Well-conditioned boundary integral formulations for the ... - Njit

Well-conditioned boundary integral equation formulations for the
solution of high-frequency electromagnetic scattering problems
Yassine Boubendir, Catalin Turc
boubendi@njit.edu catalin.c.turc@njit.edu
Abstract
We present several versions of Regularized Combined Field Integral Equation (CFIER) formulations
for the solution of three dimensional frequency domain electromagnetic scattering
problems with Perfectly Electric Conducting (PEC) boundary conditions. Just as in the Combined
Field Integral Equations (CFIE), we seek the scattered fields in the form of a combined
magnetic and electric dipole layer potentials that involves a composition of the latter type of
boundary layers with regularizing operators. The regularizing operators are of two types: (1)
modified versions of electric field integral operators with complex wavenumbers, and (2) principal
symbols of those operators in the sense of pseudodifferential operators. We show that the
boundary integral operators that enter these CFIER formulations are Fredholm of the second
kind, and invertible with bounded inverses in the classical trace spaces of electromagnetic scattering
problems. We present a spectral analysis of CFIER operators with regularizing operators
that have purely imaginary wavenumbers for spherical geometries. Under certain assumptions
on the coupling constants and the absolute values of the imaginary wavenumbers of the regularizing
operators, we show that the ensuing CFIER operators are coercive for spherical geometries.
These properties allow us to derive wavenumber explicit bounds on the condition numbers of
certain CFIER operators that have been proposed in the literature. When regularizing operators
with complex wavenumbers with non-zero real parts are used, we show numerical evidence
that those complex wavenumbers can be selected in a manner that leads to CFIER formulations
whose condition numbers can be bounded independently of frequency for spherical geometries.
In addition, the Regularized Combined Field Integral Equations that employ as regularizers
electric field integral operators with carefully chosen complex numbers possess excellent spectral
properties in the high-frequency regime for strictly convex scatterers. We provide numerical
evidence that our solvers based on fast, high-order Nyström discretization of these equations
converge in very small numbers of GMRES iterations, and the iteration counts are virtually
independent of frequency for strictly convex scatterers.
Keywords: electromagnetic scattering, Combined Field Integral Equations, pseudodifferential
operators.
1 Introduction
The simulation of frequency domain electromagnetic wave scattering gives rise to a host of computational
challenges that mostly result from oscillatory solutions, and ill-conditioning in the low
and high-frequency regimes. Computational modeling of electromagnetic scattering has been attempted
based on the classical Finite-Difference Time-Domain (FDTD) methods. However, algorithms
based on the finite-difference or finite-element discretizations require discretization of
unoccupied volumetric regions and give rise to numerical dispersion which is inevitably associated
with numerical propagation of waves across large numbers of volumetric elements [8]. An important
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computational alternative to finite-difference and finite-element approaches is found in boundary
integral methods. Numerical methods based on integral formulations of scattering problems enjoy
a number of attractive properties as they formulate the problems on lower-dimensional, bounded
computational domains and capture intrinsically the outgoing character of scattered waves. Thus,
on account of the dimensional reduction and associated small discretizations (significantly smaller
than the discretizations required by volumetric finite-element or finite-difference approximations),
in conjunction with available fast solvers [11, 13–15, 17, 18, 23, 24, 45, 46, 50], numerical algorithms
based on integral formulations, when applicable, can outperform their finite-element/difference
counterparts.
The most widely used integral equation formulations for solution of frequency domain scattering
problems from perfectly electric conducting (PEC) closed three-dimensional objects are the
Combined Field Integral Equations (CFIE) formulations [35]. The CFIE are uniquely solvable
throughout the frequency spectrum, yet the spectral properties of the boundary integral operators
associated with the CFIE formulations are not particularly suited for Krylov-subspace iterative
solvers such as GMRES [15, 30]. This is attributed to the fact that the electric field (EFIE) operator,
which is a portion of the CFIE, is a pseudodifferential operator of order 1 [48, 54]—that
is, asymptotically, the action of the operator in Fourier space amounts to multiplication by the
Fourier-transform variable. Consistent with this fact, the eigenvalues of these operators accumulate
at infinity, which causes the condition numbers of CFIE formulations to grow with the
discretization size, a property that is also shared by integral equations of the first kind. The lack of
well conditioning of the operators in CFIE is exacerbated at high frequencies, a regime where CFIE
require efficient preconditioners that should ideally control the amount of numerical work entailed
by iterative solvers. In this regard, one possibility is to use algebraic preconditioners, typically
based on multi-grid methods [21], or Frobenius norm minimizations and sparsification techniques
[22]. However, the generic algebraic preconditioning strategies are not particularly geared towards
wave scattering problems, and, in addition, they may encounter convergence breakdowns at higher
frequencies that require large discretizations [41, 56].
On the other hand, several alternative integral equation formulations for PEC scattering problems
that possess good conditioning properties have been introduced in the literature in the past
fifteen years [2–5, 15, 26, 30, 32, 33, 53]. Some of these formulations were devised to avoid the wellknown“low-frequency
breakdown” [33, 58]. For instance, the current and charge integral equation
formulation [53], although not Fredholm of the second kind, does not suffer from the low-frequency
breakdown and has reasonable properties throughout the frequency range [10]. Another class of
Fredholm boundary integral equations of the second kind for the solution of PEC electromagnetic
scattering problems can be derived using generalized Debye sources [33]. Although these
formulations targeted the low frequency case, their versions that use single layers with imaginary
wavenumbers possess good condition numbers for higher frequencies for spherical scatterers [57].
Another wide class of formulations that is directly related to the present work can be viewed
as Regularized Integral Equations as they typically involve using pseudoinverses/regularizers of
the electric field integral operators to mollify the undesirable derivative-like effects of the latter
operators. In the cases when the scattered electric fields are sought as linear combinations of
magnetic and electric dipole distributions, the former acting on tangential densities while the latter
acting on certain regularizing operators of the same tangential densities, the enforcement of the
PEC boundary conditions leads to Regularized Combined Field Integral Equations (CFIER) or
Generalized Combined Sources Integral Equations (GCSIE). In the case of smooth scatterers, the
various regularizing operators proposed in the literature one the one hand (a) Stabilize the leading
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order effect of the pseudodifferential operators of order 1 that enter CFIE, so that the integral
operators in CFIER are Fredholm and their spectra are bounded away from infinity and (b) Have
certain coercivity properties that ensure the invertibility of the CFIER operators. One way to
construct regularizing operators that achieve the objective (a) can be pursued in the framework
of approximations of admittance/Dirichlet-to-Neumann operators (that is the operators that map
the values of the vector product between the unit normal and the electric field on the surface
of the scatterer to the value of the vector product between the unit normal and the magnetic
field on the surface of the scatterer—see Section 3.1) [2–4, 32] which can be connected to onsurface
radiation conditions (OSRC) [40]. Another way to construct such operators is to start
from Calderón’s identities [5, 15, 26, 30] that establish that the square of the electric field integral
operator is a compact perturbation of the identity. All of these regularizing operators are either
electric field integral operators, its vector single layer components, or their principal symbols in
the sense of pseudodifferential operators. The regularizing operators that have property (a) can
be modified to meet the requirement (b) either via quadratic partitions of unity [3, 4] or by means
of complexification of the wavenumber in the definition of electric field integral operators or its
components [2, 15, 30, 32]. To the best of our knowledge, only the regularized formulations in [3, 4,
15] are shown rigorously to be Fredholm integral equations of the second kind and invertible in the
appropriate trace spaces of electromagnetic scattering of smooth scatterers. In the case of Lipschitz
scatterers, the situation is more complicated. The well posedness of the classical CFIE formulations
has not yet been proved for Lipschitz boundaries. Alternative boundary integral equations [37, 52]
use regularizers that act on the magnetic field integral operators and lead to formulations whose
operators are compact perturbations of coercive operators. The latter property ensures the well
posedness of the aforementioned regularized boundary integral equations in Lipschitz domains.
We present in this paper a systematic analysis of various versions of Regularized Combined Field
Integral Equations (CFIER) that are based on various regularizers of the electric field integral
operators. Starting from Calderón’s identities, we look for general regularizing operators that
consists of linear combinations of the single layer operators and the hypersingular operators with
complex wavenumbers that make up the electric field integral operator. Using Calderón’s calculus
we establish that the resulting CFIER operators are Fredhom of the second kind in the appropriate
trace spaces of electromagnetic scattering problems. The injectivity and thus the invertibility of
the CFIER operators is a result of certain coercivity properties of the imaginary and real parts of
single layer operators with complex wavenumbers. In particular, we establish the invertibility of
the CFIER operators in the case when the regularizing operators are electric field integral operators
with purely imaginary wavenumbers—a choice that was advocated in [30]. We also establish the
invertibility of the CFIER operators in the cases when the regularizing operators are given by
the electric field integral operators with complex wavenumbers. In addition, we show that using
principal symbols of the electric field integral operators with complex wavenumbers as regularizing
operators, which was introduced in [32], also leads to invertible Fredholm CFIER operators of the
second kind.
Given that all of the CFIER formulations involve several parameters (e.g. coupling constants
and values of the complex wavenumbers in the definition of the regularizing operators), we follow the
standard practice [39] and analyze the spectral properties of these operators in the case of spherical
scatterers. In the case when we use regularizing operators with purely imaginary wavenumbers
such as those introduced in [15, 30], a rigorous spectral analysis establishes that for high enough
frequencies and sufficiently large coupling constants, the CFIER operators with purely imaginary
wavenumbers are coercive in H −1/2
div
(Γ) in the case of spherical geometries Γ, and that the coercivity
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constants do not depend on the wavenumbers k. We recall that if H is a Hilbert space, an operator
A : H → H ′ (where H ′ is the dual of H) is coercive if there exists a constant γ > 0 such
that γ‖u‖ 2 H
≤ R〈Au, u〉 for all u ∈ H, where 〈·, ·〉 denotes the duality pairing between H and
H ′ . Our analysis uses and extends coercivity results introduced in [17, 34]. These coercivity results
together with wavenumber-explicit upper-bounds on the magnitude of the eigenvalues of the CFIER
operators with purely imaginary wavenumbers deliver estimates on the condition numbers of these
operators for spherical geometries of radius a: their condition numbers grow asymptotically like
(ka) 2/3 for large enough values of the wavenumber k. CFIER operators with almost optimal
spectral properties can be constructed if the values of the real and imaginary parts of the complex
wavenumbers in the definition of electric field regularizing operators are chosen carefully: our
numerical evidence suggests that these CFIER formulations are coercive and more importantly
their condition numbers is bounded by constants that are independent of frequency. The choices
of the real and imaginary parts are those advocated in [6, 7, 32]: the real part equals the (real)
wavenumber of the scattering problems and the imaginary part is proportional to the cubic roots
of the wavenumber. This selection of the complex wavenumbers in the definition of the regularizing
operator resulted from an optimization process of the norm of the difference between the exact
Dirichlet-to-Neumann map and the regularizing operator for spherical scatterers.
We present numerical results produced by our Nyström discretization implementation of various
boundary integral equation formulations considered in this text. The details of our Nyström
method were presented in our previous contribution [15]; we use overlapping partitions of unity and
analytical resolution of kernel singularities to evaluate accurately the integral operators. We use an
accelerated version of our algorithms that generalizes to the electromagnetic case the FFT “equivalent
sources” accelerated algorithms in [13, 16]. The use of fast algorithms allow us to conduct a
comparison between the performance of our solvers based on various boundary integral equation
formulations of PEC scattering problems in the high-frequency regime. Specifically, we present
results based on the classical CFIE formulations, the regularized formulations that use as regularizing
operators the vector single layer operators with purely imaginary wavenumbers introduced
in [15], and the new regularized formulations that use as regularizing operators electric field integral
operators wit carefully selected complex wavenumbers. For smooth and strictly convex scatterers,
our solvers based on the latter formulations have the remarkable property that the number of
Krylov subspace linear algebra solvers are virtually independent of frequency in the high-frequency
range. This in contrast with the behavior of our solvers based on the other two formulations considered
in our numerical experiments. Although the cost of a matrix-vector product related to
the latter formulations is on average 2.4 times more expensive than that related to the classical
CFIE formulations and 1.5 times more expensive than that related to the regularized formulations
introduced in our previous effort [15], their remarkably fast rate of convergence of the associated
Krylov-subspace linear algebra solvers garners important computational gains over the other two
formulations. Specifically, in the high-frequency regime (e.g. problems of electromagnetic size of
50 wavelengths) the computational gains of solvers based on the new formulations over solvers
based on classical CFIE formulations can be of factors 3 and of factors 2 over solvers based on the
alternative regularized formulations introduced in [15].
The paper is organized as follows: in Section 2 we introduce and analyze a wide class of Regularized
Combined Field Integral Equations, in Section 3 we analyze the spectral properties of the
boundary integral operators associated with the regularized formulations for spherical geometries,
and in Section 4 we present several numerical results that enable comparisons between the performance
of our solvers based fast Nyström discretizations of various integral formulations of PEC
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scattering problems.
2 Regularized Combined Field Integral Equations
We consider the problem of evaluating the scattered electromagnetic field (E s , H s ) that results as
an incident field (E i , H i ) impinges upon the boundary Γ of a perfectly conducting scatterer D.
Defining the total field by (E, H) = (E s + E i , H s + H i ), the scattered field is determined uniquely
by the Maxwell equations
curl E − ikH = 0, curl H + ikE = 0 in R 3 \ D (1)
together with the perfect-conductor (PEC) boundary conditions
n × E = 0 on Γ (2)
and the well known Silver-Müller radiation conditions at infinity on (E s , H s ) [28]. Here and in
what follows we assume that Γ is smooth and n denote unit normals to the surface Γ pointing into
R 3 \ D.
Integral equation formulations for electromagnetic scattering problems can be derived starting
from magnetic and electric dipole distributions. Given a tangential density m on the scatterer Γ,
the magnetic dipole distribution corresponding to the density m is defined as
∫
(Mm)(z) = curl G k (z − y)m(y)dσ(y), z ∈ R 3 \ Γ (3)
Γ
and the electric dipole distribution corresponding to the tangential density e is defined as
∫
(Ee)(z) = curl curl G k (z − y)e(y)dσ(y), z ∈ R 3 \ Γ, (4)
Γ
where G k is the outgoing fundamental solution of the Helmholtz equation, G k (z, y) = G k (z − y) =
e ik|z−y|
4π|z−y| . Both Mm and Ee are radiative solutions of the Maxwell equation curl curl u − k2 u = 0
in R 3 \ D. The limits on Γ of n × Mm and n × Ee can be expressed in the following form [36, 44]:
m(x)
lim n(x) × (Mm)(x + ɛn(x)) = − (K k m)(x)
ɛ→0+ 2
lim n(x) × (Ee)(x + ɛn(x)) = (T ke)(x), x ∈ Γ. (5)
ɛ→0+
In equations (5) K k and T k denote the magnetic and respectively the electric field integral operators.
These operators map tangential fields a on Γ into tangential fields on Γ, and are defined as [36]
∫
(K k a)(x) = n(x) × ∇ y G k (x − y) × a(y)dσ(y), (6)
and
Γ
∫
(T k a)(x) = ikn(x) × G k (x − y)a(y)dσ(y)
Γ
+ i ∫
k n(x) × ∇ x G k (x − y)div Γ a(y)dσ(y)
Γ
= (ik n × S k + i k T 1
k div Γ)a(x), x ∈ Γ. (7)
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The hyper-singular integrals in the definition of K k and T k should be interpreted in the sense of
Cauchy principal value integrals.
Regularized Combined Field Integral Equations for the solution of electromagnetic scattering
problems with perfectly electrically conducting boundary conditions seek representations of the
electromagnetic fields of the form
∫
E s (x) = curl G k (x − y)a(y)dσ(y)
Γ
+ i ∫
k curl curl G k (x − y)(Ra)(y)dσ(y), (8)
Γ
H s (x) = 1
ik curl E(x), x ∈ R3 \ D, (9)
where R denotes a tangential operator to be specified in what follows. Owing to the trace properties
(5), equations (8) and (9) define outgoing solutions to the Maxwell equations with perfectconductor
boundary conditions provided the tangential density a is a solution to the integral equation
a
2 − K ka + T k (Ra) = −n × E i . (10)
If the operator R is selected such that, modulo a compact operator, the composition of T k and
R plus I/2 is a bounded invertible operator with a bounded inverse in the appropriate functional
space setting, then the representations (8)-(9) lead to Regularized Combined Field Integral Equations
(CFIER). Operators R with the aforementioned property are referred to as right regularizing
operators for the operators T k . We note that the classical Combined Field Integral Equations assume
the choice R = ξ n × I, ξ ∈ R [29, 35], and thus those operators R are not regularizing
operators for T k . If R is chosen as a right regularizing operator for T k , the integral operator on
the left hand side of (10) is a Fredholm operator, and, thus, the unique solvability of equation
(10) is equivalent to the injectivity of the left-hand-side operator. Several operators R with the
aforementioned property have been proposed and analyzed in the literature [15]. The aim of this
paper is to present a general strategy to produce such regularizing operators R that encompasses
many of the regularizing operators previously considered.
The starting point of constructing suitable regularizing operators R is the Calderón’s identity
= I
4 − K2 k
[36]. The regularizing operators R should thus resemble the electric field operator
T k . If the choice R = T k were made, the ensuing CFIER operators would not be injective since
for certain wavenumbers k the operators I 2 − K k are not injective [29, 36]. In order to ensure the
injectivity of the resulting CFIER operators, one strategy is to modify the wavenumber in the
Tk
2
definition of the regularizing operator R = T k . Specifically, we propose a general regularizing
operator R of the form
R = η n × S K + ζ TK 1 div Γ (11)
where η and ζ are complex numbers and K is a complex wavenumber such that IK > 0. We
first establish that given the choice in equation (11), and for general closed and smooth manifolds
Γ, the composition T k ◦ R can be represented as a compact perturbation of an invertible diagonal
matrix operator. The main idea in the proof is to use the Helmholtz decomposition of tangential
vector fields on the smooth surface Γ in conjunction with the regularity properties of integral
operators with pseudo-homogeneous kernels [48] in the context of the Sobolev spaces H m− 1 2
div
(Γ) of
H m− 1 2 (T M(Γ)) tangential vector fields that admit an H m− 1 2 divergence [36].
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In what follows we use the notations and relations [44]:
−−→
curl Γ φ = ∇ Γ φ × n (12)
curl Γ a = div Γ (a × n) (13)
−−→
∆ Γ φ = div Γ ∇ Γ φ = −curl ΓcurlΓ φ (14)
where a is a tangential vector field and where φ is a scalar function defined on Γ. A few relevant
properties of the Helmholtz decomposition of tangential vector fields in Sobolev spaces are recounted
in what follows. For a given smooth tangential vector field a ∈ H m−1/2
div
(Γ) we have the Helmholtz
decomposition [36, 44]
a = ∇ Γ φ + −−→ curl Γ ψ + ω, (15)
where ω is a harmonic vector field (i.e., its divergence and curl vanish), and where, using the rightinverse
[54] ∆ −1
Γ
: Hs (Γ) → H s+2 (Γ) of the Laplace-Beltrami operator ∆ Γ , the functions φ and ψ in
the Helmholtz decomposition (15) are given by φ = ∆ −1
Γ
div Γ a and ψ = −∆ −1
Γ
curl Γ a. (Note that
for simply connected surfaces Γ we necessarily have ω = 0.) Clearly for a tangential vector field
a ∈ H m−1/2
div
(Γ) we have φ ∈ H m+3/2 (Γ) and ψ ∈ H m+1/2 (Γ) [36]; the corresponding projection
operators onto the spaces of gradients, rotationals and harmonic fields will be denoted by
Π ∇Γ = ∇ Γ ∆ −1
Γ
Π−−→ curlΓ
= − −−→ curl Γ ∆ −1
Γ
div Γ : H m−1/2
div
(Γ) → H m−1/2
div
(Γ), (16)
curl Γ : H m−1/2
div
(Γ) → H m−1/2
.div
(Γ), and (17)
Π ⋂ −−→ ∇Γ curlΓ
= I − Π ∇Γ − Π−−→ curlΓ
: H m−1/2
div
(Γ) → H m−1/2
div
(Γ). (18)
We will also make use of a more detailed version of the Helmholtz decomposition (15) that we will
review in what follows. For a smooth closed two-dimensional manifold Γ the Laplace Beltrami operator
∆ Γ admits a complete and countable sequence of eigenfunctions which form an orthonormal
basis in L 2 (Γ) [44], denoted by {Y n } 0≤n such that
−∆ Γ Y n = γ n Y n , γ n > 0 for 0 < n. (19)
These eigenfunctions of the Laplace Beltrami operator turn out to be the building block for a complete
system of eigenfunctions of the vector Laplace-Beltrami operator (or Hodge Laplace operator)
−→
∆Γ = ∇ Γ div Γ − −−→ curl Γ curl Γ . Indeed, the system {∇ Γ Y n , −−→ curl Γ Y n } 1≤n forms a system of orthogonal
nontrivial eigenvectors for −→ ∆ Γ with the same eigenvalues γ n
− −→ ∆ Γ ∇ Γ Y n = γ n ∇ Γ Y n (20)
− −→ −−→
−−→
∆ ΓcurlΓ Y n = γ ncurlΓ Y n . (21)
However, the system {∇ Γ Y n , −−→ curl Γ Y n } 1≤n may not constitute a complete basis for the space of
tangential vector fields that are square integrable, as, depending on the topology of Γ, the operator
−→
∆Γ can have a nontrivial but nevertheless finite-dimensional null-space that consists of tangential
vector fields whose tangential divergence and tangential rotational are simultaneously zero. Thus,
if we denote by {u p , p = 1, . . . , N} an orthonormal basis to the null space of −→ ∆ Γ in the case when
this is non-trivial (e.g. the genus of Γ is not zero, i.e. Γ is not simply connected, in which case N
equals the first Betti number of the manifold Γ which is two times the number of “holes” in Γ), the
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Helmholtz decomposition gives that any smooth tangential vector field v that is square integrable
can be represented in the bases previously introduced as
so that { ∇ ΓY n
√γn , −−→ curl √γn Γ Y n
v =
∞∑
n=1
v n
∇ Γ Y n
√
γn
+
∞∑
n=1
w n
−−→ curlΓ Y n
√
γn
+
N∑
c p u p (22)
, u p } 1≤n,1≤p≤N is an orthonormal basis of the space of integrable tangential
vector fields L 2 (T M(Γ)), and hence in any Sobolev space H s (T M(Γ)) of tangential vector fields [44,
pp. 206, 207].
Having reviewed the Helmholtz decomposition of tangential vector fields, we return to establishing
the unique solvability of the CFIER equations (10) with the choice of the regularizing operator
given in equation (11). Using the notation A ∼ B for two operators A and B that differ by a
compact operator from H m− 1 2
div
(Γ) to itself, we recall a result established in [15]
Lemma 2.1 The following property holds
T k ◦ (n × S K ) ∼ i
4k Π−−→ + i
curl Γ 4k Π ⋂ −−→
∇ Γ curlΓ
. (23)
Based on the result in Lemma 2.1, we establish another useful result
Lemma 2.2 The following property holds
Proof. We have that
p=1
T k ◦ (T 1 K div Γ ) ∼ ik 4 Π ∇ Γ
. (24)
T k ◦ TK 1 div Γ = ik(n × S k ) ◦ TK 1 div Γ = ik(n × S k ) ◦ Tk
1
+ ik(n × S k ) ◦ (TK 1 − Tk 1 Γ.
Let us denote by H s (T M(Γ)) the classical Sobolev space of tangential vector fields [36, 44]. Given
that S K − S k : H s−1 (Γ) → H s+2 (Γ) [4, 37], we get that (TK 1 − T k 1)
div Γ = n × ∇(S K − S k ) div Γ :
H s (T M(Γ)) → H s+1 (T M(Γ)). If we further take into account the fact that n×S k : H p (T M(Γ)) →
H p+1 (T M(Γ)) [36, 44], we obtain (n × S k ) ◦ (TK 1 − T k 1)
div Γ : H s (T M(Γ)) → H s+2 (T M(Γ)), and
hence div Γ (n × S k ) ◦ (TK 1 − T k 1)
div Γ : H s (T M(Γ)) → H s+1 (T M(Γ)). Consequently, we get that
(n×S k )◦(TK 1 −T k 1)
div Γ : H m−1/2
div
(Γ) → H m+1/2
div
(Γ). Given the compact embedding of H m+1/2
div
(Γ)
into H m−1/2
div
(Γ) [36, 44] we obtain
T k ◦ T 1 K div Γ ∼ ik(n × S k ) ◦ T 1
k div Γ.
A simple consequence of Calderón’s identity Tk
2 = I 4 − K2 k
and the compactness of the magnetic
field integral operator K k in the space H m−1/2
div
(Γ) [36, 44] is that
(n × S k ) ◦ T 1
k div Γ + T 1
k div Γ ◦ (n × S k ) ∼ I 4 .
div Γ
Given that T k = ikn × S k + i k T 1
k
div Γ, we obtain
T k ◦ (n × S k ) = ik(n × S k ) ◦ (n × S k ) + i k (T 1
k div Γ) ◦ (n × S k ).
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Since n × S k : H s (T M(Γ)) → H s+1 (T M(Γ)) we get that (n × S k ) ◦ (n × S k ) : H s (T M(Γ)) →
H s+2 (T M(Γ)) and thus on account of compact embedding of Sobolev spaces we obtain
(n × S k ) ◦ (n × S k ) : H m−1/2
div
(Γ) → H m−1/2 (Γ)
is a compact operator. Consequently, the result in Lemma 2.1 leads to the following relation
from which we obtain
T 1
k
div
div Γ ◦ (n × S k ) ∼ 1 4 Π−−→ curl Γ
+ 1 4 Π ∇ Γ
⋂ −−→ curlΓ
(n × S k ) ◦ T 1
k
and hence the result of the Lemma now follows.
div Γ ∼ 1 4 Π ∇ Γ
Unique solvability Combining the results from Lemma 2.1 and Lemma 2.2 we establish the
unique solvability of the CFIER boundary integral equations (10) with the choice of regularizing
operator R given in equation (11) under certain conditions on the coupling parameters η and ζ.
We make use of a key result concerning certain positivity properties of scalar single layer potentials
with complex wavenumbers. Specifically, we use the fact that for all κ 1 > 0, large enough κ 2 > 0,
and for all ϕ ∈ H −1/2 (Γ), ϕ ≠ 0 the following positivity relations hold [44]
∫ ∫
R G κ1 +iκ 2
(|x − y|)ϕ(x)ϕ(y)dσ(x)dσ(y) > 0
∫Γ
∫Γ
I G κ1 +iκ 2
(|x − y|)ϕ(x)ϕ(y)dσ(x)dσ(y) > 0. (25)
Γ
Γ
In what follows we use for simplicity the following notation
∫ ∫
{S κ1 +iκ 2
ϕ, ϕ} = G κ1 +iκ 2
(|x − y|)ϕ(x)ϕ(y)dσ(x)dσ(y)
Γ
Γ
and similar notations for its vector counterparts. Sufficient conditions for the validity of the
positivity relations (25) can be easily be established. Indeed, defining the Newton potential
U(z) = ∫ Γ G κ 1 +iκ 2
(|z − y|)ϕ(y)dσ(y), z ∈ R 3 \ Γ, applications of Green’s identities and properties
of the Dirichlet and Neumann traces of the potential U on Γ lead to the following relations
∫ ∫
R G κ1 +iκ 2
(|x − y|)ϕ(x)ϕ(y)dσ(x)dσ(y) = ‖U‖ 2 L 2 (R 3 ) + (κ2 2 − κ 2 1)‖∇U‖ 2 L 2 (R 3 )
Γ
Γ
and
∫ ∫
I G κ1 +iκ 2
(|x − y|)ϕ(x)ϕ(y)dσ(x)dσ(y) = 2κ 1 κ 2 ‖U‖ 2 L 2 (R 3 ) .
Γ Γ
Thus, if κ 2 ≥ κ 1 , the positivity properties (25) hold. However, this requirement can be relaxed,
see [44].
Theorem 2.3 If we take the wavenumber K in the definition of the regularizing operator R defined
in equation (11) such that K = κ 1 + iκ 2 , 0 ≤ κ 1 , 0 < κ 2 , then the boundary integral operators in
the left-hand side of equations (10) satisfy the following property
I
2 − K k + T k ◦ R ∼
+
( 1
2 + ikζ )
Π ∇Γ +
4
( 1
2 + iη
4k
9
( 1
2 + iη
4k
)
Π−−→ curlΓ
)
Π
∇Γ
⋂ −−→ curlΓ
(26)

and thus are Fredholm operators in the spaces H m− 1 2
div
(Γ), m ≥ 0. If in addition η ≠ 0 and either
Rη ≥ 0, Iη ≤ 0, Rζ ≤ 0, Imζ ≥ 0 or Rη ≤ 0, Iη ≥ 0, Rζ ≥ 0, Iζ ≤ 0, the integral equations (10)
are uniquely solvable in the spaces H m− 1 2
div
(Γ), m ≥ 0 for sufficiently large values of κ 2 .
Proof. The result in equation (26) follows from those established in Lemma 2.1 and
Lemma 2.2. Clearly, in the light of equation (26), the boundary integral operator that enters the
CFIER formulation is Fredholm in the spaces H m− 1 2
div
(Γ), m ≥ 0. Consequently, the invertibility of
this operator is equivalent to its injectivity. To establish its injectivity, let a be a solution of equation
(10) with E i = 0. It follows that the electromagnetic field (E s , H s ) defined by equations (8)-(9) is an
outgoing solution to the Maxwell equations in the unbounded domain R 3 \D whose boundary values
E s + on Γ satisfy the homogeneous conditions n × E s + = 0. In view of the uniqueness of radiating
solutions for exterior Maxwell problems [29, 44] we obtain E s = H s = 0 identically in R 3 \ D. It
follows then from the standard jump relations of vector layer potentials given in equations (5) that
the interior traces of the electric and magnetic fields defined in formulas (8) and (9) satisfy the
relations
−n × E s − = a, −n × H s − = Ra, on Γ.
Taking the scalar product of the second of these relations with the conjugate of the first one, using
standard vector relations, integrating over Γ and appealing to the divergence theorem gives
∫
∫
∫
(Ra) · n × ādσ = (Ēs − × H s −) · n dσ = −ik {|H s −| 2 − |E s −| 2 }dx. (27)
Γ
D
We have
and
Γ
∫
Γ
Γ
∫
Γ
∫
(n × S K a) · (n × ā)dσ =
Γ
S K a · ādσ (28)
∫
TKdiv 1 Γ a · (n × ā)dσ = n × ∇ Γ (S K div Γ a) · (n × ā)dσ
Γ∫
−−→
= − curl Γ (S K div Γ a) · (n × ā)dσ
∫Γ
= − (S K div Γ a) curl Γ (n × ā)dσ
∫Γ
= − (S K div Γ a) div Γ ādσ. (29)
In conclusion we obtain from equations (28) and 29 that
∫
∫
∫
(Ra) · n × ādσ = η S K a · ādσ − ζ (S K div Γ a) div Γ ādσ.
Thus, denoting by η = η R + iη I and ζ = ζ R + iζ I we get that
∫
R (Ra) · n × ādσ = η R R{S K a, a} − η I I{S K a, a}
Γ
Γ
Γ
Γ
− ζ R R{S K div Γ a, div Γ a} + ζ I I{S K div Γ a, div Γ a}. (30)
Given that for K = κ 1 + iκ 2 with κ ≥ 0 and ɛ ≥ 0 we have the following positivity relations which
follow from equations (25)
R{S K a, a} > 0, I{S K a, a} ≥ 0, R{S K div Γ a, div Γ a} ≥ 0, I{S K div Γ a, div Γ a} ≥ 0 (31)
10

for all a ∈ H −1/2
div
(Γ), a ≠ 0. Taking into account the identity (30) and the positivity relations (31)
we obtain that given the choice
η ≠ 0
we have that if
or
η R ≥ 0, η I ≤ 0, ζ R ≤ 0, ζ I ≥ 0,
η R ≤ 0, η I ≥ 0, ζ R ≥ 0, ζ I ≤ 0,
then
∫
∫
R (Ra) · n × ādσ > 0 or R (Ra) · n × ādσ < 0.
Γ
Γ
The last relations and equation (27) imply that a = 0 and thus the integral operator in the left-hand
side of equation (10) is injective under the assumptions in this Theorem.
Remark 2.4 We note that the choice K = iκ 2 , κ 2 > 0, η = ξκ 2 , and ζ = −ξκ −1
2 , where ξ > 0
leads to regularizing operators of the form R = −ξT iκ2 that have been introduced in [30], but no
rigorous proof of invertibility was provided in that reference. Also, the choice K = iκ 2 , κ 2 > 0,
η = −ξκ 2 , and ζ = 0 that amounts to choosing the regularizing operator in the form R = ξ n × S iκ2
has been proposed in the literature [15].
Another choice of regularizing operators that we will consider is given by R = −γ T κ1 +iκ 2
with
γ > 0, κ 1 > 0 and κ 2 > 0 large enough. In this case, η = −γ(κ 2 + iκ 1 ) and ζ = γ κ 2+iκ 1
, and thus
κ 2 1 +κ2 2
the unique solvability of the CFIER equations with the choice R = −γ T κ1 +iκ 2
is not guaranteed
by the result in Theorem 2.3. Nevertheless, we establish this property in the following result
Theorem 2.5 If we consider the regularizing operator R = −γ T κ1 +iκ 2
κ 2 > 0 is large enough, then the boundary integral operators
where γ > 0, κ 1 > 0 and
have the property
B k,γ,κ1 ,κ 2
∼
+
B k,γ,κ1 ,κ 2
= I 2 − K k − γT k ◦ T κ1 +iκ 2
(32)
( ) ( 1
2 + γ k
1
Π ∇Γ +
4(κ 1 + iκ 2 ) 2 + γ(κ )
1 + iκ 2 )
4k
( 1
2 + γ(κ 1 + iκ 2 )
4k
Π−−→ curlΓ
)
Π
∇Γ
⋂ −−→ curlΓ
. (33)
and in addition they are continuous with bounded inverses in the spaces H m− 1 2
div
(Γ), m ≥ 0.
Proof. We note that relation (33) follows from the results established in Lemma 2.1 and
Lemma 2.2. Just as in the proof of Theorem 2.3, the result of the Theorem is established once we
show that
∫
−R (T κ1 +iκ 2
a) · n × ādσ = κ 2 R{S κ1 +iκ 2
a, a} + κ 1 I{S κ1 +iκ 2
a, a}
Γ
κ 2
+
κ 2 1 + R{S κ1 κ2 +iκ 2
div Γ a, div Γ a} − κ 1
2
κ 2 1 + I{S κ1 κ2 +iκ 2
div Γ a, div Γ a}
2
> 0 (34)
11

for all κ 2 large enough and all a ∈ H −1/2
div
(Γ), a ≠ 0. We will make use in the proof of this result
of techniques developed in the proof of Lemma 5.6.1 in [44].
Let us assume a covering of R 3 by a collection of uniformly distributed balls of radii m log κ 2 /κ 2 .
Let us denote these balls by B j and their centers by b j . Let us denote by B i the balls that intersect
Γ, in which case we denote Γ i = Γ ∩ B i . Let us denote by N the cardinality of the set {Γ i } that
constitute a regular covering of Γ for large enough κ 2 ; obviously N ≈ κ 2 2 . We use a partition of
unity associated with the sets Γ i that consists of functions λ i that have value 1 on the ball B i and
their support included in the union of neighboring balls intersecting B i . In each of these sets we
consider a central point y i and an associated chart that is the projection ψ i on the tangent plane
to Γ at the point y i . The maps ψ i are isomorphisms from Γ i to R 2 .
Let ϕ ∈ H −1/2 (Γ); we decompose ϕ using the partition of unity {λ i } in the form
ϕ = ∑ i
ϕ i , ϕ i = ϕλ i .
We have then
∫ ∫
Γ
Γ
G κ1 +iκ 2
(|x − y|)ϕ(x)ϕ(y)dσ(x)dσ(y) = ∑ j,l
∫Γ j
∫Γ l
e (−κ 2+iκ 1 )|x−y|
4π|x − y|
ϕ j (x)ϕ l (y)dσ(x)dσ(y).
For any pair of points x and y in two different balls B j and B l that do not intersect, we get that
since |x − y| ≥ m log κ 2
κ 2
, then κ m 2 e−κ2|x−y| ≤ 1, and thus if the sets Γ j and Γ l do not intersect the
following estimate holds
∫
G κ1 +iκ
∣
2
(|x − y|)ϕ j (x)ϕ l (y)dσ(x)dσ(y)
∫Γ j Γ l
∣ ≤ cκm−1 2 ‖ϕ‖ H −1/2 (Γ j ) ‖ϕ‖ H −1/2 (Γ l ) (35)
where c is a constant independent of κ 2 . For a given number p and any chart Γ i we associate
the p neighboring charts so that the distance between the central points y i and y j is less than
p log κ m 2 /κ 2. We denote by Γ p i the union of Γ i and its p neighbors, and by s p (i) the set of indices j
such that Γ j is contained in Γ p i
. We define the diagonal and quasi-diagonal contributions D as
⎛
⎞ ⎛
⎞
D = ∑ ∫ ∫
G κ1 +iκ 2
(|x − y|) ⎝ ∑
ϕ j (x) ⎠ ⎝ ∑
ϕ l (y) ⎠ dσ(x)dσ(y)
i
Γ Γ
j∈s p(i)
l∈s p(i)
( )
= 1 ∑ ∫ ∫
card s p (i) G κ1 +iκ
N
2
(|x − y|)ϕ(x)ϕ(y)dσ(x)dσ(y) + R
i
Γ Γ
where the remainder R was shown in [44] to equal to
⎛
∫ ∫
R = ⎝ ∑ (
)
⎞
card [s p (j) ∩ s p (l)] − 1 ∑
card s p (i) ϕ j (x)ϕ l (y) ⎠
Γ Γ
N
j,l
i
× G κ1 +iκ 2
(|x − y|)dσ(x)dσ(y). (36)
Given that the system of charts Γ i is regular, it follows that card s p (i) are all bounded by cp 2 .
Furthermore, for indices j and l such that the corresponding central points y j and y l satisfy
|y j − y l | ≤ log κ m 2 /κ 2 we have the following estimate
∣ card [s p(j) ∩ s p (l)] − 1 ∑
card s p (i)
N
∣ ≤ cp.
12
i

Combining the estimate above with the estimate (35), it follows [44] that the constants m and p
can be chosen so that
|R(R)| ≤ c p ∑
‖ϕ‖
N H −1/2 (Γ j ) ‖ϕ‖ H −1/2 (Γ l ) ≤ c p ∑
κ 2 ‖ϕ‖ H −1/2 (Γ j ) ‖ϕ‖ H −1/2 (Γ l )
j,l
2 j,l
|I(R)| ≤ c p ∑
‖ϕ‖
N H −1/2 (Γ j ) ‖ϕ‖ H −1/2 (Γ l ) ≤ c p ∑
κ 2 ‖ϕ‖ H −1/2 (Γ j ) ‖ϕ‖ H −1/2 (Γ l ) . (37)
2
j,l
Estimates (37) just established show that for large enough values of κ 2 the main contributions to
the expressions R{S κ1 +iκ 2
ϕ, ϕ} and I{S κ1 +iκ 2
ϕ, ϕ} come from the real and imaginary parts of the
diagonal and quasi-diagonal terms D.
We turn our attention to the diagonal and quasi-diagonal contributions D. For each of the
domains Γ p i
we use the charts associated with Γ i , that is the projection on the tangent plane at
the central point y i . Given two points x and y in Γ p i , we denote by x p and y p their projections
onto the tangent plane at y i so that x = x p + s(x p )n(y i ) and y = y p + s(y p )n(y i ). We have that
|x − y| 2 = |x p − y p | 2 + |s(x p ) − s(y) p | 2 , ∇s(y i ) = 0, and D 2 s is bounded, from which it follows
that [44]
e (−κ 2+iκ 1 )|x−y|
|x − y|
j,l
= e(−κ 2+iκ 1 )|x p−y p|
(1 + ψ(x p , y p ))
|x p − y p |
where the function ψ is bounded by the quantity p log κ 2 /κ 2 . Thus, for large enough κ 2 , the
dominant contributions to the expression D related to Γ p i
stem from expressions of the form
∫ ∫
e (−κ 2+iκ 1 )|x p−y p|
D i =
ϕ(x p )ϕ(y p )dσ(x p )dσ(y p )
|x p − y p |
R
R
where ϕ is compactly supported in R 2 . Therefore, it suffices to establish the inequality (34) in the
case when Γ = {x 3 = 0} and a is a vector field compactly supported in R 2 . The latter inequality, in
turn, is established using Fourier transforms. In the case when Γ = {x 3 = 0}, all of the boundary
integral operators that enter equation (34) are convolutions and can be expressed in the Fourier
space in terms of the Fourier transform of G κ1 +iκ 2
(x 1 , x 2 ; 0) with respect to the first two variables.
Given the outgoing property of G κ1 +iκ 2
(x 1 , x 2 ; 0), it can be shown that [31]
1
Ĝ κ1 +iκ 2
(ξ) =
2 √ |ξ| 2 − (κ 1 + iκ 2 ) 2
where the square root is chosen so that its real and imaginary parts are both positive. We have
then that
κ 2 R{S κ1 +iκ 2
a, a} + κ 1 I{S κ1 +iκ 2
a, a} + κ 2
κ 2 1 + R{S κ1 κ2 +iκ 2
div Γ a, div Γ a}
2
κ 1
−
κ 2 1 + I{S κ1 κ2 +iκ 2
div Γ a, div Γ a}
2
= κ ∫
2
R{(|ξ| 2 − (κ 1 + iκ 2 ) 2 ) −1/2 }|â(ξ)| 2 dξ
2 R 2
+ κ ∫
1
I{(|ξ| 2 − (κ 1 + iκ 2 ) 2 ) −1/2 }|â(ξ)| 2 dξ
2 R 2 ∫
κ 2
+
2(κ 2 1 + κ2 2 ) R{(|ξ| 2 − (κ 1 + iκ 2 ) 2 ) −1/2 }|ξ · â(ξ)| 2 dξ
R
∫
2
κ 1
−
2(κ 2 1 + κ2 2 ) I{(|ξ| 2 − (κ 1 + iκ 2 ) 2 ) −1/2 }|ξ · â(ξ)| 2 dξ. (38)
R 2
13

Taking into account the fact that |ξ · â(ξ)| 2 ≤ |ξ| 2 |â(ξ)| 2 , the positivity of the expression in the
left-hand side of equations (38) follows immediately once we establish that
κ 2 R{(|ξ| 2 − (κ 1 + iκ 2 ) 2 ) −1/2 } + κ 1 I{(|ξ| 2 − (κ 1 + iκ 2 ) 2 ) −1/2 }
(
≥ |ξ| 2 κ1
κ 2 1 + I{(|ξ| 2 − (κ 1 + iκ 2 ) 2 ) −1/2 } − κ )
2
κ2 2
κ 2 1 + R{(|ξ| 2 − (κ 1 + iκ 2 ) 2 ) −1/2 } . (39)
κ2 2
The inequality (39) can be seen to be equivalent to the inequality
(
I (κ 1 + iκ 2 )[|ξ| 2 − (κ 1 + iκ 2 ) 2 ] −1/2) ( )
|ξ|
2
≥ I [|ξ| 2 − (κ 1 + iκ 2 ) 2 ] −1/2
κ 1 + iκ 2
which in turn is equivalent to the inequality
( )
−κ1 + iκ 2
I
κ 2 1 + [|ξ| 2 − (κ 1 + iκ 2 ) 2 ] 1/2 } ≥ 0. (40)
κ2 2
Given that R{[|ξ| 2 − (κ 1 + iκ 2 ) 2 ] 1/2 } > 0 and I{[|ξ| 2 − (κ 1 + iκ 2 ) 2 ] 1/2 } < 0, the inequality (40)
follows immediately, and the proof is complete.
2.1 Principal symbol regularizing operators
We present in what follows another class of regularizing operators that consists of operators that
have the same principal symbol in the sense of pseudodifferential operators [48, 54] as the operators
R defined in equations (11). The calculation of the principal symbols of the operators R defined
in equations (11) is based on the principal symbols of scalar and vector single layer operators S K
and π Γ S K for complex wavenumbers K such that IK > 0, where the projection operator onto the
tangent space of Γ is defined by π Γ = (n×·)×n. The principal symbols of these operators are equal
1
to √ and √ 1
I 2 |ξ| 2 −K 2 2 |ξ| 2 −K 2 2 respectively [4, 17, 19, 49], where we view the operator π Γ S k in terms
of the Helmholtz decomposition and I 2 stands for the identity 2 × 2 matrix—we assume without
loss of generality that Γ is simply connected. In the previous equations the variable ξ ∈ T M ∗ (Γ)
represents the Fourier symbol of the tangential gradient operator ∇ Γ , and T M ∗ (Γ) represents the
cotangent bundle of Γ [55]. Given that the principal symbol of the operators 1 2 (−∆ Γ − K 2 ) − 1 2
is equal to
1
2 √ |ξ| 2 −K 2 and the principal symbol of the operator 1 2 (−∆ Γ − K 2 ) − 1 2 I 2 is equal to
1
2 √ |ξ| 2 −K 2 I 2 [55], the previous statement can be expressed more precisely in the form [4, 17, 49]
S K = 1 2 (−∆ Γ − K 2 ) − 1 2 mod Ψ −3 (Γ), π Γ S K = 1 2 (−∆ Γ − K 2 ) − 1 2 I2 mod Ψ −3 (TM(Γ)) (41)
where Ψ −3 (Γ) denotes the operator algebra of pseudodifferential operators of order −3 on Γ and
Ψ −3 (T M(Γ)) denotes the operator algebra of pseudodifferential operators of order −3 on the space
of tangential vector fields on T M(Γ). The meaning of the notation A = B mod Ψ s (Γ) where
A and B are pseudodifferential operators defined on scalar functions on Γ is that A − B is a
pseudodifferential operator of order s, that is A − B : H p (Γ) → H p+s (Γ); the definition is similar
in the case when the operators A and B are tangential pseudodifferential operators. It follows
immediately then from equations (41) that
( ) T (
) (
n×S K = 1 ∇Γ
0 (−∆
−−→
Γ − K 2 ) − 1 2 ∆
−1
2 curl Γ −(−∆ Γ − K 2 ) − Γ
div )
Γ
1
2 0
−∆ −1
Γ
curl mod Ψ −3 (TM(Γ)).
Γ
(42)
14

We make use of the following vector calculus identities
(− −→ ∆ Γ − K 2 ) − 1 2 ∇Γ φ = ∇ Γ (−∆ Γ − K 2 ) − 1 −→
2 φ (−∆Γ − K 2 ) − 1 −−→
2 curl Γ ψ = −−→ curl Γ (−∆ Γ − K 2 ) − 1 2 ψ
which follow from representing the (smooth) scalar functions φ and ψ in the complete basis {Y n } 0≤n
of L 2 (Γ) and taking into account equations (19) and (22). Taking into account the vector calculus
identities presented above, we can rewrite the previous equation (42) in the form
n × S K = 1 2 (−−→ ∆ Γ − K 2 ) − 1 2 (n × πΓ ) mod Ψ −3 (TM(Γ)). (43)
Similarly, given that T 1 K div Γ = − −−→ curl Γ S k div Γ we obtain
T 1 Kdiv Γ = − 1 2
( ) T (
∇Γ
0 0
−−→
curl Γ (−∆ Γ − K 2 ) − 1 2 ∆ Γ 0
) ( ∆
−1
Γ
div Γ
curl Γ
−∆ −1
Γ
)
mod Ψ −1 (TM(Γ)). (44)
Using the previous vector calculus identities we get an equivalent representation of equations (44)
in the form
T 1 Kdiv Γ = − 1 2 (−−→ ∆ Γ − K 2 ) − 1 2
−−→
curl Γ curl Γ (n × π Γ ) mod Ψ −1 (TM(Γ)). (45)
We note that formulas (43) and (45) remain valid in the case when Γ is not simply connected. If
we define
P S(R) = η P S(n × S K ) + ζ P S(T 1 K div Γ )
P S(n × S K ) = 1 2 (−−→ ∆ Γ − K 2 ) − 1 2 (n × πΓ )
P S(T 1 K div Γ ) = − 1 2 (−−→ ∆ Γ − K 2 ) − 1 2
−−→
curl Γ curl Γ (n × π Γ ) (46)
then the following result is the counterpart of the result in Theorem 2.3 if we use in equations (8)
principal symbol regularizing operators P S(R) defined in equations (46) instead of the regularizing
operators R defined in equations (11)
Theorem 2.6 If we take the wavenumber K in the definition of the regularizing operator P S(R)
defined in equation (46) such that K = κ 1 + iκ 2 , 0 ≤ κ 1 , 0 ≤ κ 2 , then the following property holds
(
I
1
2 − K k + T k ◦ P S(R) ∼
2 + ikζ ) ( 1
Π ∇Γ +
4
2 + iη )
Π−−→
4k curlΓ
( 1
+
2 + iη )
Π ⋂ −−→
4k ∇Γ curlΓ
. (47)
Thus, the operators I 2 − K k + T k ◦ P S(R) are Fredholm operators in the spaces H m− 1 2
div
(Γ), m ≥ 0.
If in addition η ≠ 0 and either Rη ≥ 0, Iη ≤ 0, Rζ ≤ 0, Imζ ≥ 0 or Rη ≤ 0, Iη ≥ 0, Rζ ≥
0, Iζ ≤ 0, the operators I 2 − K k + T k ◦ P S(R) are invertible with bounded inverses in the spaces
H m− 1 2
div
(Γ), m ≥ 0.
15

Proof. We obtain from relations (43) and (45) that
ηn × S K + ζ TK 1 div Γ − η P S(n × S K ) − ζ P S(TK 1 div Γ ) : H s (T M(Γ)) → H s+1 (T M(Γ)).
Furthermore, since div Γ TK 1 div Γ = div Γ P S(TK 1 div Γ) = 0, we obtain
div Γ (R − P S(R)) = η div Γ (n × S K − P S(n × S K )) : H s (T M(Γ)) → H s+2 (T M(Γ)).
Consequently, we get that
R − P S(R) : H m− 1 2
div
(Γ) → H m+ 1 2
div
(Γ)
and hence the operator R − P S(R) : H m− 1 2
div
(Γ) → H m− 1 2
div
(Γ) is compact given the compact embedding
of the space H m+ 1 2
div
(Γ) in the space H m− 1 2
div
(Γ). Taking into account the mapping property
T k : H m− 1 2
div
(Γ) → H m− 1 2
div
(Γ) [36, 44], we established
I
2 − K k + T k ◦ P S(R) ∼ I 2 − K k + T k ◦ R.
Relation (47) follows now from the formula established above and relation (26). Also, the operators
I
2 −K k +T k ◦P S(R) : H m− 1 2
div
(Γ) → H m− 1 2
div
(Γ) are Fredholm and thus their invertibility is equivalent
to their injectivity. Just like in the proof of Theorem 2.3, the injectivity of the operators I 2 −
K k + T k ◦ P S(R) will follow once we establish the positivity/negativity of the sesquilinear form
R ∫ Γ
(P S(R)a)·n×ādσ. The main ingredient in the proof of the coercivity property of ±R(P S(R)) is
the Helmholtz decomposition (22). We write the Helmholtz decomposition of the smooth tangential
vector field v = n × a in the form
n × a =
∞∑
n=1
a n
∇ Γ Y n
√
γn
+
∞∑
n=1
b n
−−→ curlΓ Y n
√
γn
+
∞∑
c n u n (48)
where we define c j = 0 and u j = 0 for j > N. Given the Helmholtz decomposition (48), the
definition of the operator P S(n×S K ) given in equation (46), and the spectral properties recounted
in formulas (20)-(21), we obtain
∫
Γ(P S(n × S K )a) · (n × ā)dσ = 1 2
n=1
∞∑
(γ n − K 2 ) − 1 2 (|an | 2 + |b n | 2 + |c n | 2 ).
On the other hand, given the definition of the operator P S(T 1 K div Γ) in (46), we obtain
∫
n=1
Γ
P S(T 1 Kdiv Γ a) · (n × ā)dσ = − 1 2
∞∑
(γ n − K 2 ) − 1 2 γn |b n | 2 .
Consequently, if we denote η = η R + iη I and ζ = ζ R + iζ I we get that
∫
R S(R)a) · n × ādσ =
Γ(P 1 2
− 1 2
n=1
∞∑
(η R R(γ n − K 2 ) − 1 2 − ηI I(γ n − K 2 ) − 1 2 )(|an | 2 + |b n | 2 + |c n | 2 )
n=1
∞∑
(ζ R R(γ n − K 2 ) − 1 2 − ζI I(γ n − K 2 ) − 1 2 )γn |b n | 2 . (49)
n=1
16

Since γ n > 0 and RK ≥ 0 and IK > 0, then R(γ n − K 2 ) − 1 2 > 0 and I(γ n − K 2 ) − 1 2 ≥ 0, the
coercivity of ±R(P S(R)) now follows, which completes the proof of the theorem.
Another choice of regularizing operators that has been proposed in the literature [32] consists
of the following selection: K = κ 1 + iκ 2 , η = −iγ K, and ζ = −γ i K
with γ > 0 in equations (46),
that is
R = −γ P S(T κ1 +iκ 2
) = γ (κ 2 − iκ 1 )P S(n × S κ1 +iκ 2
) − γ κ 2 + iκ 1
κ 2 1 + κ2 2
P S(T 1
κ 1 +iκ 2
div Γ ) (50)
where κ 1 > 0 and κ 2 > 0. We note that these regularizing operators are outside the range of
applicability of the result in Theorem 2.6 since Rη > 0, Iη < 0, Rζ < 0, but Iζ < 0. Nevertheless,
the choice R = −γ P S(T K ) also leads to uniquely solvable CFIER formulations, as given in the
following
Theorem 2.7 If we take the wavenumber K in the definition of the regularizing operator −P S(T K )
defined in equation (50) such that K = κ 1 + iκ 2 , 0 < κ 1 , 0 < κ 2 , then the following property holds
P SB k,γ,κ1 ,κ 2
= I ( ) ( 1
2 − K k − γ T k ◦ P S(T K ) ∼
2 + γ k
1
Π ∇Γ +
4(κ 1 + iκ 2 ) 2 + γ(κ )
1 + iκ 2 )
Π−−→
4k
curlΓ
( 1
+
2 + γ(κ )
1 + iκ 2 )
Π ⋂ −−→
4k
∇Γ curlΓ
. (51)
Furthermore, the operators P SB k,γ,κ1 ,κ 2
H m− 1 2
div
(Γ), m ≥ 0 provided that γ > 0.
are invertible with bounded inverses in the spaces
Proof. The fact that the operators I 2 − K k − γ T k ◦ P S(T K ) has the property (51) was shown in
Theorem 2.6. The injectivity of these operators follows once we show that R ∫ Γ (P S(T K)a)·n×ādσ >
0 for a ≠ 0. Assuming the Helmholtz decomposition (48), we get just as in the proof of Theorem 2.6
the following identity
∫
R S(T K )a) · n × ādσ =
Γ(P 1 ∞∑
(ɛR(γ n − K 2 ) − 1 2 + κI(γn − K 2 ) − 1 2 )(|an | 2 + |c n | 2 )
2
n=1
+ 1 ∞∑
α n |b n | 2
2
n=1
A simple calculations gives
( )
i
α n = R
K (γ n − K 2 ) 1 2 =
α n = ɛR(γ n − K 2 ) − 1 2 + κI(γn − K 2 ) − 1 2 )
( ɛ
+
κ 2 + ɛ 2 R(γ n − K 2 ) − 1 2 −
κ
κ 2 + ɛ 2 I(γ n − K 2 ) − 1 2
1
κ 2 + ɛ 2 (ɛR(γ n − K 2 ) 1 2 − κI(γn − K 2 ) 1 2 ).
)
γ n . (52)
Given that for γ n > 0 we have R(γ n − K 2 ) 1 2 > 0 and I(γ n − K 2 ) 1 2 < 0, it follows that α n > 0 and
hence R ∫ Γ (P S(T K)a) · n × ādσ > 0 and the proof is complete.
Remark 2.8 The result established in Theorem 2.7 can be viewed as a principal symbol counterpart
of that in Theorem 2.5, but without any restrictions on the magnitude of the imaginary part of the
wavenumber K.
17

3 Spectral properties of the CFIER operators for spherical scatterers
We investigate in this section the spectral properties of the various CFIER operators considered
in Section 2 in the case when Γ is spherical. It turns out that in this case the eigenvalues of the
CFIER operators are related to eigenvalues of regularized combined field integral equations for
2D scattering problems with Dirichlet and Neumann boundary conditions. We recall first several
results about spectra of combined field and regularized combined field integral operators for circular
geometries in two dimensions. The scattering problem from the unit circle in two dimensions with
Dirichlet boundary conditions consists of finding scattered fields u s that are solutions of
∆u s + k 2 u s = 0 in {x : |x| > 1}
u s = −u inc on S 1
lim
|r|→∞ r1/2 (∂u s /∂r − iku s ) = 0,
where u inc denotes the incident field. If one looks for the scattered field u s in the form
u s ∂G
(z) =
∫S 2D
∫
k
(z − y)
φ(y)ds(y) − iη (z − y)φ(y)ds(y)
1 ∂n(y)
where G 2D
k
G 2D
k
S 1
(z) = i
4 H(1) 0 (k|z|), then φ is the solution of the following boundary integral equation
D k φ = −u inc on Γ, D k φ = φ 2 + K kφ − iηS k φ (53)
where the double layer operator K k is defined as (K k φ)(x) = ∫ ∂G 2D
S 1 k (x−y)
∂n(y)
φ(y)ds(y), |x| = 1,
and the single layer operator S k is defined as (S k φ)(x) = ∫ S
G 2D 1 k
(x − y)φ(y)ds(y), |x| = 1. The
operator D k is diagonalizable in the orthogonal basis {e imθ : m ∈ Z, θ ∈ [0, 2π)} and the eigenvalues
of the operator D k are given by
D k e imθ = d m (k, η)e imθ , m ∈ Z,
d m (k, η) = iπk
2 J |m| ′ (k)H(1) |m| (k) + ηπ 2 J |m|(k)H (1)
|m|
(k). (54)
The scattering problem from the unit circle in two dimensions with Neumann boundary conditions
consists of finding scattered fields u s that are solutions of
∆u s + k 2 u s = 0 in {x : |x| > 1}
∂u s
∂n = −∂uinc
∂n
on S1 , n = x
lim
|r|→∞ r1/2 (∂u s /∂r − iku s ) = 0,
where u inc denotes the incident field. If one looks for the scattered field u s in the form [17]
∫
∫
u s (z) = − (z − y)ψ(y)ds(y) + iξ k
(z − y)
(S ik ψ)(y)ds(y)
∂n(y)
G 2D
k
S 1
S 1 ∂G 2D
then ψ is the solution of the following boundary integral equation
N k ψ = − ∂uinc
∂n on Γ, N kψ = φ 2 − KT k ψ + iξ(N k ◦ S ik )ψ
18

where the operator Kk
T is defined as (KT k ψ)(x) = ∫ ∂G 2D
S 1 k (x−y)
∂n(x)
ψ(y)ds(y), |x| = 1, and the operator
N k is defined as (N k ψ)(x) = P V ∫ ∂ 2 G 2D
S 1 k (x−y)
∂n(x)∂n(y) ψ(y)ds(y), |x| = 1. The operator N k is
diagonalizable in the orthogonal basis {e imθ : m ∈ Z, θ ∈ [0, 2π)} and the eigenvalues of the
operator N k are given by [17]
N k e imθ = p m (k, ξ)e imθ , m ∈ Z,
p m (k, ξ) = 1 − iπk
2 J |m| ′ (k)H(1) |m| (k) + ξπ2 k 2
J
|m| ′ 4
(k)(H(1) |m| (k))′ [iJ |m| (ik)H (1)
|m|
(ik)]. (55)
It was proved in [17, 34] that there exists a constant C and k 0 large enough such that for all k 0 ≥ k
the following relations hold:
|d ν (k, η)| ≤ C(1 + |η|k −2/3 ), |p ν (k, ξ)| ≤ C(1 + |ξ|) for all 0 ≤ ν. (56)
Furthermore, if η = k in equations (53), then the following coercivity property was established
in [34]
R(d ν (k, k)) ≥ 1 2 for all 0 ≤ ν, k 0 ≤ k. (57)
Also, if ξ = k 1/3 in equations (55), then there exists a constant C 0 ≈ 0.377 such that [17]
R(p ν (k, k 1/3 )) ≥ C 0 for all 0 ≤ ν, k 0 ≤ k. (58)
Remark 3.1 It was also established in [17] that by selecting η = 1 in equations (53), the real part
of the eigenvalues p ν (k, 1) is still positive for sufficiently large values of k, but the corresponding
lower bounds are proportional to k −1/3 in that case.
1≤n,−n≤m≤n
Having reviewed the spectral properties of combined field integral operators for 2D scattering problems
in circular geometries, we return to the case of eigenvalues of the electromagnetic scattering
boundary integral operators. For a spherical scatterer of radius one, the spectral properties of
the electromagnetic boundary integral operators can be expressed in terms of gradients and rotationals
of the spherical harmonics (Yn m ) 0≤n,−n≤m≤n . The system (Yn m ) 0≤n,−n≤m≤n constitutes
an orthonormal basis of the space L 2 (S 2 ). The system (∇ S 2Yn m , −−→ curl S 2Yn m ) 1≤n,−n≤m≤n forms an
orthogonal basis of the space L 2 (T M(S 2 )). Here ∇ S 2 denotes the surface gradient on the sphere
S 2 , and for a scalar function b defined on S 2 , −−→ curl S 2 b = (∇ S 2
(
b) × n. Furthermore, the system
∇ S 2 Yn
√ m −−→
)
curl
, S 2 Yn
√ m forms an orthonormal basis of the space L 2 (T M(S 2 )). The spec-
n(n+1) n(n+1)
tral properties of the operators K K and T K for a wavenumber K are recounted below [39]:
K K (∇ S 2Yn m ) = −λ n (K)∇ S 2Yn m , (59)
K K ( −−→ curl S 2Yn m ) = λ n (K) −−→ curl S 2Yn m (60)
and
where
T K (∇ S 2Yn m ) = Λ (1)
n (K) −−→ curl S 2Yn m , (61)
T K ( −−→ curl S 2Yn m ) = Λ (2)
n (K)∇ S 2Yn m . (62)
λ n (K) = iK 2 {j n(K)[Kh (1)
n (K)] ′ + h (1)
n (K)[Kj n (K)] ′ }; (63)
19

and
Λ (1)
n (K) = [Kj n (K)] ′ [Kh (1)
n (K)] ′ (64)
Λ (2)
n (K) = −K 2 j n (K)h (1)
n (K). (65)
In the formulas above j n and h (1)
n denote the spherical Bessel and Hankel functions of the first
kind, respectively. From the spectral properties of the operators T K presented above, the spectral
properties of the operator n×S K can be derived immediately. Indeed, it follows from the definition
(7) (with k = K) that
(n × S K )( −−→ curl S 2Yn m ) = Λ(2) n (K)
iK ∇ S 2Y n m , (66)
and, using the definition of the Laplace-Beltrami operator ∆ S 2 = div S 2∇ S 2 and the fact that the
spherical harmonic Yn m is an eigenfunction of both the operator ∆ S 2, with eigenvalue −n(n + 1),
as well as the single layer acoustic operator corresponding to the wavenumber K [44]
∫
(S K Y n (m) )(x) = G K (x − y)Yn m (y)dσ(y) = iKj n (K)h (1)
n (K)Yn m (x), |x| = 1 (67)
S 2
we obtain
(n × S K )(∇ S 2Yn m ) = 1
iK (Λ(1) n (K) + n(n + 1)j n (K)h (1)
n (K)) −−→ curl S 2Yn m = S n
(1) (K) −−→ curl S 2Yn m . (68)
Consequently, the choice of the regularizing operators R = η κ n × S iκ , κ > 0 advocated in [15]
leads to boundary integral operators with the following spectral properties
( ( )
I 1
2 − K k + ηκT k ◦ (n × S iκ ))
∇ S 2Yn m =
2 + λ n(k) + ηκΛ (2)
n (k)S n
(1) (iκ) ∇ S 2Yn
m
( ) I −−→
2 − K k + ηκT k ◦ (n × S iκ ) curlS 2Yn m =
= A (1)
n (k, κ, η)∇ S 2Yn
m ( 1
2 − λ n(k) − ηΛ (1)
n (k)Λ (2)
n (iκ)
) −−→ curlS 2Yn
m
= A (2)
n (k, κ, η) −−→ curl S 2Yn m . (69)
On the other hand, the choice of the regularizing operators R = −ξT iκ , κ > 0, ξ > 0 introduced
in [30] leads to boundary integral operators with the following spectral properties
( )
( )
I 1
2 − K k − ξT k ◦ T iκ ∇ S 2Yn m =
2 + λ n(k) − ξΛ (2)
n (k)Λ (1)
n (iκ) ∇ S 2Yn
m
( I
2 − K k − ξT k ◦ T iκ
) −−→ curlS 2Y m n =
= B n
(1) (k, κ, ξ)∇ S 2Yn
m ( 1
2 − λ n(k) − ξΛ (1)
n (k)Λ (2)
n (iκ)
) −−→ curlS 2Yn
m
= B n
(2) (k, κ, ξ) −−→ curl S 2Yn
m
. (70)
We note that the operators ( I
2 − K k + ηκT k ◦ (n × S iκ ) ) and ( I
2 − K )
(
k − ξT k ◦ T iκ are diagonal in
∇
the orthonormal basis S 2 Yn
√ m −−→
)
curl
, S 2 Yn
√ m of the space L 2 (T M(S 2 )), having the same
n(n+1) n(n+1)
1≤n,−n≤m≤n
eigenvalues (A (1)
n (k, κ, η), A (2)
n (k, κ, η) and (B n (1) (k, κ, ξ), B n
(2) (k, κ, ξ)) in that basis. We investigate
20

next the behavior of the eigenvalues (A (1)
n (k, κ, η), A (2)
n (k, κ, η) and (B n (1) (k, κ, ξ), B n
(2) (k, κ, ξ)) for
suitable choices of the parameters κ, η, and ξ. The numerical evidence suggests that the choice
κ = ck (where c = 1/2 or c = 1) leads to boundary integral operators with very good spectral
properties [15] and [30]. We will consider next therefore κ = k—the general case κ = ck with c
being a constant independent of k can be treated analogously. We start by stating a useful result
whose proof follows the same lines as the proof of Lemma 3.1 in [17] and it will be presented in the
Appendix:
Lemma 3.2 There exist constants C j > 0, j = 1, . . . , 4 and a number ˜k 0 > 0 such that
(i) C 1 k −2 (n 2 + k 2 ) 1/2 ≤ iJ ′ n+1/2 (ik)(H(1) n+1/2 )′ (ik) ≤ C 2 k −2 (n 2 + k 2 ) 1/2
(ii) 1 4 (n2 + k 2 ) −1/2 ≤ −S (1)
n (ik) ≤ C 3 (n 2 + k 2 ) −1/2 + C 3 k −2 ,
(iii)|J ′ n+1/2 (ik)H(1) n+1/2 (ik)| ≤ C 4k −1 , |J n+1/2 (ik)(H (1)
n+1/2 )′ (ik)| ≤ C 4 k −1
for all k > ˜k 0 and all n ≥ 0.
We investigate next the properties of the eigenvalues (A (1)
n (k, ɛ, η), A (2)
n (k, ɛ, η)) 1≤n with ɛ = k.
We have
A (1)
n (k, k, η) = iπk
2 J n+1/2 ′ (k)H(1) ηπk2
n+1/2
(k) −
2 J n+1/2(k)H (1)
n+1/2 (k)S(1) n (ik) + a (1)
n
n+1/2 (k) + ηπ2 k 2
A (2)
n (k, k, η) = 1 − iπk
2 J ′ n+1/2 (k)H(1)
where
4
+ a (2)
n (k, η) = p n+1/2 (k, η) + a (2)
n (k, η)
a (1)
n (k) = iπ 4 J n+1/2(k)H (1)
n+1/2 (k)
a (2)
n (k, η) = − iπ 4 J n+1/2(k)H (1) ηπ2
n+1/2
(k) +
4 [iJ n+1/2(ik)H (1)
n+1/2 (ik)]
(k)
J ′ n+1/2 (k)(H(1) n+1/2 )′ (k)[iJ n+1/2 (ik)H (1)
n+1/2 (ik)]
× [ 1 4 J n+1/2(k)H (1)
n+1/2 (k) + k 2 J n+1/2(k)(H (1)
n+1/2 )′ (k) + k 2 J n+1/2 ′ (k)H(1) n+1/2
(k)] (72)
and p ν (k, ξ) are defined in equations (55). We use the following two estimates which were established
in [9] (Proposition 3.10): there exists a constant C > 0 independent of k such that |J ν (k)H ν
(1) (k)| ≤
Ck − 2 3 and |kJ ν(k)H ′ ν
(1) (k)| ≤ C for sufficiently large k and for all ν ≥ 0. We obtain immediately
that |a (1)
n (k)| ≤ Ck −2/3 for all 1 ≤ n and k sufficiently large. Using the Wronskian identity
J ν (k)Y ν(k) ′ − J ν(k)Y ′ ν (k) = 2
πk we get that |kJ ν(k)(H ν (1) ) ′ (k)| ≤ C as well. We established in [17],
that there exists a constant C independent of k such that 0 < iJ ν (ik)H ν (1) (ik) ≤ C(ν 2 + k 2 ) −1/2
for sufficiently large k and all ν ≥ 0. Using this result together with the estimates recounted above
and estimate (ii) in Lemma 3.2 we get that
Theorem 3.3 There exists a constant C 5 > 0 and a wavenumber k 4 sufficiently large so that
|A (1)
n (k, k, η)| ≤ C 5 |η|k 1/3 |A (2)
n (k, k, η)| ≤ C 5 (1 + |η|) for all k 4 ≤ k, and all 1 ≤ n.
(71)
21

The same arguments that were used to derive the result in Theorem 3.3 also lead to the following
estimate: there exists a constant C 6 independent of k and a wavenumber k 5 sufficiently large such
that
|a (2)
n (k, η)| ≤ C 6 k − 2 3 (1 + |η|k
− 1 3 ) for all k5 ≤ k, 1 ≤ n. (73)
The next result concerns the positivity of the real part of the eigenvalues A (1)
n (k, k, η). Given
formulas (71) and the previously established estimates (58) and (73), we consider the choice η = k 1/3
in the definition of A (j)
n (k, k, η), j = 1, 2. In this case, it turns out that the positivity of the
eigenvalues A (1)
n (k, k, k 1/3 ) is a consequence of the positivity of the Dirichlet eigenvalues d n (k, k)
defined in equation (54), while the positivity of the eigenvalues A (2)
n (k, k, k 1/3 ) is a consequence of
the positivity of the Neumann eigenvalues p n (k, k 1/3 ) defined in equation (55). We begin with the
following result:
Theorem 3.4 There exists a wavenumber ˜k large enough so that for all k ≥ ˜k and all 1 ≤ n the
following estimate holds:
R(A (1)
n (k, k, k 1/3 )) ≥ 1 2 .
Proof. Using the estimate (ii) in Lemma 3.2, the previously established estimate |a (1)
n (k)| ≤
Ck −2/3 for all 1 ≤ n and k sufficiently large, and the fact that |J ν (k)| ≤ Ck −1/3 , 0 ≤ ν and k
sufficiently large [9], we get that there exists a number ˜k 1 such that
where
R(A (1)
n (k, k, k 1/3 )) ≥ ˜d n (k) − Ck −2/3 , for all k ≥ ˜k 1
˜d n (k) = − πk
2 J n+1/2 ′ (k)Y n+1/2(k) + πk7/3
8 √ n 2 + k J 2 2 n+1/2
(k), 1 ≤ n.
We will establish that ˜d n (k) ≥ 1 2 for all ˜k ≤ k, and all 1 ≤ n. Using the Wronskian identity
Y ν(k)J ′ ν (k) − Y ν (k)J ν(k) ′ = 2
πk , we can re-express ˜d n (k) as
(
)
˜d n (k) = 1 2 + πk k 4/3
2 4 √ n 2 + k J 2 2 n+1/2 (k) − 1 2 (J n+1/2(k)Y n+1/2 (k)) ′ .
It was established in [34] (Lemma 4.5) that there exists δ > 0 and m 0 > 0 such that for all m ≥ m 0
we have (J m (k)Y m (k)) ′ < 0 for all k ∈ (0, m − δm 1/3 ). It follows immediately then that for all
n ≥ max{m 0 − 1/2, 0} we have
˜d n (k) ≥ 1 2 , k ∈ (0, n + 1/2 − δ(n + 1/2)1/3 ). (74)
It remains to consider two cases, that is Case 1: n < m 0 − 1/2 and k large enough, and Case 2:
max{m 0 − 1/2, 0} ≤ n and k ≥ (n + 1/2) − δ(n + 1/2) 1/3 . In both cases we will compare ˜d n (k)
with R(d n+1/2 (k, k)), where d m (k, k) was defined in equation (54). A simple application of the
Wronskian identity for Bessel functions allows us to rewrite R(d n+1/2 (k, k)) in the following form
R(d n+1/2 (k, k)) = 1 2 + πk (Jn+1/2 2 2
(k) − 1 )
2 (J n+1/2(k)Y n+1/2 (k)) ′ .
Case 1: n < m 0 − 1/2 and k large enough: We require that in this case ˜d n (k) ≥ R(d n+1/2 (k, k)),
which is equivalent to 4 √ n 2 + k 2 ≤ k 4/3 . For the last inequality, it suffices to require that
22

4 √ m 2 0 + k2 ≤ k 4/3 . It follows immediately that 4 √ m 2 0 + k2 ≤ k 4/3 provided that k ≥ max{m 0 , 2 15/2 }.
Case 2: max{m 0 −1/2, 0} ≤ n and k ≥ (n+1/2)−δ(n+1/2) 1/3 : Again we require that 4 √ n 2 + k 2 ≤
k 4/3 in this case. Clearly we can choose m 0 large enough so that if both max{m 0 − 1/2, 0} ≤ n and
(n + 1/2) − δ(n + 1/2) 1/3 ≤ k hold, then it implies that n 2 ≤ (n + 1/2) − δ(n + 1/2)1/3 ≤ k and
16(n 2 + k 2 ) ≤ 16(4k 2 + k 2 ) = 80k 2 ≤ k 8/3 .
In conclusion, in both Case 1 and Case 2 we have
˜d n (k) ≥ R(d n+1/2 (k, k)). (75)
Given that R(d ν (k, k)) ≥ 1 2
for all large enough k and all ν ≥ 0 [34], the result of the Theorem now
follows if we take into account estimates (74) and (75).
Having established the positivity of R(A (1)
n (k, k, k 1/3 )), we establish next the positivity of
R(A (2)
n (k, k, k 1/3 )):
Theorem 3.5 There exists a constant C 7 and a wavenumber k ′ such that for all k ≥ k ′ we have
that
R(A (2)
n (k, k, k 1/3 )) ≥ C 7 for all n ≥ 1.
Proof. It follows from the definition of A (2)
n (k, k, k 1/3 ) given in equations (71) that
R(A (2)
n (k, k, k 1/3 )) ≥ R(p n+1/2 (k, k 1/3 )) − |a (2)
n (k, k 1/3 )|, which implies that there exists a wavenumber
k ′ such that R(A (2)
n (k, k, k 1/3 )) ≥ C 0
2
= C 7 for all n ≥ 1 and all k ≥ k ′ if we take into account
the estimates (58) and (73).
(
∇
Given the fact that S 2 Yn
√ m −−→
)
curl
, S 2 Yn
√ m constitute an orthonormal basis for the
n(n+1) n(n+1)
1≤n,−n≤m≤n
space L 2 (T M(S 2 )) (and hence H m−1/2
div
(S 2 ) for all 0 ≤ m), we derive from the results in Theorem 3.3,
Theorem 3.4, and Theorem 3.5 the following
Corollary 3.6 Let A = I 2 − K k + k 4/3 T k ◦ (n × S ik ). There exists a constant C A independent of k
and m such that for all sufficiently large k and all 0 ≤ m
‖A‖ H
m−1/2
div
(S 2 )→H m−1/2
div (S 2 ) ‖A−1 ‖ m−1/2 H div (S 2 )→H m−1/2
div (S 2 ) ≤ C Ak 2/3 .
If we use P S(n×S ik ) instead of n×S ik as regularizing operators in the CFIER formulations (10)
we obtain boundary integral operators P SA = I 2 − K k + k 4/3 T k ◦ (P S(n × S ik )) whose spectral
properties are qualitatively similar to those of the operators A defined above. Indeed, we have
( I
2 − K k + k 4/3 T k ◦ (P S(n × S ik )))
∇ S 2Y m n =
(
1
2 + λ n(k) −
)
k 4/3 Λ (2)
n (k)
2(n(n + 1) + k 2 ) 1/2 ∇ S 2Yn
m
= P SA (1)
n (k, k, k 1/3 )∇ S 2Yn
m
( ) (
)
I −−→
2 − K k + k 4/3 T k ◦ (P S(n × S ik )) curlS 2Yn m 1
=
2 − λ k 4/3 Λ (1)
n (k) −−→
n(k) +
2(n(n + 1) + k 2 ) 1/2 curl S 2Yn
m
= P SA (2)
n (k, k, k 1/3 ) −−→ curl S 2Yn m . (76)
23

Similarly to equations (71) we get
P SA (1)
n (k, k, k 1/3 ) = iπk
2 J n+1/2 ′ (k)H(1) n+1/2 (k) + πk7/3 J n+1/2 (k)H (1)
n+1/2 (k)
4(n(n + 1) + k 2 ) 1/2 + a (1)
n (k)
n+1/2 )′ (k)
P SA (2)
n (k, k, k 1/3 ) = 1 − iπk
2 J n+1/2 ′ (k)H(1) n+1/2 (k) + πk7/3 J
n+1/2 ′ (k)(H(1)
4(n(n + 1) + k 2 ) 1/2
+ P Sa (2)
n (k, k)
(77)
where
P Sa (2)
n (k, η) = − iπ 4 J n+1/2(k)H (1)
n+1/2 (k) + k 1/3 π
4(n(n + 1) + k 2 ) 1/2
× [ 1 4 J n+1/2(k)H (1)
n+1/2 (k) + k 2 J n+1/2(k)(H (1)
n+1/2 )′ (k) + k 2 J ′ n+1/2 (k)H(1) n+1/2 (k)].
Given the results in Lemma 3.2, it follows immediately that qualitatively similar results on upper
bounds and coercivity properties hold for the eigenvalues (P SA (1)
n (k, k, k 1/3 ), P SA (2)
n (k, k, k 1/3 )) 1≤n
and the eigenvalues (A (1)
n (k, k, k 1/3 ), A (2)
n (k, k, k 1/3 )) 1≤n ; in particular, the condition number of the
integral equation formulations based on the operators P SA grow like k 2/3 in the high-frequency
regime, just like those based on the operators A.
Remark 3.7 In the light of the result mentioned in Remark 3.1 and the results established in
Theorem 3.3, Theorem 3.4, and Theorem 3.5, we can derive that the same asymptotic bounds
would hold for the condition numbers of the CFIER formulations based on the operators I 2 − K k +
kT k ◦ (n × S ik/2 ) and I 2 − K k + kT k ◦ (P S(n × S ik/2 )).
We investigate next the properties of the eigenvalues (B n (1) (k, ɛ, ξ), B n
(2) (k, ɛ, ξ)) 0≤n with the
choice ɛ = k previously advocated. We have
B (1)
n (k, k, ξ) = iπk
2 J ′ n+1/2 (k)H(1)
4
n+1/2 (k) + ξπ2 k 2
(78)
J n+1/2 (k)H (1)
n+1/2 (k)[iJ ′ n+1/2 (ik)(H(1) n+1/2 )′ (ik)]
+ b (1)
n (k, ξ)
B n (2) (k, k, ξ) = A (2)
n (k, k, ξ) (79)
where
b (1)
n (k, ξ) = iπ 4 J n+1/2(k)H (1) ξπ2
n+1/2
(k) +
4i J n+1/2(k)H (1)
n+1/2 (k)
× [ 1 4 J n+1/2(ik)H (1)
n+1/2 (ik) + ik 2 J n+1/2(ik)(H (1)
n+1/2 )′ (ik) + ik 2 J ′ n+1/2 (ik)H(1) n+1/2 (ik)]
and p ν (k, ξ) are defined in equations (55). We make use again of the estimate established in [9]
(Proposition 3.10): there exists a constant C > 0 independent of k such that |J ν (k)H ν (1) (k)| ≤ Ck − 2 3
for sufficiently large k and for all ν ≥ 0. We established in [17], that there exists a constant C
24
(80)

0.62
0.6
k −1/3 max k
b
(k)
0.58
0.56
0.54
0.52
0.5
50 100 150 200 250 300 350 400 450 500
k
Figure 1: Plot of k −1/3 max k≤ν b ν (k) for 5041 values of k from k = 8 to k = 512.
independent of k such that 0 < iJ ν (ik)H ν
(1) (ik) ≤ C(ν 2 + k 2 ) −1/2 ≤ Ck −1 for sufficiently large k
and all ν ≥ 0. Using these two results in conjunction with estimates (iii) in Lemma 3.2 we get that
there exist C > 0 and k 6 > 0 such that
|b (1)
n (k, ξ)| ≤ Ck − 2 3 (1 + |ξ|) for all k ≥ k6 , 0 ≤ n. (81)
We investigate next the possibility to find upper bounds for the quantities |B n
(1) (k, k, ξ)| for all
values of n and large enough values of the wavenumber k. We make use one more time of the
estimate established in [9] (Proposition 3.10): there exists a constant C > 0 independent of k such
that |kJ ν(k)H ′ ν
(1) (k)| ≤ C for sufficiently large k and for all ν ≥ 0. If we further take into account
estimates (i) in Lemma 3.2, we see that upper bounds for the quantities |B n
(1) (k, k, ξ)| for all values
of n and large enough values of the wavenumber k can be obtained once upper bounds for the
expressions
b ν (k) = |J ν (k)H (1)
ν (k)| √ ν 2 + k 2
were established for all values of ν ≥ 0 and large enough k. Given the estimate |J ν (k)H ν
(1) (k)| ≤
Ck − 2 3 for sufficiently large k and for all ν ≥ 0, it follows immediately that for large enough values
of k we have
b ν (k) ≤ C √ 2 k 1/3 , ν ≤ k.
We plot in Figure 1 the values of k −1/3 max k≤ν b ν (k) for 5041 values of k from k = 8 to k = 512.
In view of the estimate above and the results illustrated in Figure 1, we are led to the following
heuristic result
b ν (k) ≤ Ck 1/3 , for all k 7 ≤ k, 0 ≤ ν
which would in turn imply that there exists a constant C 8 such that
|B (1)
n (k, k, ξ)| ≤ C 8 |ξ|k 1/3 for all k 7 ≤ k, and all 0 ≤ n. (82)
Although a rigorous proof of the heuristic results above is outside the scope of this paper, we give
further asymptotic evidence in two important regimes (ν ∼ k, ν → ∞ and k fixed and ν → ∞)
25

that the heuristic bounds are true.
(9.3.31)–(9.3.32))
We use the following asymptotic expansions [1] (Formulas
J ν (ν) = aν −1/3 + O(ν −5/3 ) Y ν (ν) = − √ 3aν −1/3 + O(ν −5/3 )
2
which are valid as ν → ∞, and where a = 1/3
the following estimate is valid
3 2/3 Γ(2/3)
b ν (k) ∼ 2 √ 2a 2 k 1/3 , ν ∼ k, k → ∞.
, we obtain that for large enough values of k
The plot of k −1/3 max k≤ν b ν (k) in Figure 1 seems to be consistent with the estimate just derived
above, given that 2 √ 2a 2 ≈ 0.56592. Furthermore, we use the result in Formula 9.3 in [1] which is
valid for fixed k and ν → ∞
J ν (k)H (1)
ν (k) ∼ 1
2πν
( ) ek 2ν
− i 1
2ν πν = O(ν−1 )
and we get that for a fixed and large enough k there exists a constant C(k) such that for all ν large
enough we have
b ν (k) ≤ C(k).
The next result is a coercivity result that establishes that given the choice ξ = k 1/3 in the
definition of the regularizing operators R = −ξT ik , the real parts of the eigenvalues B n (1) (k, k, k 1/3 )
and B n
(2) (k, k, k 1/3 ) are bounded from below by strictly positive constants for all values of n for
large enough values of the wavenumber k. More precisely, we have that
Theorem 3.8 There exists a constant C 9 and a wavenumber k ′ such that for all k ≥ k ′ we have
that
min{R(B n (1) (k, k, k 1/3 )), R(B n (2) (k, k, k 1/3 ))} ≥ C 9 for all n ≥ 1.
Proof. We take into account the estimate (i) established in Lemma 3.2 to obtain
R(B n
(1) (k, k, k 1/3 )) ≥ − πk
2 J n+1/2 ′ (k)Y n+1/2(k) + C 1π 2 k 1/3
4
On the other hand, the estimate (57) can be written explicitly in the form
R(d n+1/2 (k, k)) = − πk
2 J ′ n+1/2 (k)Y n+1/2(k) + kπ 2 J 2 n+1/2 (k) ≥ 1 2
J 2 n+1/2 (k)√ n 2 + k 2 − |b (1)
n (k, k 1/3 )|.
for all 1 ≤ n
and sufficiently large k. Since √ n 2 + k 2 ≥ k and C 1 πk 1/3 ≥ 2 for sufficiently large k, it follows
from the previous two estimates that
R(B (1)
n (k, k, k 1/3 )) ≥ R(d n+1/2 (k, k)) − |b (1)
n (k, k 1/3 )| ≥ 1 2 − |b(1) n (k, k 1/3 )| for all n ≥ 1
and k sufficiently large. The result of the theorem now follows if we take into account estimates (81)
and the result established in Theorem 3.4.
(
∇
Given the fact that S 2 Yn
√ m −−→
)
curl
, S 2 Yn
√ m constitute an orthonormal basis for the
n(n+1) n(n+1)
1≤n,−n≤m≤n
space L 2 (T M(S 2 )) (and hence H m−1/2
div
(S 2 ) for all 0 ≤ m), we derive from the results in Theorem 3.3,
equation (82), and Theorem 3.8 the following
26

Corollary 3.9 Let B = I 2 − K k − k 1/3 T k ◦ T ik . There exists a constant C B independent of k and
m such that for all sufficiently large k and all 0 ≤ m
‖B‖ H
m−1/2
div
(S 2 )→H m−1/2
div (S 2 ) ‖B−1 ‖ m−1/2 H div (S 2 )→H m−1/2
div (S 2 ) ≤ C Bk 2/3 .
If we use P S(T ik ) instead of T ik as regularizing operators in the CFIER formulations (10) we
obtain boundary integral operators P SB = I 2 − K k − k 1/3 T k ◦ (P S(T ik )) whose spectral properties
are qualitatively similar to those of the operators B defined above. Given that for a complex
wavenumber K = κ + iɛ, κ ≥ 0, ɛ > 0 we have that
P S(T K )∇ S 2Y m n =
i
2K (n(n + 1) − K2 1/2 −−→
) curl S 2Yn
m
P S(T K ) −−→ curl S 2Y m n = iK 2 (n(n + 1) − K2 ) −1/2 ∇ S 2Y m n (83)
we obtain
( I
2 − K k − k 1/3 T k ◦ (P S(T ik )))
∇ S 2Y m n =
( ) I −−→
2 − K k − k 1/3 T k ◦ (P S(T ik ))
(
)
1
2 + λ n(k) − Λ(2) n (k)(n(n + 1) + k 2 ) 1/2
2k 2/3 ∇ S 2Yn
m
= P SB n
(1) (k, k, k 1/3 )∇ S 2Yn
m
curlS 2Yn m = P SB n
(2) (k, k, k 1/3 ) −−→ curl S 2Yn
m
= P SA (2)
n (k, k, k 1/3 ) −−→ curl S 2Yn m . (84)
Taking into account the results in Lemma 3.2, it follows immediately that qualitatively similar
results on upper bounds and coercivity properties hold for the eigenvalues
(P SB n (1) (k, k, k 1/3 ), P SB n (2) (k, k, k 1/3 )) 1≤n and the eigenvalues (B n (1) (k, k, k 1/3 ), B n (2) (k, k, k 1/3 )) 1≤n ;
in particular, the condition number of the integral equation formulations based on the operators
P SB grow like k 2/3 in the high-frequency regime, just like those based on the operators B.
3.1 Dirichlet to Neumann maps and nearly optimal choices of regularizing operators
for spherical scatterers
We present in this subsection the remarkable spectral properties that the CFIER integral operators
B k,γ,κ1 ,κ 2
and and P SB k,γ,κ1 ,κ 2
defined in equations (32) and (51) respectively possess in the case
when γ = 2, κ 1 = k, and κ 2 = 0.4k 1/3 . This choice of parameters γ, κ 1 , and κ 2 is motivated by
considerations on the Dirichlet-to-Neumann map of the electromagnetic scattering problem [32].
Specifically, the Dirichlet-to-Neumann (DtN) map is defined as Y (n × E s ) = n × H s . Using the
Stratton-Chu representation formula [28]
∫
E s (z) = curl G k (z − y)(n(y) × E s (y))dσ(y) + i ∫
Γ
k curl curl G k (z − y)(n(y) × H s (y))dσ(y)
∫
Γ
= curl G k (z − y)(n × E s )(y)dσ(y) + i ∫
k curl curl G k (z − y)Y (n × E s )(y)dσ(y)
Γ
we get if we apply the n × · trace that T k Y = I 2 + K k. Clearly, if the regularizing operator R in
equations (8) were chosen such that R = Y , then the ensuing CFIER boundary integral operators
would be equal to the identity operator. From this point of view, the better the regularizing operator
Γ
27

R approximates the DtN operator, the closer the regularized combined field integral operators
would be to the identity operator. The DtN operators Y cannot be computed analytically but
for simple scatterers (e.g. spherical, planes); for general surfaces Γ, the computation of the DtN
operators, if at all possible, is typically as expensive as the solution of the scattering problem since
Y = T −1 ( I
k 2 + K )
k . Using Calderón’s identities and the previous formula we get that Y ∼ −2Tk .
However, as previously argued, the choice R = −2T k does not lead to uniquely solvable regularized
formulations for all wavenumbers k. Nevertheless, we can seek an approximation to the DtN
operator Y in the form R = −2T k+iκ2 , which does lead to invertible operators B k,2,k,κ2 = I 2 −
K k − 2T k ◦ T k+iκ2 —see Theorem 2.5. On the other hand, using Fourier transforms, the operators
DtN can be computed as Fourier multipliers in the case when Γ is a plane in R 3 . Standard
techniques of tangent plane approximations lead to approximations of the DtN operators Y for
general smooth surfaces Γ in the form R = −2P S(T k+iκ2 ) [32]; the complexification in the definition
of the latter operator is needed in order to ensure the injectivity of the operators P SB k,2,k,κ2 =
I
2 − K k − 2T k ◦ P S(T k+iκ2 )—see Theorem 2.6. The selection of the parameters κ 2 is guided by
considerations on the spectral properties of the DtN operators for spherical scatterers. The spectral
properties of the DtN operators in the case Γ = S 2 [44] are given below
where z n (k) = k(h(1)
Y (∇ S 2Yn m ) = i(z n(k) + 1) −−→
curl
k
S 2Yn m = Z n
(1)
Y ( −−→ curl S 2Yn m ) =
n (k)) ′
h (1)
n (k)
(k) −−→ curl S 2Y m n , (85)
ik
z n (k) + 1 ∇ S 2Y m n = Z (2)
n (k)∇ S 2Y m n , (86)
. The value of the parameter κ 2 can be selected by minimizing the expression
max 1≤n {|Z n (1) (k)+2Λ (1)
n (k+iκ 2 )|, |Z n (2) (k)+2Λ (2)
n (k+iκ 2 )|}, where the values {Z n (1) (k), Z n
(2) (k)} 1≤n
were defined in equations (85)-(86) and the values {Λ (1)
n (k + iκ 2 ), Λ (2)
n (k + iκ 2 )} 1≤n defined in equations
(64)-(65) respectively. For large values of the wavenumber k, the maximum sought after
occurs for values of the index n such that n ≈ k. Just as in [32], using standard asymptotic formulas
of Hankel functions [1] (Formulas (9.3.31)–(9.3.34)), the optimal value κ 2 = 0.4k 1/3 can be
derived in the case when Γ = S 2 . We present in Figure 2 the quotients |Z(1) n (k)+2Λ (1)
n (k+0.4 ik 1/3 )|
(black),
|Z n (2) (k)+2Λ (2)
n (k+0.4 ik 1/3 )|
|Z n
(2) (k)|
|Z n
(1) (k)|
(red) for k = 32 (top) and k = 160 (bottom) and and n =
1, . . . , 320 (bottom). Similar results are obtained if we replace in the formulas above the values
{Λ (1)
n (k +iκ 2 ), Λ (2)
n (k +iκ 2 )} 1≤n by {P SΛ (1)
n (k +iκ 2 ), P SΛ (2)
n (k +iκ 2 )} 1≤n defined in equation (83).
Furthermore, in the case when Γ = R S 2 (that is spheres of radius R) similar calculations lead to
the almost optimal choices of the regularizing operators in the form R = −2T k+0.4iR −2/3 k 1/3 and
R = −2P S(T k+0.4iR −2/3 k1/3) [6, 7, 32].
In addition, the eigenvalues of the operators B k,2,k,0.4k 1/3 and P SB k,2,k,0.4k 1/3 can be computed
easily from equations (63), (64), (65), and (83). We display in Figure 3 and Figure 4 the coercivity
constants (that is the minimum value of the real parts of the eigenvalues of these operators) and
the condition numbers of the operators B k,2,k,0.4k 1/3 and P SB k,2,k,0.4k 1/3 respectively for 5041 values
of k from k = 8 to k = 512. The numerical results depicted in Figure 3 and Figure 4 suggest that
both operators B k,2,k,0.4k 1/3 and P SB k,2,k,0.4k 1/3 are coercive for large enough values of k in the case
of spherical scatterers, and their condition numbers appear to be bounded independently of the
wavenumber k for large enough values of k. A rigorous proof of these results is outside the scope
of the present effort.
28

0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
20 40 60 80 100 120
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
50 100 150 200 250 300
Figure 2: Plots of |Z(1) n (k)+2Λ (1)
n (k+0.4 ik 1/3 )|
|Z n
(1) (k)|
(black), |Z(2) n (k)+2Λ (2)
n (k+0.4 ik 1/3 )|
|Z n
(2) (k)|
and k = 160 (bottom) for n = 1, . . . , 128 (top) and n = 1, . . . , 320 (bottom).
(red) for k = 32 (top)
29

0.4
0.39
0.38
Coercivity constant B k,k,0.4k
1/3
0.37
0.36
0.35
0.34
0.33
0.32
0.31
50 100 150 200 250 300 350 400 450 500
k
3.2
3.1
3
Condition numbers B k,k,0.4k
1/3
2.9
2.8
2.7
2.6
2.5
2.4
2.3
2.2
50 100 150 200 250 300 350 400 450 500
k
Figure 3: Plots of the coercivity constants (top) and condition numbers (bottom) of the operators
B k,2,k,0.4k 1/3 for 5041 values of the wavenumber k from k = 8 to k = 512.
30

0.515
0.51
Coercivity constant PSB k,k,0.4k
1/3
0.505
0.5
0.495
0.49
0.485
0.48
50 100 150 200 250 300 350 400 450 500
k
2.1
2.08
Condition numbers PSB k,k,0.4k
1/3
2.06
2.04
2.02
2
1.98
1.96
1.94
50 100 150 200 250 300 350 400 450 500
k
Figure 4: Plots of the coercivity constants (top) and condition numbers (bottom) of the operators
P SB k,2,k,0.4k 1/3 for 5041 values of the wavenumber k from k = 8 to k = 512.
31

4 High-frequency numerical experiments
Our strategy for evaluation of the relevant discrete integro-differential operators that enter the
integral equation formulations presented in this text relies on use of local coordinate charts together
with fixed and floating partitions of unity (POU), as proposed in [13, 15, 16]. First, following the
prescriptions in [15], we express all the electromagentic boundary integral operators as integral
operators with weakly singular kernels, that is
and
∫
(K K a)(x) =
Γ
(
(n(x) − n(y)) · a(y)∇ y G K (x − y) + ∂G )
K(x − y)
a(y) dσ(y), (87)
∂n(x)
∫
(T K a)(x) = iK n(x) × G K (x − y)a(y)dσ(y)
Γ
−
i ∫
(n(y) − n(x)) × ∇ x G K (x − y)div Γ a(y)dσ(y)
K Γ
−
i ∫
G K (x − y) −−→ curl Γ div Γ a(y)dσ(y), (88)
K
Γ
where a is a tangential vector field, and K is a wavenumber, possibly complex. The various integral
operators that enter equations (87) and (88) are computed in two stages which consist of (a) the
evaluation of the adjacent/singular interactions of sources (i.e. the values of the density a(y) or its
derivatives for y on the surface Γ) via the Green’s functions (i.e. the terms that involve G K (x−y))
when the target points x are close to the integration points y and (b) the accelerated evaluation
of far-away interactions of sources that are well-separated. The separation of these contributions
is effected by floating POUs which are pairs of functions of the form (η x (y), 1 − η x (y)) (where
η x is a function with a “small” support which equals 1 in a neighborhood of the point x). The
approach for the solution of the integration problems (a) and (b) recounted above relies on use
of smooth surface parametrizations of the surface Γ via a family of overlapping two-dimensional
patches P l , l = 1, . . . P together with smooth mappings to coordinate sets H l in two-dimensional
space (where actual integrations are performed) and subordinated partitions of unity i.e smooth
functions w l supported on P l such that ∑ l w l = 1 throughout Γ. This framework allows us
to (i) reduce the integration of the tangential densities a over the surface Γ to calculations of
integrals of smooth vector fields a l compactly supported in the planar sets H l and (ii) compute
the derivatives of the density a via derivatives of smooth and periodic functions only [15, 16]. The
first part (a) requires the analytic resolution of weakly singular Green’s functions (i.e. the order of
the singularity is O(|x − y| −1 )) which is performed via polar changes of variables whose Jacobian
cancels the singularity and interpolation procedures that allow for evaluations of the densities at
radial integration points. On the other hand, part (b) corresponds to the acceleration procedure
performed based on two-face planar arrays of “equivalent sources” that allow for 3D FFT based
fast evaluations of cartesian grid convolutions of the equivalent source intensities with Green’s
functions [13, 16]. The various boundary integral operators that are components of the operators
described in equations (87) and (88) are vector versions of the scalar operators presented in [16].
A lengthy discussion was given in [16] on the accelerated evaluation of those boundary integral
operators and on the specific choices of the various parameters that are needed in the accelerators.
The accelerated evaluation of electromagentic integral operators of the type (87) and (88) use the
following four parameters: (1) the number L 3 of small boxes that make up the partition of a box
32

enclosing the scatterer; (2) the number M eq of equivalent sources placed on faces of each of the
small boxes so that the far-field contributions of the equivalent sources matches in the least square
sense the one of the real surface sources; (3) the number n coll of collocation points needed in the
solution of the least-square problem related to equivalent sources expansions; and (4) the number
n w of plane waves needed for the evaluation of the fields on the scatterer from the FFTs based
computations of the fields arising from the Cartesian grids placed equivalent sources. As a result
of the acceleration procedure, and for all of the formulations considered in this text the cost of
one matrix-vector product is O(N 4/3 log N), where N is the number of discretization points—see
Table 3.
We present in this section a variety of numerical results that demonstrate the properties of
various versions of regularized combined field integral equations constructed in the previous sections.
Specifically, we consider two choices of regularized combined field integral equations that involve
the following boundary integral operators:
A k,Sik/2 = I/2 − K k + kT k ◦ (n × S ik/2 ) (89)
and B k,2,k,0.4ck 1/3, c > 0 defined in equations (32). As discussed in [15], accurate evaluations of
the operators B k,2,k,0.4ck 1/3 require use of the cancellation of the composition of the hypersingular
terms that enter the definition of the operators T k and T k+i 0.4ck 1/3 respectively. Accordingly, the
operators B k,2,k,0.4ck 1/3 are evaluated using their equivalent definition
B k,2,k,0.4ck 1/3 = I/2 − K k − 2i(k + i 0.4ck 1/3 )T k ◦ (n × S k+i 0.4ck 1/3)
+
2k
k + i 0.4ck 1/3 (n × S k) ◦ T 1
k+i 0.4ck 1/3 div Γ . (90)
In addition, we compare the performance of our solvers based o regularized combined field integral
equations involving the integral operators defined in equations (89) and (32) to those based on the
classical combined field integral equation based on the integral operator [39]
C k = I/2 − K k + T k ◦ (n × I). (91)
Although the combined field integral equations are not integral equations of the second kind, it was
argued in [39] that the operator C k defined in (91) leads to optimally conditioned combined field
integral equations. The properties of the solvers based on the integral operators A k,Sik/2 defined in
equations (89) were investigated in [15] in the low and medium-frequency range. It was discussed
in [15] that the choice of the operator A k,Sik/2 leads to solvers that converge in small numbers of
GMRES iterations. The same reference [15] provides ample numerical comparisons with solvers
based on the integral operators B k,1,0,k that were proposed in [30]. The numerical evidence therein
suggests that solvers based on the operators B k,1,0,k lead to somewhat larger iteration counts than
those based on the operators A k,Sik/2 , and the cost of evaluating a matrix-vector product associated
with the former operators is on average about 1.6 times more expensive than the that associated
with the latter operators. We present further numerical evidence in Figure 5 on the wavenumber
dependence of the condition numbers of the operators A k,Sik/2 , B k,1,0,k , and B k,2,k,0.4k 1/3 for
spherical geometries. As it can be seen from the results in Figure 5, the condition numbers of
the operators A k,Sik/2 and B k,1,0,k behave asymptotically as O(k 2/3 ) for spherical scatterers, with
smaller proportionality constants for the former operators. In contrast, the condition numbers of
the operators B k,2,k,0.4k 1/3 are bounded independently of frequency for spherical scatterers. In light
of these results, and given that the computational costs entailed by the evaluation of a matrix
33

50
45
40
35
condition number
30
25
20
15
10
5
0
50 100 150 200 250 300 350 400 450 500
k
Figure 5: Plot of the condition numbers of the three regularized combined field integral operators
A k,Sik/2 (black), B k,1,0,k (magenta), and B k,2,k,0.4k 1/3 (red) as a function of the wavenumber k for
5041 values of k ranging from k = 8 to k = 512. A fairly sharp upper bound on the condition
number of the operators A k,Sik/2 is given by 0.38 k 2/3 (plotted in green) and a fairly sharp upper
bound on the condition number of the operators B k,1,0,k is given by 0.75 k 2/3 (plotted in green).
Both upper bounds are in agreement with the theoretical predictions presented in Section 3.
vector product related to the integral operators B k,1,0,k is higher than the one associated with the
integral operators A k,Sik/2 , we present comparisons between solvers based on the integral operators
C k , A k,Sik/2 , and B k,2,k,0.4ck 1/3. Given that for spheres of radius R, the nearly optimal choice of the
constant c in the definition of the operators B k,2,k,0.4ck 1/3 is given by c = R −2/3 (see Section 3.1),
we choose following well establish practice c = H 2/3 , where H is the maximum mean curvature of
the surface Γ for strictly convex scatterers. The optimal choice of the parameter c for non-convex
scatterers is subject of ongoing research and is left for future consideration.
In what follows we present the major steps of the accelerated algorithms for the evaluation of the
matrix vector products associated with each of the three operators C k , A k,Sik/2 , and B k,2,k,0.4H 2/3 k 1/3.
Algorithm I: Accelerated CFIE algorithm for the evaluation of the operator C k (91)
1. For a given tangential density a, evaluate n×a and compute the surface derivatives div Γ (n×a),
−−→
curl Γ div Γ (n × a) and combine the densities a, n × a, div Γ (n × a), and −−→ curl Γ div Γ (n × a) into
an extended vector density ã;
2. Apply the operators K k and T k to the corresponding components of the vector density ã—
using accelerated algorithms with wavenumber k and lumping common kernel components
that enter in the definitions of K k and T k .
Algorithm II: Accelerated CFIER algorithm for the evaluation of the operator A k,Sik/2 (89)
1. For a given tangential density a, evaluate n × S ik/2 a using accelerated algorithms with imaginary
wavenumbers ik/2;
34

2. Compute the surface derivatives div Γ (n × S ik/2 a), −−→ curl Γ div Γ (n × S ik/2 a) and combine the
densities a, n × S ik/2 a, div Γ (n × S ik/2 a), and −−→ curl Γ div Γ (n × S ik/2 a) into an extended vector
density ã;
3. Apply the operators K k and T k to the corresponding components of the vector density ã—
using accelerated algorithms with wavenumber k and lumping common kernel components
that enter in the definitions of K k and T k .
Algorithm III: Accelerated CFIER algorithm for the evaluation of the operator B k,2,k,0.4ck 1/3 (90)
1. For a given tangential density a, compute the surface derivatives div Γ a, −−→ curl Γ div Γ a and
combine the densities a, div Γ a, and −−→ curl Γ div Γ a into an extended vector density ã 1 ;
2. Apply the operators n × S k+ick 1/3 and T 1
k+ick 1/3 div Γ to the corresponding components of
the extended vector density ã 1 using accelerated algorithms with complex wavenumbers k +
ick 1/3 and lumping common kernel components that enter the definitions of the operators
n × S k+ick 1/3 and T 1
k+ick 1/3 div Γ ;
3. Compute the surface derivatives div Γ (n×S k+ick 1/3a), −−→ curl Γ div Γ (n×S k+ick 1/3a) and combine
the densities a, n × S k+ick 1/3a, div Γ (n × S k+ick 1/3a), and −−→ curl Γ div Γ (n × S k+ick 1/3a) into an
extended vector density ã 2 ;
4. Apply the operators K k and T k to the corresponding components of the vector density ã 2 —
using accelerated algorithms with wavenumber k and lumping common kernel components
that enter in the definitions of K k and T k ; at the same time apply the operator n × S k to the
density T 1
k+ick 1/3 div Γ a using accelerated algorithms with wavenumber k;
5. Perform a linear combination of the results obtained in the previous step according to the
formula (90).
Solutions of the linear systems arising from the discretization of the boundary integral equations
under consideration, namely those based on the boundary integral operators A k,Sik/2 , B k,2,k,0.4H 2/3 k 1/3,
and C k defined in equations (89), (32), and (91) are obtained by means of the fully complex version of
the iterative solver GMRES [47] without restart. All of the results contained in the tables presented
in this section were obtained by prescribing a GMRES residual tolerance equal to 10 −4 . We present
results for three scattering surfaces: a sphere of radius one, an elongated ellipsoid of principal axes
2, 0.5 and 0.5, and an ellipsoid of principal axes 2, 0.5, and 2. We consider scattering problems involving
wavenumbers that correspond to scatterers whose diameters D correspond to 10.2λ, 20.4λ,
30.6λ, 40.8λ, and 51.0λ respectively. All of the results contained in the tables presented in this
section were obtained by using discretizations corresponding to 6 points/wavelength, leading to discretizations
of size N. Given the coercivity results established in Section 3 for spherical scatterers,
and which are probably true for smooth strictly convex scatterers (see reference [17] for numerical
evidence of similar claims in the scalar case), discretizations corresponding to 6 points/wavelength
seem reasonable. For every scattering experiment we present the maximum relative error amongst
all directions ˆx = x
|x|
of the far field E ∞ (ˆx):
E(x) = eik|x|
|x|
(
( )) 1
E ∞ (ˆx) + O
|x|
35
as |x| → ∞. (92)

The maximum relative far-field error, which we denote by ε ∞ ,
ε ∞ = maxˆx |E calc
∞ (ˆx) − E ref
∞ (ˆx)|
maxˆx |E ref , (93)
∞ (ˆx)|
was evaluated in our numerical examples as the maximum difference evaluated at sufficiently many
points between far fields E calc
∞ obtained from our numerical solutions and corresponding far fields
E ref
∞ associated with reference solutions. The reference solutions E ref
∞ were computed by Mie series
in the case of spherical scatterers and by use of Combined Field Integral Equations based on
the boundary integral operator C k defined in equation (91) and the same levels of discretization.
We used accelerator parameters that lead to small computational times and memory usage, while
delivering results with about three digits of accuracy in the far-field metrics ε ∞ . Specifically, in all
our numerical experiments we have used the following three accelerator parameter values: M eq = 6,
n coll = 10, and n w = 6. The values of the remaining accelerator parameter were chosen as follows:
L = 6 for scattering problems with D = 10.2λ; L = 12 for scattering problems with D = 20.4λ;
L = 16 for scattering problems with D = 30.6λ; L = 24 for scattering problems with D = 40.8λ; and
L = 30 for scattering problems with D = 51.0λ. The computational times and the memory usage
reported resulted from a C++ numerical implementation of our various accelerated algorithms on
a workstation with 16 cores, 24 GB RAM and each processor is 2.27 GHz Intel (R) Xeon. All of the
times reported correspond to runs performed on a single processor running GNU/Linux, and using
the GNU/gcc compiler, the PETSC 3.0 library for the fully complex implementation of GMRES,
and the FFTW3 library for evaluation of FFTs.
We illustrate in Tables 1-3 the performance of our accelerated solvers based on formulations
that involve the boundary integral operators C k , A k,Sik/2 , and B k,2,k,0.4H 2/3 k1/3. In all the numerical
experiments presented in those tables we assumed x-polarized plane wave normal incidence. As it
can be seen from the results presented in Tables 1-3, the numbers of GMRES iterations associated
with solvers based on the operators C k and A k,Sik/2 grow with the frequency, which is an accord with
the theoretical predictions for spherical scatterers presented in Section 3 for spherical scatterers.
In contrast, solvers based on the operators B k,2,k,0.4H 2/3 k 1/3 appear to require numbers of GMRES
iterations that do not depend on the frequency for strictly convex scatterers. Thus, solvers based
on the operators B k,2,k,0.4H 2/3 k1/3 should eventually outperform in computing times those based on
the operators C k and A k,Sik/2 . Specifically, in the high-frequency range presented in Tables 1-3,
the solvers based on the operators B k,2,k,0.4H 2/3 k 1/3 outperform the solvers based on the classical
CFIE operators C k in each numerical experiment performed, and the gains provided by the use
of the former formulations over those using the latter formulations become more significant with
increased problem size. These gains can be up to factors of 3.3 for problems of 51 wavelengths in
electromagnetic size. Also, solvers based on the operators A k,Sik/2 consistently outperform those
based on the classical CFIE operators C k in terms of computational times, and the gains can be up
to factors of 2.6 for high-frequencies. Furthermore, solvers based on the operators B k,2,k,0.4H 2/3 k 1/3
outperform those based on the operators A k,Sik/2 in terms of computational times, in some cases for
all frequencies considered (e.g. Table 1 and Table 3), and in the other case for higher frequencies
(e.g. Table 2). Given that the levels of accuracy reached by our solvers seems to be commensurate
for all the formulations considered, formulations based on the operators B k,2,k,0.4H 2/3 k 1/3 appear to
be very suitable for high-frequency simulations.
We illustrate in Table 4 statistics of the memory requirements and costs of one matrix-vector
product associated to solvers based on formulations that involve the boundary integral operators C k ,
A k,Sik/2 , and B k,2,k,0.4H 2/3 k1/3. As it can be seen, the cost of one matrix-vector product of all of these
36

D N C k A k,Sik/2 B k,2,k,0.4H 2/3 k 1/3
It/ Total time ɛ ∞ It/ Total time ɛ ∞ It/ Total time ɛ ∞
10.2 λ 47,628 37/0.38 h 8.0 × 10 −4 16/0.25 h 1.0 × 10 −3 8/0.18 h 1.0 × 10 −3
20.4 λ 193,548 43/2.48 h 2.5 × 10 −3 21/1.89 h 3.0 × 10 −3 8/1.09 h 1.1 × 10 −3
30.6 λ 437,772 48/7.71 h 1.9 × 10 −3 25/6.60 h 2.7 × 10 −3 8/3.17 h 2.4 × 10 −3
40.8 λ 780,300 54/17.6 h 3.0 × 10 −3 29/15.0 h 3.8 × 10 −3 8/6.34 h 2.6 × 10 −3
51.0 λ 1,175,628 58/35.8 h 4.0 × 10 −3 32/33.4 h 4.5 × 10 −3 8/12.1 h 3.1 × 10 −3
Table 1: Performance of our solvers based on formulations that involve the boundary integral
operators C k , A k,Sik/2 , and B k,2,k,0.4H 2/3 k1/3. Accelerated computations for spheres of diameters D,
x-polarized plane wave normal incidence.
D N C k A k,Sik/2 B k,2,k,0.4H 2/3 k 1/3
It/ Total time It/ Total time ɛ ∞ It/ Total time ɛ ∞
10.2 λ 47,628 69/0.70 h 14/0.21 h 6.2 × 10 −4 11/0.25 h 7.6 × 10 −4
20.4 λ 193,548 75/4.32 h 15/1.35 h 9.3 × 10 −3 11/1.48 h 1.0 × 10 −3
30.6 λ 437,772 80/12.8 h 16/4.22 h 6.7 × 10 −4 11/4.33 h 7.7 × 10 −4
40.8 λ 780,300 86/28.0 h 18/9.30 h 8.6 × 10 −4 11/8.72 h 9.2 × 10 −4
51.0 λ 1,175,628 89/54.9 h 20/20.8 h 1.3 × 10 −3 11/16.6 h 1.1 × 10 −3
Table 2: Performance of our solvers based on formulations that involve the boundary integral
operators C k , A k,Sik/2 , and B k,2,k,0.4H 2/3 k1/3. Accelerated computations for ellipsoids of size D ×
D/4 × D/4, x-polarized plane wave normal incidence.
D N C k A k,Sik/2 B k,2,k,0.4H 2/3 k 1/3
It/ Total time It/ Total time ɛ ∞ It/ Total time ɛ ∞
10.2 λ 47,628 68/0.69 h 19/0.29 h 6.8 × 10 −4 11/0.25 h 1.1 × 10 −3
20.4 λ 193,548 73/4.21 h 20/1.80 h 6.9 × 10 −4 11/1.48 h 9.5 × 10 −4
30.6 λ 437,772 76/12.16 h 21/5.54 h 6.6 × 10 −4 11/4.33 h 7.0 × 10 −4
40.8 λ 780,300 78/25.39 h 22/11.3 h 7.5 × 10 −4 10/7.93 h 9.7 × 10 −4
51.0 λ 1,175,628 79/48.7 h 23/24.0 h 7.9 × 10 −4 10/15.1 h 8.8 × 10 −4
Table 3: Performance of our solvers based on formulations that involve the boundary integral
operators C k , A k,Sik/2 , and B k,2,k,0.4H 2/3 k1/3. Accelerated computations for ellipsoids of size D ×
D/4 × D, x-polarized plane wave normal incidence.
37

D N C k A k,Sik/2 B k,2,k,0.4H 2/3 k 1/3
Mem Time/It Mem Time/It Mem Time/It
10.2 λ 47,628 0.1 Gb 0.61 m 0.16 Gb 0.93 m 0.22 Gb 1.38 m
20.4 λ 193,548 0.4 Gb 3.46 m 0.64 Gb 5.40 m 0.88 Gb 8.10 m
30.6 λ 437,772 0.9 Gb 9.63 m 1.44 Gb 15.84 m 1.98 Gb 23.64 m
40.8 λ 780,300 1.6 Gb 19.55 m 2.5 Gb 31.03 m 3.5 Gb 47.58 m
51.0 λ 1,175,628 2.5 Gb 37.03 m 3.9 Gb 62.62 m 5.5 Gb 90.60 m
Table 4: Statistics of the memory requirements and costs of one matrix-vector product associated
to solvers based on formulations that involve the boundary integral operators C k , A k,Sik/2 , and
B k,2,k,0.4H 2/3 k 1/3. The cost of one matrix-vector product of all of these formulations is O(N 4/3 log N).
formulations is O(N 4/3 log N). The computational times required by one matrix-vector product for
the cases of the sphere and the ellipsoid are virtually identical given that the parametrizations of
these scatterers was constructed identically. Two more conclusions can be drawn from the results
in Table 4. A matrix-vector product resulted from discretization of the formulations based on the
operators A k,Sik/2 is on average at most 1.6 more computationally expensive than the matrix-vector
product for the CFIE formulation based on the operators C k at the same level of discretization.
A matrix-vector product resulted from discretization of the formulations based on the operators
B k,2,k,0.4H 2/3 k 1/3 is on average at most 2.5 more computationally expensive than the matrix-vector
product for the CFIE formulation based on the operators C k at the same level of discretization.
We conclude this section with an illustration in Figure 6 of the iteration counts required by our
solvers based on the integral operators A k,Sik/2 and B k,2,k,0.4H 2/3 k1/3 respectively to reach GMRES
residuals of 10 −4 for two scatterers, namely a unit sphere and an ellipsoid with principal axes 2,
0.5, and 2, and 121 wavenumbers k = 8, 9, . . . , 127, 128. The corresponding electromagnetic sizes
of the scattering problems range from 2.5 to 40.8 wavelengths. As it can be seen, the number of
GMRES iterations required by our solvers based on the operators B k,2,k,0.4H 2/3 k1/3 are independent
of frequency. In the case of the spherical scatterers, the runtimes of our solvers based on the
operators B k,2,k,0.4H 2/3 k 1/3 are always smaller than those based on the operators A k,S ik/2
. For
ellipsoid scatterers, the runtimes of our solvers based on the operators B k,2,k,0.4H 2/3 k 1/3 are smaller
than those based on the operators A k,Sik/2 for scattering problems corresponding to wavenumbers
k that are greater than equal to the value k c = 98.
5 Conclusions
We presented several versions of Regularized Combined Field Integral Equations formulations for
the solution of electromagnetic scattering equations with PEC boundary conditions. These formulations
consist of combined field representations where the electric field integral operators act
on certain regularizing operators. The construction of the regularizing operators, in turn, is based
on Calderón’s calculus and complexification techniques. These integral equation formulations are
well conditioned on account of the choice of the regularizing operators. Some of the resulting integral
operators possess excellent eigenvalues clustering properties, which translate in the case of
strictly convex scatterers into very small numbers of iterations necessary to obtain the solution of
the ensuing linear systems, regardless of the discretization size and the frequency of the scattering
38

30
25
20
15
10
5
20 40 60 80 100 120
20
19
18
17
16
15
14
13
12
11
10
9
20 40 60 80 100 120
Figure 6: Plot of the numbers of iterations required by our solvers based on the operators A k,Sik/2
(black) and B k,2,k,0.4H 2/3 k 1/3 (red) as a function of the wavenumber k for 121 values of k ranging
from k = 8 to k = 128 for a unit sphere scatterer (top) and an ellipsoid of of principal axes 2,
0.5, and 2 (bottom). The vertical green line in the figure on the bottom represent the cut-off
frequency k c = 98: for values of the wavenumber larger than k c our solvers based on the operators
B k,2,k,0.4H 2/3 k 1/3 require less computational times than our solvers based on the operators A k,S ik/2
.
39

problems. In addition, the far field errors incurred by our implementation are of the same order as
those resulting from the classical Combined Field Integral Equation counterparts. Thus, the important
gains in computational complexity make the Regularized Combined Field Integral Equations
discussed in this text a viable method of solution to PEC scattering problems.
Acknowledgments
Yassine Boubendir gratefully acknowledge support from NSF through contract DMS-1016405.
Catalin Turc gratefully acknowledge support from NSF through contract DMS-1312169.
6 Appendix
We present in this section a proof of the technical Lemma 3.2
Lemma 6.1 There exist constants C j > 0, j = 1, . . . , 4 and a number ˜k 0 > 0 such that
(i) C 1 k −2 (n 2 + k 2 ) 1/2 ≤ iJ ′ n+1/2 (ik)(H(1) n+1/2 )′ (ik) ≤ C 2 k −2 (n 2 + k 2 ) 1/2
(ii) 1 4 (n2 + k 2 ) −1/2 ≤ −S (1)
n (ik) ≤ C 3 (n 2 + k 2 ) −1/2 + C 4 k −2 ,
(iii)|J ′ n+1/2 (ik)H(1) n+1/2 (ik)| ≤ C 4k −1 , |J n+1/2 (ik)(H (1)
n+1/2 )′ (ik)| ≤ C 4 k −1
for all k > ˜k 0 and all n ≥ 0.
Proof. (i) We begin by using the representation of the functions J n+1/2 (ik) and H (1)
terms of the Bessel and Hankel functions of the third kind
iJ ′ n+1/2 (ik)(H(1) n+1/2 )′ (ik) = − 2 π I′ n+1/2 (k)K′ n+1/2 (k)
n+1/2
(ik) in
where I ′ ν(k) > 0 and K ′ ν(k) < 0 for ν ≥ 0. Using the uniform asymptotic expansions as ν → ∞
(Formulas 9.7.9 and 9.7.10 in [1])
where µ = √ 1 + z 2 + ln
I ν(νz)
′ 1 (1 + z 2 ) 1 4
∼ √ e νµ (1 + O(ν −1 ))
2πν z
√ π
K ν(νz) ′ (1 + z 2 ) 1 4
∼ −√ e −νµ (1 + O(ν −1 )) (94)
2ν z
z
1+ √ 1+z 2 , we get that there exists a constant N 0 > 0 such that
1
4 k−2 ((n + 1/2) 2 + k 2 ) 1/2 ≤ −I ′ n+1/2 (k)K′ n+1/2 (k) ≤ k−2 ((n + 1/2) 2 + k 2 ) 1/2 ,
for all n > N 0 , k > 0. Given that we can choose the constant N 0 above to further satify (n +
1/2) 2 + k 2 < 2(n 2 + k 2 ) for all n > N 0 and all k > 0, we obtain the following estimate
1
4 k−2 (n 2 + k 2 ) 1/2 ≤ −I ′ n+1/2 (k)K′ n+1/2 (k) ≤ √ 2k −2 (n 2 + k 2 ) 1/2 , for all n > N 0 , k > 0. (95)
40

Using the asymptotic expansion (Formula 9.7.6 in [1]) which is valid for ν fixed and z → ∞
−I ν(z)K ′ ν(z) ′ ∼ 1 (
1 + 1 ( ))
µ − 3 µ
2
2z 2 (2z) 2 + O z 4 , µ = 4ν 2 (96)
we get that for a fixed n there exists k ′ n such that
1
4k ≤ −I′ n+1/2 (k)K′ n+1/2 (k) ≤ 1 k , for all k ≥ k′ n. (97)
It follows from the estimate (97) that there exist k 1 = max n≤N0 k n ′ and c = (1 + N0 2/k2 1 )− 1 2 such
that
1
4 ck−2 (n 2 + k 2 ) 1/2 ≤ 1
4k ≤ −I′ n+1/2 (k)K′ n+1/2 (k) ≤ 1 k ≤ √ 2k −2 (n 2 + k 2 ) 1/2 , 0 ≤ n ≤ N 0 , k ≥ k 1 .
We obtain by combining estimates (95) and (98) that there exist constants C 1 = 1
2π
N0 2/k2 1 )− 1 2 ) and C 2 = 2√ 2
π
such that
(98)
max(1, (1 +
C 1 k −2 (n 2 +k 2 ) 1/2 ≤ iJ ′ n+1/2 (ik)(H(1) n+1/2 )′ (ik) = − 2 π I′ n+1/2 (k)K′ n+1/2 (k) ≤ C 2k −2 (n 2 +k 2 ) 1/2 (99)
for all n ≥ 0 and all k ≥ k 1 .
(ii) In order to prove the estimate concerning S (1)
n (ik), we use the identities (formulas 9.1.27
in [1])
J
n+1/2 ′ m + 1/2
(K) =
K J n+1/2(K) − J n+3/2 (K), (H (1)
n+1/2 )′ (K) = n + 1/2
K
H(1) n+1/2
(K) − H(1)
n+3/2 (K)
to derive
−S n (1)
π
(ik) =
2ik 2 [(2n2 + 3n + 1)J n+1/2 (ik)H (1)
n+1/2 (ik) − k2 J n+3/2 (ik)H (1)
n+3/2 (ik)
− ik(n + 1)(J n+1/2 (ik)H (1)
n+3/2 (ik) + J n+3/2(ik)H (1)
n+1/2 (ik))].
We use Bessel and Hankel functions of the third kind K n+1/2 and I n+1/2 together with the Wronskian
identity I ν (k)K ν+1 (k) + I ν+1 (k)K ν (k) = 1 k
(formula 9.6.15 in [1]) to reexpress the previous
identity in the form
−S (1)
n (ik) = − 1 k 2 [(2n2 + 3n + 1)I n+1/2 (k)K n+1/2 (k) − k 2 I n+3/2 (k)K n+3/2 (k)
+ k(n + 1)(2I n+3/2 (k)K n+1/2 (k) − 1/k)].
Using the fact that I
n+1/2 ′ n+1/2
(k) =
k
I n+1/2 (k) + I n+3/2 (k) ( (formulas 9.6.26 in [1])) we get
−S (1)
n (ik) = n + 1
k 2 (2kI ′ n+1/2 (k)K n+1/2(k) − 1) + I n+3/2 (k)K n+3/2 (k). (100)
Using the uniform asymptotic expansions as ν → ∞ (Formulas 9.7.8 and 9.7.9 in [1])
√ π e −νµ
K ν (νz) ∼ √ (1 + O(ν −1 ))
2ν (1 + z 2 ) 1 4
I ′ ν(νz)
∼
1 (1 + z 2 ) 1 4
√ e νµ (1 + O(ν −1 )) (101)
2πν z
41

where µ = √ 1 + z 2 z
+ ln
1+ √ , we get that there exist constants N 1+z 2 1 > 0 and c 1 > 0 such that
1
2 c 1k −2 ≤ n + 1
k 2 (2kI
n+1/2 ′ (k)K n+1/2(k) − 1) ≤ c 1 k −2 , for all n > N 1 , k > 0. (102)
Using the asymptotic expansions (Formula 9.7.2 and 9.7.3 in [1]) which are valid for ν fixed and
z → ∞
√ ( π
K ν (z) ∼ √ e −z 1 + µ − 1 ( )) µ
2
+ O
2z 8z z 2
(
I ν(z)
′ 1
∼ √ e z 1 − µ + 3 ( )) µ
2
+ O
2πz 8z z 2 , µ = 4ν 2 (103)
we get that for a fixed n ≥ 0 there exists ˜k n such that
n + 1
4k 3 ≤ m + 1
k 2 (2kI ′ n+1/2 (k)K n+1/2(k) − 1) ≤ n + 1
k 3 , for all k ≥ ˜k n . (104)
We conclude that there exists a constant ˜c = max(c 1 , N 1+1
k 2
) where ˜k 2 = max 0≤n≤N1 ˜kn , such that
0 ≤ n + 1
k 2 (2kI
n+1/2 ′ (k)K n+1/2(k) − 1) ≤ ˜c
k 2 , for all k ≥ ˜k 2 , for all n ≥ 0. (105)
Furthermore, if we use the following result established in Lemma 3.1 in [17], namely there exists a
constant ˜C 1 and a number ˆk 2 > 0 such that
1
4 (n2 + k 2 ) −1/2 ≤ I n+3/2 (k)K n+3/2 (k) ≤ ˜C 1 (n 2 + k 2 ) −1/2 for all k ≥ ˆk 2 , for all n ≥ 0
together with the estimate established in equation (107), we obtain if we take into account equation
(100) that there exists a constant C 3 and a number k 2 = max{˜k 2 , ˆk 2 } such that
1
4 (n2 + k 2 ) −1/2 ≤ −S (1)
n (ik) ≤ C 3 (n 2 + k 2 ) −1/2 + C 4 k −2 for all k ≥ k 2 , for all n ≥ 0.
(iii) We get from the asymptotic expansions (101) that there exist constants N 1 > 0 and c 1 > 0
such that
1
4 k−1 ≤ I ′ n+1/2 (k)K n+1/2(k) ≤ k −1 , for all n > N 1 , k > 0. (106)
On the other hand, using the asymptotic expansions (103) we get that for a fixed n ≥ 0 there exists
˜k n such that
1
4 k−1 ≤ I ′ n+1/2 (k)K n+1/2(k) ≤ k −1 , for all k ≥ ˜k n . (107)
If we let k 3 = max 0≤n≤N1 ˜kn we obtain that
|J ′ n+1/2 (ik)H(1) n+1/2 (ik)| = 2 π I′ n+1/2 (k)K n+1/2(k) ≤ 2 π k−1 , for all k ≥ k 3 , 0 ≤ n. (108)
The estimate concerning |J n+1/2 (ik)(H (1)
n+1/2 )′ (ik)| follows form the previous estimate and the Wronskian
identity (formula 9.6.15 in [1]). Finally, if we take ˜k 0 = max{k 1 , k 2 , k 3 }, the result of the
Lemma follows.
42

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