Eigenfunction Concentration via Geodesic Beams

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