| Abstract: |
| The spectral properties of operators associated with scattering phenomena carry critical information about the scattering media. The theory of scattering resonances is a rich and elegant area of scattering theory. Although resonances are inherently dynamic in nature, they can be mathematically formulated as the poles of the meromorphic extension of the scattering operator. Scattering poles, which are complex with a negative imaginary part, encapsulate physical information: the real part of a pole corresponds to the rate of oscillation, while the imaginary part reflects the rate of decay. At a scattering pole, a non-zero scattered field exists even in the absence of an incident field. On the other side of this characterization, one could ask if there are frequencies for which an incident field does not scatter from the scattering object. For inhomogeneous media, this question leads to the concept of transmission eigenvalues, or interior eigenvalues in the case of obstacles.
In this talk, we present a conceptually unified approach for characterizing and determining scattering poles and transmission eigenvalues for the scattering problem for inhomogeneous media. Our approach explores a duality that arises by interchanging the roles of incident and scattered fields. Both sets -the scattering poles and transmission eigenvalues- are connected to the kernel of the relative scattering operator, which maps incident fields to scattered fields. This operator corresponds to the exterior scattering problem for transmission eigenvalues and the interior scattering problem for scattering poles. We will conclude with numerical examples for the scattering by an obstacle as a proof of concept. |
|