In this article we develop new techniques for studying concentration of
Laplace eigenfunctions ϕλ as their frequency, λ, grows. The
method consists of controlling ϕλ(x) by decomposing ϕλ
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−21. We control
ϕλ(x) by the L2-mass of ϕλ on each geodesic tube and
derive a purely dynamical statement through which ϕλ(x) can be
studied. In particular, we obtain estimates on ϕλ(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∞ norms, Lp norms, pointwise Weyl
laws, and averages over submanifolds.Comment: 61 pages, 2 figures. Improved exposition and includes new explanatory
material in the introduction as well as an examples section (1.5) and a full
section on comparison with previous work (1.6). Appendices A.1 (Index of
notation) and B were also adde