2.43 in [17]) using
$r\hat{r}h_{\ell}^{(1)}(\lambda r)$ instead of $r\hat{r}j_{\ell}^{1}(\lambda
r)$ we have a free space solution of the Maxwell system 7.1, as remarked in
[17]. Now we need to compare this to the exact solution. The free space kernel
for the scalar Helmholtz equation is

$\displaystyle
R_{0,\lambda}(x,y)=\frac{\mathrm{i}}{4}\left(\frac{\lambda}{2\pi|x-y|}\right)^{\frac{1}{2}}H_{\frac{1}{2}}^{(1)}(\lambda|x-y|).$
(12.23)

In three dimensions this simplifies to

$\displaystyle
R_{0,\lambda}(x,y)=\frac{e^{\mathrm{i}\lambda|x-y|}}{4\pi|x-y|}.$ (12.24)

The leading order asymptotic terms coincide with the free space kernel of
$\mathrm{curl}(xR_{0,\lambda})$ and
$\mathrm{curl}\mathrm{curl}(xR_{0,\lambda})$ using the expansions in the
previous part of this appendix subsection with $j_{\ell}(\lambda r)$ replaced
by $h_{\ell}^{(1)}(\lambda r)$ and (12.3). ∎

###  12.5 Vector Spherical Coordinates

This short section is a review of some necessary facts about spherical
coordinate systems. It is taken directly from example A.17 in [17], page 328.
We consider the sphere of radius $r$. We can parametrize the boundary of a
ball of radius $r$ in local coordinates as follows

$\displaystyle\sigma(\theta,\phi)=r(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)^{t}.$
(12.25)

The surface gradient and the surface divergence respectively on the boundary
of the sphere are given by

$\displaystyle\mathrm{Grad}f(\theta,\phi)=\frac{1}{r}\frac{\partial
f}{\partial\theta}(\theta,\phi)\hat{\theta}+\frac{1}{r\sin\theta}\frac{\partial
f}{\partial\varphi}(\theta,\phi)\hat{\varphi}$
$\displaystyle\mathrm{Div}F(\theta,\phi)=\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}(\sin\theta
F_{\theta}(\theta,\phi))+\frac{1}{r\sin\theta}\frac{\partial
F_{\phi}}{\partial\phi}(\theta,\phi)$

where

$\hat{\theta}=(\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta)^{t}$

and

$\hat{\phi}=(-\sin\phi,\cos\phi,0)^{t}$

are the vectors which span the tangent plane and $F_{\theta},F_{\phi}$ are the
components of $F$ with respect to these vectors, that is
$F=F_{\theta}\hat{\theta}+F_{\phi}\hat{\phi}$.

When spherical coordinates are used $x=r\hat{r}\in\mathbb{R}^{3}$, the surface
differential operators with respect to the unit sphere are used instead of the
operators on the boundary, $\partial D$ of

$D=\\{x\in\mathbb{R}^{3}:|x|=r\\}.$

Therefore we indicate this by using the index $\mathbb{S}^{2}$ for the
differential operators with respect to the sphere with $r=1$. By scaling
invariance we have

$\displaystyle\mathrm{Grad}_{\mathbb{S}^{2}}f(r,\hat{r})=r\mathrm{Grad}f(r\hat{r})\qquad\mathrm{Div}_{\mathbb{S}^{2}}F(r,\hat{r})=r\mathrm{Div}F(r\hat{r})$
(12.26)

It is understood that $f(r,\cdot)$ is a function of the unit sphere on the
right hand side and on the left hand side it is a function on $\partial D$ on
the right hand side. This scaling allows us to create divergence free