Bounding the number of nodal domains of eigenfunctions without singular points on the square
Junehyuk Jung
math.SP

Bounding the number of nodal domains on the square]Bounding the number of nodal domains of eigenfunctions without singular points on the square

Department of Mathematics, Texas A&M University, College Station, TX 77840
junehyuk@math.tamu.edu

We prove Polterovich's conjecture concerning the growth of the number of nodal domains for eigenfunctions on a unit square domain, under the assumption that the eigenfunctions do not have any singular points.

We would like to thank Jean Bourgain for introducing the problem, and encouragement. We appreciate Stefan Steinerberger, Van Vu, Igor Wigman, Steve Zelditch, and Joon-Hyeok Yim for many helpful discussions. We also thank Bernard Helffer and the anonymous referee for detailed comments on the earlier version of the manuscript.
[
    Junehyuk Jung
    December 30, 2023
=====================



§ INTRODUCTION

Let Ω⊂ℝ^2 be a domain with a piecewise smooth boundary. Let {ϕ_j}_j≥ 1 be an orthonormal Dirichlet eigenbasis with the eigenvalues 0<λ_1 < λ_2 ≤λ_3 …, i.e.,

    -Δϕ_j    = λ_j ϕ_j
    ϕ_j (x)    = 0  for  x∈∂Ω
    ⟨ϕ_j , ϕ_k ⟩   = δ_jk

where Δ = ∂_x^2+∂_y^2 is the Laplacian on ℝ^2, and δ_jk is the Kronecker delta. We assume that all ϕ_j's are real-valued.

Denoting by N(ϕ_j) the number of nodal domains (connected components of Ω minus the zero set of ϕ_j), Courant's nodal domain theorem <cit.> implies that

    N(ϕ_j) ≤ j

for all j≥ 1. This inequality is sharp, and we say that λ_j a Courant-sharp Dirichlet eigenvalue, whenever the equality is satisfied by ϕ_j. For instance, λ_1, λ_2 are Courant-sharp Dirichlet eigenvalues.

For large values of j, (<ref>) is not sharp anymore. In particular, Pleijel's theorem <cit.> states that

    lim sup_j→∞N(ϕ_j)/j≤(2/j_0,1)^2 = 0.69167…,

where j_0,1 is the smallest positive zero of the Bessel function, J_0(x).

Note that when Ω is a unit square domain D:= 0,1  ^2 ⊂ℝ^2, one can prove that

    lim sup_j→∞N(ϕ_j)/j≥2/π =0.63661…

by considering a sequence of Dirichlet eigenfunctions {sin (k π x)sin (k π y)}_k=1,2,….

In <cit.>, Polterovich observed that Pleijel's inequality should not be sharp, because it uses Faber–Krahn inequality which is only sharp on a disk, whereas not all nodal domains can simultaneously be disks. He also conjectured that the maximum number of nodal domains are obtained by the eigenfunctions sin (k π x)sin (k π y) on Ω=D, i.e.,

    lim sup_j→∞N(ϕ_j)/j≤2/π.


The idea that not all nodal domains can simultaneously be disks
was used by Bourgain <cit.> to improve  Pleijel's inequality, see  also <cit.>. In <cit.>, the packing density of disks, and a refined Faber–Krahn inequality are used to prove that

    lim sup_j→∞N(ϕ_j)/j≤(2/j_0,1)^2 - 3· 10^-9.

<cit.> utilizes a “geometric uncertainty principle” that quantifies the fact that if a partition is given by sets of equal measure, then the sets cannot be disks, and proves the existence of a small constant η>0, such that

    lim sup_j→∞N(ϕ_j)/j≤(2/j_0,1)^2 - η.


The main purpose of this article is to present a new way of counting the number of nodal domains. In particular, we give a proof of Polterovich's conjecture (<ref>) for certain eigenfunctions on a unit square domain D. (We refer the readers to <cit.> for the case of irrational rectangles.)


Let 0<λ_1 < λ_2 ≤λ_3 ≤… be the complete spectrum of Laplacian on a unit square domain D=  0,1 ^2 ⊂ℝ^2. Let ϕ_j be a real valued Dirichlet eigenfunction on D with the eigenvalue λ_j. Assume that ϕ_j does not have any interior singular point, where both the value and the gradient vanishes. Then we have

    N(ϕ_j)/j≤2/π + O(j^-1/2),

as j →∞.


For any coprime integers a,b, and a real parameter t∈ℝ, sin (π a x) sin (π b y) + tsin (π b x) sin (π a y) is a real valued Dirichlet eigenfunction on D with the eigenvalue π^2(a^2+b^2). It has no interior singular point for all but finitely many t∈ℝ.

The main idea of the proof is to construct a mesh, where most of the connected components of the nodal set have to intersect at least twice. By counting the number of zeros of the eigenfunctions on the mesh, we find an upper bound for the number of connected components of the nodal set. When an eigenfunction does not have any singular point, the number of connected components of the nodal set +1 is the number of nodal domains on D, from which Theorem <ref> follows. We describe details in the following section.

In Section <ref>, we demonstrate how one can improve the counting and go beyond (<ref>) by considering different types of meshes for eigenfunctions on a flat torus. 

In Section <ref>, we consider nodal domains of eigenfunctions on a closed surface. In particular, we prove a formula that relates the number of nodal domains with the number of singular points and the number of connected components of the nodal set. As an application, we give a quick proof of an upper bound for the number of singular points that is first obtained in <cit.> and <cit.>.



§ PROOF OF THE MAIN THEOREM

Let Z_ϕ = {0<x,y<1 : ϕ(x,y) = 0} be the interior nodal set. Then each connected component of Z_ϕ is either a closed curve or a segment touching the boundary ∂ D. We label connected components of Z_ϕ those are segments  by Z_ϕ,s (1) , … , Z_ϕ,s (N_s), and those are closed curves by Z_ϕ,c (1) , …, Z_ϕ,c (N_c). Firstly, note that we have the following equality for the number of nodal domains of ϕ

    N(ϕ) = N_s+N_c+1.


Now we consider a mesh M in D given by

    M_τ = ⋃_k=1^⌊√(τ-π^2)/π⌋{(π k/√(τ-π^2),y)  :  0≤ y ≤ 1},

which is the nodal set of sin√(τ-π^2) x sinπ y. We may choose τ > λ such that M_τ intersect Z_ϕ transversally. (Note that τ can be chosen arbitrarily close to λ.)

Any Z_ϕ,c(j) should intersect M_τ at least twice.


If some Z_ϕ,c(j) does not intersect M_τ, then we have a connected component of D - M_τ that contains a nodal domain of ϕ. This is contradiction to τ>λ. Since Z_ϕ,c(j) is a closed curve and any connected component of M_τ divides D into two,  the intersection has to occur at least twice.



ϕ has at most √(λ)/π sign changes on any fixed component of M_τ.


ϕ restricted to any connected component of M_τ is a linear combination of 2i sin (π m y)=e^π i m y-e^-π i m y where 1≤ m ≤√(λ)/π, which is a degree < √(λ)/π polynomial in e^π i y+e^-π i y when divided by e^π i  y-e^-π i  y. Therefore it has at most √(λ)/π sign changes.


From these two lemma, we conclude that

    2N_c ≤√(τ-π^2)/π√(λ)/π,

for any τ>λ.

In order to bound N_s, we first note that each Z_ϕ,s has two end points on ∂ D. Also note that there exists a sufficiently small ϵ>0, such that the number of end points on

    {(0,y) : 0 ≤ y ≤ 1}

is equal to the number of zeros of ϕ on

    {(ϵ,y) : 0 < y < 1}.

It follows from the same argument in Lemma <ref> applied to this segment that there are at most √(λ)/π end points of Z_ϕ,s on

    {(0,y) : 0 ≤ y ≤ 1},

hence there are at most 4√(λ)/π end points on ∂ D. Therefore

    N_s ≤2√(λ)/π,

and combining with (<ref>), we have

    N(ϕ) ≤λ/2π^2 + 2√(λ)/π+1.

Now Theorem <ref> follows from the Weyl law j = λ/4π+O(√(λ)).





§ FURTHER DEVELOPMENT

To begin with, we define 𝒩(θ) to be the set of positive integer n such that there exists a sector S_α, θ:={(r,φ) ∈ℝ^2 : α<φ <α+θ, r≥ 0} for some α so that no solution (a,b) ∈ℤ^2 to a^2+b^2=n is contained in S_α, θ. For instance, if the number of distinct solutions to a^2+b^2=n is less than 2π/θ, then n is in 𝒩(θ).

Fix θ and ϵ, such that θ>ϵ>0. Let 0<λ_1 < λ_2 ≤λ_3 ≤… be the complete spectrum of Laplacian on a flat torus 𝕋= ℝ^2/ℤ^2. Let ϕ_j be a real valued Laplacian eigenfunction on 𝕋 with the eigenvalue λ_j. Assume that ϕ_j does not have any singular point, and assume further that n_j:=λ_j/(4π^2) ∈𝒩(2θ). Then we have that

    N(ϕ_j)/j≤2/πcos (θ-ϵ) +O_ϵ(j^-1/2).



For any pair of integers a and b, cos 2π(ax+by) is an eigenfunction that has no singular point. (In fact, in each eigenspace E_λ, the set of eigenfunctions with at least one singular point has codimension ≥ 1 <cit.>.)  So the Theorem applies to cos 2π (ax+by) with θ = π/9, whenever a^2+b^2=p is a prime ≡ 1 4, since x^2+y^2=p has exactly 8 integral solutions.


The existence of a sequence of eigenfunctions on 𝕋 for which we have stronger upper bound than (<ref>) was known in <cit.>. In <cit.>, Bourgain adopted the idea of Nazarov–Sodin <cit.> to a certain sequence of eigenfunctions on 𝕋 and proved that

    N(ϕ_j)/j∼κ

for such eigenfunctions ϕ_j, where κ>0 is the constant given in <cit.> (it is known that κ≤1/√(2)π <2/π <cit.>). One of the necessary conditions that the eigenfunctions ϕ_j should satisfy is that the set

    {(a/√(a^2+b^2),b/√(a^2+b^2)) : a,b∈ℤ,  n_j =a^2+b^2}

being equidistributed on the unit circle as j→∞.  Note that this is mutually exclusive to the condition that the eigenfunctions in Theorem <ref> satisfy.



We first fix a finite set of points P(ϵ) in ℤ^2 such that S_α, ϵ contains at least one element of P(ϵ) for any α. We may assume without loss of generality that n_j>x^2+y^2 for any (x,y) ∈ P(ϵ).

From the assumption that n_j ∈𝒩(2θ), there exists a point (p,q) ∈ P(ϵ) such that

    |(a,b)(p,q)| ≤√(n_j)√(p^2+q^2)cos (θ -ϵ)

for any (a,b) with a^2+b^2 = n_j.

Writing

    ϕ_j (x,y)= ∑_a^2+b^2 =n_jα_ab e^2π i (a,b)(x,y),

we see that for any fixed τ≥ 0,

    ϕ_j (p(t-τ), qt) = ∑_a^2+b^2 =n_jα_abe^ - 2π i aτ p e^2π i (a,b)(p,q) t

is a trigonometric polynomial of length ≤ 2√(n_j)√(p^2+q^2)cos (θ-ϵ) +1.

Now let C(p,q) be a closed geodesic on 𝕋, that is an image of

    {(pt, qt)  :  0≤ t≤ 1}∈ℝ^2.

For any small τ>0 we let M(p,q)_τ be the union of shifts of C(p,q),

    C(p,q)+ (τ,0),  C(p,q)+ (2τ,0) , ⋯,  C(p,q) + (⌈1/qτ⌉τ,0),

and the image of

    E={(x,0) : 0≤ x ≤ 1}∪{(0,y) : 0≤ y ≤ 1}⊂ℝ^2

in 𝕋.

Considering

    sinπ/τ(x - p/qy),

we see that the region in 𝕋 bounded by C(p,q)+kτ and C(p,q)+(k+1)τ has the first Dirichlet eigenvalue

    π^2/τ^2 (1+p^2/q^2).

Therefore if we choose τ such that

    π^2/τ^2 (1+p^2/q^2) = λ_j,

then any connected component of the nodal set of ϕ_j should intersect M(p,q)_τ at least twice.

Denoting by C(ϕ_j) the number of connected components of the nodal set of ϕ_j, we have

    2C(ϕ_j) ≤ 2 √(n_j)√(p^2+q^2)cos (θ-ϵ) ⌈1/qτ⌉ + 4√(n_j)

where the latter term amounts to the intersection between the nodal set and E.

Note that when ϕ_j has no singular points, we have C(ϕ_j)+1 ≥ N(ϕ_j), we therefore conclude that

    N(ϕ_j)    ≤√(n_j)√(p^2+q^2)cos (θ-ϵ) ⌈1/qτ⌉ + 2√(n_j)
       < √(n_j)√(p^2+q^2)cos (θ-ϵ) ( 1/qτ +1)+2√(n_j)
       <√(n_j λ_j)/πcos (θ-ϵ) + O_ϵ (n_j^1/2)
       =λ_j/2π^2cos (θ-ϵ) +O_ϵ (λ_j^1/2),

and the theorem follows from the Weyl law j = λ_j/4π+O(√(λ_j)).



§ ON CLOSED SURFACES

We first prove an inequality for a graph embedded on a closed surface.

Let G be a graph embedded on a closed surface M with genus 𝔤. Let v, e, f, and c be the number of vertices, edges, faces, and connected components of G. Then we have

    1 ≥ v-e+f-c ≥ 1-2𝔤.



The right hand side is (6.1) of <cit.>. To prove the left hand side inequality, we first note that when 𝔤=0, we have the equality

    v-e+f-c=1.

We are going to prove the other cases by induction on c+𝔤. Assume that the inequality is true for all c and 𝔤 such that c+𝔤 =N. For a pair of a graph G and a closed surface M, we consider two cases:

  
  * Every face is homeomorphic to a disk.
  
  * Otherwise.

In the first case, we have c=1, and v-e+f =2-2𝔤 (Euler's theorem), so the inequality is satisfied.

In the second case, we can find a face of G that contains a closed curve C that is not contractible within the face. M-C is a surface with two boundary components, so we consider a closed surface M̃ obtained by attaching two caps on each boundary component. If M̃ is connected, then we have v=ṽ, e=ẽ, c=c̃,  f̃≥ f, and 𝔤̃=𝔤-1, hence by the induction hypothesis ṽ-ẽ+f̃-c̃≤ 1, we see that v-e+f-c ≤ 1.

If M̃ is disconnected, then we have two graphs embedded on surfaces: (G_1, M̃_1) and (G_2,M̃_2). If both G_1 and G_2 are non-empty, then c_j+𝔤_𝔧≤ N for j=1,2, hence

    v_j -e_j+f_j - c_j ≤ 1

for j=1,2 by the induction hypothesis. Now because v=v_1+v_2, e=e_1+e_2, f=f_1+f_2-1, and c=c_1+c_2, we conclude that

    v-e+f-c ≤ 1.

Now, we consider the case that G_2 is empty. In this case, 𝔤_2 has to be greater than equal to 1, because of the assumption that C is not contractible. Hence 𝔤_1 ≤𝔤 - 1, and so we can apply the induction hypothesis to (G_1, M̃_1). Because v=v_1, e=e_1, f=f_1, and c=c_1, this completes the proof.


This lemma allows one to express the number of nodal domains in terms of the number of connected components of the nodal set, and the order of vanishing of singular points. 

Let M be a smooth compact Riemannian surface. Let ϕ be a real valued Laplacian eigenfunction on M. We denote by C(ϕ) the number of connected components of the nodal set Z_ϕ, by Σ(ϕ) the set of all singular points of ϕ. Then we have

    N(ϕ) = C(ϕ) +∑_p ∈Σ(ϕ) (ord_x=pϕ(x)-1) + O(1)



As done in <cit.>, we may give a graph structure to the nodal set. In this context, we have N(ϕ) = f, C(ϕ) = c. To handle v and e, observe that

    2e = ∑_x ∈ V deg x = 2v+∑_p ∈Σ(ϕ) 2(ord_x=pϕ(x)-1),

hence

    e-v = ∑_p ∈Σ(ϕ) (ord_x=pϕ(x)-1).

Now the theorem follows by Lemma <ref>.

As a direct application, we give a short proof of an upper bound for the number of singular points of an eigenfunction on a surface, that is first obtained in <cit.> and <cit.>:

For a Laplacian eigenfunction ϕ on a surface with eigenvalue λ, we have

    ∑_p ∈Σ(ϕ) (ord_x=pϕ(x)-1) =O(λ),

as λ→∞.


Courant's general nodal domain theorem implies that N(ϕ_j) ≤ j, and Weyl's law implies that j is asymptotically proportional to λ_j. Therefore we have

    ∑_p ∈Σ(ϕ) (ord_x=pϕ(x)-1) ≤ N(ϕ) + O(1) = O(λ).


Another application is the following

For an eigenfunction on a generic surface, we have

    N(ϕ) = C(ϕ) + O(1)

as λ→∞.


It is proven in <cit.> that eigenfunctions on a generic surface do not have any singular points.

In particular, as done in previous sections, we may bound the number of nodal domains by counting the number of connected components of the nodal set on generic surfaces. However, the input that is missing in this case is the sharp upper bound for the number of intersection between the nodal set and a curve.

alpha



