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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2506.17426 |
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| _version_ | 1866912443132805120 |
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| author | Derkach, Alexey Sobolev, Alexander V. |
| author_facet | Derkach, Alexey Sobolev, Alexander V. |
| contents | We study discrete spectrum of self-adjoint Weyl pseudodifferential operators with discontinuous symbols of the form $1_Ωϕ$ where $1_Ω$ is the indicator of a domain in $Ω\subset\mathbb R^2$, and $ϕ\in C^\infty_0(\mathbb R^2)$ is a real-valued function. It was known that in general, the singular values $s_k$ of such an operator satisfy the bound $s_k = O(k^{-3/4})$, $k = 1, 2, \dots$. We show that if $Ω$ is a polygon, the singular values decrease as $O(k^{-1}\log k)$. In the case where $Ω$ is a sector, we obtain an asymptotic formula which confirms the sharpness of the above bound.
Our main technical tool is the reduction to another symbol that we call \textit{dual}, which is automatically smooth. To analyse the dual symbol we find new bounds for singular values of pseudodifferential operators with smooth symbols in $L^2(\mathbb R^d)$ for arbitrary dimension $d\ge 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17426 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral asymptotics of pseudodifferential operators with discontinuous symbols Derkach, Alexey Sobolev, Alexander V. Analysis of PDEs Mathematical Physics Spectral Theory Primary 47G30, Secondary 35P20, 81S30 We study discrete spectrum of self-adjoint Weyl pseudodifferential operators with discontinuous symbols of the form $1_Ωϕ$ where $1_Ω$ is the indicator of a domain in $Ω\subset\mathbb R^2$, and $ϕ\in C^\infty_0(\mathbb R^2)$ is a real-valued function. It was known that in general, the singular values $s_k$ of such an operator satisfy the bound $s_k = O(k^{-3/4})$, $k = 1, 2, \dots$. We show that if $Ω$ is a polygon, the singular values decrease as $O(k^{-1}\log k)$. In the case where $Ω$ is a sector, we obtain an asymptotic formula which confirms the sharpness of the above bound. Our main technical tool is the reduction to another symbol that we call \textit{dual}, which is automatically smooth. To analyse the dual symbol we find new bounds for singular values of pseudodifferential operators with smooth symbols in $L^2(\mathbb R^d)$ for arbitrary dimension $d\ge 1$. |
| title | Spectral asymptotics of pseudodifferential operators with discontinuous symbols |
| topic | Analysis of PDEs Mathematical Physics Spectral Theory Primary 47G30, Secondary 35P20, 81S30 |
| url | https://arxiv.org/abs/2506.17426 |