Dynamical formulation of low-frequency scattering in two and three dimensions
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arXiv
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| Formato: | Preprint |
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2024
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| _version_ | 1866915208718450688 |
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| author | Loran, Farhang Mostafazadeh, Ali |
| author_facet | Loran, Farhang Mostafazadeh, Ali |
| contents | The transfer matrix of scattering theory in one dimension can be expressed in terms of the time-evolution operator for an effective non-unitary quantum system. In particular, it admits a Dyson series expansion which turns out to facilitate the construction of the low-frequency series expansion of the scattering data. In two and three dimensions, there is a similar formulation of stationary scattering where the scattering properties of the scatterer are extracted from the evolution operator for a corresponding effective quantum system. We explore the utility of this approach to scattering theory in the study of the scattering of low-frequency time-harmonic scalar waves, $e^{-iωt}ψ(\mathbf{r})$, with $ψ(\mathbf{r})$ satisfying the Helmholtz equation, $[\nabla^2+k^2\hat\varepsilon(\mathbf{r};k)]ψ(\mathbf{r})=0$, $ω$ and $k$ being respectively the angular frequency and wavenumber of the incident wave, and $\hat\varepsilon(\mathbf{r};k)$ denoting the relative permittivity of the carrier medium which in general takes complex values. We obtain explicit formulas for low-frequency scattering amplitude, examine their effectiveness in the study of a class of exactly solvable scattering problems, and outline their application in devising a low-frequency cloaking scheme. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16906 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamical formulation of low-frequency scattering in two and three dimensions Loran, Farhang Mostafazadeh, Ali Quantum Physics Mathematical Physics Optics The transfer matrix of scattering theory in one dimension can be expressed in terms of the time-evolution operator for an effective non-unitary quantum system. In particular, it admits a Dyson series expansion which turns out to facilitate the construction of the low-frequency series expansion of the scattering data. In two and three dimensions, there is a similar formulation of stationary scattering where the scattering properties of the scatterer are extracted from the evolution operator for a corresponding effective quantum system. We explore the utility of this approach to scattering theory in the study of the scattering of low-frequency time-harmonic scalar waves, $e^{-iωt}ψ(\mathbf{r})$, with $ψ(\mathbf{r})$ satisfying the Helmholtz equation, $[\nabla^2+k^2\hat\varepsilon(\mathbf{r};k)]ψ(\mathbf{r})=0$, $ω$ and $k$ being respectively the angular frequency and wavenumber of the incident wave, and $\hat\varepsilon(\mathbf{r};k)$ denoting the relative permittivity of the carrier medium which in general takes complex values. We obtain explicit formulas for low-frequency scattering amplitude, examine their effectiveness in the study of a class of exactly solvable scattering problems, and outline their application in devising a low-frequency cloaking scheme. |
| title | Dynamical formulation of low-frequency scattering in two and three dimensions |
| topic | Quantum Physics Mathematical Physics Optics |
| url | https://arxiv.org/abs/2410.16906 |