A short proof of a strong Weyl law in dimension 1
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910358372876288 |
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| author | Bjerg, August |
| author_facet | Bjerg, August |
| contents | For the Dirichlet realization of $-d^2/dx^2-λ^2V$ on a bounded interval, with $V$ a positive $C^2$ potential bounded away from $0$ and $λ>0$ a large parameter, we prove an asymptotic law for the values $λ_n$ of $λ$ at the $n^{\text{th}}$ appearance of a new negative eigenvalue. This approximation is correct up to an error of order $1/n$, thus making the result strictly stronger than the classical Weyl law for the number of negative eigenvalues for these operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_05137 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A short proof of a strong Weyl law in dimension 1 Bjerg, August Mathematical Physics Spectral Theory 34L20 (Primary) 34L40, 81Q20 (Secondary) For the Dirichlet realization of $-d^2/dx^2-λ^2V$ on a bounded interval, with $V$ a positive $C^2$ potential bounded away from $0$ and $λ>0$ a large parameter, we prove an asymptotic law for the values $λ_n$ of $λ$ at the $n^{\text{th}}$ appearance of a new negative eigenvalue. This approximation is correct up to an error of order $1/n$, thus making the result strictly stronger than the classical Weyl law for the number of negative eigenvalues for these operators. |
| title | A short proof of a strong Weyl law in dimension 1 |
| topic | Mathematical Physics Spectral Theory 34L20 (Primary) 34L40, 81Q20 (Secondary) |
| url | https://arxiv.org/abs/2403.05137 |