Helmholtz quasi-resonances are unstable under most single-signed perturbations of the wave speed
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911967135924224 |
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| author | Spence, Euan A. Wunsch, Jared Zou, Yuzhou |
| author_facet | Spence, Euan A. Wunsch, Jared Zou, Yuzhou |
| contents | We consider Helmholtz problems with a perturbed wave speed, where the single-signed perturbation is linear in a parameter $z$. Both the wave speed and the perturbation are allowed to be discontinuous (modelling a penetrable obstacle). We show that there exists a polynomial function of frequency such that, for any frequency, for most values of $z$, the norm of the solution operator is bounded by that function.
This solution-operator bound is most interesting for Helmholtz problems with strong trapping; recall that here there exists a sequence of real frequencies, tending to infinity, through which the solution operator grows superalgebraically, with these frequencies often called $\textit{quasi-resonances}$. The result of this paper then shows that, at every fixed frequency in the quasi-resonance, the norm of the solution operator becomes much smaller for most single-signed perturbations of the wave speed, i.e., quasi-resonances are unstable under most such perturbations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00843 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Helmholtz quasi-resonances are unstable under most single-signed perturbations of the wave speed Spence, Euan A. Wunsch, Jared Zou, Yuzhou Analysis of PDEs We consider Helmholtz problems with a perturbed wave speed, where the single-signed perturbation is linear in a parameter $z$. Both the wave speed and the perturbation are allowed to be discontinuous (modelling a penetrable obstacle). We show that there exists a polynomial function of frequency such that, for any frequency, for most values of $z$, the norm of the solution operator is bounded by that function. This solution-operator bound is most interesting for Helmholtz problems with strong trapping; recall that here there exists a sequence of real frequencies, tending to infinity, through which the solution operator grows superalgebraically, with these frequencies often called $\textit{quasi-resonances}$. The result of this paper then shows that, at every fixed frequency in the quasi-resonance, the norm of the solution operator becomes much smaller for most single-signed perturbations of the wave speed, i.e., quasi-resonances are unstable under most such perturbations. |
| title | Helmholtz quasi-resonances are unstable under most single-signed perturbations of the wave speed |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2402.00843 |