A Rigorous Time-Domain Analysis of Full–Wave
Electromagnetic Cloaking (Invisibility) ∗†
Ricardo Weder‡
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas
Departamento de Métodos Matemáticos y Numéricos
Universidad Nacional Autónoma de México
Apartado Postal 20-726, México DF 01000
weder@servidor.unam.mx
Abstract
There is currently a great deal of interest in the theoretical and practical possibility
of cloaking objects from the observation by electromagnetic waves. The basic idea
of these invisibility devices [8, 9, 13], [18] is to use anisotropic transformation media
whose permittivity and permeability ελν , µλν, are obtained from the ones, ελν0 , µ
of isotropic media, by singular transformations of coordinates.
In this paper we study electromagnetic cloaking in the time-domain using the for-
malism of time-dependent scattering theory [23]. This formalism provides us with
a rigorous method to analyze the propagation of electromagnetic wave packets
with finite energy in transformation media. In particular, it allows us to settle
in an unambiguous way the mathematical problems posed by the singularities of
the inverse of the permittivity and the permeability of the transformation media
on the boundary of the cloaked objects. Von Neumann’s theory of self-adjoint
extensions of symmetric operators plays an important role on this issue. We
write Maxwell’s equations in Schrödinger form with the electromagnetic propa-
gator playing the role of the Hamiltonian. We prove that the electromagnetic
propagator outside of the cloaked objects is essentially self-adjoint. This means
that it has only one self-adjoint extension, AΩ, and that this self-adjoint extension
generates the only possible unitary time evolution, with constant energy, for finite
energy electromagnetic waves, propagating outside of the cloaked objects.
∗PACS classification scheme 2006: 41.20.Jb, 02.30.Tb,02.30.Zz, 02.60.Lj.
†Research partially supported by CONACYT under Project P42553F.
‡Fellow Sistema Nacional de Investigadores.
http://arxiv.org/abs/0704.0248v4
Moreover, AΩ is unitarily equivalent to the electromagnetic propagator in the
medium ελν0 , µ
0 . Using this fact, and since the coordinate transformation is the
identity outside of a ball, we prove that the scattering operator is the identity.
This implies that for any incoming finite-energy electromagnetic wave packet the
outgoing wave packet is precisely the same. In other words, it is not possible
to detect the cloaked objects in any scattering experiment where a finite-energy
wave packet is sent towards the cloaked objects, since the outgoing wave packet
that is measured after interaction is the same as the incoming one. Our results
give a rigorous proof that the construction of [8, 9, 13], [18] cloaks passive and
active devices from observation by electromagnetic waves. Actually, the cloaking
outside is independent of what is inside the cloaked objects.
As is well known, self-adjoint extensions can be understood in terms of bound-
ary conditions. Actually, for the electromagnetic fields in the domain of AΩ the
component tangential to the exterior of the boundary of the cloaked objects of
both, the electric and the magnetic field have to be zero. This boundary condi-
tion is self-adjoint in our case because the permittivity and the permeability are
degenerate on the boundary of the cloaked objects.
Furthermore, we prove cloaking for general anisotropic materials. In particular,
our results prove that it is possible to cloak objects inside general crystals.
1 Introduction
There is currently a great deal of interest in the theoretical and practical possibility of
cloaking objects from the observation by electromagnetic fields. The basic idea of these in-
visibility devices [8, 9, 13], [18] is to use anisotropic transformation media whose permittivity
and permeability, ελν , µλν , are obtained from the ones, ελν0 , µ
0 , of isotropic media, by sin-
gular transformations of coordinates. The singularities lie on the boundary of the objects to
be cloaked. Here the material interpretation is taken. Namely, the ελν , µλν and the ελν0 , µ
represent the components in flat Cartesian space of the permittivity and the permeability of
physical media with different material properties. It appears that with existing technology it
is possible to construct media as described above using artificially structured metamaterials.
In [8, 9] a proof of cloaking was given for the conductivity equation -i.e., in the case of zero
frequency- from detection by measurement of the Dirichlet to Neumann map that relates
the value of the electric potential on the boundary to its normal derivative. The papers [13]
and [18] consider electromagnetic waves in the geometrical optics approximation, i.e. for
large frequencies. In [24] a experimental verification of cloaking is presented and [4] and
[5] give a numerical simulation. A rigorous prof of cloaking has already been given by [7]
where fixed frequency waves were studied, i.e., in the frequency domain. They consider a
class of finite energy solutions to Maxwell’s equations in a bounded set, O, that contains the
cloaked object on its interior, and they prove cloaking, at any frequency, with respect to the
measurement of the Cauchy data of these solutions on the boundary of O. We give further
comments on this paper below. For other results on this problem see [25] and [15]. In [16]
cloaking of elastic waves is considered, and the history of invisibility is discussed.
In this paper we study electromagnetic cloaking in the time-domain using the formalism of
time-dependent scattering theory [23]. This formalism provides us with a rigorous method to
analyze the propagation of electromagnetic wave packets with finite energy in transformation
media. In particular, it allows us to settle in an unambiguous way the mathematical problems
posed by the singularities of the inverse of the permittivity and the permeability of the
transformation media on the boundary of the cloaked objects. Von Neumann’s theory of self-
adjoint extensions of symmetric operators plays an important role on this issue. We write
Maxwell’s equations in Schrödinger form with the electromagnetic propagator playing the
role of the Hamiltonian. We prove that the electromagnetic propagator outside of the cloaked
objects is essentially self-adjoint. This means that it has only one self-adjoint extension,
AΩ, and that this self-adjoint extension generates the only possible unitary time evolution,
with constant energy, for finite energy electromagnetic waves propagating outside of the
cloaked objects. Moreover, AΩ is unitarily equivalent to the electromagnetic propagator in
the medium ελν0 , µ
0 . Using this fact, and since the coordinate transformation is the identity
outside of a ball, we prove that the scattering operator is the identity. This implies that for
any incoming finite-energy electromagnetic wave packet the outgoing wave packet is precisely
the same. In other words, it is not possible to detect the cloaked objects in any scattering
experiment where a finite-energy wave packet is sent towards the cloaked objects, since the
outgoing wave packet that is measured after interaction is the same as the incoming one.
Our results give a rigorous proof that the construction of [8, 9, 13], [18] cloaks passive and
active devices from observation by electromagnetic waves. Actually, the cloaking outside is
independent of what is inside the cloaked objects.
As is well known, self-adjoint extensions can be understood in terms of boundary condi-
tions. Actually, for the electromagnetic fields in the domain of AΩ the component tangential
to the exterior of the boundary of the cloaked objects of both, the electric and the mag-
netic field have to be zero. This boundary condition is self-adjoint in our case because the
permittivity and the permeability are degenerate on the boundary of the cloaked objects.
Furthermore, we prove cloaking for general anisotropic materials. In particular, our
results prove that it is possible to cloak objects inside general crystals.
Even though, as mentioned above, the cloaking is independent of the cloaked objects,
and in particular, the cloaking outside is not affected by the presence of passive and/or
active devices inside the cloaked objects, we discuss the dynamics of electromagnetic waves
inside the cloaked objects for completeness, since it helps to understand the above mentioned
independence of cloaking from the properties of the cloaked objects.
We prove that every self-adjoint extension of the electromagnetic propagator in a trans-
formation medium is the direct sum of the unique self-adjoint extension in the exterior of
the cloaked objects, AΩ, with some self-adjoint extension of the electromagnetic propagator
in the interior of the cloaked objects. Each of these self-adjoint extensions corresponds to a
possible unitary time evolution for finite energy electromagnetic waves. As is well known,
the fact that time evolution is unitary assures us that energy is conserved. This results
implies that the electromagnetic waves inside and outside of the cloaked objects completely
decouple from each other. Actually, the electromagnetic waves inside the cloaked objects
are not allowed to leave them, and viceversa, the electromagnetic waves outside can not go
inside.
In terms of boundary conditions, this means that transmission conditions that link the
electromagnetic fields inside and outside the cloaked objects are not allowed, since they do
not correspond to self-adjoint extensions of the electromagnetic propagator, and then, they
do not lead to a unitary dynamics that conserves energy. Furthermore, choosing a particular
self-adjoint extension of the electromagnetic propagator of the cloaked objects amounts to
choosing some boundary condition on the inside of the boundary of the cloaked objects. In
other words, any possible unitary dynamics implies the existence of some boundary condition
on the inside of the boundary of the cloaked objects.
The fact that there is a large class of self-adjoint extensions -or boundary conditions-
that can be taken inside the cloaked objects could be useful in order to enhance cloaking
in practice, where one has to consider approximate transformation media as well as in the
analysis of the stability of cloaking.
Actually, we consider a slightly more general construction than the one of [8, 9, 13], [18]
since we allow for a finite number of cloaked objects.
In [7] a very general construction for cloaking is introduced. In the case of Maxwell’s
equations all their constructions are made within the context of the permittivity and the
permeability tensor densities being conformal to each other, i.e., multiples of each other by a
positive scalar function. In particular, all isotropic media are included in this category. They
mention that both for mathematical and practical reasons it would be very interesting to
understand cloaking for general anisotropic materials in the absence of this assumption. In
this paper we actually solve this problem, since we prove cloaking for all general anisotropic
materials. In particular, our results prove that it is possible to cloak objects inside general
crystals.
Note, moreover, that [7] also considers the cases of the Helmholtz equation. We do not
discuss this problems here.
Furthermore, remark that the existing theorems in the uniqueness of inverse scattering
do not apply under the present conditions.
In [7] cloaking is proven with respect to the Cauchy data at any fixed frequency given
on a surface that encloses the cloaked object. In the case where the permittivity and the
permeability are bounded above and below it is well known that the Cauchy data at a fixed
frequency is equivalent to the scattering matrix at the same frequency. See for example
[17] and [26]. This equivalence is, however, not proven in the case where the permittivity
and the permeability are degenerate on the boundary of the objects. In fact, it is perhaps
even not true for general degenerate media that are not transformation media since in this
case it is possible that there are finite energy electromagnetic waves that are absorbed by
the boundary of the objects as t → ±∞. If this is true, the equivalence will not hold
since the Cauchy data in a surface that encloses the objects will not contain information
on the waves that are asymptotically absorbed by the boundary of the objects. It is a
problem of independent interest to see if this actually happens or not for general degenerate
permittivities and permeabilities. For an example of scattering by a bounded obstacle with
a singular boundary and Neumann boundary condition, where this happens see [10]. For a
similar situation in the scattering of electromagnetic waves by a Schwarzschild black-hole see
[2]. Note that in our approach we directly consider the scattering operator that is measured
in scattering experiments.
In the analysis of Maxwell’s equations with permittivity and permeability that are inde-
pendent of frequency the dispersion of the medium is not taken into account. This means
that cloaking will hold for electromagnetic wave packets with a narrow enough range of
frequencies, such that this assumption is valid.
The paper is organized as follows. In Section 2 we prove our results in electromagnetic
cloaking. In Section 3 we consider the propagation of electromagnetic waves in the interior
of the cloaked objects. In Section 4 we formulate cloaking as a boundary value problem
outside of the cloaked objects for the Maxwell equations at a fixed frequency, following
our analysis of the self-adjoint extensions of the electromagnetic propagator. In particular,
we give the appropriate boundary condition on the outside of the boundary of the cloaked
objects. Finally, in Section 5 we prove cloaking of infinite cylinders. This is of interest
since this is the case considered in the experimental verification in [24] and in the numerical
simulations of [4] and [5]. Of course, [24] only consider a slice of the cylinder. In Sections 3
and 4 we give further comments on the results of [7].
Addendum
After the previous version of this paper was posted in the arXiv we published the paper
[29] where we generalized the results of this paper on spherical cloaks to the case of high-
order cloaks, and where we also discussed cloaking in the frequency domain. Moreover, in
our paper [30] we identified the cloaking boundary condition that has to be satisfied in the
inside of the boundary of the cloaked objects, in the case where the permittivity and the
permeability are bounded above and below inside the cloaked objects.
2 Electromagnetic Cloaking
Let us consider Maxwell’s equations,
∇× E = −
B, ∇×H =
D, (2.1)
∇ ·B = 0,∇ ·D = 0, (2.2)
in a domain, Ω ⊂ R3, as follows,
Ω := R3 \ ∪Nj=1Kj, Kj ∩Kl = ∅, j 6= l (2.3)
where Kj, j = 1, 2, · · · , N, are the objects to be cloaked. We assume that each Kj is a ball
with center cj and radius aj, i.e.,
x ∈ R3 : |x− cj | ≤ aj
, j = 1, 2, · · · , N. (2.4)
The cloaked objects are denoted by
K := ∪Nj=1Kj .
We designate the Cartesian coordinates of x by xλ, λ = 1, 2, 3 and by Eλ, Hλ, B
λ, Dλ, λ =
1, 2, 3, respectively, the components of E,H,B, and D. As usual, we denote by ελν and µλν ,
respectively, the permittivity and the permeability. We have that,
Dλ = ελνEν , B
λ = µλνHν , (2.5)
where we use the standard convention of summing over repeated lower and upper indices.
We consider now a transformation from Ω0 := R
3 \ {c1, c2, · · · , cN} onto Ω that was first
used to obtain cloaking for the conductivity equation, i.e. at zero frequency, by [8, 9] and
then by [18] for cloaking electromagnetic waves (for a related result in two dimensions using
conformal mappings see [13]).
For any y ∈ R3 we denote, ŷ := y/|y|. Let yλ, λ = 1, 2, 3, designate the cartesian
coordinates of y ∈ Ω0. Take bj > aj, j = 1, 2, · · · , N . Then, for 0 < |y− cj| ≤ bj , we define,
x = x(y) = f(y) := cj +
bj − aj
|y − cj |+ aj
ŷ − cj. (2.6)
Note that this transformation blows up the point cj onto ∂Kj and that it sends the punctu-
ated ball B̃cj (bj) := {y ∈ R
3 : 0 < |y − cj | ≤ bj} onto the spherical shell, aj < |x− cj | ≤ bj .
We assume that,
B̃cj (bj) ∩ B̃cl(bl) = ∅, j 6= l, 1 ≤ j, l ≤ N. (2.7)
For y ∈ R3\∪Nj=1B̃cj (bj) we define the transformation to be the identity, x = x(y) = f(y) :=
y. Our transformation is a bijection from Ω0 onto Ω. By y = y(x) := f
−1(x) we designate
the inverse transformation. We denote the elements of the Jacobian matrix by Aλλ′ ,
Aλλ′ :=
. (2.8)
Note that the Aλλ′ ∈ C
Ω0 \ ∪
j=1∂B̃cj (bj)
. We designate by Aλ
λ the elements of the
Jacobian of the inverse bijection, y = y(x) := f−1(x),
Ω \ ∪Nj=1∂B̃cj (bj)
. (2.9)
The papers [8, 9] and [18] considered the case where N = 1, c1 = 0.
We take here the so called material interpretation and we consider our transformation
as a bijection between two different spaces, Ω0 and Ω. However, our transformation can be
considered, as well, as a change of coordinates in Ω0. Of course, these two point of view
are mathematically equivalent. This means, in particular, that under our transformation the
Maxwell equations in Ω0 and in Ω will have the same invariance that they have under change
of coordinates in three-space. See, for example, [21]. Let us denote by ∆ the determinant of
the Jacobian matrix (2.8). Then,
bj − aj
( bj−aj
|y− cj|+ aj
|y − cj|
, for 0 < |y − cj | ≤ bj . (2.10)
This result is easily obtained rotating into a coordinate system such that, y − cj = (|y −
cj|, 0, 0) [25]. For y ∈ Ω0 \ ∪
j=1B̃cj (bj),∆ ≡ 1.
Let us denote by E0,H0,B0,D0, ε
0 , µ
0 , respectively, the electric and magnetic fields,
the magnetic induction, the electric displacement, and the permittivity and permeability of
Ω0. ε
0 , µ
0 , are positive, Hermitian matrices that are constant in Ω0.
The electric field is a covariant vector that transforms as,
Eλ(x) = A
λ (y)E0,λ′(y). (2.11)
The magnetic fieldH is a covariant pseudo-vector, but as we only consider space transfor-
mations with positive determinant, it also transforms as in (2.11). The magnetic induction
B and the electric displacement D are contravariant vector densities of weight one that
transform as
Bλ(x) = (∆(y))
Aλλ′(y)B
0 (y), (2.12)
with the same transformation for D. The permittivity and permeability are contravariant
tensor densities of weight one that transform as,
ελν(x) = (∆(y))
Aλλ′(y)A
ν′(y) ε
0 (y), (2.13)
with the same transformation for µλν . The Maxwell equations (2.1, 2.2) are the same in
both spaces Ω and Ω0. Let us denote by ελν , µλν, ε0λν , µ0λν , respectively, the inverses of the
corresponding permittivity and permeability. They are covariant tensor densities of weight
minus one that transform as,
ελν(x) = ∆(y)A
λ (y)A
ν (y) ε0λ′ν′(y), µλν(x) = ∆(y)A
λ (y)A
ν (y)µ0λ′ν′(y). (2.14)
Note that
det ελν = ∆−1 det ελν0 , detµ
λν = ∆−1 detµλν0 , (2.15)
det ελν = ∆det ε0λν , detµλν = ∆detµ0λν . (2.16)
We now introduce the Hilbert spaces of electric and magnetic fields with finite energy.
The E0,H0,B0,D0, were defined in Ω0, but since R
3 \ Ω0 = {cj}
j=1 is of measure zero, we
can consider them as defined in R3, what we do below.
We denote by H0E the Hilbert space of all measurable, square integrable, C
3− valued
functions defined on R3 with the scalar product,
0ν dy
3. (2.17)
We similarly define the Hilbert space,H0H , of all measurable, square integrable, C
3− valued
functions defined on R3 with the scalar product,
0ν dy
3. (2.18)
The Hilbert space of finite energy fields in R3 is the direct sum
H0 := H0E ⊕H0H . (2.19)
Moreover, we designate byHΩE the Hilbert space of all measurable, C
3− valued functions
defined on Ω that are square integrable with the weight ελν , with the scalar product,
(1),E(2)
3. (2.20)
Finally, we denote by HΩH the Hilbert space of all measurable, C
3− valued functions
defined on Ω that are square integrable with the weight µλν , with the scalar product,
(1),H(2)
3. (2.21)
The Hilbert space of finite energy fields in Ω is the direct sum
HΩ := HΩE ⊕HΩH . (2.22)
We now write the Maxwell’s equations (2.1) in Schrödinger form. We first consider the
case of R3. We denote by ε0 and µ0, respectively, the matrices with entries ε0λν and µ0λν .
Recall that (∇×E)
= sλνρ ∂
Eρ where s
λνρ is the permutation contravariant pseudo-
density of weight −1 (see section 6 of chapter II of [21], where a different notation is used).
By a0 we denote the following formal differential operator,
ε0∇×H0
−µ0∇× E0
. (2.23)
Here, as usual, we denote, ε0∇ × H0 := ε0λν(∇ × H0)
ν , and µ0∇ × E0 = µ0λν(∇ × E0)
Then, equations (2.1) are equivalent to,
. (2.24)
Let us denote by C10(R
3) the set of all C6−valued continuously differentiable functions
on R3 that have compact support. Then, a0 with domain C
3) is a symmetric operator in
H0, i.e., a0 ⊂ a
0. Moreover, it is essentially self-adjoint in H0, i.e., it has only one self-adjoint
extension, that we denote by A0. Its domain is given by,
D(A0) =
, (2.25)
∈ D(A0), (2.26)
where the derivatives are taken in distribution sense. These results follow easily from the
fact that -via the Fourier transform- a0 is unitarily equivalent to multiplication by a matrix
valued function that is symmetric with respect to the scalar product of H0. Moreover, it
follows from explicit computation that the only eigenvalue of A0 is zero, that it has infinite
multiplicity, and that,
H0⊥ := (kernelA0)
∈ H0 :
ελν0 E0ν = 0,
µλν0 H0ν = 0
. (2.27)
Furthermore, A0 has no singular-continuous spectrum and its absolutely-continuous spec-
trum is R. See, for example, [27, 28].
Taking any
∈ H0⊥ ∩D(A0) (2.28)
we obtain a finite energy solution to the Maxwell equations (2.1, 2.2) as follows
(t) = e−itA0
. (2.29)
This is the unique finite energy solution with initial value at t = 0 given by (2.28). Note
that as e−itA0H0⊥ ⊂ H0⊥ equations (2.2) are satisfied for all times if they are satisfied at
t = 0.
Let us now consider the case of Ω. We denote by ε and µ, respectively, the matrices with
entries ελν and µλν .
We now define the following formal differential operator,
−µ∇× E
. (2.30)
Equations (2.1) are equivalent to,
Let us denote by C10(Ω) the set of all C
6−valued continuously differentiable functions on
Ω that have compact support. Then, aΩ with domain C
0(Ω) is a symmetric operator in HΩ.
To construct a unitary dynamics that preserves energy we have to analyse the self-adjoint
extensions of aΩ.
We denote by UE the following unitary operator from H0E onto HΩE ,
(UEE0)λ (x) := A
λ E0λ′(y), (2.31)
and by UH the unitary operator from H0H onto HΩH ,
(UHH0)λ (x) := A
λ H0λ′(y). (2.32)
Then,
U := UE ⊕ UH (2.33)
is a unitary operator from H0 onto HΩ.
We denote by a00 the restriction of a0 to C
0(Ω0). The operator a00 is essentially
self-adjoint and its only self-adjoint extension is A0. This follows from the essential self-
adjointness of a0 and from the fact that any function in C
3) can be approximated in the
graph norm of a0 by functions in C
0(Ω0). To prove this take any continuously differentiable
real-valued function, φ, defined on R such that, φ(y) = 0, |y| ≤ 1 and φ(y) = 1, |y| ≥ 2.
Then, for any
∈ C10(R
we have that,
φ(n|y− cj|)
∈ C10(Ω0)
and moreover,
s- lim
φ(n|y− cj|)
s- lim
φ(n|y− cj|)
where by s- lim we designate the strong limit in H0.
As a00 is essentially self-adjoint, it follows from the invariance of Maxwell equations that
aΩ is essentially self-adjoint, and that its unique self-adjoint extension, that we denote by
AΩ, satisfies
AΩ = U A0 U
∗. (2.34)
For the proof of these facts see [29]. Hence, we have the following theorem.
THEOREM 2.1. The operator aΩ is essentially self-adjoint, and its unique self-adjoint
extension, AΩ, satisfies (2.34).
The unitary equivalence given by (2.34) implies that AΩ has the same spectral prop-
erties that A0. Namely, it has no singular-continuous spectrum, the absolutely-continuous
spectrum is R and the only eigenvalue is zero and it has infinite multiplicity. Moreover,
HΩ⊥ := (kernelAΩ)
∈ HΩ :
ελνEν = 0,
µλνHν = 0
. (2.35)
Furthermore, taking any
∈ HΩ⊥ ∩D(AΩ) (2.36)
we obtain a finite energy solution to the Maxwell equations (2.1, 2.2) as follows
(t) = e−itAΩ
. (2.37)
This is the unique finite energy solution with initial value at t = 0 given by (2.36). Note
that as e−itAΩHΩ⊥ ⊂ HΩ⊥ equations (2.2) are satisfied for all times if they are satisfied at
t = 0. We can consider more general solutions by considering the scale of spaces associated
with AΩ, but we do not go into this direction here.
The facts that aΩ is essentially self-adjoint and that its unique self-adjoint extension AΩ is
unitarily equivalent to the propagator A0 of the homogeneous medium are strong statements.
They mean that the only possible unitary dynamics in Ω that preserves energy is given by
(2.37) and that this dynamics is unitarily equivalent to the free dynamics in R3 given by
(2.29). In fact, ∂Ω acts like a horizon for electromagnetic waves propagating in Ω in the
sense that the dynamics is uniquely defined without any need to consider the cloaked objects
K = ∪Nj=1Kj . As we will prove below this implies electromagnetic cloaking for all frequencies
in the strong sense that the scattering operator is the identity.
Since D(AΩ) = UD(A0), for any (E,H)
T ∈ D(AΩ) there is a (E0,H0)
T ∈ D(A0) such
. (2.38)
Then, it follows from (2.31, 2.32, 2.33) that
E× n = 0,H× n = 0, in ∂K+, (2.39)
where ∂K+ denotes the outside of the boundary of the cloaked objects, K, and n is the
normal vector to ∂K+, if (E0,H0) are, for example, bounded near ∂K+. That is to say, for
electromagnetic fields in the domain of AΩ the tangential components of both, the electric
and the magnetic field vanish in the exterior of the boundary of the cloaked objects. This is a
self-adjoint boundary condition because the permittivity and the permeability are degenerate
on ∂K+.
Let χΩ be the characteristic function of Ω, i.e., χΩ(x) = 1,x ∈ Ω, χΩ(x) = 0,x ∈ R
3 \ Ω.
We define,
(x) := χΩ(x)
(x). (2.40)
By (2.6, 2.10, 2.13),
∣ελν(x)
∣ ≤ C,
∣µλν(x)
∣ ≤ C, x ∈ Ω.
Then, J is a bounded operator from H0 into HΩ.
The wave operators are defined as follows,
W± = s- lim
eitAΩ Je−itA0P0⊥, (2.41)
where P0⊥ denotes the projector onto H0⊥.
Let us designate by W1,2(R3) the Sobolev space of C6 valued functions. We denote by I
the identity operator on H0. Then,
LEMMA 2.2.
W± = UP0⊥. (2.42)
Proof: Denote,
W (t) := eitAΩ J e−itA0P0⊥.
By (2.34), for any ϕ ∈ H0
W (t)ϕ = ψ(t) + UP0⊥ϕ, (2.43)
ψ(t) := U eitA0 (U∗J − I) e−itA0P0⊥ϕ. (2.44)
Let BR denote the ball of center zero and radius R in R
3. Since for |y| ≥ R, with R
large enough, our transformation, x = f(y), is the identity, x = y, and in consequence,
Aλλ′(y) = δ
λ′ for |y| ≥ R, we have that,
(U∗J − I) = (U∗J − I)χBR
. (2.45)
It follows that,
s- lim
ψ(t) = U s- lim
eitA0ϑ(t) (2.46)
with,
ϑ(t) := (U∗J − I)χBR
e−itA0P0⊥ϕ. (2.47)
We have that,
‖ϑ(t)‖
e−itA0P0⊥ϕ
e−itA0P0⊥ϕ
e−itA0P0⊥ϕ
. (2.48)
Then, as (A0 + i)
−1P0⊥ is bounded from H0 into W
1,2(R3) [27] [28], it follows from the
Rellich local compactness theorem that
(A0 + i)
−1P0⊥
is a compact operator in H0. Suppose that ϕ ∈ D(A0) ∩ H0⊥. Then,
s- lim
e−itA0P0⊥ϕ = s- lim
(A0 + i)
−1P0⊥e
−itA0(A0 + i)ϕ = 0, (2.49)
and whence, by (2.48),
s- lim
ϑ(t) = 0, (2.50)
and it follows that in this case,
s- lim
ψ(t) = 0. (2.51)
By continuity this is also true for ϕ ∈ H0⊥.
Then, (2.42) follows from (2.43) and (2.51).
The scattering operator is defined as
S := W ∗+W−. (2.52)
COROLLARY 2.3.
S = P0⊥. (2.53)
Proof: This is immediate from (2.42) because U∗ U = I.
Let us denote by S⊥ the restriction of S to H0⊥. S⊥ is the physically relevant scattering
operator that acts in the Hilbert space H0⊥ of finite energy fields that satisfy equations (2.2).
We designate by I⊥ the identity operator on H0⊥. We have that,
COROLLARY 2.4.
S⊥ = I⊥. (2.54)
Proof: This follows from Corollary 2.3.
The fact that S⊥ is the identity operator on H0⊥ means that there is perfect cloaking for
all frequencies. Suppose that for very negative times we are given an incoming wave packet
e−itA0ϕ−, with ϕ− ∈ H0⊥. Then, for large positive times the outgoing wave packet is given
by e−itA0ϕ+ with ϕ+ = S⊥ϕ−. But, as S⊥ = I, we have that ϕ+ = ϕ− and then,
e−itA0ϕ− = e
−itA0ϕ+.
Since the incoming and the outgoing wave packets are the same there is no way to detect
the cloaked objects K from scattering experiments performed in Ω.
In this paper we considered transformation media obtained from a singular transformation
that blows up a finite number of points, by simplicity, and since this is the situation in the
applications. Suppose that we have a transformation that is singular in a set of points
that we call M and denote now Ω0 := R
3 \M . What we really used in the proofs is that
W1,2(R3) = W
0 (Ω0) where W
0 (Ω0) denotes the completion of C
0 (Ω0) in the norm of
W1,2(R3). We also assumed that ελν0 , µ
0 are constant. What was actually needed is that a0
is essentially self-adjoint. All our results hold under this more general conditions provided
that in (2.41, 2.42) and (2.53) we replace P0⊥ by the projector onto the absolutely-continuous
subspace of A0 and that we assume that D(A0) ∩H0ac ⊂ W
1,2(R3), where we have denoted
the absolutely-continuous subspace of A0 by H0ac. Moreover, S⊥ has to be defined as the
restriction of S to H0ac and in (2.54) I⊥ has to be the identity operator on H0ac. Note that
under these general assumptions A0 could have non-zero eigenvalues and singular-continuous
spectrum.
For example, W1,2(R3) = W
0 (Ω0) if M has zero Sobolev one capacity [1, 11, 12].
Moreover, assume that the permittivity and the permeability tensor densities ελν0 , µ
0 are
bounded below and above. Under this condition a0 is essentially self-adjoint. Furthermore,
let us denote by Ĥ0 the Hilbert space of finite energy solutions defined as in (2.19) but
with ελν0 = µ
0 = δ
λµ. Let Â0, Ĥ0⊥ be, respectively, the electromagnetic propagator in Ĥ0
and the orthogonal complement of its kernel. We have that H0 and Ĥ0 are the same set
of functions with equivalent norms. Furthermore, D(A0) = D(Â0), kernel Â0 = kernelA0.
Moreover, (E0,H0)
T ∈ H0⊥ if and only if E0 = ε0Ê0,H0 = µ0Ĥ0 for some (Ê0, Ĥ0) ∈ Ĥ0⊥.
As [27, 28] D(Â0) ∩ Ĥ0⊥ ⊂ W
1,2(R3) we have that D(A0) ∩ H0⊥ ⊂ W
1,2(R3) if ε0, µ0 are
bounded operators on W1,2(R3) and this is true if the derivatives ∂
µ0 are bounded
operators on Ĥ0 for ρ = 1, 2, 3. Note, furthermore, that H0ac ⊂ H0⊥.
3 Electromagnetic Waves Inside the Cloaked Objects
Let us now consider the propagation of electromagnetic waves in the cloaked objects. For
this purpose we assume that in each Kj the permittivity and the permeability are given
by ελνj , µ
j , with inverses εjλν, µjλν and where εj, µj are the matrices with entries εjλν, µjλν.
Furthermore, we assume that 0 < ελνj , µ
j ≤ C,x ∈ Kj and that for any compact set Q con-
tained in the interior of Kj there is a positive constant CQ such that det ε
j > CQ, detµ
CQ,x ∈ Q. In other words, we only allow for possible singularities of εj, µj on the boundary
of Kj.
We designate by HjE the Hilbert space of all measurable, C
3− valued functions defined
on Kj that are square integrable with the weight ε
j , with the scalar product,
jν dx
3. (3.1)
Similarly, we denote by HjH the Hilbert space of all measurable, C
3− valued functions
defined on Kj that are square integrable with the weight µ
j , with the scalar product,
jν dx
3. (3.2)
The Hilbert space of finite energy fields in Kj is the direct sum
Hj := HjE ⊕HjH, (3.3)
and the Hilbert space in the cloaked objects K is the direct sum,
HK := ⊕
j=1Hj .
The complete Hilbert space of finite energy fields including the cloaked objects is,
H := HΩ ⊕HK . (3.4)
We now write (2.1) as a Schrödinger equation in each Kj as before. We define the
following formal differential operator,
εj∇×Hj
−µj∇× Ej
. (3.5)
Equation (2.1) in Kj is equivalent to
. (3.6)
Let us denote by C10(K̂j) the set of all C
6−valued continuously differentiable functions on
Kj that have compact support in the interior of Kj, that we denote by K̂j := Kj \ ∂Kj .
Then, aj with domain C
0(K̂j) is a symmetric operator in Hj . We denote,
a := aΩ ⊕
j=1 aj , (3.7)
with domain,
D(a) :=
⊕Nj=1
∈ C10(Ω)⊕
j=1 C
0(K̂j)
. (3.8)
The operator a is symmetric in H. The possible unitary dynamics that preserve energy for
the whole system including the cloaked objects K are given by the self-adjoint extensions of
a. Let us denote a the closure of a, with similar notation for aΩ, aj , j = 1, · · · , N . Then,
a = AΩ ⊕
j=1 aj,
where we used the fact that as aΩ is essentially self-adjoint, aΩ = AΩ. The adjoint of a is
given by,
D(a∗) =
⊕Nj=1
∈ H :
∈ D(AΩ), aj
, (3.9)
⊕Nj=1
⊕Nj=1 aj
, (3.10)
⊕Nj=1
∈ D(a∗). (3.11)
Let us denote by KΩ± := kernel(i∓ a
Ω),Kj± := kernel(i∓ a
j) the deficiency subspaces of
aΩ and aj, j = 1, · · · , N . Since aΩ is essentially self-adjoint KΩ± = {0}. Let K± := ⊕
j=1Kj±
be the deficiency subspaces of aK := ⊕
j=1aj . Suppose that K± have the same dimension.
Then, it follows from Corollary 1 in page 141 of [22] that there is a one-to-one correspondence
between self-adjoint extensions of aK and unitary maps from K+ into K−. If V is such a
unitary, then the corresponding self-adjoint extension AKV is given by,
D(AKV ) = {ϕ+ ϕ+ + V ϕ+ : ϕ ∈ D(aK), ϕ+ ∈ K+} ,
AKϕ = aKϕ+ iϕ+ − iV ϕ+.
Hence, since KΩ± = {0} and a = AΩ⊕aK there is a one-to-one correspondence between self-
adjoint extensions of a and unitary maps, V , from K+ into K−. The self-adjoint extension
AV corresponding to V is given by,
AV = AΩ ⊕ AKV .
Thus, we have proven the following theorem.
THEOREM 3.1. Every self-adjoint extension, A, of a is the direct sum of AΩ and of some
self-adjoint extension, AK of aK , i.e.,
A = AΩ ⊕AK . (3.12)
This theorem tells us that the cloaked objects K and the exterior Ω are completely
decoupled and that we are free to choose any boundary condition inside the cloaked objectsK
that makes aK self-adjoint without disturbing the cloaking effect in Ω. Boundary conditions
that make AK self-adjoint are well known. See for example, [19], [20], [14] and [6].
It follows from explicit computation that zero is an eigenvalue of every AK with infinite
multiplicity and that,
HK⊥ := (kernelAK)
∈ HK :
ελνK Eν = 0,
µλνK Hν = 0
, (3.13)
where by ελνK (x) := ε
j (x) for x ∈ Kj, and µ
K (x) := µ
j (x) for x ∈ Kj , j = 1, 2, · · · , N . It
follows that zero is an eigenvalue of A with infinite multiplicity and that,
H⊥ := (kernelA)
= HΩ⊥ ⊕HK⊥. (3.14)
For any ϕ = ϕΩ ⊕ ϕK ∈ H⊥ ∩D(A),
e−itAϕ = e−itAΩ ϕΩ ⊕ e
−itAK ϕK (3.15)
is the unique solution of Maxwell’s equations (2.1, 2.2) with finite energy that is equal to ϕ
at t = 0. This shows once again that the dynamics in Ω and in K are completely decoupled.
If at t = 0 the electromagnetic fields are zero in Ω, they remain equal to zero for all times,
and viceversa. Actually, electromagnetic waves inside the cloaked objects are not allowed to
leave them, and viceversa, electromagnetic waves outside can not go inside. This implies, in
particular, that the presence of active devices inside the cloaked objects has no effect on the
cloaking outside. In terms of boundary conditions, this means that transmission conditions
that link the electromagnetic fields inside and outside the cloaked objects are not allowed.
Furthermore, choosing a particular self-adjoint extension of the electromagnetic propagator
of the cloaked objects amounts to choosing some boundary condition on the inside of the
boundary of the cloaked objects. In other words, any possible unitary dynamics implies
the existence of some boundary condition on the inside of the boundary of the cloaked
objects. The particular boundary condition that nature will take depends on the specific
properties of the metamaterial used to build the transformation media as well us on the
properties of the media inside the cloaked objects. Note that this does not mean that we
have to put any physical surface, a lining, on the surface of the cloaked object to enforce any
particular boundary condition on the inside, since as we already mentioned this plays no role
in the cloaking outside. It would be, however, of theoretical interest to see what the interior
boundary condition turns out to be for specific cloaked objects and metamaterials. These
results apply to the exact transformation media that we consider on this paper. However,
the fact that there is a large class of self-adjoint extensions -or boundary conditions- that can
be taken inside the cloaked objects could be useful in order to enhance cloaking in practice,
where one has to consider approximate transformation media as well as in the analysis of
the stability of cloaking.
The fact that for the single coating there has to be boundary conditions on the inside
of ∂K has already been observed by [7]. In Definition 4.1 of [7] a definition of finite energy
solutions is given. Furthermore, is proven in Theorem 6.1 that in the case of the single
coating -where the permittivity and the permeability are bounded above and below inside
the cloaked object- the tangential components of the electric and the magnetic field of these
solutions have to vanish in the inside of the boundary of the cloaked object. Note that in
this case in order to have a self-adjoint extension of the electromagnetic propagator inside
the cloaked object we are only allowed to require that either the tangential component of E
or the tangential component of H vanishes, but not both.
These boundary conditions are called hidden boundary conditions in [7] where also the
case of the Helmholtz equation is considered. In the case of Maxwell’s equations they propose
two solutions to this issue. One of them is a lining, i.e., a physical material on the boundary of
the cloaked object that enforces a particular boundary condition, for example, they propose
a lining by a perfect electric conductor. Note that this raises now the question of what is the
boundary condition between the lining and the cloaking metamaterial. In fact, we face the
same problem as before, since we can always consider that the lining is part of the cloaked
objects, and then, the question of what is the appropriate boundary condition remains. The
second proposal of [7] is a double coating that corresponds to surrounding both the inner
and the outersurface of the cloaked objects with appropriately matched metamaterials. As
our permittivities and permeabilities inside K are allowed to vanish as they approach ∂K
the double coating fits in our formalism.
In Theorem 5.1 of [7] cloaking is proven for all frequencies and active devices, with the
double coating, with respect to the Cauchy data of the finite energy solutions that they define
in Definition 4.1.
Remark that there is no real contradiction between our results and the ones of [7]. Our
results imply that there is always a hidden boundary condition on the inside of the boundary
of the cloaked objects, that is imposed upon us by the fundamental principle of the con-
servation of the energy of the electromagnetic waves, that implies that time evolution has
to be given by a unitary group generated by a self-adjoint extension of the electromagnetic
propagator, and this amounts to a boundary condition at the inside of the boundary of
the cloaked objects. Note that we do not exclude here the possibility that in some cases
the electromagnetic propagator of the cloaked objects could be essentially self-adjoint, and
in this situation the dynamics inside the cloaked objects will be uniquely defined. In this
case the hidden boundary condition will be uniquely determined by the boundary conditions
satisfied by the functions in the domain of the unique self-adjoint realization of the elec-
tromagnetic propagator in the cloaked objects. Note, however, that we have proven that
for exact transformation media the cloaking outside is actually independent of the cloaked
objects.
4 Cloaking as a Boundary Value Problem
It is a question of independent interest to consider cloaking as a boundary value problem for
the Maxwell’ system at a fixed frequency
∇× E = iλB, ∇×H = −iλD, λ 6= 0, (4.1)
∇ ·B = 0,∇ ·D = 0. (4.2)
As we have already shown, cloaking is independent of the cloaked object, and this means
that we only have to consider these equation in Ω. The main question now is to decide what
is an appropriate class of solutions with locally finite energy. Our analysis of the self-adjoint
extensions of the electromagnetic propagator shows that we have to take solutions that are
locally in the domain of AΩ, that is to say that they are given by (2.38)
with (E0,H0)
T locally in the domain of A0, i.e., (E0,H0)
T are in the domain of A0 when
multiplied by any function in C∞0 (R
3). It follows from (2.39) that the solutions with locally
finite energy have to satisfy the boundary condition,
E× n = 0,H× n = 0, in ∂K+,
where ∂K+ is the outside of the boundary of the cloaked object. This is the only self-
adjoint boundary condition on ∂K+. Note that we define in the same way solutions with
(locally) finite energy in a bounded subset of Ω. In [7] a different definition of solutions with
(locally) finite energy is given in Definition 4.1.
5 Cloaking an Infinite Cylinder
We discuss now the case of an infinite cylinder. For simplicity we consider one cylinder
centered at zero and with its axis the vertical line L := (0, 0, x3), x3 ∈ R. Then,
x = (x1, x2, x3) ∈ R3 :
|x1|2 + |x2|2 ≤ a, x3 ∈ R
, Ω := R3 \K. (5.3)
The set Ω0 is now given by,
Ω0 = R
3 \ L. (5.4)
Let us denote by x := (x,1 , x2) the vectors in the x1 − x2 plane and x̂ := x/|x|. The
transformation (2.6) is replaced by
x = x(y) = f(y) :=
x = ( b−a
|y|+ a)ŷ,
x3 = y3,
(5.5)
for 0 < |y| ≤ b and with b > a. This transformation blows up the line L onto ∂K and it
sends Kb \ L onto Kb \K where
Kb :=
y = (y1, y2, y3) ∈ R3 :
|y1|2 + |y2|2 ≤ b, y3 ∈ R
For y ∈ R3 \Kb we define the transformation to be the identity, x = y.
The Hilbert spaces of finite energy electromagnetic fields, the unitary operator U and a0,
A0, aΩ, aK , a, are defined as in Section 2.
THEOREM 5.1. The operator aΩ is essentially self-adjoint, and its unique self-adjoint
extension, AΩ, satisfies
AΩ = U A0 U
∗. (5.6)
Proof: The theorem is proven as Theorem 2.1 observing that W 1,2(R2) = W
2 \ 0) since
{0} has zero Sobolev one capacity in R2 [1, 11, 12].
We now consider the wave and the scattering operators. For simplicity we assume below
that ελν0 = ε̃ δ
λν , µλν0 = µ̃ δ
λν . The wave operators are defined as in (2.41) but now the
operator J is defined as follows,
(x) := χΩ(x)φ(x)
where φ is continuous and it satisfies φ(x) = (|x| − a), a ≤ |x| ≤ a + δ and φ(x) = 1 for
|x| ≥ a+ 2δ, for some δ > 0.
LEMMA 5.2.
W± = UP0⊥. (5.7)
Proof: The lemma is proven as in the proof of Lemma 2.2, but in (2.45, 2.46, 2.47, 2.48) we
have to replace χBR
, by χCR
where, CR := {y ∈ R
3 : |y| ≤ R} for R large enough. Now we
can not prove (2.49, 2.50, 2.51) by compactness arguments because K is unbounded. Instead
we use propagation estimates for A0. The following results are well known. See for example
[3, 27, 28, 31] where the general anisotropic case is considered. For any ϕ ∈ H0⊥,
e−itA0ϕ =
(2π)3/2
eik·y
e−iω+(k)tP+(k)ϕ̂(k) + e
−iω−(k)tP−(k)ϕ̂(k)
d3k (5.8)
where ϕ̂ is the Fourier transform of ϕ, ω±(k) = ±|k|c with c := (ε̃µ̃)
−1/2, and P±(k) are
projectors on R3 that are infinitely differentiable for k ∈ R3\0. Suppose that ϕ̂ ∈ C∞0 (R
and let O be a bounded open set such that O ⊂ R3 \ L and support ϕ̂ ⊂ O. Denote
Ô :=
: k ∈ O
Then by the (non) stationary phase Theorem (see the Corollary to Theorem XI.14 of [23] ),
for any n = 1, 2, · · · there is a constant Cn such that
e−itA0ϕ
∣ ≤ Cn (1 + |y|+ |t|)
/∈ Ô. (5.9)
We write,
−itA0ϕ = φ1 + φ2 (5.10)
φ1 := χ(±y/(ct)/∈Ô)χCRe
−itA0ϕ (5.11)
φ2 := χ(±y/(ct)∈Ô)χCRe
−itA0ϕ. (5.12)
by (5.9)
s- lim
φ1 = 0. (5.13)
Note, furthermore, that there is an ǫ > 0 such that |k| ≥ ǫ for any k ∈ Ô. Then, for any
∈ Ô, |y| ≥ c|t|ǫ. It follows that there is a T such that
φ2 = 0, for|t| ≥ T. (5.14)
By (5.10, 5.13, 5.14)
s- lim
−itA0ϕ = 0, (5.15)
and (2.50, 2.51) follow. Note that P0⊥ is not needed because ϕ ∈ H0⊥. By continuity
this is true for all ϕ ∈ H0⊥. Then, (5.7) follows from (2.43, 2.51).
The scattering operator is defined as in (2.52).
COROLLARY 5.3.
S = P0⊥. (5.16)
Proof: This is immediate from (5.7) because U∗ U = I.
Let us denote by S⊥ the restriction of S to H0⊥. S⊥ is the physically relevant scattering
operator that acts in the Hilbert space H0⊥ of finite energy fields that satisfy equations (2.2).
We designate by I⊥ the identity operator on H0⊥. We have that,
COROLLARY 5.4.
S⊥ = I⊥. (5.17)
Proof: This follows from Corollary 5.3.
Again, the fact that S⊥ is the identity operator on H0⊥ means that there is cloaking for
all frequencies.
In Theorem 7.1 of [7] cloaking is proven for all frequencies with respect to the Cauchy
data of the finite energy solutions that they define in Definition 4.1 and furthermore, in
Theorem 8.2, they prove cloaking for all frequencies with the SHS boundary condition with
respect to the Cauchy data of the finite energy solutions that they define in Definition 8.1.
Theorem 3.1 remains true in the case of the cylinder. The proof is the same. Furthermore,
all the remarks about finite energy solutions, and cloaking and that we made in Sections
2, 3, are true in the case of a cylinder. We do not repeat them here. Moreover, equations
(2.38) hold. However, since now the transformation (5.5) only acts on the plane orthogonal
to the axis of the cylinder equations (2.39) has to be replaced by
E× x̂ = 0, H× x̂ = 0, in ∂K+, (5.18)
where E := (E1, E2),H := (H1, H2).
As in Section 4 we define solutions to (4.1, 4.2) with locally finite energy as solutions
that are locally in the domain of AΩ, that is to say that they are given by (2.38)
with (E0,H0)
T locally in the domain of A0, i.e., (E0,H0)
T are in the domain of A0 when
multiplied by any function in C∞0 (R
3). It follows from (5.18) that the solutions with locally
finite energy have to satisfy the boundary condition,
E× x̂ = 0, H× x̂ = 0, in ∂K+. (5.19)
Note that (5.19) is the SHS boundary condition considered in [7]. We have proven here
that (5.19) is the only self-adjoint boundary condition on ∂K+ allowed by energy conserva-
tion.
Acknowledgement
This work was partially done while I was visiting the Institut für Theoretische Physik,
Eidgenössische Techniche Höchschule Zurich. I thank professors Gian Michele Graf and
Jürg Fröhlich for their kind hospitality.
References
[1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Second Edition, Academic Press,
Amsterdam, 2003
[2] A. Bachelot, Gravitational scattering of electromagnetic fields by Schwarzschild black-
holes, Ann. Inst. H. Poincaré 54 261-320 (1991).
[3] R. Courant and D. Hilbert, Methoden der Mathematischen Physik II, Zweite Auflage,
Springer-Verlag, Berlin, 1968.
[4] W. Cai, U.K. Chettiar, A.V. Kildishev and V.M. Shalaev, Optical Cloaking with meta-
materials, Nature Photonics 1 224-226 (2007).
[5] S.A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith and J. Pendry, Full-wave simulation
of electromagnetic cloaking structures, Phys. Rev. E 74 036621 (2006).
[6] M. Sh. Birman and M.Z. Solomyak, The self-adjoint Maxwell operator in arbitrary
domains, Algebra i Analiz. 1 96-110 (1989). English transl. Leningrad Math. J. 1 99-
115 (1989).
[7] A. Greenleaf, Y. Kurylev, M. Lassas and G. Ulhmann, Full-wave invisibility of active
devices at all frequencies, Comm. Math. Phys. 275 749-789 (2007).
[8] A. Greenleaf, M. Lassas, and G. Uhlmann, Anisotropic conductivities that cannot be
detected by EIT, Physiol. Meas. 24 413-419 (2003).
[9] A. Greenleaf, M. Lassas, and G. Uhlmann, On nonuniqueness for Calderón’s inverse
problem, Math. Res. Let. 10 685-693 (2003).
[10] R. Hempel and R. Weder, On the completeness of wave operators under loss of local
compactness, Journal of Functional Analysis, 113 391-412 (1993).
[11] T. Kilpeläinen, J. Kinnunen and O. Martio, Sobolev spaces with zero boundary values
on metric spaces, Potential Anal. 12 233-247 (2000).
[12] J. Kinnunen and O. Martio, The Sobolev capacity on metric spaces, Ann. Acad. Sci.
Fenn. Math. 21 367-382 (1997).
[13] U. Leonhardt, Optical conformal mapping, Science 312 1777-1780 (2006).
[14] R. Leis, Initial Boundary Value Problems in Mathematical Physics, John Wiley & Sons,
New York, 1986.
[15] U. Leonhardt and T. G. Philbin, General relativity in electrical engineering, New J.
Phys. 8 247 (2006).
[16] G. W. Milton, M. Briane, and J. R. Willis, On cloaking for elasticity and physical
equations with transformation invariant form, New J. Phys. 8 248 (2006).
[17] A. Nachman, Reconstruction from boundary measurements, Ann. of Math. 128 71-96
(1988).
[18] J. B. Pendry, D. Schurig, and D. R. Smith, Controlling electromagnetic fields, Science
312 1780-1782 (2006).
[19] R. Picard, Ein Randwertproblem für die zeitunabhängigen Maxwellschen Gleichungen
mit der Randbedingung n · εE = n · µH = 0 in beschränken Gebieten beliebigen
Zusammenhangs Appl. Anal. 6 207-221 (1977).
[20] R. Picard, On the low frequency asymptotics in electromagnetic theory, J. Reine Angew.
Math. 354 50-73 (1984).
[21] E.J. Post, Formal Structure of Electromagnetics General Covariance and Electromag-
netics, Dover Publications, Mineola, New York, 1997.
[22] M. Reed and B. Simon, Methods of Modern Mathematical Physics II Fourier Analysis
and Self-Adjointness, Academic Press, New York, 1975.
[23] M. Reed and B. Simon, Methods of Modern Mathematical Physics III Scattering Theory,
Academic Press, New York, 1979.
[24] D. Schurig, , J.J. Mock, B.J. Justice, S.A. Cummer, J.B.. Pendry, A.F. Starr and D.
R. Smith, Metamaterial electromagnetic cloak at microwave frequencies Science 314
977-980 (2006).
[25] D. Schurig, J.B. Pendry, and D. R. Smith, Calculation of material properties and ray
tracing in transformation media, Opt. Exp. 14 9794-9804 (2006).
[26] G. Uhlmann, Scattering by a metric, Chap. 6.1.5 in Encyclopedia on Scvattering, Aca-
demic Press, R. Pike and P. Sabatier, eds (2002), 1668-1677.
[27] R. Weder, Analyticity of the scattering matrix for wave propagation in crystals, J. Math.
Pures et Appl. 64 121-148 (1985).
[28] R. Weder, Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified
Media. Applied Mathematical Sciences 87 Springer-Verlag, New York, 1991.
[29] R. Weder, A rigorous analysis of high-order electromagnetic invisibility cloaks, arXiv:
0711.0507, 2007, J. Phys. A: Mathematical and Theoretical 41 065207 (2008). IOP-
Select.
[30] R. Weder, The boundary conditions for electromagnetic invisibility cloaks, arXiv:
0801.3611, 2008.
[31] C.H. Wilcox, Asymptotic wave functions and energy distributions in strongly propaga-
tive media, J. Math. Pures et Appl. 57 275-231 (1978).
	Introduction
	Electromagnetic Cloaking
	Electromagnetic Waves Inside the Cloaked Objects 
	Cloaking as a Boundary Value Problem
	Cloaking an Infinite Cylinder

  There is currently a great deal of interest in the theoretical and practical
possibility of cloaking objects from the observation by electromagnetic waves.
The basic idea of these invisibility devices \cite{glu1, glu2, le},\cite{pss1}
is to use anisotropic {\it transformation media} whose permittivity and
permeability $\var^{\lambda\nu}, \mu^{\lambda\nu}$, are obtained from the ones,
$\var_0^{\lambda\nu}, \mu^{\lambda\nu}_0$, of isotropic media, by singular
transformations of coordinates. In this paper we study electromagnetic cloaking
in the time-domain using the formalism of time-dependent scattering theory.
This formalism allows us to settle in an unambiguous way the mathematical
problems posed by the singularities of the inverse of the permittivity and the
permeability of the {\it transformation media} on the boundary of the cloaked
objects. We write Maxwell's equations in Schr\"odinger form with the
electromagnetic propagator playing the role of the Hamiltonian. We prove that
the electromagnetic propagator outside of the cloaked objects is essentially
self-adjoint. Moreover, the unique self-adjoint extension is unitarily
equivalent to the electromagnetic propagator in the medium
$\var_0^{\lambda\nu}, \mu^{\lambda\nu}_0$. Using this fact, and since the
coordinate transformation is the identity outside of a ball, we prove that the
scattering operator is the identity. Our results give a rigorous proof that the
construction of \cite{glu1, glu2, le}, \cite{pss1} perfectly cloaks passive and
active devices from observation by electromagnetic waves. Furthermore, we prove
cloaking for general anisotropic materials. In particular, our results prove
that it is possible to cloak objects inside general crystals.

Introduction
There is currently a great deal of interest in the theoretical and practical possibility of
cloaking objects from the observation by electromagnetic fields. The basic idea of these in-
visibility devices [8, 9, 13], [18] is to use anisotropic transformation media whose permittivity
and permeability, ελν , µλν , are obtained from the ones, ελν0 , µ
0 , of isotropic media, by sin-
gular transformations of coordinates. The singularities lie on the boundary of the objects to
be cloaked. Here the material interpretation is taken. Namely, the ελν , µλν and the ελν0 , µ
represent the components in flat Cartesian space of the permittivity and the permeability of
physical media with different material properties. It appears that with existing technology it
is possible to construct media as described above using artificially structured metamaterials.
In [8, 9] a proof of cloaking was given for the conductivity equation -i.e., in the case of zero
frequency- from detection by measurement of the Dirichlet to Neumann map that relates
the value of the electric potential on the boundary to its normal derivative. The papers [13]
and [18] consider electromagnetic waves in the geometrical optics approximation, i.e. for
large frequencies. In [24] a experimental verification of cloaking is presented and [4] and
[5] give a numerical simulation. A rigorous prof of cloaking has already been given by [7]
where fixed frequency waves were studied, i.e., in the frequency domain. They consider a
class of finite energy solutions to Maxwell’s equations in a bounded set, O, that contains the
cloaked object on its interior, and they prove cloaking, at any frequency, with respect to the
measurement of the Cauchy data of these solutions on the boundary of O. We give further
comments on this paper below. For other results on this problem see [25] and [15]. In [16]
cloaking of elastic waves is considered, and the history of invisibility is discussed.
In this paper we study electromagnetic cloaking in the time-domain using the formalism of
time-dependent scattering theory [23]. This formalism provides us with a rigorous method to
analyze the propagation of electromagnetic wave packets with finite energy in transformation
media. In particular, it allows us to settle in an unambiguous way the mathematical problems
posed by the singularities of the inverse of the permittivity and the permeability of the
transformation media on the boundary of the cloaked objects. Von Neumann’s theory of self-
adjoint extensions of symmetric operators plays an important role on this issue. We write
Maxwell’s equations in Schrödinger form with the electromagnetic propagator playing the
role of the Hamiltonian. We prove that the electromagnetic propagator outside of the cloaked
objects is essentially self-adjoint. This means that it has only one self-adjoint extension,
AΩ, and that this self-adjoint extension generates the only possible unitary time evolution,
with constant energy, for finite energy electromagnetic waves propagating outside of the
cloaked objects. Moreover, AΩ is unitarily equivalent to the electromagnetic propagator in
the medium ελν0 , µ
0 . Using this fact, and since the coordinate transformation is the identity
outside of a ball, we prove that the scattering operator is the identity. This implies that for
any incoming finite-energy electromagnetic wave packet the outgoing wave packet is precisely
the same. In other words, it is not possible to detect the cloaked objects in any scattering
experiment where a finite-energy wave packet is sent towards the cloaked objects, since the
outgoing wave packet that is measured after interaction is the same as the incoming one.
Our results give a rigorous proof that the construction of [8, 9, 13], [18] cloaks passive and
active devices from observation by electromagnetic waves. Actually, the cloaking outside is
independent of what is inside the cloaked objects.
As is well known, self-adjoint extensions can be understood in terms of boundary condi-
tions. Actually, for the electromagnetic fields in the domain of AΩ the component tangential
to the exterior of the boundary of the cloaked objects of both, the electric and the mag-
netic field have to be zero. This boundary condition is self-adjoint in our case because the
permittivity and the permeability are degenerate on the boundary of the cloaked objects.
Furthermore, we prove cloaking for general anisotropic materials. In particular, our
results prove that it is possible to cloak objects inside general crystals.
Even though, as mentioned above, the cloaking is independent of the cloaked objects,
and in particular, the cloaking outside is not affected by the presence of passive and/or
active devices inside the cloaked objects, we discuss the dynamics of electromagnetic waves
inside the cloaked objects for completeness, since it helps to understand the above mentioned
independence of cloaking from the properties of the cloaked objects.
We prove that every self-adjoint extension of the electromagnetic propagator in a trans-
formation medium is the direct sum of the unique self-adjoint extension in the exterior of
the cloaked objects, AΩ, with some self-adjoint extension of the electromagnetic propagator
in the interior of the cloaked objects. Each of these self-adjoint extensions corresponds to a
possible unitary time evolution for finite energy electromagnetic waves. As is well known,
the fact that time evolution is unitary assures us that energy is conserved. This results
implies that the electromagnetic waves inside and outside of the cloaked objects completely
decouple from each other. Actually, the electromagnetic waves inside the cloaked objects
are not allowed to leave them, and viceversa, the electromagnetic waves outside can not go
inside.
In terms of boundary conditions, this means that transmission conditions that link the
electromagnetic fields inside and outside the cloaked objects are not allowed, since they do
not correspond to self-adjoint extensions of the electromagnetic propagator, and then, they
do not lead to a unitary dynamics that conserves energy. Furthermore, choosing a particular
self-adjoint extension of the electromagnetic propagator of the cloaked objects amounts to
choosing some boundary condition on the inside of the boundary of the cloaked objects. In
other words, any possible unitary dynamics implies the existence of some boundary condition
on the inside of the boundary of the cloaked objects.
The fact that there is a large class of self-adjoint extensions -or boundary conditions-
that can be taken inside the cloaked objects could be useful in order to enhance cloaking
in practice, where one has to consider approximate transformation media as well as in the
analysis of the stability of cloaking.
Actually, we consider a slightly more general construction than the one of [8, 9, 13], [18]
since we allow for a finite number of cloaked objects.
In [7] a very general construction for cloaking is introduced. In the case of Maxwell’s
equations all their constructions are made within the context of the permittivity and the
permeability tensor densities being conformal to each other, i.e., multiples of each other by a
positive scalar function. In particular, all isotropic media are included in this category. They
mention that both for mathematical and practical reasons it would be very interesting to
understand cloaking for general anisotropic materials in the absence of this assumption. In
this paper we actually solve this problem, since we prove cloaking for all general anisotropic
materials. In particular, our results prove that it is possible to cloak objects inside general
crystals.
Note, moreover, that [7] also considers the cases of the Helmholtz equation. We do not
discuss this problems here.
Furthermore, remark that the existing theorems in the uniqueness of inverse scattering
do not apply under the present conditions.
In [7] cloaking is proven with respect to the Cauchy data at any fixed frequency given
on a surface that encloses the cloaked object. In the case where the permittivity and the
permeability are bounded above and below it is well known that the Cauchy data at a fixed
frequency is equivalent to the scattering matrix at the same frequency. See for example
[17] and [26]. This equivalence is, however, not proven in the case where the permittivity
and the permeability are degenerate on the boundary of the objects. In fact, it is perhaps
even not true for general degenerate media that are not transformation media since in this
case it is possible that there are finite energy electromagnetic waves that are absorbed by
the boundary of the objects as t → ±∞. If this is true, the equivalence will not hold
since the Cauchy data in a surface that encloses the objects will not contain information
on the waves that are asymptotically absorbed by the boundary of the objects. It is a
problem of independent interest to see if this actually happens or not for general degenerate
permittivities and permeabilities. For an example of scattering by a bounded obstacle with
a singular boundary and Neumann boundary condition, where this happens see [10]. For a
similar situation in the scattering of electromagnetic waves by a Schwarzschild black-hole see
[2]. Note that in our approach we directly consider the scattering operator that is measured
in scattering experiments.
In the analysis of Maxwell’s equations with permittivity and permeability that are inde-
pendent of frequency the dispersion of the medium is not taken into account. This means
that cloaking will hold for electromagnetic wave packets with a narrow enough range of
frequencies, such that this assumption is valid.
The paper is organized as follows. In Section 2 we prove our results in electromagnetic
cloaking. In Section 3 we consider the propagation of electromagnetic waves in the interior
of the cloaked objects. In Section 4 we formulate cloaking as a boundary value problem
outside of the cloaked objects for the Maxwell equations at a fixed frequency, following
our analysis of the self-adjoint extensions of the electromagnetic propagator. In particular,
we give the appropriate boundary condition on the outside of the boundary of the cloaked
objects. Finally, in Section 5 we prove cloaking of infinite cylinders. This is of interest
since this is the case considered in the experimental verification in [24] and in the numerical
simulations of [4] and [5]. Of course, [24] only consider a slice of the cylinder. In Sections 3
and 4 we give further comments on the results of [7].
Addendum
After the previous version of this paper was posted in the arXiv we published the paper
[29] where we generalized the results of this paper on spherical cloaks to the case of high-
order cloaks, and where we also discussed cloaking in the frequency domain. Moreover, in
our paper [30] we identified the cloaking boundary condition that has to be satisfied in the
inside of the boundary of the cloaked objects, in the case where the permittivity and the
permeability are bounded above and below inside the cloaked objects.
2 Electromagnetic Cloaking
Let us consider Maxwell’s equations,
∇× E = −
B, ∇×H =
D, (2.1)
∇ ·B = 0,∇ ·D = 0, (2.2)
in a domain, Ω ⊂ R3, as follows,
Ω := R3 \ ∪Nj=1Kj, Kj ∩Kl = ∅, j 6= l (2.3)
where Kj, j = 1, 2, · · · , N, are the objects to be cloaked. We assume that each Kj is a ball
with center cj and radius aj, i.e.,
x ∈ R3 : |x− cj | ≤ aj
, j = 1, 2, · · · , N. (2.4)
The cloaked objects are denoted by
K := ∪Nj=1Kj .
We designate the Cartesian coordinates of x by xλ, λ = 1, 2, 3 and by Eλ, Hλ, B
λ, Dλ, λ =
1, 2, 3, respectively, the components of E,H,B, and D. As usual, we denote by ελν and µλν ,
respectively, the permittivity and the permeability. We have that,
Dλ = ελνEν , B
λ = µλνHν , (2.5)
where we use the standard convention of summing over repeated lower and upper indices.
We consider now a transformation from Ω0 := R
3 \ {c1, c2, · · · , cN} onto Ω that was first
used to obtain cloaking for the conductivity equation, i.e. at zero frequency, by [8, 9] and
then by [18] for cloaking electromagnetic waves (for a related result in two dimensions using
conformal mappings see [13]).
For any y ∈ R3 we denote, ŷ := y/|y|. Let yλ, λ = 1, 2, 3, designate the cartesian
coordinates of y ∈ Ω0. Take bj > aj, j = 1, 2, · · · , N . Then, for 0 < |y− cj| ≤ bj , we define,
x = x(y) = f(y) := cj +
bj − aj
|y − cj |+ aj
ŷ − cj. (2.6)
Note that this transformation blows up the point cj onto ∂Kj and that it sends the punctu-
ated ball B̃cj (bj) := {y ∈ R
3 : 0 < |y − cj | ≤ bj} onto the spherical shell, aj < |x− cj | ≤ bj .
We assume that,
B̃cj (bj) ∩ B̃cl(bl) = ∅, j 6= l, 1 ≤ j, l ≤ N. (2.7)
For y ∈ R3\∪Nj=1B̃cj (bj) we define the transformation to be the identity, x = x(y) = f(y) :=
y. Our transformation is a bijection from Ω0 onto Ω. By y = y(x) := f
−1(x) we designate
the inverse transformation. We denote the elements of the Jacobian matrix by Aλλ′ ,
Aλλ′ :=
. (2.8)
Note that the Aλλ′ ∈ C
Ω0 \ ∪
j=1∂B̃cj (bj)
. We designate by Aλ
λ the elements of the
Jacobian of the inverse bijection, y = y(x) := f−1(x),
Ω \ ∪Nj=1∂B̃cj (bj)
. (2.9)
The papers [8, 9] and [18] considered the case where N = 1, c1 = 0.
We take here the so called material interpretation and we consider our transformation
as a bijection between two different spaces, Ω0 and Ω. However, our transformation can be
considered, as well, as a change of coordinates in Ω0. Of course, these two point of view
are mathematically equivalent. This means, in particular, that under our transformation the
Maxwell equations in Ω0 and in Ω will have the same invariance that they have under change
of coordinates in three-space. See, for example, [21]. Let us denote by ∆ the determinant of
the Jacobian matrix (2.8). Then,
bj − aj
( bj−aj
|y− cj|+ aj
|y − cj|
, for 0 < |y − cj | ≤ bj . (2.10)
This result is easily obtained rotating into a coordinate system such that, y − cj = (|y −
cj|, 0, 0) [25]. For y ∈ Ω0 \ ∪
j=1B̃cj (bj),∆ ≡ 1.
Let us denote by E0,H0,B0,D0, ε
0 , µ
0 , respectively, the electric and magnetic fields,
the magnetic induction, the electric displacement, and the permittivity and permeability of
Ω0. ε
0 , µ
0 , are positive, Hermitian matrices that are constant in Ω0.
The electric field is a covariant vector that transforms as,
Eλ(x) = A
λ (y)E0,λ′(y). (2.11)
The magnetic fieldH is a covariant pseudo-vector, but as we only consider space transfor-
mations with positive determinant, it also transforms as in (2.11). The magnetic induction
B and the electric displacement D are contravariant vector densities of weight one that
transform as
Bλ(x) = (∆(y))
Aλλ′(y)B
0 (y), (2.12)
with the same transformation for D. The permittivity and permeability are contravariant
tensor densities of weight one that transform as,
ελν(x) = (∆(y))
Aλλ′(y)A
ν′(y) ε
0 (y), (2.13)
with the same transformation for µλν . The Maxwell equations (2.1, 2.2) are the same in
both spaces Ω and Ω0. Let us denote by ελν , µλν, ε0λν , µ0λν , respectively, the inverses of the
corresponding permittivity and permeability. They are covariant tensor densities of weight
minus one that transform as,
ελν(x) = ∆(y)A
λ (y)A
ν (y) ε0λ′ν′(y), µλν(x) = ∆(y)A
λ (y)A
ν (y)µ0λ′ν′(y). (2.14)
Note that
det ελν = ∆−1 det ελν0 , detµ
λν = ∆−1 detµλν0 , (2.15)
det ελν = ∆det ε0λν , detµλν = ∆detµ0λν . (2.16)
We now introduce the Hilbert spaces of electric and magnetic fields with finite energy.
The E0,H0,B0,D0, were defined in Ω0, but since R
3 \ Ω0 = {cj}
j=1 is of measure zero, we
can consider them as defined in R3, what we do below.
We denote by H0E the Hilbert space of all measurable, square integrable, C
3− valued
functions defined on R3 with the scalar product,
0ν dy
3. (2.17)
We similarly define the Hilbert space,H0H , of all measurable, square integrable, C
3− valued
functions defined on R3 with the scalar product,
0ν dy
3. (2.18)
The Hilbert space of finite energy fields in R3 is the direct sum
H0 := H0E ⊕H0H . (2.19)
Moreover, we designate byHΩE the Hilbert space of all measurable, C
3− valued functions
defined on Ω that are square integrable with the weight ελν , with the scalar product,
(1),E(2)
3. (2.20)
Finally, we denote by HΩH the Hilbert space of all measurable, C
3− valued functions
defined on Ω that are square integrable with the weight µλν , with the scalar product,
(1),H(2)
3. (2.21)
The Hilbert space of finite energy fields in Ω is the direct sum
HΩ := HΩE ⊕HΩH . (2.22)
We now write the Maxwell’s equations (2.1) in Schrödinger form. We first consider the
case of R3. We denote by ε0 and µ0, respectively, the matrices with entries ε0λν and µ0λν .
Recall that (∇×E)
= sλνρ ∂
Eρ where s
λνρ is the permutation contravariant pseudo-
density of weight −1 (see section 6 of chapter II of [21], where a different notation is used).
By a0 we denote the following formal differential operator,
ε0∇×H0
−µ0∇× E0
. (2.23)
Here, as usual, we denote, ε0∇ × H0 := ε0λν(∇ × H0)
ν , and µ0∇ × E0 = µ0λν(∇ × E0)
Then, equations (2.1) are equivalent to,
. (2.24)
Let us denote by C10(R
3) the set of all C6−valued continuously differentiable functions
on R3 that have compact support. Then, a0 with domain C
3) is a symmetric operator in
H0, i.e., a0 ⊂ a
0. Moreover, it is essentially self-adjoint in H0, i.e., it has only one self-adjoint
extension, that we denote by A0. Its domain is given by,
D(A0) =
, (2.25)
∈ D(A0), (2.26)
where the derivatives are taken in distribution sense. These results follow easily from the
fact that -via the Fourier transform- a0 is unitarily equivalent to multiplication by a matrix
valued function that is symmetric with respect to the scalar product of H0. Moreover, it
follows from explicit computation that the only eigenvalue of A0 is zero, that it has infinite
multiplicity, and that,
H0⊥ := (kernelA0)
∈ H0 :
ελν0 E0ν = 0,
µλν0 H0ν = 0
. (2.27)
Furthermore, A0 has no singular-continuous spectrum and its absolutely-continuous spec-
trum is R. See, for example, [27, 28].
Taking any
∈ H0⊥ ∩D(A0) (2.28)
we obtain a finite energy solution to the Maxwell equations (2.1, 2.2) as follows
(t) = e−itA0
. (2.29)
This is the unique finite energy solution with initial value at t = 0 given by (2.28). Note
that as e−itA0H0⊥ ⊂ H0⊥ equations (2.2) are satisfied for all times if they are satisfied at
t = 0.
Let us now consider the case of Ω. We denote by ε and µ, respectively, the matrices with
entries ελν and µλν .
We now define the following formal differential operator,
−µ∇× E
. (2.30)
Equations (2.1) are equivalent to,
Let us denote by C10(Ω) the set of all C
6−valued continuously differentiable functions on
Ω that have compact support. Then, aΩ with domain C
0(Ω) is a symmetric operator in HΩ.
To construct a unitary dynamics that preserves energy we have to analyse the self-adjoint
extensions of aΩ.
We denote by UE the following unitary operator from H0E onto HΩE ,
(UEE0)λ (x) := A
λ E0λ′(y), (2.31)
and by UH the unitary operator from H0H onto HΩH ,
(UHH0)λ (x) := A
λ H0λ′(y). (2.32)
Then,
U := UE ⊕ UH (2.33)
is a unitary operator from H0 onto HΩ.
We denote by a00 the restriction of a0 to C
0(Ω0). The operator a00 is essentially
self-adjoint and its only self-adjoint extension is A0. This follows from the essential self-
adjointness of a0 and from the fact that any function in C
3) can be approximated in the
graph norm of a0 by functions in C
0(Ω0). To prove this take any continuously differentiable
real-valued function, φ, defined on R such that, φ(y) = 0, |y| ≤ 1 and φ(y) = 1, |y| ≥ 2.
Then, for any
∈ C10(R
we have that,
φ(n|y− cj|)
∈ C10(Ω0)
and moreover,
s- lim
φ(n|y− cj|)
s- lim
φ(n|y− cj|)
where by s- lim we designate the strong limit in H0.
As a00 is essentially self-adjoint, it follows from the invariance of Maxwell equations that
aΩ is essentially self-adjoint, and that its unique self-adjoint extension, that we denote by
AΩ, satisfies
AΩ = U A0 U
∗. (2.34)
For the proof of these facts see [29]. Hence, we have the following theorem.
THEOREM 2.1. The operator aΩ is essentially self-adjoint, and its unique self-adjoint
extension, AΩ, satisfies (2.34).
The unitary equivalence given by (2.34) implies that AΩ has the same spectral prop-
erties that A0. Namely, it has no singular-continuous spectrum, the absolutely-continuous
spectrum is R and the only eigenvalue is zero and it has infinite multiplicity. Moreover,
HΩ⊥ := (kernelAΩ)
∈ HΩ :
ελνEν = 0,
µλνHν = 0
. (2.35)
Furthermore, taking any
∈ HΩ⊥ ∩D(AΩ) (2.36)
we obtain a finite energy solution to the Maxwell equations (2.1, 2.2) as follows
(t) = e−itAΩ
. (2.37)
This is the unique finite energy solution with initial value at t = 0 given by (2.36). Note
that as e−itAΩHΩ⊥ ⊂ HΩ⊥ equations (2.2) are satisfied for all times if they are satisfied at
t = 0. We can consider more general solutions by considering the scale of spaces associated
with AΩ, but we do not go into this direction here.
The facts that aΩ is essentially self-adjoint and that its unique self-adjoint extension AΩ is
unitarily equivalent to the propagator A0 of the homogeneous medium are strong statements.
They mean that the only possible unitary dynamics in Ω that preserves energy is given by
(2.37) and that this dynamics is unitarily equivalent to the free dynamics in R3 given by
(2.29). In fact, ∂Ω acts like a horizon for electromagnetic waves propagating in Ω in the
sense that the dynamics is uniquely defined without any need to consider the cloaked objects
K = ∪Nj=1Kj . As we will prove below this implies electromagnetic cloaking for all frequencies
in the strong sense that the scattering operator is the identity.
Since D(AΩ) = UD(A0), for any (E,H)
T ∈ D(AΩ) there is a (E0,H0)
T ∈ D(A0) such
. (2.38)
Then, it follows from (2.31, 2.32, 2.33) that
E× n = 0,H× n = 0, in ∂K+, (2.39)
where ∂K+ denotes the outside of the boundary of the cloaked objects, K, and n is the
normal vector to ∂K+, if (E0,H0) are, for example, bounded near ∂K+. That is to say, for
electromagnetic fields in the domain of AΩ the tangential components of both, the electric
and the magnetic field vanish in the exterior of the boundary of the cloaked objects. This is a
self-adjoint boundary condition because the permittivity and the permeability are degenerate
on ∂K+.
Let χΩ be the characteristic function of Ω, i.e., χΩ(x) = 1,x ∈ Ω, χΩ(x) = 0,x ∈ R
3 \ Ω.
We define,
(x) := χΩ(x)
(x). (2.40)
By (2.6, 2.10, 2.13),
∣ελν(x)
∣ ≤ C,
∣µλν(x)
∣ ≤ C, x ∈ Ω.
Then, J is a bounded operator from H0 into HΩ.
The wave operators are defined as follows,
W± = s- lim
eitAΩ Je−itA0P0⊥, (2.41)
where P0⊥ denotes the projector onto H0⊥.
Let us designate by W1,2(R3) the Sobolev space of C6 valued functions. We denote by I
the identity operator on H0. Then,
LEMMA 2.2.
W± = UP0⊥. (2.42)
Proof: Denote,
W (t) := eitAΩ J e−itA0P0⊥.
By (2.34), for any ϕ ∈ H0
W (t)ϕ = ψ(t) + UP0⊥ϕ, (2.43)
ψ(t) := U eitA0 (U∗J − I) e−itA0P0⊥ϕ. (2.44)
Let BR denote the ball of center zero and radius R in R
3. Since for |y| ≥ R, with R
large enough, our transformation, x = f(y), is the identity, x = y, and in consequence,
Aλλ′(y) = δ
λ′ for |y| ≥ R, we have that,
(U∗J − I) = (U∗J − I)χBR
. (2.45)
It follows that,
s- lim
ψ(t) = U s- lim
eitA0ϑ(t) (2.46)
with,
ϑ(t) := (U∗J − I)χBR
e−itA0P0⊥ϕ. (2.47)
We have that,
‖ϑ(t)‖
e−itA0P0⊥ϕ
e−itA0P0⊥ϕ
e−itA0P0⊥ϕ
. (2.48)
Then, as (A0 + i)
−1P0⊥ is bounded from H0 into W
1,2(R3) [27] [28], it follows from the
Rellich local compactness theorem that
(A0 + i)
−1P0⊥
is a compact operator in H0. Suppose that ϕ ∈ D(A0) ∩ H0⊥. Then,
s- lim
e−itA0P0⊥ϕ = s- lim
(A0 + i)
−1P0⊥e
−itA0(A0 + i)ϕ = 0, (2.49)
and whence, by (2.48),
s- lim
ϑ(t) = 0, (2.50)
and it follows that in this case,
s- lim
ψ(t) = 0. (2.51)
By continuity this is also true for ϕ ∈ H0⊥.
Then, (2.42) follows from (2.43) and (2.51).
The scattering operator is defined as
S := W ∗+W−. (2.52)
COROLLARY 2.3.
S = P0⊥. (2.53)
Proof: This is immediate from (2.42) because U∗ U = I.
Let us denote by S⊥ the restriction of S to H0⊥. S⊥ is the physically relevant scattering
operator that acts in the Hilbert space H0⊥ of finite energy fields that satisfy equations (2.2).
We designate by I⊥ the identity operator on H0⊥. We have that,
COROLLARY 2.4.
S⊥ = I⊥. (2.54)
Proof: This follows from Corollary 2.3.
The fact that S⊥ is the identity operator on H0⊥ means that there is perfect cloaking for
all frequencies. Suppose that for very negative times we are given an incoming wave packet
e−itA0ϕ−, with ϕ− ∈ H0⊥. Then, for large positive times the outgoing wave packet is given
by e−itA0ϕ+ with ϕ+ = S⊥ϕ−. But, as S⊥ = I, we have that ϕ+ = ϕ− and then,
e−itA0ϕ− = e
−itA0ϕ+.
Since the incoming and the outgoing wave packets are the same there is no way to detect
the cloaked objects K from scattering experiments performed in Ω.
In this paper we considered transformation media obtained from a singular transformation
that blows up a finite number of points, by simplicity, and since this is the situation in the
applications. Suppose that we have a transformation that is singular in a set of points
that we call M and denote now Ω0 := R
3 \M . What we really used in the proofs is that
W1,2(R3) = W
0 (Ω0) where W
0 (Ω0) denotes the completion of C
0 (Ω0) in the norm of
W1,2(R3). We also assumed that ελν0 , µ
0 are constant. What was actually needed is that a0
is essentially self-adjoint. All our results hold under this more general conditions provided
that in (2.41, 2.42) and (2.53) we replace P0⊥ by the projector onto the absolutely-continuous
subspace of A0 and that we assume that D(A0) ∩H0ac ⊂ W
1,2(R3), where we have denoted
the absolutely-continuous subspace of A0 by H0ac. Moreover, S⊥ has to be defined as the
restriction of S to H0ac and in (2.54) I⊥ has to be the identity operator on H0ac. Note that
under these general assumptions A0 could have non-zero eigenvalues and singular-continuous
spectrum.
For example, W1,2(R3) = W
0 (Ω0) if M has zero Sobolev one capacity [1, 11, 12].
Moreover, assume that the permittivity and the permeability tensor densities ελν0 , µ
0 are
bounded below and above. Under this condition a0 is essentially self-adjoint. Furthermore,
let us denote by Ĥ0 the Hilbert space of finite energy solutions defined as in (2.19) but
with ελν0 = µ
0 = δ
λµ. Let Â0, Ĥ0⊥ be, respectively, the electromagnetic propagator in Ĥ0
and the orthogonal complement of its kernel. We have that H0 and Ĥ0 are the same set
of functions with equivalent norms. Furthermore, D(A0) = D(Â0), kernel Â0 = kernelA0.
Moreover, (E0,H0)
T ∈ H0⊥ if and only if E0 = ε0Ê0,H0 = µ0Ĥ0 for some (Ê0, Ĥ0) ∈ Ĥ0⊥.
As [27, 28] D(Â0) ∩ Ĥ0⊥ ⊂ W
1,2(R3) we have that D(A0) ∩ H0⊥ ⊂ W
1,2(R3) if ε0, µ0 are
bounded operators on W1,2(R3) and this is true if the derivatives ∂
µ0 are bounded
operators on Ĥ0 for ρ = 1, 2, 3. Note, furthermore, that H0ac ⊂ H0⊥.
3 Electromagnetic Waves Inside the Cloaked Objects
Let us now consider the propagation of electromagnetic waves in the cloaked objects. For
this purpose we assume that in each Kj the permittivity and the permeability are given
by ελνj , µ
j , with inverses εjλν, µjλν and where εj, µj are the matrices with entries εjλν, µjλν.
Furthermore, we assume that 0 < ελνj , µ
j ≤ C,x ∈ Kj and that for any compact set Q con-
tained in the interior of Kj there is a positive constant CQ such that det ε
j > CQ, detµ
CQ,x ∈ Q. In other words, we only allow for possible singularities of εj, µj on the boundary
of Kj.
We designate by HjE the Hilbert space of all measurable, C
3− valued functions defined
on Kj that are square integrable with the weight ε
j , with the scalar product,
jν dx
3. (3.1)
Similarly, we denote by HjH the Hilbert space of all measurable, C
3− valued functions
defined on Kj that are square integrable with the weight µ
j , with the scalar product,
jν dx
3. (3.2)
The Hilbert space of finite energy fields in Kj is the direct sum
Hj := HjE ⊕HjH, (3.3)
and the Hilbert space in the cloaked objects K is the direct sum,
HK := ⊕
j=1Hj .
The complete Hilbert space of finite energy fields including the cloaked objects is,
H := HΩ ⊕HK . (3.4)
We now write (2.1) as a Schrödinger equation in each Kj as before. We define the
following formal differential operator,
εj∇×Hj
−µj∇× Ej
. (3.5)
Equation (2.1) in Kj is equivalent to
. (3.6)
Let us denote by C10(K̂j) the set of all C
6−valued continuously differentiable functions on
Kj that have compact support in the interior of Kj, that we denote by K̂j := Kj \ ∂Kj .
Then, aj with domain C
0(K̂j) is a symmetric operator in Hj . We denote,
a := aΩ ⊕
j=1 aj , (3.7)
with domain,
D(a) :=
⊕Nj=1
∈ C10(Ω)⊕
j=1 C
0(K̂j)
. (3.8)
The operator a is symmetric in H. The possible unitary dynamics that preserve energy for
the whole system including the cloaked objects K are given by the self-adjoint extensions of
a. Let us denote a the closure of a, with similar notation for aΩ, aj , j = 1, · · · , N . Then,
a = AΩ ⊕
j=1 aj,
where we used the fact that as aΩ is essentially self-adjoint, aΩ = AΩ. The adjoint of a is
given by,
D(a∗) =
⊕Nj=1
∈ H :
∈ D(AΩ), aj
, (3.9)
⊕Nj=1
⊕Nj=1 aj
, (3.10)
⊕Nj=1
∈ D(a∗). (3.11)
Let us denote by KΩ± := kernel(i∓ a
Ω),Kj± := kernel(i∓ a
j) the deficiency subspaces of
aΩ and aj, j = 1, · · · , N . Since aΩ is essentially self-adjoint KΩ± = {0}. Let K± := ⊕
j=1Kj±
be the deficiency subspaces of aK := ⊕
j=1aj . Suppose that K± have the same dimension.
Then, it follows from Corollary 1 in page 141 of [22] that there is a one-to-one correspondence
between self-adjoint extensions of aK and unitary maps from K+ into K−. If V is such a
unitary, then the corresponding self-adjoint extension AKV is given by,
D(AKV ) = {ϕ+ ϕ+ + V ϕ+ : ϕ ∈ D(aK), ϕ+ ∈ K+} ,
AKϕ = aKϕ+ iϕ+ − iV ϕ+.
Hence, since KΩ± = {0} and a = AΩ⊕aK there is a one-to-one correspondence between self-
adjoint extensions of a and unitary maps, V , from K+ into K−. The self-adjoint extension
AV corresponding to V is given by,
AV = AΩ ⊕ AKV .
Thus, we have proven the following theorem.
THEOREM 3.1. Every self-adjoint extension, A, of a is the direct sum of AΩ and of some
self-adjoint extension, AK of aK , i.e.,
A = AΩ ⊕AK . (3.12)
This theorem tells us that the cloaked objects K and the exterior Ω are completely
decoupled and that we are free to choose any boundary condition inside the cloaked objectsK
that makes aK self-adjoint without disturbing the cloaking effect in Ω. Boundary conditions
that make AK self-adjoint are well known. See for example, [19], [20], [14] and [6].
It follows from explicit computation that zero is an eigenvalue of every AK with infinite
multiplicity and that,
HK⊥ := (kernelAK)
∈ HK :
ελνK Eν = 0,
µλνK Hν = 0
, (3.13)
where by ελνK (x) := ε
j (x) for x ∈ Kj, and µ
K (x) := µ
j (x) for x ∈ Kj , j = 1, 2, · · · , N . It
follows that zero is an eigenvalue of A with infinite multiplicity and that,
H⊥ := (kernelA)
= HΩ⊥ ⊕HK⊥. (3.14)
For any ϕ = ϕΩ ⊕ ϕK ∈ H⊥ ∩D(A),
e−itAϕ = e−itAΩ ϕΩ ⊕ e
−itAK ϕK (3.15)
is the unique solution of Maxwell’s equations (2.1, 2.2) with finite energy that is equal to ϕ
at t = 0. This shows once again that the dynamics in Ω and in K are completely decoupled.
If at t = 0 the electromagnetic fields are zero in Ω, they remain equal to zero for all times,
and viceversa. Actually, electromagnetic waves inside the cloaked objects are not allowed to
leave them, and viceversa, electromagnetic waves outside can not go inside. This implies, in
particular, that the presence of active devices inside the cloaked objects has no effect on the
cloaking outside. In terms of boundary conditions, this means that transmission conditions
that link the electromagnetic fields inside and outside the cloaked objects are not allowed.
Furthermore, choosing a particular self-adjoint extension of the electromagnetic propagator
of the cloaked objects amounts to choosing some boundary condition on the inside of the
boundary of the cloaked objects. In other words, any possible unitary dynamics implies
the existence of some boundary condition on the inside of the boundary of the cloaked
objects. The particular boundary condition that nature will take depends on the specific
properties of the metamaterial used to build the transformation media as well us on the
properties of the media inside the cloaked objects. Note that this does not mean that we
have to put any physical surface, a lining, on the surface of the cloaked object to enforce any
particular boundary condition on the inside, since as we already mentioned this plays no role
in the cloaking outside. It would be, however, of theoretical interest to see what the interior
boundary condition turns out to be for specific cloaked objects and metamaterials. These
results apply to the exact transformation media that we consider on this paper. However,
the fact that there is a large class of self-adjoint extensions -or boundary conditions- that can
be taken inside the cloaked objects could be useful in order to enhance cloaking in practice,
where one has to consider approximate transformation media as well as in the analysis of
the stability of cloaking.
The fact that for the single coating there has to be boundary conditions on the inside
of ∂K has already been observed by [7]. In Definition 4.1 of [7] a definition of finite energy
solutions is given. Furthermore, is proven in Theorem 6.1 that in the case of the single
coating -where the permittivity and the permeability are bounded above and below inside
the cloaked object- the tangential components of the electric and the magnetic field of these
solutions have to vanish in the inside of the boundary of the cloaked object. Note that in
this case in order to have a self-adjoint extension of the electromagnetic propagator inside
the cloaked object we are only allowed to require that either the tangential component of E
or the tangential component of H vanishes, but not both.
These boundary conditions are called hidden boundary conditions in [7] where also the
case of the Helmholtz equation is considered. In the case of Maxwell’s equations they propose
two solutions to this issue. One of them is a lining, i.e., a physical material on the boundary of
the cloaked object that enforces a particular boundary condition, for example, they propose
a lining by a perfect electric conductor. Note that this raises now the question of what is the
boundary condition between the lining and the cloaking metamaterial. In fact, we face the
same problem as before, since we can always consider that the lining is part of the cloaked
objects, and then, the question of what is the appropriate boundary condition remains. The
second proposal of [7] is a double coating that corresponds to surrounding both the inner
and the outersurface of the cloaked objects with appropriately matched metamaterials. As
our permittivities and permeabilities inside K are allowed to vanish as they approach ∂K
the double coating fits in our formalism.
In Theorem 5.1 of [7] cloaking is proven for all frequencies and active devices, with the
double coating, with respect to the Cauchy data of the finite energy solutions that they define
in Definition 4.1.
Remark that there is no real contradiction between our results and the ones of [7]. Our
results imply that there is always a hidden boundary condition on the inside of the boundary
of the cloaked objects, that is imposed upon us by the fundamental principle of the con-
servation of the energy of the electromagnetic waves, that implies that time evolution has
to be given by a unitary group generated by a self-adjoint extension of the electromagnetic
propagator, and this amounts to a boundary condition at the inside of the boundary of
the cloaked objects. Note that we do not exclude here the possibility that in some cases
the electromagnetic propagator of the cloaked objects could be essentially self-adjoint, and
in this situation the dynamics inside the cloaked objects will be uniquely defined. In this
case the hidden boundary condition will be uniquely determined by the boundary conditions
satisfied by the functions in the domain of the unique self-adjoint realization of the elec-
tromagnetic propagator in the cloaked objects. Note, however, that we have proven that
for exact transformation media the cloaking outside is actually independent of the cloaked
objects.
4 Cloaking as a Boundary Value Problem
It is a question of independent interest to consider cloaking as a boundary value problem for
the Maxwell’ system at a fixed frequency
∇× E = iλB, ∇×H = −iλD, λ 6= 0, (4.1)
∇ ·B = 0,∇ ·D = 0. (4.2)
As we have already shown, cloaking is independent of the cloaked object, and this means
that we only have to consider these equation in Ω. The main question now is to decide what
is an appropriate class of solutions with locally finite energy. Our analysis of the self-adjoint
extensions of the electromagnetic propagator shows that we have to take solutions that are
locally in the domain of AΩ, that is to say that they are given by (2.38)
with (E0,H0)
T locally in the domain of A0, i.e., (E0,H0)
T are in the domain of A0 when
multiplied by any function in C∞0 (R
3). It follows from (2.39) that the solutions with locally
finite energy have to satisfy the boundary condition,
E× n = 0,H× n = 0, in ∂K+,
where ∂K+ is the outside of the boundary of the cloaked object. This is the only self-
adjoint boundary condition on ∂K+. Note that we define in the same way solutions with
(locally) finite energy in a bounded subset of Ω. In [7] a different definition of solutions with
(locally) finite energy is given in Definition 4.1.
5 Cloaking an Infinite Cylinder
We discuss now the case of an infinite cylinder. For simplicity we consider one cylinder
centered at zero and with its axis the vertical line L := (0, 0, x3), x3 ∈ R. Then,
x = (x1, x2, x3) ∈ R3 :
|x1|2 + |x2|2 ≤ a, x3 ∈ R
, Ω := R3 \K. (5.3)
The set Ω0 is now given by,
Ω0 = R
3 \ L. (5.4)
Let us denote by x := (x,1 , x2) the vectors in the x1 − x2 plane and x̂ := x/|x|. The
transformation (2.6) is replaced by
x = x(y) = f(y) :=
x = ( b−a
|y|+ a)ŷ,
x3 = y3,
(5.5)
for 0 < |y| ≤ b and with b > a. This transformation blows up the line L onto ∂K and it
sends Kb \ L onto Kb \K where
Kb :=
y = (y1, y2, y3) ∈ R3 :
|y1|2 + |y2|2 ≤ b, y3 ∈ R
For y ∈ R3 \Kb we define the transformation to be the identity, x = y.
The Hilbert spaces of finite energy electromagnetic fields, the unitary operator U and a0,
A0, aΩ, aK , a, are defined as in Section 2.
THEOREM 5.1. The operator aΩ is essentially self-adjoint, and its unique self-adjoint
extension, AΩ, satisfies
AΩ = U A0 U
∗. (5.6)
Proof: The theorem is proven as Theorem 2.1 observing that W 1,2(R2) = W
2 \ 0) since
{0} has zero Sobolev one capacity in R2 [1, 11, 12].
We now consider the wave and the scattering operators. For simplicity we assume below
that ελν0 = ε̃ δ
λν , µλν0 = µ̃ δ
λν . The wave operators are defined as in (2.41) but now the
operator J is defined as follows,
(x) := χΩ(x)φ(x)
where φ is continuous and it satisfies φ(x) = (|x| − a), a ≤ |x| ≤ a + δ and φ(x) = 1 for
|x| ≥ a+ 2δ, for some δ > 0.
LEMMA 5.2.
W± = UP0⊥. (5.7)
Proof: The lemma is proven as in the proof of Lemma 2.2, but in (2.45, 2.46, 2.47, 2.48) we
have to replace χBR
, by χCR
where, CR := {y ∈ R
3 : |y| ≤ R} for R large enough. Now we
can not prove (2.49, 2.50, 2.51) by compactness arguments because K is unbounded. Instead
we use propagation estimates for A0. The following results are well known. See for example
[3, 27, 28, 31] where the general anisotropic case is considered. For any ϕ ∈ H0⊥,
e−itA0ϕ =
(2π)3/2
eik·y
e−iω+(k)tP+(k)ϕ̂(k) + e
−iω−(k)tP−(k)ϕ̂(k)
d3k (5.8)
where ϕ̂ is the Fourier transform of ϕ, ω±(k) = ±|k|c with c := (ε̃µ̃)
−1/2, and P±(k) are
projectors on R3 that are infinitely differentiable for k ∈ R3\0. Suppose that ϕ̂ ∈ C∞0 (R
and let O be a bounded open set such that O ⊂ R3 \ L and support ϕ̂ ⊂ O. Denote
Ô :=
: k ∈ O
Then by the (non) stationary phase Theorem (see the Corollary to Theorem XI.14 of [23] ),
for any n = 1, 2, · · · there is a constant Cn such that
e−itA0ϕ
∣ ≤ Cn (1 + |y|+ |t|)
/∈ Ô. (5.9)
We write,
−itA0ϕ = φ1 + φ2 (5.10)
φ1 := χ(±y/(ct)/∈Ô)χCRe
−itA0ϕ (5.11)
φ2 := χ(±y/(ct)∈Ô)χCRe
−itA0ϕ. (5.12)
by (5.9)
s- lim
φ1 = 0. (5.13)
Note, furthermore, that there is an ǫ > 0 such that |k| ≥ ǫ for any k ∈ Ô. Then, for any
∈ Ô, |y| ≥ c|t|ǫ. It follows that there is a T such that
φ2 = 0, for|t| ≥ T. (5.14)
By (5.10, 5.13, 5.14)
s- lim
−itA0ϕ = 0, (5.15)
and (2.50, 2.51) follow. Note that P0⊥ is not needed because ϕ ∈ H0⊥. By continuity
this is true for all ϕ ∈ H0⊥. Then, (5.7) follows from (2.43, 2.51).
The scattering operator is defined as in (2.52).
COROLLARY 5.3.
S = P0⊥. (5.16)
Proof: This is immediate from (5.7) because U∗ U = I.
Let us denote by S⊥ the restriction of S to H0⊥. S⊥ is the physically relevant scattering
operator that acts in the Hilbert space H0⊥ of finite energy fields that satisfy equations (2.2).
We designate by I⊥ the identity operator on H0⊥. We have that,
COROLLARY 5.4.
S⊥ = I⊥. (5.17)
Proof: This follows from Corollary 5.3.
Again, the fact that S⊥ is the identity operator on H0⊥ means that there is cloaking for
all frequencies.
In Theorem 7.1 of [7] cloaking is proven for all frequencies with respect to the Cauchy
data of the finite energy solutions that they define in Definition 4.1 and furthermore, in
Theorem 8.2, they prove cloaking for all frequencies with the SHS boundary condition with
respect to the Cauchy data of the finite energy solutions that they define in Definition 8.1.
Theorem 3.1 remains true in the case of the cylinder. The proof is the same. Furthermore,
all the remarks about finite energy solutions, and cloaking and that we made in Sections
2, 3, are true in the case of a cylinder. We do not repeat them here. Moreover, equations
(2.38) hold. However, since now the transformation (5.5) only acts on the plane orthogonal
to the axis of the cylinder equations (2.39) has to be replaced by
E× x̂ = 0, H× x̂ = 0, in ∂K+, (5.18)
where E := (E1, E2),H := (H1, H2).
As in Section 4 we define solutions to (4.1, 4.2) with locally finite energy as solutions
that are locally in the domain of AΩ, that is to say that they are given by (2.38)
with (E0,H0)
T locally in the domain of A0, i.e., (E0,H0)
T are in the domain of A0 when
multiplied by any function in C∞0 (R
3). It follows from (5.18) that the solutions with locally
finite energy have to satisfy the boundary condition,
E× x̂ = 0, H× x̂ = 0, in ∂K+. (5.19)
Note that (5.19) is the SHS boundary condition considered in [7]. We have proven here
that (5.19) is the only self-adjoint boundary condition on ∂K+ allowed by energy conserva-
tion.
Acknowledgement
This work was partially done while I was visiting the Institut für Theoretische Physik,
Eidgenössische Techniche Höchschule Zurich. I thank professors Gian Michele Graf and
Jürg Fröhlich for their kind hospitality.
References
[1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Second Edition, Academic Press,
Amsterdam, 2003
[2] A. Bachelot, Gravitational scattering of electromagnetic fields by Schwarzschild black-
holes, Ann. Inst. H. Poincaré 54 261-320 (1991).
[3] R. Courant and D. Hilbert, Methoden der Mathematischen Physik II, Zweite Auflage,
Springer-Verlag, Berlin, 1968.
[4] W. Cai, U.K. Chettiar, A.V. Kildishev and V.M. Shalaev, Optical Cloaking with meta-
materials, Nature Photonics 1 224-226 (2007).
[5] S.A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith and J. Pendry, Full-wave simulation
of electromagnetic cloaking structures, Phys. Rev. E 74 036621 (2006).
[6] M. Sh. Birman and M.Z. Solomyak, The self-adjoint Maxwell operator in arbitrary
domains, Algebra i Analiz. 1 96-110 (1989). English transl. Leningrad Math. J. 1 99-
115 (1989).
[7] A. Greenleaf, Y. Kurylev, M. Lassas and G. Ulhmann, Full-wave invisibility of active
devices at all frequencies, Comm. Math. Phys. 275 749-789 (2007).
[8] A. Greenleaf, M. Lassas, and G. Uhlmann, Anisotropic conductivities that cannot be
detected by EIT, Physiol. Meas. 24 413-419 (2003).
[9] A. Greenleaf, M. Lassas, and G. Uhlmann, On nonuniqueness for Calderón’s inverse
problem, Math. Res. Let. 10 685-693 (2003).
[10] R. Hempel and R. Weder, On the completeness of wave operators under loss of local
compactness, Journal of Functional Analysis, 113 391-412 (1993).
[11] T. Kilpeläinen, J. Kinnunen and O. Martio, Sobolev spaces with zero boundary values
on metric spaces, Potential Anal. 12 233-247 (2000).
[12] J. Kinnunen and O. Martio, The Sobolev capacity on metric spaces, Ann. Acad. Sci.
Fenn. Math. 21 367-382 (1997).
[13] U. Leonhardt, Optical conformal mapping, Science 312 1777-1780 (2006).
[14] R. Leis, Initial Boundary Value Problems in Mathematical Physics, John Wiley & Sons,
New York, 1986.
[15] U. Leonhardt and T. G. Philbin, General relativity in electrical engineering, New J.
Phys. 8 247 (2006).
[16] G. W. Milton, M. Briane, and J. R. Willis, On cloaking for elasticity and physical
equations with transformation invariant form, New J. Phys. 8 248 (2006).
[17] A. Nachman, Reconstruction from boundary measurements, Ann. of Math. 128 71-96
(1988).
[18] J. B. Pendry, D. Schurig, and D. R. Smith, Controlling electromagnetic fields, Science
312 1780-1782 (2006).
[19] R. Picard, Ein Randwertproblem für die zeitunabhängigen Maxwellschen Gleichungen
mit der Randbedingung n · εE = n · µH = 0 in beschränken Gebieten beliebigen
Zusammenhangs Appl. Anal. 6 207-221 (1977).
[20] R. Picard, On the low frequency asymptotics in electromagnetic theory, J. Reine Angew.
Math. 354 50-73 (1984).
[21] E.J. Post, Formal Structure of Electromagnetics General Covariance and Electromag-
netics, Dover Publications, Mineola, New York, 1997.
[22] M. Reed and B. Simon, Methods of Modern Mathematical Physics II Fourier Analysis
and Self-Adjointness, Academic Press, New York, 1975.
[23] M. Reed and B. Simon, Methods of Modern Mathematical Physics III Scattering Theory,
Academic Press, New York, 1979.
[24] D. Schurig, , J.J. Mock, B.J. Justice, S.A. Cummer, J.B.. Pendry, A.F. Starr and D.
R. Smith, Metamaterial electromagnetic cloak at microwave frequencies Science 314
977-980 (2006).
[25] D. Schurig, J.B. Pendry, and D. R. Smith, Calculation of material properties and ray
tracing in transformation media, Opt. Exp. 14 9794-9804 (2006).
[26] G. Uhlmann, Scattering by a metric, Chap. 6.1.5 in Encyclopedia on Scvattering, Aca-
demic Press, R. Pike and P. Sabatier, eds (2002), 1668-1677.
[27] R. Weder, Analyticity of the scattering matrix for wave propagation in crystals, J. Math.
Pures et Appl. 64 121-148 (1985).
[28] R. Weder, Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified
Media. Applied Mathematical Sciences 87 Springer-Verlag, New York, 1991.
[29] R. Weder, A rigorous analysis of high-order electromagnetic invisibility cloaks, arXiv:
0711.0507, 2007, J. Phys. A: Mathematical and Theoretical 41 065207 (2008). IOP-
Select.
[30] R. Weder, The boundary conditions for electromagnetic invisibility cloaks, arXiv:
0801.3611, 2008.
[31] C.H. Wilcox, Asymptotic wave functions and energy distributions in strongly propaga-
tive media, J. Math. Pures et Appl. 57 275-231 (1978).
	Introduction
	Electromagnetic Cloaking
	Electromagnetic Waves Inside the Cloaked Objects 
	Cloaking as a Boundary Value Problem
	Cloaking an Infinite Cylinder
