#### 2.3.1. Null Spaces of the Integral Equations for the Interior Resonance Problem

[8] All the above inhomogeneous integral equations do have nonunique solutions if the corresponding homogeneous equations have nontrivial solutions. But homogeneous equations account for the interior resonance problem, hence, the corresponding integral equations for the interior problem and their pertinent null spaces should be related to the integral equations for the exterior scattering problem.

[9] From the proper formulation of the representation theorem for the interior Dirichlet and Neumann resonance problems (a single-layer potential for the Dirichlet- and a double-layer potential for the Neumann-problem), the integral equations of the first kind

can be deduced for the physically equivalent sources (upper index D^{(i)}, N^{(i)}). It is obvious that the operators agree with the respective exterior operators, hence, if there exist nontrivial null spaces *N*() = *V*, *N*() = *U* for the interior problems, they exist for the exterior problems as well. As a matter of fact, *V* is nontrivial for all ω = ω_{j}^{D}, *j* = 1, 2, 3, …, for which the homogeneous Helmholtz-equation has nonzero solutions for *R* ∈ *V*_{c} satisfying a Dirichlet boundary condition on *S*_{c}; in 2-D electromagnetic terminology: *V* contains axial electric surface current densities on the perfectly electric conducting surface at the pertinent resonant frequencies. Interesting enough: These current densities do not contribute to the exterior Helmholtz-representation of the scattered field, hence, they do not affect the unique solvability of the exterior Dirichlet-problem (this is not recognized by *Cho* [1990]). As for *N*() = *U*, this null space is nontrivial for the Neumann resonance frequencies ω = ω_{j}^{N}, *j* = 1, 2, 3, …, and, in 2-D electromagnetics, its elements correspond to circumferential electric current densities on the perfectly electric conducting surface *S*_{c}. These current densities to neither radiate into the exterior of the scatterer.

[10] So far with the integral equations of the first kind. To investigate the nonuniqueness of the solutions of the integral equations of the second kind, we must once more formulate them for the (homogeneous) interior resonance problem. Based on the representation theorem we find

and based on the potential layers we find

for the Dirichlet double-layer and the Neumann single-layer secondary sources.

[11] Now we use physical intuition instead of a mathematical proof (which, as already mentioned, can be found in [*Colton and Kress*, 1983]) to postulate null spaces

for the representation theorem integral equations (20), (21): Interior resonances and their respective secondary sources are deduced from the homogeneous Helmholtz-equation, which has the physical Huygens-type Helmholtz-integral (5) as a solution defining physically relevant secondary sources for the respective boundary conditions. These can either be found from the integral equations of the first kind (18), (19) or from the integral equations of the second kind (20), (21), whence nontrivial null spaces (22), (23) for ω = ω_{j}^{D,N} must exist for the integral equations of the second kind.

#### 2.3.2. Null Spaces of the Integral Equations for the Exterior Scattering Problem

[12] We have already mentioned that solutions of the integral equations of the first kind for the exterior scattering problem may be contaminated by elements of the null spaces *V* and *U* of the interior resonance problem. If integral equations of the second kind are considered, these null spaces appear as well for the interior resonance problem, leaving the question of null spaces of the integral equations (8), (9) for the exterior scattering problem involving the operators and still open. At that point we have to recognize that the operator for the exterior Dirichlet scattering problem is adjoint to the operator for the interior Neumann resonance problem, and a similar “cross-wise” adjoint relationship is observed for the exterior Neumann scattering and the interior Dirichlet resonance problem. Consulting Fredholm's alternative we must conclude, that null spaces *U*′ and *V*′ exist for the representation-type second kind integral equations for the exterior Dirichlet- and Neumann-problem

which, for ω = ω_{j}^{N,D}, have the same dimension as the null spaces *U* and *V* for the respective interior problem second kind integral equations, i.e., *U* and *U*′ have the same dimension for ω = ω_{j}^{N}, and *V* and *V*′ have the same dimension for ω = ω_{j}^{D}. For all other frequencies all null spaces are trivial.

[13] Obviously, it is of interest now to investigate the physical relevance of the null spaces *V*′ and *U*′, and, in particular to answer that question, a look at the potential-layer-type integral equations of the second kind is mandatory: We find that the respective interior resonance operator for the Dirichlet-problem is adjoint to the interior resonance representation-type operator for the Dirichlet-problem, and a similar finding holds for the Neumann-problems. So, if *V* is nontrivial, *V*′ is nontrivial having the same dimension, and, if *U* is nontrivial, *U*′ is nontrivial having the same dimension. The elements of *V*′ therefore comprise mathematical secondary sources on a Dirichlet boundary related to a double-layer potential, and the elements of *U*′ comprise mathematical secondary sources on a Neumann boundary related to a single-layer potential, and, as we already pointed out for 2-D electromagnetics, these are “mathematically adjoint secondary sources” with no physical relevance [*Cho*, 1990]. Hence, the solutions of the representation-type second kind integral equations for the exterior Dirichlet and Neumann scattering problems may be contaminated by elements of the nonphysical null spaces *U*′ and *V*′ (and these contaminate the Helmholtz-representation of the scattered field as well, because they produce a nonzero field in the exterior when inserted into this representation).

[14] Since the exterior potential-layer-type operators of the second kind are adjoint to the exterior representation-type operators of the second kind — Dirichlet adjoint to Dirichlet, Neumann adjoint to Neumann — we find, on behalf of Fredholm's alternative, *U* and *V* as their pertinent null spaces, i.e., the elements of neither *U* nor *V* do contribute to the exterior scattered field, making the field representation unique. But: Also due to Fredholm's alternative, we have to recognize a solvability condition for the corresponding integral equations: The incident field must be orthogonal to the mathematical secondary sources of the interior resonance problem, i.e., the interior Neumann mathematical secondary sources as elements of *U*′ for the exterior Dirichlet problem, and the interior Dirichlet mathematical secondary sources as elements of *V*′ for the exterior Neumann problem. This seems to contradict the unconstrained unique solvability of the exterior scattering problems, but a reformulation of integral equations expressing a balanced contribution of physical and mathematical secondary sources fixes this contradiction. It should be mentioned that solvability conditions of the above type for the exterior representation-type integral equations of the second kind can be shown to hold for any incident field, and this is possibly the reason, why they are not explicitly considered in the electric engineering literature [*Cho*, 1990].