Modified transmission eigenvalues in inverse scattering theory

Abstract

A recent problem of interest in inverse problems has been the study of eigenvalue problems arising from scattering theory and their potential use as target signatures in nondestructive testing of materials. Towards this pursuit we introduce a new eigenvalue problem related to Maxwell's equations that is generated from a comparison of measured scattering data to that of a non-standard auxiliary scattering problem. This choice of auxiliary problem permits the application of regularity results for Maxwell's equations in order to show that a related interior transmission problem possesses the Fredholm property, which is used to establish that the eigenvalues are discrete. We investigate the properties of this new class of eigenvalues and show that the eigenvalues may be determined from measured scattering data, concluding with a simple demonstration of this result.

New sets of eigenvalues in inverse scattering for inhomogeneous media and their determination from scattering data

“…Remark 1. We observe that the operator T given by (11) is coercive if k is not an transmission eigenvalue for (5) and a fixed sign assumption is made on the coefficients A − I and n − 1 in a neighborhood of the boundary of ∂D (see e.g Theorem 2.42 in [5]). We also indicate that for more complex configurations, e.g.…”

“…We shall consider two different possibilities for the construction of F λ b , the one introduced in [6] in the isotropic case leading to the so-called Steklov eignevalue problem, and the another one based on the artificial scattering problem for inhomogeneous metamaterial media. The latter is related to the one discussed in [11], but here we use different sign combination for the parameters. Considering a metamaterial artificial background leads to an eigenvalue problem that has similar structure as the Steklov eignevalue problem.…”

5. Eigenvalue problems in inverse electromagnetic scattering theory

“…This domain has been used for numerical testing of Stekloff eigenvalues and modified transmission eigenvalues previously (cf. [3] and [8], respectively), and we see that the shift in the eigenvalues due to a circular flaw located at (x c , y c ) = (0.1, 0.4) of radius r c = 0.05 is much more pronounced for modified transmission eigenvalues than Stekloff eigenvalues. It should be noted that for the case of Stekloff eigenvalues there exist peaks in the GLSM indicator corresponding to some of the other exact eigenvalues shown, but the height of these peaks is considerably less than the one visible.…”

“…It should be noted that for the case of Stekloff eigenvalues there exist peaks in the GLSM indicator corresponding to some of the other exact eigenvalues shown, but the height of these peaks is considerably less than the one visible. We remark that the modified transmission eigenvalues correspond to the choice γ = 0.5 in [8] and that instead using γ = 2 produces poor results.…”

Inside–outside duality with artificial backgrounds

“…Some monotonicity results have been obtained in [10] but only for some of the TEs (see also [7] for other results related to transmission eigenvalues). Recently, the idea of using an artificial background for which the associated TEs have a more direct connection with the material index of the inclusion has been introduced [8,2,13,3].…”