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A duality between scattering poles and transmission eigenvalues in scattering theory.
Fioralba Cakoni1, David Colton2, Houssem Haddar3
1Department of Mathematics, Rutgers University, New Brunswick, NJ, USA.
This study introduces a unified method to find scattering poles and interior eigenvalues by exploring field duality. This approach offers new insights into scattering problems and numerical computation methods.
Area of Science:
- Mathematical physics
- Inverse problems
- Wave phenomena
Background:
- Scattering problems involve understanding how waves interact with objects.
- Characterizing scattering poles and interior eigenvalues is crucial for analyzing wave behavior.
- Existing methods often treat these properties separately.
Purpose of the Study:
- To develop a unified theoretical framework for characterizing scattering poles and interior eigenvalues.
- To explore the duality between incident and scattered fields in scattering problems.
- To propose a novel numerical method for computing scattering poles.
Main Methods:
- Utilizing a duality principle by interchanging incident and scattered fields.
- Analyzing the kernel of the relative scattering operator.
- Applying the approach to Dirichlet and inhomogeneous scattering problems.
Main Results:
- Established a conceptually unified approach for scattering poles and interior eigenvalues.
- Demonstrated duality between scattering poles and Dirichlet/transmission eigenvalues.
- Proposed a new characterization of scattering poles for numerical computation.
Conclusions:
- The unified approach simplifies the analysis of scattering problems.
- The identified duality provides deeper theoretical understanding.
- The proposed numerical method facilitates practical computation of scattering poles.
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