Weyl laws for exponentially small singular values of the $\overline{\partial}$ operator
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910969488211968 |
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| author | Hitrik, Michael Sjöstrand, Johannes Vogel, Martin |
| author_facet | Hitrik, Michael Sjöstrand, Johannes Vogel, Martin |
| contents | We study the number of exponentially small singular values of the semiclassical $\overline{\partial}$ operator on exponentially weighted $L^2$ spaces on the two-dimensional torus. Accurate upper and lower bounds on the number of such singular values are established with the help of auxiliary notions of upper and lower bound weights. Assuming that the Laplacian of the exponential weight changes sign along a curve, we construct optimal such weights by solving a free boundary problem, which yields a Weyl asymptotics for the counting function of the singular values in an interval of the form $[0,\mathrm{e}^{-τ/h}]$, for $τ>0$ smaller than the oscillation of the weight. We also provide a precise description of the leading term in the Weyl asymptotics, in the regime of small $τ> 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07292 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weyl laws for exponentially small singular values of the $\overline{\partial}$ operator Hitrik, Michael Sjöstrand, Johannes Vogel, Martin Spectral Theory Analysis of PDEs Complex Variables We study the number of exponentially small singular values of the semiclassical $\overline{\partial}$ operator on exponentially weighted $L^2$ spaces on the two-dimensional torus. Accurate upper and lower bounds on the number of such singular values are established with the help of auxiliary notions of upper and lower bound weights. Assuming that the Laplacian of the exponential weight changes sign along a curve, we construct optimal such weights by solving a free boundary problem, which yields a Weyl asymptotics for the counting function of the singular values in an interval of the form $[0,\mathrm{e}^{-τ/h}]$, for $τ>0$ smaller than the oscillation of the weight. We also provide a precise description of the leading term in the Weyl asymptotics, in the regime of small $τ> 0$. |
| title | Weyl laws for exponentially small singular values of the $\overline{\partial}$ operator |
| topic | Spectral Theory Analysis of PDEs Complex Variables |
| url | https://arxiv.org/abs/2505.07292 |