This paper solves Maxwell’s equations on the wave-number side. I apply tools from classical functional analysis to the Electromagnetic Helmholtz Reduced, Vector Wave Equation. These tools have been used successfully to understand solutions of second order, linear, ordinary differential equations with Dirichlet or Neumann boundary conditions, but to the author’s knowledge do not apply to more general settings.
I follow techniques described by Chew  using , and the spectral representation for . The spectral or side motivates the study of solutions in wave-number space.
To the author’s knowledge, there has not been a significant amount of research on solving electromagnetic problems on the wave-number side. This could be because students in electromagnetics do not take courses in Functional Analysis or Applied Analysis.
Electromagnetics (EM) and Quantum Mechanics (QM) usually yield very singular solutions. All EM problems are scattering problems solved using singular
Green’s functions. The Operators in both EM ( ) and QM ( ) problems
are singular. As an example, the solution of the homogeneous vector wave equation is unbounded. If we consider, with a norm that is complete w.r.t. , the solution is compact using weak derivatives.
2. The Reduced, Vector Helmholtz Wave Equation
2.1. Homogeneous Case, Radiation Condition at ∞
We derive the vector Helmholtz wave equation assuming harmonic time dependence, , from Maxwell’s curl equations, and the divergence for in a source-free region. To introduce later on a Fourier transform for , consider the domain for to be , . Then our problem is given by
is a root of a homogeneous polynomial of degree 2, with constant coefficients; i.e., the last equation in (2) is the Dispersion equation,
From the last of Equation (2), we have
From Equation (3), if , then then the homogeneous system in Equation (1) has a non-zero solution in the form of a spherical wave traveling in and satisfying
To construct a solution for , go to the Fourier side where
where is the symbol of the operator, and the plus sign in the exponential corresponds to waves traveling in the −z direction and the negative sign for waves traveling in the +z direction. The symbol of the operator will be constructed to satisfy
Parseval’s theorem on the Fourier side gives
The symbol of the operator satisfies
If , the solution for when is to the plane of propagation containing , the so-called TM case, as in Figure 1, is
Figure 1. Geometry showing plane of incident wave and plane containing perfect conductor.
2.2. Inhomogeneous Case, Dirichlet Boundary Conditions, Radiation Condition at ∞
with Dirichlet boundary conditions on the green-black plane in Figure 1,
and a radiation condition at .
Two independent vectors span the red-yellow plane in Figure 1. The basis vectors on the Fourier side, are given by
and a basis vector perpendicular to the incident plane as
Tangential satisfies the boundary condition on the surface at , i.e.,
The perfectly conducting plane in Figure 1 where , is translationally invariant in the z-direction, so the arguments for in the z-direction in the inverse transform of Equation (6) are given by
Equation (11) can be written in matrix form as
Equation (17) consists of 3 equations in 3 unknowns yielding a consistent system. On the Fourier side the EM problem for a region containing an object with induced currents, the solution reduces to solving a linear algebra problem. The inverse operator, exists if and only if
which occurs when we are “sitting on an eigenvalue” i.e., . The properties of the resolvent in the next section will demonstrate the different descriptions of the spectrum.
The inverse operator is given by
The inverse is singular when or . The eigenvalues are
where Equation (20) has algebraic multiplicity of 2 since the geometric multiplicity is
3. The Resolvent
The operator in this study is where
a complex variable. With we associate the operator
where is a complex varible. is called the Resolvent Operator (Kreyszig  ) because it solves the equation
Figure 2. Variation of versus eigenvalue, .
Table 1. The values of and .
The determinant of is a polynomial of Degree 4, with coefficients in consisting of 78 monomials. After several attempts, I was able to factor the polynomial as
The denominator is a quadratic in which factors as
Figure 2 shows a plot of the denominator versus , and as stated above , , .
We now give the 9 elements in the matrix for ;
In Figure 3, I show a plot of the resolvent versus for the parameters in Table 1. The singularity at = = −1 is part of the point spectrum, . The eigenvalues = = 0 do not produce a singularity because their algebraic is 2.
In general, the spectrum  , is partitioned into three disjoint sets:
Figure 3. The resolvent versus λ.
In our problem we have 3 eigenvalues and an unbounded spectrum.
The Fourier transform of the resolvent is
The Fourier transform of the current, is
Using the Convolution Theorem the general solution to our problem on the wave number side for arbitrary currents in the inhomogeneous equation is:
which is integrated one row at a time. The integration in the direction in required for the geometry in Figure 1.
The integrand in Equation (31) without J the Green’s function,
and the solution is
For the geometry in Figure 1, and the resolvent simplifies to
and Equation (29) becomes
The integration can be evaluated in closed form and the first row of Equation (35) becomes
with the integration of the form
4. The Resolvent as a Projector, P
The matrix , is the projector onto along . If , then and , for an arbitrary . Then
Let be a simple contour in (the spectrum) which encloses the eigenvalues , and form the integral where encloses the first r eigenvalues, as
From Equation (20), . The residue at yields,
and the residue at yields,
and is projected to .
5. Sobolev Estimates
Weak derivatives  , allow a periodic solution where the operators live in finite space. In order to introduce the Fourier series expansion for , consider the domain of , a torus, ; i.e.,
where is a Sobolev space  of dimension 2 and has 2 derivatives in . is a Hilbert space with an inner product, . In order to develop the Sobolev estimates, define the map as
Defintion: weak derivatives of order n in are derived using integration by parts.
The derivatives in Equation (40) are evaluated as:
If , then the linear functional , in , is bounded. If , then the weak derivative is f. Here is the space , and is a test function of compact support, such that
, in .
We want to show that gains 2 derivatives in the Sobolev space .
Weak Derivatives in L2 1.
gains 2 derivatives in the Sobolev space , where . (41)
#1 Proof: From the definition of a finite Sobolev space ,
where denotes the space of infinitely differentiable functions with compact support in . These functions, , are called test functions, and the derivatives are called weak derivatives. The proof is independent of the choice of . Use the following
Consider the first row of the matrix since the other 2 rows are similar, and depending on may not come into play. Differentiating under the integral sign is permissible since the integral is uniformly convergent,
Equation (43) is substituted into 41 and the norms are then computed. The norms are locally integrable. The calculation of the norms requires additional work.
The resolvent provides insight into the spectrum of the operator  . The spectrum can have discrete eigenvalues and a continuous part. The discrete part corresponds to the eigenvalues and if is not an eigenvalue, the Green’s function can be expanded in an eigenfunction expansion. These eigenfunctions corresponding to nonzero eigenvalues form a complete set.
I have shown that solving the vector Helmholtz equation on the wave-number side yields a relatively simple approach to the spectra and . We demonstrated this approach by example and simple mathematical expressions for the resolvent. The Sobolev approach using a finite series expansion for is complex mathematically and requires as much work as the Green’s function approach for . Also, the singularities are present in the Sobolev approach as in the Green’s function. EM and QM problems yield highly singular solutions. This is just the nature of these disciplines.
Future research should examine numerical evaluations of the norms using any , used in the Sololev estimates.
An example testing function is
Also, future work could involve the study of the resolvent as a function of the given wave-number .