Spectral asymptotics and estimates for matrix Birman-Schwinger operators with singular measures
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911112007516160 |
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| author | Rozenblum, Grigori Tashchiyan, Grigory |
| author_facet | Rozenblum, Grigori Tashchiyan, Grigory |
| contents | We consider operators of the form $\mathbf{T}=\mathbf{A^*}(Vμ)\mathbf{A}$ in $\mathbb{R}^\mathbf{N}$, where $\mathbf{A}$ is a pseudodifferential operator of order $-l$, $μ$ is a compactly supported singular measure, order $s>0$ Ahlfors-regular, and $V$ is a weight function on the support of $μ$. The scalar type operator $\mathbf{A}$ and the weight function $V$ are supposed to be $m\times m$ matrix valued. We establish Weyl type asymptotic formulas for singular numbers and eigenvalues of $\mathbf{T}$ for $μ$ being the natural measure on a compact Lipschitz surface. For a general Ahlfors-regular measure $μ$, we prove that the previously found upper spectral estimates are order sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14517 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral asymptotics and estimates for matrix Birman-Schwinger operators with singular measures Rozenblum, Grigori Tashchiyan, Grigory Spectral Theory Functional Analysis 47G30, 47G40 We consider operators of the form $\mathbf{T}=\mathbf{A^*}(Vμ)\mathbf{A}$ in $\mathbb{R}^\mathbf{N}$, where $\mathbf{A}$ is a pseudodifferential operator of order $-l$, $μ$ is a compactly supported singular measure, order $s>0$ Ahlfors-regular, and $V$ is a weight function on the support of $μ$. The scalar type operator $\mathbf{A}$ and the weight function $V$ are supposed to be $m\times m$ matrix valued. We establish Weyl type asymptotic formulas for singular numbers and eigenvalues of $\mathbf{T}$ for $μ$ being the natural measure on a compact Lipschitz surface. For a general Ahlfors-regular measure $μ$, we prove that the previously found upper spectral estimates are order sharp. |
| title | Spectral asymptotics and estimates for matrix Birman-Schwinger operators with singular measures |
| topic | Spectral Theory Functional Analysis 47G30, 47G40 |
| url | https://arxiv.org/abs/2508.14517 |