تقرير
Embedded eigenvalues of the Neumann problem in a strip with a box-shaped perturbation
| العنوان: | Embedded eigenvalues of the Neumann problem in a strip with a box-shaped perturbation |
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| المؤلفون: | Cardone, G., Durante, T., Nazarov, S. A. |
| المصدر: | Journal de Math\'ematiques Pures Appliqu\'ees, 112 (2018), 1-40 |
| سنة النشر: | 2015 |
| المجموعة: | Mathematics Mathematical Physics |
| مصطلحات موضوعية: | Mathematics - Spectral Theory, Mathematical Physics, Mathematics - Analysis of PDEs, 35P05, 47A75, 49R50, 78A50 |
| الوصف: | We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide $\Pi_{l}^{\varepsilon}$ obtained from a straight unit strip by a low box-shaped perturbation of size $2l\times\varepsilon,$ where $\varepsilon>0$ is a small parameter. We prove the existence of the length parameter $l_{k}^{\varepsilon}=\pi k+O\left( \varepsilon\right) $ with any $k=1,2,3,...$ such that the waveguide $\Pi_{l_{k}^{\varepsilon}}^{\varepsilon }$ supports a trapped mode with an eigenvalue $\lambda_{k}^{\varepsilon}% =\pi^{2}-4\pi^{4}l^{2}\varepsilon^{2}+O\left( \varepsilon^{3}\right) $ embedded into the continuous spectrum. This eigenvalue is unique in the segment $\left[ 0,\pi^{2}\right] $ and is absent in the case $l\neq l_{k}^{\varepsilon}.$ The detection of this embedded eigenvalue is based on a criterion for trapped modes involving an artificial object, the augmented scattering matrix. The main technical difficulty is caused by corner points of the perturbed wall $\partial\Pi_{l}^{\varepsilon}$ and we discuss available generalizations for other piecewise smooth boundaries. Comment: 36 pages, 6 figures |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1016/j.matpur.2018.01.002 |
| URL الوصول: | http://arxiv.org/abs/1512.06891 |
| رقم الأكسشن: | edsarx.1512.06891 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.1016/j.matpur.2018.01.002 |
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