Bounds for quasimodes with polynomially narrow bandwidth on surfaces of revolution
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910060341362688 |
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| author | Chabert, Ambre |
| author_facet | Chabert, Ambre |
| contents | Given a compact surface of revolution with Laplace-beltrami operator $Δ$, we consider the spectral projector $P_{λ,δ}$ on a polynomially narrow frequency interval $[λ-δ,λ+ δ]$, which is associated to the self-adjoint operator $\sqrt{-Δ}$. For a large class of surfaces of revolution, and after excluding small disks around the poles, we prove that the $L^2 \to L^{\infty}$ norm of $P_{λ,δ}$ is of order $λ^{\frac{1}{2}} δ^{\frac{1}{2}}$ up to $δ\geq λ^{-\frac{1}{32}}$. We adapt the microlocal approach introduced by Sogge for the case $δ= 1$, by using the Quantum Completely Integrable structure of surfaces of revolution introduced by Colin de Verdière. This reduces the analysis to a number of estimates of explicit oscillatory integrals, for which we introduce new quantitative tools.This is the first sharp result in the case $δ\ll 1$ beyond the case of locally symmetric surfaces (torus, sphere, arithmetic hyperbolic surfaces). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00143 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds for quasimodes with polynomially narrow bandwidth on surfaces of revolution Chabert, Ambre Spectral Theory Analysis of PDEs Classical Analysis and ODEs Given a compact surface of revolution with Laplace-beltrami operator $Δ$, we consider the spectral projector $P_{λ,δ}$ on a polynomially narrow frequency interval $[λ-δ,λ+ δ]$, which is associated to the self-adjoint operator $\sqrt{-Δ}$. For a large class of surfaces of revolution, and after excluding small disks around the poles, we prove that the $L^2 \to L^{\infty}$ norm of $P_{λ,δ}$ is of order $λ^{\frac{1}{2}} δ^{\frac{1}{2}}$ up to $δ\geq λ^{-\frac{1}{32}}$. We adapt the microlocal approach introduced by Sogge for the case $δ= 1$, by using the Quantum Completely Integrable structure of surfaces of revolution introduced by Colin de Verdière. This reduces the analysis to a number of estimates of explicit oscillatory integrals, for which we introduce new quantitative tools.This is the first sharp result in the case $δ\ll 1$ beyond the case of locally symmetric surfaces (torus, sphere, arithmetic hyperbolic surfaces). |
| title | Bounds for quasimodes with polynomially narrow bandwidth on surfaces of revolution |
| topic | Spectral Theory Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2502.00143 |