论文标题
zoll歧管上光谱函数的渐近学
Asymptotics for the spectral function on Zoll manifolds
论文作者
论文摘要
令$（m，g）$为zoll歧管，即平滑，紧凑的riemannian歧管，没有边界，其所有大地测量学都以最小的公共时期$ t $关闭。积极的确定Laplace-Beltrami操作员具有eigenvalues $ \ {λ_j^2 \} _ j $，该_ j $将围绕$ν^2_ \ ell $集中在某些序列$ν_\ ell \ ell \ to \ infty $中。本文涉及固定尺寸$ \ mathrm {w} $周围$ν_\ ell $的$λ_j$的数量，由$ \ m \ m mathbf {n}表示（ν_\ ell，\ ell，\ ell，\ alsrm {w}） λ_j\ in [ν_\ ell- \ Mathrm {w}，ν_\ Ell+\ ell+\ \ \ \ \ \ \ \ \}] \}。 \ Mathbf {n}（ν_\ ell，\ Mathrm {w}）= C_N \ operatorname {vol} _g（m）ν_ {\ ell}^{n-1}+O（n-1}+o（v {n-1}+O（v _ {\ ell}但是，对于一般的Zoll歧管，情况可能并非如此。我们表明，尽管如此，有$ n> 0 $，独立于$ \ ell $，因此$$ \ sum_ {j = 0}^{n-1} \ mathbf {n}（ν_ {\ ell+j}，\ mathrm {w}）= {w}）= c_nn \ operatorName {vol} _g（m）ν_ {\ ell}^{n-1}+o（ν_{\ ell}^{n-1}），$ \ as as $ \ ell \ to \ to \ infty $。除了用于计数函数的渐近学外，我们还研究了laplacian光谱投影仪的内核，$π_ {\ ell，\ mathrm {w}}（x，y）$ $ {\ bigCup_ {j = 0}^{n-1} [ν_ {\ ell+j} - \ mathrm {w}，ν_ {\ ell+j}+j}+\ mathrm {w}]}} $。我们表明，对于$ x $和$ y $的$ y $，在一个缩小的社区中，其长度小于$ t $，$π_ {\ ell，\ mathrm {w}}}（x，y）$及其衍生物具有与圆形球体和平坦的圆环的衍生物相同的衍生物。
Let $(M,g)$ be a Zoll manifold, i.e., a smooth, compact, Riemannian manifold without boundary all of whose geodesics are closed with a minimal common period $T$. The positive definite Laplace-Beltrami operator has eigenvalues $\{λ_j^2\}_j$ which cluster around $ν^2_\ell$ for some sequence $ν_\ell\to \infty$. This article is concerned with the number of $λ_j$ in a window of fixed size $\mathrm{w}$ around $ν_\ell$, denoted by $\mathbf{N}(ν_\ell,\mathrm{w}):=\#\{j\,:\, λ_j\in[ν_\ell-\mathrm{w},ν_\ell+\mathrm{w}]\}.$ When the set of trajectories with period smaller than $T$ has zero measure, there is $c_{n}>0$, depending only on $n=\operatorname{dim} M$, such that $$ \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.
