Eigenfunction concentration via geodesic beams

Yaiza Canzani; Jeffrey Galkowski

doi:10.1515/crelle-2020-0039

Published by De Gruyter November 7, 2020

Eigenfunction concentration via geodesic beams

Yaiza Canzani and Jeffrey Galkowski

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2020-0039

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Abstract

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-12. 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.

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 Appendix

A.1 Index of notation

In general we denote points in T*⁢M by ρ. When position and momentum need to be distinguished, we write ρ=(x,ξ) for x∈M and ξ∈Tx*⁢M. Sets of indices are denoted in calligraphic font (e.g., ℐ). Next, we list symbols that are used repeatedly in the text along with the location where they are first defined.

𝒞xr,t	(1.4)	ℋΣ	(2.9)	βδ	(3.1)
ΣH,p	(2.1)	τinj	(2.10)	𝔇n	Proposition 3.3
φt	(2.2)	ΛAτ⁢(r)	(2.11)	Ψδk	(A.1)
𝒦α	(2.3)	Λρτ⁢(r)	(2.12)	Sδk	(A.1)
rH	(2.7)	𝒥h⁢(w)	(2.13)	Hsclk	(A.3)
Kp	(2.6)	Te⁢(h)	(2.14)	MSh	Definition 5
ℑ0	(2.8)	Λmax	(2.14)

For the definition of [t,T] non-self-looping, see (1.2). For that of (𝔇,τ,r)-good covers, see Definition 4.

A.2 Notation from semiclassical analysis

We refer the reader to [65] or [22, Appendix E] for a complete treatment of semiclassical analysis, but recall some of the relevant notation here. We say a∈C∞⁢(T*⁢M) is a symbol of order m and class 0≤δ<12, writing a∈Sδm⁢(T*⁢M) if there exists Cα⁢β>0 so that

(A.1)|∂xα⁡∂ξβ⁡a⁢(x,ξ)|≤Cα⁢β⁢h-δ⁢(|α|+|β|)⁢〈ξ〉m-|β|,〈ξ〉:=(1+|ξ|g2)12.

Note that we implicitly allow a to also depend on h, but omit it from the notation. We then define Sδ∞⁢(T*⁢M):=⋃mSδm⁢(T*⁢M). We sometimes write Sm⁢(T*⁢M) for S0m⁢(T*⁢M). We also sometimes write Sδ for Sδm. Next, we say that a∈Sδcomp⁢(T*⁢M) if a is supported in an h-independent compact subset of T*⁢M.

Next, there is a quantization procedure O⁢ph:Sδm→ℒ⁢(C∞⁢(M),𝒟′⁢(M)) and we say A∈Ψδm⁢(M) if there exists a∈Sδm⁢(T*⁢M) so that O⁢ph⁢(a)-A=O⁢(h∞)Ψ-∞, where we say an operator is O⁢(hk)Ψ-∞ if for all N>0 there exists CN>0 so that

∥A⁢u∥HN⁢(M)≤CN⁢hk⁢∥u∥H-N⁢(M),

and say an operator, A, is O⁢(h∞)Ψ-∞ if for all N>0 there exists CN>0 so that

∥A⁢u∥HN⁢(M)≤CN⁢hN⁢∥u∥H-N⁢(M).

For a∈Sδm1⁢(T*⁢M) and b∈Sδm2⁢(T*⁢M), we have that

(A.2)O⁢ph⁢(a)⁢O⁢ph⁢(b)=O⁢ph⁢(c),c⁢(x,ξ)∼∑jhj⁢L2⁢j⁢(a⁢(x,ξ)⁢b⁢(y,η))|x=yξ=η,

where L2⁢j is a differential operator of order j in (x,ξ) and order j in (y,η).

There is a symbol map σ:Ψδm⁢(M)→Sδm⁢(T*⁢M)/h1-2⁢δ⁢Sδm-1⁢(T*⁢M) so that

σ⁢(O⁢ph⁢(a))=a,σ⁢(O⁢ph⁢(a)*)=a¯,σ⁢(O⁢ph⁢(a)⁢O⁢ph⁢(b))=a⁢b,σ⁢([O⁢ph⁢(a),O⁢ph⁢(b)])=-i⁢h⁢{a,b},

and

0→h1-2⁢δ⁢Ψδm-1⁢(M)→Ψδm⁢(M)⁢→𝜎⁢Sδm⁢(M)/h1-2⁢δ⁢Sδm-1⁢(M)→0

is exact.

The main consequence of (A.2) that we will use is that if p∈Sm⁢(M) and a∈Sδk⁢(T*⁢M), then

[O⁢ph⁢(p),O⁢ph⁢(a)]=hi⁢O⁢ph⁢(Hp⁢a)+h2-2⁢δ⁢O⁢ph⁢(r)

with r∈Sδm+k-2⁢(T*⁢M).

We define the semiclassical Sobolev spaces Hscls⁢(M) by

(A.3)Hscls⁢(M):={u∈𝒟′⁢(M):∥u∥Hscls⁢(M)<∞},

where

∥u∥Hscls⁢(M):=∥O⁢ph⁢(〈ξ〉s)⁢u∥L2⁢(M).

A.3 Background on microsupports and Egorov’s Theorem

Definition 5.

For a pseudodifferential operator A∈Ψδcomp⁢(M), we say that A is microsupported in a family of sets {V⁢(h)}h and write MSh⁡(A)⊂V⁢(h) if

A=O⁢ph⁢(a)+O⁢(h∞)Ψ-∞

and for all α,N, there exists Cα,N>0 so that

sup(x,ξ)∈T*⁢M∖V⁢(h)⁡|∂x,ξα⁡a⁢(x,ξ)|≤Cα,N⁢hN.

For B⁢(h)⊂T*⁢M, will also write MSh⁡(A)∩B⁢(h)=∅ for MSh⁡(A)⊂(B⁢(h))c.

Note that the notation MSh⁡(A)⊂V⁢(h) is a shortening for MSh⁡(A)⊂{V⁢(h)}h.

Lemma A.1.

Let 0≤δ<12 and δ′>δ, c>0. Suppose that A∈Ψδcomp⁢(M) and that MSh⁡(A)⊂V⁢(h). Then

MSh⁡(A)⊂{(x,ξ):d⁢((x,ξ),V⁢(h)c)≤c⁢hδ′}.

Proof.

Let A=O⁢ph⁢(a)+O⁢(h∞)Ψ-∞. Suppose that 2⁢r⁢(h):=d⁢(ρ1,V⁢(h)c)≤c⁢hδ′ and let ρ0∈V⁢(h)c with d⁢(ρ1,ρ0)≤r⁢(h). Then, for any N>0,

|∂α⁡a⁢(ρ1)|≤∑|β|≤N-1|∂α+β⁡a⁢(ρ0)|⁢r⁢(h)|β|+C|α|+N⁢sup|k|≤|α|+N,T*⁢M⁡|∂k⁡a|⁢r⁢(h)N

≤∑|β|≤N-1supVc⁡|∂α+β⁡a⁢(ρ)|⁢r⁢(h)|β|+Cα⁢N⁢h-N⁢δ⁢r⁢(h)N

≤Cα⁢N⁢M⁢hM+Cα⁢N⁢h-N⁢δ⁢r⁢(h)N.

So, letting N≥M⁢(δ′-δ)-1,

|∂α⁡a⁢(ρ1)|≤Cα⁢M⁢hM.∎

Lemma A.2.

Let 0≤δ<12 and A,B∈Ψδcomp⁢(M). Suppose that MSh⁡(A)⊂V⁢(h) and MSh⁡(B)⊂W⁢(h).

The statement MSh⁡(A)⊂V⁢(h) is well defined. In particular, it does not depend on the choice of quantization procedure.
MSh⁡(A⁢B)⊂V⁢(h)∩W⁢(h)
MSh⁡(A*)⊂V⁢(h)
If V⁢(h)=∅, then WFh⁡(A)=∅.
If A=O⁢ph⁢(a)+O⁢(h∞)Ψ-∞, then MSh⁡(a)⊂supp⁡a.

Proof.

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 A=O⁢ph⁢(a)+O⁢(h∞)Ψ-∞ and B=O⁢ph⁢(b)+O⁢(h∞)Ψ-∞. Then

O⁢ph⁢(a)⁢O⁢ph⁢(b)=O⁢ph⁢(#⁡ab)+O⁢(h∞)Ψ-∞,

where

#⁡ab⁢(x,ξ)∼∑jhj⁢L2⁢j⁢a⁢(x,ξ)⁢b⁢(y,η)|x=yξ=η

and L2⁢j are differential operators of order 2⁢j. Suppose that MSh⁡(A)⊂V. Then, for any N>0,

supVc⁡|∂α⁡a|≤Cα⁢N⁢hN.

So, choosing M>(N+δ⁢|α|)⁢(1-2⁢δ)-1,

|∂αa#b|≤|∂α∑j<MhjL2⁢ja(x,ξ)b(y,η)|x=yξ=η|+Cα⁢MhM⁢(1-2⁢δ)-|α|⁢δ≤Cα⁢NhN.

In particular,

supVc⁡|∂α⁡#⁡ab|≤Cα⁢N⁢hN.

An identical argument shows

supWc⁡|∂α⁡#⁡ab|≤Cα⁢N⁢hN.

Statement (4) follows from the definition since if V⁢(h)=∅, a∈h∞⁢Sδ, and (5) follows easily from the definition. ∎

Lemma A.3.

Let φt:=exp⁡(t⁢Hp) and Σ⊂T*⁢M compact. There exist δ>0 small enough and C1>0 so that uniformly for t∈[0,δ], and (xi,ξi)∈Σ,

12⁢d⁢((x1,ξ1),(x2,ξ2))-C1⁢d⁢((x1,ξ1),(x2,ξ2))2

≤d⁢(φt⁢(x1,ξ2),φt⁢(x2,ξ1))

≤2⁢d⁢((x1,ξ1),(x2,ξ2))+C1⁢d⁢((x1,ξ1),(x2,ξ2))2,

where d is the distance induced by the Sasaki metric. Furthermore, if φt⁢(xi,ξi)=(xi⁢(t),ξi⁢(t)), then

dM⁢(x1⁢(t),x2⁢(t))≤dM⁢(x1,x2)+C1⁢d⁢((x1,ξ1),(x2,ξ2))⁢δ,

where dM is the distance induced by the metric on M.

Proof.

By Taylor’s theorem,

φt⁢(x1,ξ1)-φt⁢(x2,ξ2)=dx⁢φt⁢(x2,ξ2)⁢(x1-x2)+dξ⁢φt⁢(x2,ξ2)⁢(ξ1-ξ2)

+OC∞⁢(supq∈Σ⁡|d2⁢φt⁢(q)|⁢(|ξ1-ξ2|2+|x1-x2|2)).

Now,

φt⁢(x,ξ)=(x,ξ)+(∂ξ⁡p⁢(x,ξ)⁢t,-∂x⁡p⁢(x,ξ)⁢t)+O⁢(t2)

dξ⁢φt⁢(x,ξ)=(0,I)+t⁢(∂ξ2⁡p,-∂ξ⁢x2⁡p)+O⁢(t2)

dx⁢φt⁢(x,ξ)=(I,0)+t⁢(∂x⁢ξ2⁡p,-∂x2⁡p)+O⁢(t2).

In particular,

φt⁢(x1,ξ1)-φt⁢(x2,ξ2)=((0,I)+O⁢(t))⁢(ξ1-ξ2)+((I,0)+O⁢(t))⁢(x1-x2)

+O⁢((ξ1-ξ2)2+(x1-x2)2)

and choosing δ>0 small enough gives the result. ∎

B Proofs of Lemmas 1.2 and 1.3

Lemma B.1.

Let t,T>0 and suppose that G⊂Sx*⁢M is a closed set that is [t,T] non-self-looping. Then there is R>0 such that BT*⁢M⁢(G,R) is [t,T] non-self-looping.

Proof.

We will assume that φs⁢(G)∩G=∅ for s∈[t,T], the case of s∈[-T,-t] being similar. Let q∈G. We claim there is Rq>0 such that

⋃s∈[t,T]φt⁢(BT*⁢M⁢(q,Rq))∩BT*⁢M⁢(G,Rq)=∅.

Suppose not. Then there are qn→q and sn∈[t,T] such that d⁢(φsn⁢(qn),G)→0. Extracting subsequences, we may assume sn→s∈[t,T] and φsn⁢(qn)→ρ∈G. But then φs⁢(q)=ρ and, in particular, G is not [t,T] non-self-looping.

Now, G⊂⋃q∈GB⁢(q,Rq) and hence, by compactness, there are qi, i=1,…,N, such that

G⊂⋃i=1NB⁢(qi,Rqi).

In particular, there is 0<R<mini⁡Rqi such that

B⁢(G,R)⊂⋃i=1NB⁢(qi,Rqi).

This implies that B⁢(G,R) is [t,T] non-self-looping. ∎

Lemma B.2.

Let τ,D,t,T>0, R⁢(h)≥8⁢hδ, and {Λρjτ⁢(R⁢(h))}j∈G a (D,τ,R⁢(h))-good cover of Sx*⁢M. Suppose that G⊂Sx*⁢M is closed and [t,T] non-self-looping. Then, for all ε>0, there is R>0 small enough such that for R⁢(h)<R,

𝒢:={j∈𝒥:Λρjτ⁢(R⁢(h))∩BSx*⁢M⁢(G,R)≠∅}

satisfies

(B.1)⋃j∈𝒢Λρjτ⁢(R⁢(h))⁢ is ⁢[max⁡(t,3⁢τ),max⁡(t,3⁢τ,T)]⁢ non-self-looping

and

(B.2)|𝒢|≤𝔇⁢R⁢(h)1-n⁢(volSx*⁢M⁡(G)+ε).

Proof.

By Lemma B.1, there is R0>0 such that B⁢(G,R0) is [t,T] non-self-looping. Furthermore, since G is closed, there is R1>0 such that

volSx*⁢M⁡(B⁢(G,R1))<volSx*⁢M⁡(G)+ε.

Therefore, putting R=min⁡(R04,R14), for R⁢(h)≤R, and j∈𝒢,

⋃j∈𝒢Λρjτ⁢(R⁢(h))∩Sx*⁢M⊂BT*⁢M⁢(G,min⁡(R0,R1)).

In particular, (B.1) and (B.2) hold. ∎

Proof of Lemma 1.2.

Suppose that x non-self-focal. Let ℒxT:=T+-1⁢([0,T]) and note that for all T>0, ℒxT is closed. Thus, by Lemma B.2 for all T>0 there is R0=R0⁢(T)>0 such that for R⁢(h)≤R0, with ℬ~:={j:Λρjτ⁢(R⁢(h))∩BSx*⁢M⁢(ℒxT,R0)}, one has

|ℬ~|≤R⁢(h)1-nT.

Next, since G:=Sx*⁢M∖B⁢(ℒxT,R0) is closed and [inj⁡M2,T] non-self-looping, there is R1=R1⁢(T)>0 such for R⁢(h)≤R1 and

𝒢={j:Λρjτ⁢(R⁢(h))∩B⁢(G,R1)},

equation (B.1) holds with t=inj⁡M2 and T=T. Putting

R⁢(T):=min⁡(R1⁢(T),R2⁢(T)),ℬ:=ℬ~∖𝒢,

and defining

h0⁢(T)=inf⁡{h>0:R⁢(h)>R⁢(T)},T⁢(h)=sup⁡{T>0:h0⁢(T)>h},

we have shown that x is (inj⁡M2,T⁢(h)) non-looping. ∎

Proof of Lemma 1.3.

Let ℛx±,δ,S be the set of points ρ∈Sx*⁢M for which there exists 0<±t≤S such that φt⁢(ρ)∈Sx*⁢M and d⁢(φt⁢(ρ),ρ)≤δ. Then

ℛx=⋂δ>0⋃S>0ℛxδ,S,ℛxδ,S:=⋂±ℛx±,δ,S.

Note that ℛxδ,S is closed for all δ,S, and that for all ε>0 there is δ>0 such that for all S>0,

volSx*⁢M⁡(ℛxS,δ)≤volSx*⁢M⁡(ℛx)+ε.

Now, assume that x is non-recurrent. Then, for all ε>0, there is a constant δ=δ⁢(ε)>0 such that for all S>0,

volSx*⁢M⁡(ℛxS,δ)≤ε.

Let {ρi}i=1N⁢(δ)⊂Sx*⁢M be such that Sx*⁢M⊂⋃iB⁢(ρi,δ4) and N⁢(δ)≤C⁢δ1-n.

Letting G0:=ℛxS,δ, by Lemma B.2 there is a constant R0=R0⁢(ε,S)>0 such that for R⁢(h)≤R0, defining 𝒢~0:={j:Λρjτ⁢(R⁢(h))∩BSx*⁢M⁢(Gi,R0)}, we have

|𝒢~0|≤𝔇⁢R⁢(h)1-n⁢ε.

Next, let Gi:=BSx*⁢M⁢(ρi,δ4)¯∖BSx*⁢M⁢(ℛxT,δ,R0) so that Gi is closed and [inj⁡M2,S] non-self-looping. By Lemma B.2, there are Ri=Ri⁢(ε,S)>0 such that for R⁢(h)≤mini⁡Ri, if we set 𝒢~i:={j:Λρjτ⁢(R⁢(h))∩BSx*⁢M⁢(Gi,Ri)}, then

|𝒢~i|≤R⁢(h)1-n⁢𝔇⁢δn-1,i≥1,

and for i≥1,

⋃j∈𝒢~iΛρjτ⁢(R⁢(h))⁢ is ⁢[inj⁡M/2,S]⁢ non-self-looping.

Then we have

∑i=0N|𝒢~i|⁢R⁢(h)n-1⁢inj⁡M2⁢S≤N⁢(δ)⁢δn-12⁢𝔇⁢inj⁡M2⁢S+𝔇⁢ε.

Now, for ε:=14⁢𝔇⁢T let δ:=δ⁢(ε) and set

S:=2⁢N2⁢(δ)⁢δn-1⁢𝔇⁢inj⁡M.

Working with Ri=Ri⁢(ε,S)=Ri⁢(T) as defined before, we have

∑i=0N|𝒢~i|⁢R⁢(h)n-1⁢inj⁡M2⁢S≤1T.

Defining

h0⁢(T)=inf⁡{h>0:R⁢(h)>mini⁡Ri⁢(T)},T⁢(h)=sup⁡{T>0:h0⁢(T)>h},

we have shown that x is (inj⁡M2,T⁢(h)) non-recurrent. ∎

Acknowledgements

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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Received: 2020-03-20

Revised: 2020-09-15

Published Online: 2020-11-07

Published in Print: 2021-06-01

Eigenfunction concentration via geodesic beams

Abstract

A Appendix

A.1 Index of notation

A.2 Notation from semiclassical analysis

A.3 Background on microsupports and Egorov’s Theorem

Definition 5.

Lemma A.1.

Proof.

Lemma A.2.

Proof.

Lemma A.3.

Proof.

B Proofs of Lemmas 1.2 and 1.3

Lemma B.1.

Proof.

Lemma B.2.

Proof.

Proof of Lemma 1.2.

Proof of Lemma 1.3.

Acknowledgements

References

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