We develop new techniques for studying concentration of Laplace eigenfunctions ϕλ{\phi_{\lambda}} as their frequency, λ, grows. The method consists of controlling ϕλ⁢(x){\phi_{\lambda}(x)} by decomposing ϕλ{\phi_{\lambda}} into a superposition of geodesic beams that run through the point x . Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than λ-12{\lambda^{-\frac{1}{2}}}. We control ϕλ⁢(x){\phi_{\lambda}(x)} by the L2{L^{2}}-mass of ϕλ{\phi_{\lambda}} on each geodesic tube and derive a purely dynamical statement through which ϕλ⁢(x){\phi_{\lambda}(x)} can be studied. In particular, we obtain estimates on ϕλ⁢(x){\phi_{\lambda}(x)} by decomposing the set of geodesic tubes into those that are non-self-looping for time T and those that are. This approach allows for quantitative improvements, in terms of T , on the available bounds for L∞{L^{\infty}}-norms, Lp{L^{p}}-norms, pointwise Weyl laws, and averages over submanifolds.

中文翻译：

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通过测地线束的特征函数集中

我们开发了研究拉普拉斯特征函数 ϕλ{\phi_{\lambda}} 随着频率 λ 增长的浓度的新技术。该方法包括通过将 ϕλ{\phi_{\lambda}} 分解为穿过点 x 的测地线束的叠加来控制 ϕλ⁢(x){\phi_{\lambda}(x)} 。每个光束都位于以测地线为中心的管上的相空间中，该测地线的半径收缩速度略慢于 λ-12{\lambda^{-\frac{1}{2}}}。我们通过每个测地线管上 ϕλ{\phi_{\lambda}} 的 L2{L^{2}} 质量控制 ϕλ⁢(x){\phi_{\lambda}(x)} 并推导出纯动力学陈述通过它可以研究 ϕλ⁢(x){\phi_{\lambda}(x)}。特别是，我们通过将测地线管组分解为时间 T 的非自环和非自环的管来获得对 ϕλ⁢(x){\phi_{\lambda}(x)} 的估计。