Asymptotics for the spectral function on Zoll manifolds
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arXiv
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2022
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| _version_ | 1866909248741441536 |
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| author | Canzani, Yaiza Galkowski, Jeffrey Keeler, Blake |
| author_facet | Canzani, Yaiza Galkowski, Jeffrey Keeler, Blake |
| contents | Let $(M,g)$ be a Zoll manifold, i.e., a smooth, compact, Riemannian manifold without boundary all of whose geodesics are closed with a minimal common period $T$. The positive definite Laplace-Beltrami operator has eigenvalues $\{λ_j^2\}_j$ which cluster around $ν^2_\ell$ for some sequence $ν_\ell\to \infty$. This article is concerned with the number of $λ_j$ in a window of fixed size $\mathrm{w}$ around $ν_\ell$, denoted by $\mathbf{N}(ν_\ell,\mathrm{w}):=\#\{j\,:\, λ_j\in[ν_\ell-\mathrm{w},ν_\ell+\mathrm{w}]\}.$ When the set of trajectories with period smaller than $T$ has zero measure, there is $c_{n}>0$, depending only on $n=\operatorname{dim} M$, such that $$ \mathbf{N}(ν_\ell,\mathrm{w}) =c_n\operatorname{vol}_g(M)ν_{\ell}^{n-1}+o(ν_{\ell}^{n-1}), $$ as $\ell \to \infty$. However, for a general Zoll manifold this may not be the case. We show that, nevertheless, there is $N>0$, independent of $\ell$, such that $$ \sum_{j=0}^{N-1}\mathbf{N}(ν_{\ell+j},\mathrm{w})= c_nN\operatorname{vol}_g(M)ν_{\ell}^{n-1}+o(ν_{\ell}^{n-1}), $$ as $\ell \to \infty$. In addition to asymptotics for the counting function, we study the kernel of the spectral projector for the Laplacian, $Π_{\ell,\mathrm{w}}(x,y)$ onto the spectrum in ${\bigcup_{j=0}^{N-1}[ν_{\ell+j}-\mathrm{w},ν_{\ell+j}+\mathrm{w}]}$. We show that for $x$ and $y$ in a shrinking neighborhood of a point with few loops of length smaller than $T$, $Π_{\ell,\mathrm{w}}(x,y)$ and its derivatives have the same asymptotics as those on the round sphere and flat torus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_09644 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Asymptotics for the spectral function on Zoll manifolds Canzani, Yaiza Galkowski, Jeffrey Keeler, Blake Analysis of PDEs Spectral Theory 35P20, 58J50 Let $(M,g)$ be a Zoll manifold, i.e., a smooth, compact, Riemannian manifold without boundary all of whose geodesics are closed with a minimal common period $T$. The positive definite Laplace-Beltrami operator has eigenvalues $\{λ_j^2\}_j$ which cluster around $ν^2_\ell$ for some sequence $ν_\ell\to \infty$. This article is concerned with the number of $λ_j$ in a window of fixed size $\mathrm{w}$ around $ν_\ell$, denoted by $\mathbf{N}(ν_\ell,\mathrm{w}):=\#\{j\,:\, λ_j\in[ν_\ell-\mathrm{w},ν_\ell+\mathrm{w}]\}.$ When the set of trajectories with period smaller than $T$ has zero measure, there is $c_{n}>0$, depending only on $n=\operatorname{dim} M$, such that $$ \mathbf{N}(ν_\ell,\mathrm{w}) =c_n\operatorname{vol}_g(M)ν_{\ell}^{n-1}+o(ν_{\ell}^{n-1}), $$ as $\ell \to \infty$. However, for a general Zoll manifold this may not be the case. We show that, nevertheless, there is $N>0$, independent of $\ell$, such that $$ \sum_{j=0}^{N-1}\mathbf{N}(ν_{\ell+j},\mathrm{w})= c_nN\operatorname{vol}_g(M)ν_{\ell}^{n-1}+o(ν_{\ell}^{n-1}), $$ as $\ell \to \infty$. In addition to asymptotics for the counting function, we study the kernel of the spectral projector for the Laplacian, $Π_{\ell,\mathrm{w}}(x,y)$ onto the spectrum in ${\bigcup_{j=0}^{N-1}[ν_{\ell+j}-\mathrm{w},ν_{\ell+j}+\mathrm{w}]}$. We show that for $x$ and $y$ in a shrinking neighborhood of a point with few loops of length smaller than $T$, $Π_{\ell,\mathrm{w}}(x,y)$ and its derivatives have the same asymptotics as those on the round sphere and flat torus. |
| title | Asymptotics for the spectral function on Zoll manifolds |
| topic | Analysis of PDEs Spectral Theory 35P20, 58J50 |
| url | https://arxiv.org/abs/2211.09644 |