We develop new techniques for studying concentration of Laplace eigenfunctions as their frequency, λ, grows. The method consists of controlling by decomposing 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 . We control by the -mass of on each geodesic tube and derive a purely dynamical statement through which can be studied. In particular, we obtain estimates on 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 -norms, -norms, pointwise Weyl laws, and averages over submanifolds.
Funding statement: Jeffrey Galkowski is grateful to the National Science Foundation for support under the Mathematical Sciences Postdoctoral Research Fellowship DMS-1502661. Yaiza Canzani is grateful to the Alfred P. Sloan Foundation.
A.1 Index of notation
In general we denote points in by ρ. When position and momentum need to be distinguished, we write for and . Sets of indices are denoted in calligraphic font (e.g., ). Next, we list symbols that are used repeatedly in the text along with the location where they are first defined.
A.2 Notation from semiclassical analysis
We refer the reader to  or [22, Appendix E] for a complete treatment of semiclassical analysis, but recall some of the relevant notation here. We say is a symbol of order m and class , writing if there exists so that
Note that we implicitly allow a to also depend on h, but omit it from the notation. We then define . We sometimes write for . We also sometimes write for . Next, we say that if a is supported in an h-independent compact subset of .
Next, there is a quantization procedure and we say if there exists so that , where we say an operator is if for all there exists so that
and say an operator, A, is if for all there exists so that
For and , we have that
where is a differential operator of order j in and order j in .
There is a symbol map so that
The main consequence of (A.2) that we will use is that if and , then
We define the semiclassical Sobolev spaces by
A.3 Background on microsupports and Egorov’s Theorem
For a pseudodifferential operator , we say that A is microsupported in a family of sets and write if
and for all , there exists so that
For , will also write for .
Note that the notation is a shortening for .
Let and , . Suppose that and that . Then
Let . Suppose that and let with . Then, for any ,
So, letting ,
Let and . Suppose that and .
The statement is well defined. In particular, it does not depend on the choice of quantization procedure.
If , then .
If , then .
The proofs of (1)–(3) are nearly identical, relying on the asymptotic expansion for, respectively, the change of quantization, composition, and adjoint so we write the proof for only (2). Write and . Then
and are differential operators of order . Suppose that . Then, for any ,
So, choosing ,
An identical argument shows
Statement (4) follows from the definition since if , , and (5) follows easily from the definition. ∎
Let and compact. There exist small enough and so that uniformly for , and ,
where is the distance induced by the Sasaki metric. Furthermore, if , then
where is the distance induced by the metric on M.
By Taylor’s theorem,
and choosing small enough gives the result. ∎
Let and suppose that is a closed set that is non-self-looping. Then there is such that is non-self-looping.
We will assume that for , the case of being similar. Let . We claim there is such that
Suppose not. Then there are and such that . Extracting subsequences, we may assume and . But then and, in particular, G is not non-self-looping.
Now, and hence, by compactness, there are , , such that
In particular, there is such that
This implies that is non-self-looping. ∎
Let , , and a -good cover of . Suppose that is closed and non-self-looping. Then, for all , there is small enough such that for ,
By Lemma B.1, there is such that is non-self-looping. Furthermore, since G is closed, there is such that
Therefore, putting , for , and ,
Proof of Lemma 1.2.
Suppose that x non-self-focal. Let and note that for all , is closed. Thus, by Lemma B.2 for all there is such that for , with , one has
Next, since is closed and non-self-looping, there is such for and
equation (B.1) holds with and . Putting
we have shown that x is non-looping. ∎
Proof of Lemma 1.3.
Let be the set of points for which there exists such that and . Then
Note that is closed for all , and that for all there is such that for all ,
Now, assume that x is non-recurrent. Then, for all , there is a constant such that for all ,
Let be such that and .
Letting , by Lemma B.2 there is a constant such that for , defining , we have
Next, let so that is closed and non-self-looping. By Lemma B.2, there are such that for , if we set , then
and for ,
Then we have
Now, for let and set
Working with as defined before, we have
we have shown that x is non-recurrent. ∎
Thanks to Pat Eberlein, John Toth, Andras Vasy, and Maciej Zworski for many helpful conversations and comments on the manuscript. Thanks also to the anonymous referees for many suggestions which improved the exposition.
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