Nodal lengths of eigenfunctions in the disc
Xiaolong Han Michael Murray Chuong Tran

 Novel Sensor Scheduling Scheme for Intruder 

	Tracking in Energy Efficient Sensor Networks 
    Raghuram Bharadwaj D.^1,
        Prabuchandran K.J.^1,
        and Shalabh Bhatnagar^1
^1 Authors are with the Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India


    December 30, 2023
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In this paper, we derive the sharp lower and upper bounds of nodal lengths of Laplacian eigenfunctions in the disc. Furthermore, we observe a geometric property of the eigenfunctions whose nodal curves maximize the nodal length.





§ INTRODUCTION

In an n- smooth and compact Riemannian manifold (,g), let Δ=Δ_g be the Laplacian and u be an eigenfunction with eigenvalue λ, i.e. -Δ u=λ u. If  has smooth boundary, we impose Dirchlet or Neumann boundary condition. Yau <cit.> conjectured that

    c_1√(λ)≤^n-1((u))≤ c_2√(λ)

for some constants 0<c_1,c_2<∞ depending on (,g) and independent of the eigenvalues λ→∞. Here, ^n-1 denotes the (n-1)- Hausdorff measure and (u)={x∈:u(x)=0} denotes the nodal set of the function u.

If the metric g is analytic, then (<ref>) was proved by Donnelly-Fefferman <cit.> (see also Lin <cit.> for the upper bound); if g is smooth, then Logunov <cit.> recently showed that 

    c_1√(λ)≤^n-1((u))≤ c_2λ^α,

in which α>1/2 depends only on n=. There are partial results in this direction <cit.> etc, c.f. the survey by Zelditch <cit.>.

In this paper, we are concerned with the precise and sharp dependence of the constants c_1 and c_2 in (<ref>) on the geometry (,g). That is, write

    H_1()=lim inf_λ→∞^n-1((u))/√(λ)  and   H_2()=lim sup_λ→∞^n-1((u))/√(λ).

Thanks to <cit.>, we know that 0<H_1≤ H_2<∞ on analytic manifolds and H_1>0 on smooth manifolds. 

Regarding the two limits in (<ref>), we also pursue the categorization of the eigenfunctions (in relation to the geometry of the manifold) which saturate lim inf and lim sup of ^n-1((u))/√(λ) as λ→∞. That is, what geometric properties do the nodal sets of these eigenfunctions achieving the limits in (<ref>) have? Our primary interest is to categorize the sequence of eigenfunctions whose nodal curves are geodesics in the manifold, if these eigenfunctions exist. See Corollary <ref> and Problem <ref>.

We begin from the easiest case, an interval 𝕀_L=[0,L]. The j-th Dirichlet eigenfunction is

    u_j(x)=sin(jπ/Lx)   with eigenvalue λ_j=(jπ/L)^2, j=1,2,3,...

The nodal set (in the interior of the domain) of u_j is a collection of nodal points

    (u_j)={L/j· l:l=1,...,j-1}.

Hence, the size of the nodal set of the j-th Dirichlet eigenfunction u_j on [0,L], i.e. the number of nodal points, is 

    ^0((u_j))=j-1=L√(λ_j)/π-1,   j=1,2,3,...

Therefore, in (<ref>) we have that

    H_1(𝕀_L)=H_2(𝕀_L)=lim_λ→∞^0((u))/√(λ)=1/πL.

One can similarly show the same results for Neumann eigenfunctions in [0,L].

If ≥2, then very little is known about (<ref>). To the authors' knowledge, the first result in this direction is due to Brüning-Gromes <cit.>: They remarked that in an irrational rectangle R with side-lengths a and b for which a^2/b^2 is irrational, 

    H_1(R)=1/π(R)  and   H_2(R)=√(2)/π(R).

However, H_1 and H_2 are not known in more general rectangles. In <ref>, we discuss finding H_1 and H_2 in the rectangles and the tori.

Gichev <cit.> proved that on the n- unit sphere ^n,

    H_2(^n)=lim sup_λ→∞^n-1((u))/√(λ)=(^n-1)=Γ((n+1)/2)/Γ(n/2)√(π)(^n),

in which (^n-1) is the volume of ^n-1. For example, H_2(^2)=2π=(^2)/2. In fact, the eigenfunctions on the sphere are spherical harmonics (i.e. homogeneous harmonic polynomials restricted to the sphere) and Gichev proved a stronger result that 

    ^n-1((u))≤ k(^n-1),

in which k is the homogeneous degree of u. Moreover, the equation in the above inequality is obtained by the Gaussian beams (i.e. highest weight spherical harmonics). One then deduces (<ref>) by observing that λ=k(k+n-1). In the same paper <cit.>, Gichev conjectured that on ^2,

    H_1(^2)=lim inf_λ→∞^1((u))/√(λ)=4=1/π(^2),

and the limit is achieved by the zonal harmonics.

Our main result is to provide the case in the disc for which both of the sharp constants H_1 and H_2 in (<ref>) are explicitly proved, see Theorem <ref>; moreover, we observe the geometric properties of the eigenfunctions which achieve the lim sup^1((u))/√(λ) as λ→∞, see Corollary <ref>.

Let _1={(x,y)∈^2:x^2+y^2<1} be the unit disc. Consider the eigenfunctions with Dirichlet boundary condition. In polar coordinates {(r,θ):0≤ r≤1,0≤θ<2π}, the real-valued Dirichlet eigenfunctions are

    u_k,s(r,θ)=J_k(√(λ_k,s)· r)sin(kθ+θ_0),  where k=0,1,2,3,... and s=1,2,3,...

Here, J_k is the k-th Bessel function and λ_k,s=j_k,s^2, where j_k,s is the s-th nonnegative zero of J_k, and θ_0∈[0,2π). So all the Dirichlet eigenvalues are the squares of zeros of Bessel functions. In particular, the eigenvalues j^2_k,s are distinct for different values of k and s, (c.f. <cit.>) and the multiplicity of j^2_k,s is two with eigenspace spanned by 

    J_k(√(λ_k,s)· r)sin(kθ)  and   J_k(√(λ_k,s)· r)cos(kθ).

The nodal set (in the interior of the disc) of the eigenfunction with eigenvalue λ_k,s=j_k,s^2 is a collection of 2k radials (i.e. k diameters) and s-1 concentric circles with radii j_k,l/j_k,s, l=1,...,s-1.  Our main theorem states that

In the disc _1, let u be a Dirichlet eigenfunction with eigenvalue λ. Then

    H_1(_1)=lim inf_λ→∞^1((u))/√(λ)=1,

in which the limit is achieved by the eigenfunctions u_0,s as s→∞; and

    H_2(_1)=lim sup_λ→∞^1((u))/√(λ)=2,

in which the limit is achieved by the eigenfunctions u_k,1 as k→∞.



The same results as in Theorem <ref> hold for Neumann eigenfunctions in the disc. See the discussion in <ref>.


A simple dilation gives

In the disc _R={(x,y)∈^2:x^2+y^2<R^2}, let u be a Dirichlet eigenfunction with eigenvalue λ. Then

    H_1(_R)=lim inf_λ→∞^1((u))/√(λ)=R^2=1/π (_R)

and

    H_2(_R)=lim sup_λ→∞^1((u))/√(λ)=2R^2=2/π (_R).




Some bounds of c_1 in (<ref>) on Riemannian surfaces are previously known, with which one can have some non-sharp estimates of H_1. In a Euclidean domain Ω⊂^2, Brüning-Gromes <cit.> proved that

    H_1(Ω)≥1/2j_0,1 (Ω),

where, as before, j_0,1≈2.4048 is the first zero of the Bessel function J_0. On a smooth Riemannian surface , Savo <cit.> proved that 

    H_1()≥1/11 ().

So our calculation of H_1 in Corollary <ref> can be regarded as the sharp improvement of these results applied to the disc.


The other problem in question is to characterize the (possible) geometric properties of the eigenfunctions that achieve  lim inf or lim sup of ^1((u))/√(λ) as λ→∞. Here, we make the observation that the nodal set of 

    u_k,1=J_k(√(λ_k,1)· r)sin(kθ+θ_0),
 
is a collection of k diameters that pass through the origin. Hence,

In the disc _R, 

    H_2(_R)=lim sup_λ→∞^1((u))/√(λ)

is saturated by a sequence of eigenfunctions whose nodal curves in the interior are geodesics, i.e. pieces of straight lines.


Recall that on ^n proved by Gichev <cit.>, lim sup^n-1((u))/√(λ) is saturated by the Gaussian beams, whose nodal sets are totally geodesic. Based on these evidence, we propose the following problem.

In what manifold (,g), one has that

    H_2()=lim sup_λ→∞^n-1((u))/√(λ)=lim_k→∞^n-1((u_k))/√(λ_k)

for a sequence of eigenfunctions {u_k}_k=1^∞ with eigenvalues λ_k such that the nodal sets of u_k are totally geodesic in the interior of ?



The answer to Problem <ref> is positive on the spheres by Gichev <cit.> and in the disc by Corollary <ref>. In the irrational rectangles (see <ref>), the nodal curves of all eigenfunctions are geodesics so the answer to Problem <ref> is trivially positive. It would be interesting to see if Problem <ref> holds in other rectangles (or tori). On a general manifold, the answer to Problem <ref> is not known and in fact it is not even known whether there exists a sequence of eigenfunctions whose nodal sets are totally geodesic. 


In all the manifolds that we consider in this paper (irrational rectangles and tori, spheres, and discs), we have that H_1()<H_2(). So a natural question follows as


In what manifold (,g), H_1()=H_2()?



In the case when H_1()=H_2(), there is a unique limit of ^n-1((u))/√(λ) as λ→∞ for all the eigenfunctions. This is not known to be positive on any manifold with dimension higher than one. 

On an analytic manifold, one can extend the Laplacian eigenfunctions to a complex neighborhood of the manifold. In <cit.>, Zelditch showed that on an analytic manifold with ergodic geodesic flow, there is a full density subsequence of eigenfunctions for which [(u^)]/√(λ)
has a unique limit as λ→∞. Here, u^ denotes the complex extension of the eigenfunctions u and [(u^)] denotes the complex hypersurface measure of the nodal set of u^. Even though this result is for the complex extensions of a full density subsequence of eigenfunctions, it suggests that on manifolds with ergodic geodesic flow (e.g. negatively curved manifolds), the answer to Problem <ref> might be positive.





§ PROOF OF THEOREM <REF>



 §.§ The irrational rectangles and tori

Before proving Theorem <ref>, we discuss the proof of (<ref>) in an irrational rectangle R={(x,y)∈^2:0≤ x≤ a,0≤ y≤ b}, where a^2/b^2 is irrational. (This is observed in <cit.>.) We then make some remarks about finding H_1 and H_2 in more general rectangles and tori. 

The Dirichlet eigenfunctions in R have the form

    u_k,j(x,y)=sin(π k/ax)sin(π j/by)

with the eigenvalues

    λ_k,j=(π k/a)^2+(π j/b)^2,  where k,j=1,2,3...

Notice that if a^2/b^2 is irrational, then all the eigenvalues are simple. Indeed, if λ_k̃,j̃=λ_k,j for another pair (k̃,j̃), k̃,j̃=1,2,3..., then 

    k^2-k̃^2/a^2=j̃^2-j^2/b^2,

which forces (k̃,j̃)=(k,j) since a^2/b^2 is irrational. The nodal set (u_k,j) (in the interior of R) consists of (k-1) line segments of length b and (j-1) line segments of length a. So the nodal length of u_k,j is

    ^1((u_k,j))=(k-1)b+(j-1)a.

Hence,

    ^1((u_k,j))/√(λ_k,j)=(R)/π×(k-1)b+(j-1)a/√((kb)^2+(ja)^2).

One then sees from √(p^2+q^2)≤ p+q≤√(2)√(p^2+q^2) for p,q≥0 that 

    H_1(R)=lim inf_λ→∞^1((u_k,j))/√(λ_k,j)=1/π(R),

in which the limit is achieved by u_1,j as j→∞ and by u_k,1 as k→∞, and

    H_2(R)=lim sup_λ→∞^1((u_k,j))/√(λ_k,j)=√(2)/π(R),

in which the limit is achieved by u_k,j such that k/j→ a/b as k,j→∞.



  * The above proof holds with little modification for Neumann eigenfunctions in these irrational rectangles.

  * In a more general rectangles, the eigenvalues may not be simple and can have high multiplicity, e.g. in the rectangle [0,π]×[0,π], there are eigenvalues λ with multiplicity of the order √(logλ) as λ→∞. In the case of high multiplicity, one has to estimate the precise nodal lengths of linear combinations of eigenfunctions of the form (<ref>). These linear combinations have complex nodal portraits and the problem of finding H_1 and H_2 becomes challenging.

  * In the disc, the eigenvalues have multiplicity two. Therefore, in the following subsection, we can use explicit formulae to deduce H_1 and H_2.



If we identify the two opposing sides of the rectangle R=[0,a]×[0,b] and define the torus  (i.e. without boundary), then the real-valued eigenfunctions are spanned by

    sin(2π k/ax±2π j/by)  and  cos(2π k/ax±2π j/by)

with eigenvalue

    (2π k/a)^2+(2π j/b)^2,  where k,j=0,1,2,3...

Given that a^2/b^2 is irrational, the eigenvalues are not simple (except when k=j=0) but their multiplicity is uniformly bounded by 4. A similar argument as in the corresponding irrational rectangle shows that the same results of H_1 and H_2 in (<ref>) and (<ref>) hold on the torus. However, H_1 and H_2 remain unknown on other tori, for the same reason as described in the above remark.



 §.§ Proof of Theorem <ref>

Now we prove Theorem <ref>. Recall that the Dirichlet eigenfunction with eigenvalue λ_k,s=j_k,s^2 has the form

    u_k,s=J_k(√(λ_k,s)· r)sin(kθ+θ_0),  where k=0,1,2,3,... and s=1,2,3,...

Since the nodal length ^1((u_k,s)) is independent of θ_0 here, we assume θ_0=0 without loss of generality. The nodal set of u_k,s is a collection of k diameters and s-1 concentric circles with radii j_k,l/j_k,s, l=1,...,s-1. In particular, (u_0,s) consists of circles only and (u_k,1) consists of diameters only. 

Here, we provide the graphs of the nodal curves of some eigenfunctions with different nodal portrait.


From left to right: u_0,5, u_4,3, and u_10,1. Their eigenvalues are approximately 14^2 and one can check that ^1((u_0,5))<^1((u_4,3))<^1((u_10,1)), which is a reflection of Theorem <ref>.

To estimate 

    H_1(_1)=lim inf_λ→∞^1((u))/√(λ)  and     H_2(_2)=lim sup_λ→∞^1((u))/√(λ),

we pick any subsequence of {u_k,s} such that

    lim_λ→∞^1((u))/√(λ)  exists,

and divide into three cases.



  * Case 1: k tends to infinity and s is bounded;

  * Case 2: s tends to infinity and k is bounded;

  * Case 3: k and s both tend to infinity.


Case 1. As λ=j_k,s^2→∞, s is bounded so k→∞. First set s=1. Then the nodal set of u_k,1 is the union of 2k radials, that is,

    ^1((u_k,1))=2k,

and

    lim_k→∞^1((u_k,1))/√(λ_k,1)=lim_k→∞2k/j_k,1=2.

Here, we use the fact that from <cit.>,

    j_k,1=k+O(k^1/3)  as k→∞.

This argument works for all the subsequences of {u_k,s} for which s is bounded and k→∞. Indeed, If s is bounded by M, then

    2k≤^1((u_k,s))=2π∑_l=1^s-1j_k,l/j_k,s+2k≤2π M+2k.

So by squeezing,

    lim_k→∞, s bounded^1((u_k,s))/√(λ_k,s)=2.

Here, we need to use a formula <cit.> that if s is bounded, then

    j_k,s=k+o(k)  as k→∞.
 

Case 2. As λ=j_k,s^2→∞, k is bounded so s→∞. First set k=0. Then the nodal set of u_0,s is the union of s-1 concentric circles with radii j_0,l/j_0,s, l=1,...,s-1, that is,

    ^1((u_0,s))=2π∑_l=1^s-1j_0,l/j_0,s,

and

    lim_s→∞^1((u_0,s))/√(λ_0,s)=2πlim_s→∞∑_l=1^s-1j_0,l/j_0,s^2=2πlim_s→∞∑_l=1^s(l-1/4)π/[(s-1/4)π]^2=1.

Here, we use the fact that from <cit.>: If k≪ s, then

    j_k,s=(s+k/2-1/4)π+O(s^-1)  as s→∞.

This argument works for all the subsequences of {u_k,s} for which k is bounded and s→∞. Indeed, If k is bounded by M, then (u_k,s) contains s-1 circles and at most M diameters. Hence,

    2π∑_l=1^s-1j_k,l/j_k,s≤^1((u_k,s))=2π∑_l=1^s-1j_k,l/j_k,s+2k≤2π∑_l=1^sj_k,l/j_k,s+2M.

So by squeezing, using (<ref>) again, we have that

    lim_s→∞, k bounded^1((u_k,s))/√(λ_k,s)=1.


Case 3. As λ=j_k,s^2→∞, k,s→∞. Suppose that in such a subsequence

    lim_λ→∞^1((u_k,s))/√(λ_k,s)=p.

Our goal is then to prove that 1≤ p≤2.

We need the uniform bounds of the zeros j_k,s of Bessel functions as k,s→∞. By <cit.>, we have that

    j_k,s>k+2/3|a_s-1|^3/2  for k=0,1,2,3,... and s=1,2,3,...

Here, a_s is the s-th negative zero of the Airy function. By <cit.>, we have that as s→∞,

    a_s=-[3π(4s-1)/8]^2/3[1+O(s^-2)].

Hence,

    j_k,s>k+2/3|a_s-1|^3/2=k+π s+O(s^-1).

We now proceed to prove the upper bound that p≤2. Using (<ref>),

    ^1((u_k,s))/√(λ_k,s)=2π∑_l=1^s-1j_k,l/j_k,s+2k/j_k,s<2π s+2k/k+π s+O(s^-1)≤2  as k,s→∞.

We then prove the lower bound that p≥1. By <cit.>, we have that

    j_k,s<π/2k+2/3|a_s|^3/2=π/2k+π s+O(s^-1)  for k=0,1,2,3,... and s=1,2,3,...

Using (<ref>) and (<ref>), we compute that

    ∑_l=1^s-1j_k,l/j_k,s   ≥   1/π/2k+π s+O(s^-1)∑_l=1^s-1[k+π l+O(l^-1)]
       ≥   π/2(s-1)s+(s-1)k+O(s)/π/2k+π s+O(s^-1)
       ≥   s/2,  if s and k are large enough.



Notice that j_k,l/j_k,s, l=1,...,s-1, are fractions which are distributed in the interval [0,1]. If k≪ s, then by the asymptotic formula (<ref>), these fractions are rather equidistributed. So the above inequality is natural in this case. If s≪ k, then by (<ref>) and (<ref>), one sees that j_k,1/j_k,s≳ k/(π k/2+π s)>1/2 and therefore j_k,l/j_k,s>1/2 for all l=1,...,s-1. So the above inequality is natural in this case as well. The above inequality in fact shows that it is true for all sufficiently large k and s.


Now by (<ref>) again,

    ^1((u_k,s))/√(λ_k,s)=2π∑_l=1^s-1j_k,l/j_k,s+2k/j_k,s≥π s+2k/π/2k+π s+O(s^-1)≥1  as k,s→∞.

Hence, the lower bound is proved.



 §.§ Neumann eigenfunctions

The Neumann eigenfunctions in _1 can be written as

    v_k,s(r,θ)=J_k(√(μ_k,s)· r)sin(kθ+θ_0),  where k=0,1,2,3,... and s=1,2,3,...

Here, J_k is the k-th Bessel function and μ_k,s=(j'_k,s)^2, where j'_k,s is the s-th nonnegative zero of J_k'. So all the Neumann eigenvalues are the squares of zeros of the derivatives of Bessel functions. 

Here for Neumann eigenfunctions, we could repeat the argument in the previous subsection, using instead the estimates of j_k,s'. However, notice that Neumann eigenfunctions in _1 extends to ^2 and in fact defines a Dirichlet eigenfunction in a slightly larger disc. So we can estimate the nodal set of Neumann eigenfunctions by Corollary <ref> for Dirichlet eigenfunctions.

Indeed, the zeros j'_k,s and j_k,s interlace according to

    k≤ j_k,s'<j_k,s<j_k,s+1'.

See <cit.>. Using this relation, we see that v_k,s extends from _1 to _R as a Dirichlet eigenfunction u_k,s with

    R=j_k,s/j'_k,s→1  as k or s→∞.

Now the nodal set of v_k,s in _1 and the nodal set of u_k,s in _R differ by the 2k radials in _R∖_1. That is,

    ^1((v_k,s))=^1((u_k,s))-2k(R-1).

Hence,

    ^1((v_k,s))/j'_k,s=^1((u_k,s))-2k(R-1)/j_k,s· R.

By (<ref>) and (<ref>), we see that

    2k(R-1)/j_k,s→0  as j_k,s→∞.

Then applying Corollary <ref> and again (<ref>), we have that

    lim inf_λ→∞^1((v))/√(μ)=1  and  lim sup_λ→∞^1((v))/√(μ)=2.




§ ACKNOWLEDGMENTS

XH wants to thank Stephen Breen, Andrew Hassell, Hamid Hezari, and Steve Zelditch for all the discussions that are related to this article, in particular, Problem <ref>; XH also wants to thank Zeév Rudnick for informing him the results in Gichev <cit.> and Werner Horn for his translation of Brüning-Gromes <cit.>.

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