Nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces

Iosif Polterovich; David A. Sher; John A. Toth

doi:10.1515/crelle-2017-0018

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Nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces

Iosif Polterovich , David A. Sher and John A. Toth

Published/Copyright: April 20, 2017

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2019 Issue 754

Abstract

We prove sharp upper and lower bounds for the nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces with boundary. The argument involves frequency function methods for harmonic functions in the interior of the surface as well as the construction of exponentially accurate approximations for the Steklov eigenfunctions near the boundary.

Funding source: National Science Foundation

Award Identifier / Grant number: EMSW21-RTG 1045119

Funding source: Agence Nationale de la Recherche

Award Identifier / Grant number: Gerasic-ANR-13-BS01-0007-0

Funding statement: The research of Iosif Polterovich was partially supported by NSERC, FRQNT and Canada Research Chairs Program. The research of David A. Sher was partially supported by NSF EMSW21-RTG 1045119. The research of John A. Toth was partially supported by NSERC and FRQNT. Iosif Polterovich and John A. Toth were also supported by the French National Research Agency project Gerasic-ANR-13-BS01-0007-0.

A Proofs of Lemma 3.1 and Proposition 3.3

Throughout the appendix, for simplicity, we work with a real basis of eigenfunctions.

A.1 An auxiliary lemma

Lemma 3.1 depends on the presence of recurring spectral gaps, so in order to prove it, we must first prove the following lemma.

Lemma A.1.

Fix any sufficiently small ε>0. Then there exist pairwise disjoint closed intervals Ii=[Ai,Bi]⊂R+, i=1,…,∞, with the following properties:

σ⁢(P)∪σ⁢(Mj)⊆⋃i=1∞ℐi,
Ai+1≥Bi+ε for all i,
for i≥2 and all j, each ℐi contains at most one distinct element of σ⁢(Mj),
for each i, there exists n such that λn∈ℐi and μn∈ℐi.

Proof.

Pick ε<π2⁢k⁢L, where L is the maximum of the boundary lengths L1,…,Lk. Since |λn-μn|→0, there exists N such that if n≥N, then |λn-μn|<ε2.

Observe that since {μn} is a union of k arithmetic progressions, each with period ≥2⁢πL, there are at most k elements of {μn} in any interval of length πL. Thus any such interval must contain a gap of length at least πk⁢L>2⁢ε with no elements of {μn}. We may therefore choose some m2 so that m2>N and μm2≥μm2-1+2⁢ε. Consider the interval [μm2,μm2+πL] and observe that it must itself contain a gap of length at least 2⁢ε. Let n2≥m2 be such that μn2 is the left endpoint of the first such gap; we have μn2-μm2≤πL. Then let m3=n2+1, so that μm3 is the right endpoint of that gap. Choosing n3 as with n2 and iterating this process, we produce m2,n2,m3,n3,….

Now let

ℐ1=[0,μm2-3⁢ε2],ℐi=[μmi-ε2,μni+ε2] for all ⁢j≥2.

We claim that these intervals satisfy each property we want. Indeed, by construction, property (2) is automatic. Property (1) follows immediately from the fact that |λn-μn|<ε2 whenever n≥m2-1≥N. To see property (3), note that each ℐi with i≥2 has length at most πL+ε<2⁢πL, and hence contains at most one element of each arithmetic progression with period at least 2⁢πL. Finally, property (4) is also immediate by construction (note that ℐ1 contains μ0=0). This completes the proof. ∎

We now use these intervals to split the sequence {λn} into pieces with gaps of size at least ε between each. Specifically, fix some ε and let ℐi, i=1,2,…, be the intervals constructed int Lemma A.1. For each i, we let ℒi be the set of all j for which λj∈ℐi, and we say that

(A.2)j1∼j2⇔there exists i∈ℕ such that ⁢j1∈ℐi⁢ and ⁢j2∈ℐi.

Since each interval ℐi for i≥2 contains at most k eigenvalues λn (and each interval, including ℐi, contains at least one), there exists a universal constant C such that

(A.3)j∈ℒi⟹j≤C⁢i⁢ and ⁢i≤j.

A.2 Completing the proofs

Proof of Lemma 3.1.

Let {e¯n} be an orthonormal basis of eigenfunctions for M, with eigenvalues μn, and note that each e¯n,j=e¯n|∂⁡Dj is either a trigonometric polynomial with frequency μn or identically zero. Write aj,k=〈φλj,e¯k〉. By orthonormality and completeness of the eigenbases, we have

(A.1)e¯k=∑j=1∞aj,k⁢φλj,φλk=∑j=1∞ak,j⁢e¯j.

We claim that if we set

ψλn=∑j∼nan,j⁢e¯j,fλn=∑j≁nan,j⁢e¯j,

where ∼ is defined by (A.2), then the conditions of the lemma are satisfied. Indeed, condition (1) is obvious, since by the definition of our intervals ℐi, the frequency of each e¯j with j∼n is within 2⁢πL of λn. It remains to prove (2).

In what follows, we let C and τ>0 be universal constants (which may depend on the geometry of D and on the conformal map from D to Ω) and re-label at will. From Proposition 2.2 and the Weyl asymptotics, we know that there exists τ>0 such that

|P⁢e¯j-μj⁢e¯j|≤C⁢e-τ⁢j.

Plugging in the first equation in (A.1), we see that

∥∑n=1∞an,j⁢(λn-μj)⁢φλn∥∞≤C⁢e-τ⁢j

and hence that the same is the true of the L2-norm. Summing only over the n with n≁j, we obtain

(A.2)(∑n≁jan,j2⁢(λn-μj)2)12≤C⁢e-τ⁢j.

By property (2) of Lemma A.1, we know |λn-μj|≥ε whenever n≁j, so

(A.3)∑n≁jan,j2≤C⁢e-τ⁢j.

However, by (A.1), for each j we have ∑nan,j2=1. So

(A.4)1-C⁢e-τ⁢j≤∑n∼jan,j2≤1.

In addition, suppose j1∼j2 with j1<j2. Then, again using (A.1) and the orthonormality of φk, as well as Cauchy–Schwarz,

(A.5)|∑n∼j1an,j1⁢an,j2|=|-∑n≁j1an,j1⁢an,j2|

≤(∑n≁j1an,j12)12⁢(∑n≁j1an,j22)12≤C⁢e-τ⁢j1.

Thinking of A={an,j} as an infinite matrix, let Mi be the square submatrix of A with n and j in ℒi; note that by condition (3) of Lemma A.1, Mi is of size at most 2⁢k×2⁢k. We interpret (A.4) and (A.5) as saying that the columns of Mi have almost unit length and are almost orthogonal to each other. In particular, for each i, we may write MiT⁢Mi=I+Ri. By (A.4) and (A.5), there exists a constant τ>0 such that ∥Ri∥∞≤C⁢e-τ⁢i (the norm ∥M∥∞ denotes the supremum of the entries of M). Therefore, for large enough i, (I+Ri) is always invertible, with inverse of the form I+Si, with ∥Si∥∞≤C⁢e-τ⁢i as well. In fact,

(I+Ri)-1⁢MiT⁢Mi=I,

so Mi-1 exists and equals (I+Ri)-1⁢MiT=MiT+Si⁢MiT. Multiply the equation

MiT⁢Mi=I+Ri

on the left by Mi and on the right by Mi-1 to yield

Mi⁢MiT=Mi⁢(I+Ri)⁢Mi-1=I+Mi⁢Ri⁢(I+Si)⁢MiT.

We see now that Mi⁢MiT has the form I+Qi, where Qi satisfies ∥Qi∥∞≤C⁢e-τ⁢i (note that the entries of Mi and MiT are bounded by 1).

This now tells us that the rows of Mi have almost unit length, i.e. that

1-C⁢e-τ⁢i≤∑j∼n,j∈ℒian,j2≤1.

By subtraction and (A.3), there exists τ>0 such that

(A.6)∑j≁nan,j2≤C⁢e-τ⁢n,

which is the analogue of (A.3), but summing in j instead of in n. Additionally, it is an immediate consequence of (A.3) and (A.6) that if j≁n,

(A.7)an,j2≤C⁢e-τ⁢j,an,j2≤C⁢e-τ⁢n.

By the Sobolev embedding theorem, for each k∈ℕ0, we have

∥⋅∥Ck⁢(∂⁡D)≤C∥⋅∥Hk+1⁢(∂⁡D).

We apply this to fλn, knowing that we can compute Sobolev norms of the e¯j directly:

∥fλn∥Ck2≤C⁢∥fn∥Hk+12≤C⁢∑j≁n(1+μj)k+1⁢|an,j|2.

By Weyl asymptotics of the μj,

∥fλn∥Ck2≤C⁢∑j≁n(1+j)k+1⁢an,j2

=C⁢∑j≁n,j<n(1+j)k+1⁢an,j2+C⁢∑j≁n,j>n(1+j)k+1⁢an,j2.

Using (A.7), then the Weyl asymptotics again, we have

∥fλn∥Ck2≤C⁢∑j≁n,j<n(1+n)k+1⁢e-τ⁢n+C⁢∑j≁n,j>n(1+j)k+1⁢e-τ⁢j

≤C⁢n⁢(1+n)k+1⁢e-τ⁢n+C⁢∫n∞(1+t)k+1⁢e-τ⁢t⁢𝑑t

≤C⁢λnk+2⁢e-τ⁢n.

Choosing τ slightly less than the current τ2, we can absorb λn2 and take square roots, completing the proof. ∎

Proof of Proposition 3.3.

From the discussion before (A.2), considering each term individually, we see that for every n and j,

(A.8)|an,j⁢(λn-μj)|≤C⁢e-τ⁢j.

Using the matrix notation from the previous proof, let Vi and Wi be the diagonal matrices whose entries are {λj:j∈ℒi} and {μj:j∈ℒi}, respectively. Using now (A.3), (A.8) implies the statement

∥Mi⁢Vi-Wi⁢Mi∥∞≤C⁢e-τ⁢i.

For sufficiently large i, Mi is invertible with inverse uniformly bounded in the ∥⋅∥∞ matrix norm. We deduce that

Mi⁢Vi⁢Mi-1=Wi+R¯i,∥R¯i∥∞≤C⁢e-τ⁢i.

So Wi+R¯i is diagonalized by Mi and has eigenvalues {λj:j∈ℒi}. By the Bauer–Fike theorem (see, e.g., [18, Observation 6.3.1]), the eigenvalues of Wi=(Wi+R¯i)-R¯i lie in disks centered at each λj of radius ∥Mi∥∞⋅∥Mi∥∞-1⋅∥Ri∥∞≤C⁢e-τ⁢i. Although the perturbed eigenvalues may move from one disk to another if there is overlap, there are at most k disks, so they can move at most 2⁢k⁢C⁢e-τ⁢i. Relabeling C=2⁢k⁢C, this shows that |λn-μn|≤C⁢e-τ⁢i, where λn∈ℒi. The result follows immediately by another use of implication (A.3), replacing τ with τ1=τC. ∎

Acknowledgements

The authors are grateful to M. Sodin and M. Taylor for useful discussions, and to the anonymous referee for many helpful suggestions on the presentation of the paper.

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Received: 2015-08-24

Revised: 2016-12-20

Published Online: 2017-04-20

Published in Print: 2019-09-01

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