# Eigenfunction concentration via geodesic beams

Yaiza Canzani and Jeffrey Galkowski

## 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 xM and ξTx*M. 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.

 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 aC(T*M) is a symbol of order m and class 0δ<12, writing aSδm(T*M) if there exists Cαβ>0 so that

(A.1)|xαξβa(x,ξ)|Cαβh-δ(|α|+|β|)ξm-|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 aSδcomp(T*M) if a is supported in an h-independent compact subset of T*M.

Next, there is a quantization procedure Oph:Sδm(C(M),𝒟(M)) and we say AΨδm(M) if there exists aSδm(T*M) so that Oph(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

AuHN(M)CNhkuH-N(M),

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

AuHN(M)CNhNuH-N(M).

For aSδm1(T*M) and bSδm2(T*M), we have that

(A.2)Oph(a)Oph(b)=Oph(c),c(x,ξ)jhjL2j(a(x,ξ)b(y,η))|x=yξ=η,

where L2j 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)/h1-2δSδm-1(T*M) so that

σ(Oph(a))=a,σ(Oph(a)*)=a¯,σ(Oph(a)Oph(b))=ab,σ([Oph(a),Oph(b)])=-ih{a,b},

and

0h1-2δΨδm-1(M)Ψδm(M)𝜎Sδm(M)/h1-2δSδm-1(M)0

is exact.

The main consequence of (A.2) that we will use is that if pSm(M) and aSδk(T*M), then

[Oph(p),Oph(a)]=hiOph(Hpa)+h2-2δOph(r)

with rSδm+k-2(T*M).

We define the semiclassical Sobolev spaces Hscls(M) by

(A.3)Hscls(M):={u𝒟(M):uHscls(M)<},

where

uHscls(M):=Oph(ξs)uL2(M).

### Definition 5.

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

A=Oph(a)+O(h)Ψ-

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

sup(x,ξ)T*MV(h)|x,ξαa(x,ξ)|Cα,NhN.

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 0δ<12 and δ>δ, c>0. Suppose that AΨδcomp(M) and that MSh(A)V(h). Then

MSh(A){(x,ξ):d((x,ξ),V(h)c)chδ}.

### Proof.

Let A=Oph(a)+O(h)Ψ-. Suppose that 2r(h):=d(ρ1,V(h)c)chδ and let ρ0V(h)c with d(ρ1,ρ0)r(h). Then, for any N>0,

|αa(ρ1)||β|N-1|α+βa(ρ0)|r(h)|β|+C|α|+Nsup|k||α|+N,T*M|ka|r(h)N
|β|N-1supVc|α+βa(ρ)|r(h)|β|+CαNh-Nδr(h)N
CαNMhM+CαNh-Nδr(h)N.

So, letting NM(δ-δ)-1,

|αa(ρ1)|CαMhM.

### Lemma A.2.

Let 0δ<12 and A,BΨδcomp(M). Suppose that MSh(A)V(h) and MSh(B)W(h).

1. The statement MSh(A)V(h) is well defined. In particular, it does not depend on the choice of quantization procedure.

2. MSh(AB)V(h)W(h)

3. MSh(A*)V(h)

4. If V(h)=, then WFh(A)=.

5. If A=Oph(a)+O(h)Ψ-, then MSh(a)suppa.

### 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=Oph(a)+O(h)Ψ- and B=Oph(b)+O(h)Ψ-. Then

Oph(a)Oph(b)=Oph(#ab)+O(h)Ψ-,

where

#ab(x,ξ)jhjL2ja(x,ξ)b(y,η)|x=yξ=η

and L2j are differential operators of order 2j. Suppose that MSh(A)V. Then, for any N>0,

supVc|αa|CαNhN.

So, choosing M>(N+δ|α|)(1-2δ)-1,

|αa#b||αj<MhjL2ja(x,ξ)b(y,η)|x=yξ=η|+CαMhM(1-2δ)-|α|δCαNhN.

In particular,

supVc|α#ab|CαNhN.

An identical argument shows

supWc|α#ab|CαNhN.

Statement (4) follows from the definition since if V(h)=, ahSδ, and (5) follows easily from the definition. ∎

### Lemma A.3.

Let φt:=exp(tHp) and ΣT*M compact. There exist δ>0 small enough and C1>0 so that uniformly for t[0,δ], and (xi,ξi)Σ,

12d((x1,ξ1),(x2,ξ2))-C1d((x1,ξ1),(x2,ξ2))2
d(φt(x1,ξ2),φt(x2,ξ1))
2d((x1,ξ1),(x2,ξ2))+C1d((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)+C1d((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)=dxφt(x2,ξ2)(x1-x2)+dξφt(x2,ξ2)(ξ1-ξ2)
+OC(supqΣ|d2φt(q)|(|ξ1-ξ2|2+|x1-x2|2)).

Now,

φt(x,ξ)=(x,ξ)+(ξp(x,ξ)t,-xp(x,ξ)t)+O(t2)

so

dξφt(x,ξ)=(0,I)+t(ξ2p,-ξx2p)+O(t2)
dxφt(x,ξ)=(I,0)+t(xξ2p,-x2p)+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. ∎

## Lemma B.1.

Let t,T>0 and suppose that GSx*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 qG. 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 qnq and sn[t,T] such that d(φsn(qn),G)0. Extracting subsequences, we may assume sns[t,T] and φsn(qn)ρG. But then φs(q)=ρ and, in particular, G is not [t,T] non-self-looping.

Now, GqGB(q,Rq) and hence, by compactness, there are qi, i=1,,N, such that

Gi=1NB(qi,Rqi).

In particular, there is 0<R<miniRqi 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)8hδ, and {Λρjτ(R(h))}jG a (D,τ,R(h))-good cover of Sx*M. Suppose that GSx*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:={j𝒥:Λρjτ(R(h))BSx*M(G,R)}

satisfies

(B.1)j𝒢Λρjτ(R(h)) is [max(t,3τ),max(t,3τ,T)] non-self-looping

and

(B.2)|𝒢|𝔇R(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 j𝒢,

j𝒢Λρjτ(R(h))Sx*MBT*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 xT:=T+-1([0,T]) and note that for all T>0, xT is closed. Thus, by Lemma B.2 for all T>0 there is R0=R0(T)>0 such that for R(h)R0, with ~:={j:Λρjτ(R(h))BSx*M(xT,R0)}, one has

|~|R(h)1-nT.

Next, since G:=Sx*MB(xT,R0) is closed and [injM2,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=injM2 and T=T. Putting

R(T):=min(R1(T),R2(T)),:=~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 (injM2,T(h)) non-looping. ∎

## Proof of Lemma 1.3.

Let x±,δ,S be the set of points ρSx*M for which there exists 0<±tS such that φt(ρ)Sx*M and d(φt(ρ),ρ)δ. Then

x=δ>0S>0xδ,S,xδ,S:=±x±,δ,S.

Note that xδ,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(xS,δ)volSx*M(x)+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(xS,δ)ε.

Let {ρi}i=1N(δ)Sx*M be such that Sx*MiB(ρi,δ4) and N(δ)Cδ1-n.

Letting G0:=xS,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|𝔇R(h)1-nε.

Next, let Gi:=BSx*M(ρi,δ4)¯BSx*M(xT,δ,R0) so that Gi is closed and [injM2,S] non-self-looping. By Lemma B.2, there are Ri=Ri(ε,S)>0 such that for R(h)miniRi, if we set 𝒢~i:={j:Λρjτ(R(h))BSx*M(Gi,Ri)}, then

|𝒢~i|R(h)1-n𝔇δn-1,i1,

and for i1,

j𝒢~iΛρjτ(R(h)) is [injM/2,S] non-self-looping.

Then we have

i=0N|𝒢~i|R(h)n-1injM2SN(δ)δn-12𝔇injM2S+𝔇ε.

Now, for ε:=14𝔇T let δ:=δ(ε) and set

S:=2N2(δ)δn-1𝔇injM.

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

i=0N|𝒢~i|R(h)n-1injM2S1T.

Defining

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

we have shown that x is (injM2,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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