Nodal lengths of eigenfunctions in the disc
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.
Key words and phrases:Laplacian eigenfunctions, nodal sets, geodesics
2010 Mathematics Subject Classification:58J50, 35J05, 35P15
In an - smooth and compact Riemannian manifold , let be the Laplacian and be an eigenfunction with eigenvalue , i.e. . If has smooth boundary, we impose Dirchlet or Neumann boundary condition. Yau [Y] conjectured that
for some constants depending on and independent of the eigenvalues . Here, denotes the - Hausdorff measure and denotes the nodal set of the function .
In this paper, we are concerned with the precise and sharp dependence of the constants and in (1.1) on the geometry . That is, write
Regarding the two limits in (1.2), we also pursue the categorization of the eigenfunctions (in relation to the geometry of the manifold) which saturate and of as . That is, what geometric properties do the nodal sets of these eigenfunctions achieving the limits in (1.2) 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 3 and Problem 4.
We begin from the easiest case, an interval . The -th Dirichlet eigenfunction is
The nodal set (in the interior of the domain) of is a collection of nodal points
Hence, the size of the nodal set of the -th Dirichlet eigenfunction on , i.e. the number of nodal points, is
Therefore, in (1.2) we have that
One can similarly show the same results for Neumann eigenfunctions in .
If , then very little is known about (1.2). To the authors’ knowledge, the first result in this direction is due to Brüning-Gromes [BG, Equation (8)]: They remarked that in an irrational rectangle with side-lengths and for which is irrational,
However, and are not known in more general rectangles. In §2.1, we discuss finding and in the rectangles and the tori.
Gichev [G, Theorem 3] proved that on the - unit sphere ,
in which is the volume of . For example, . 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
in which is the homogeneous degree of . Moreover, the equation in the above inequality is obtained by the Gaussian beams (i.e. highest weight spherical harmonics). One then deduces (1.4) by observing that . In the same paper [G, Page 563], Gichev conjectured that on ,
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 and in (1.2) are explicitly proved, see Theorem 1; moreover, we observe the geometric properties of the eigenfunctions which achieve the as , see Corollary 3.
Let be the unit disc. Consider the eigenfunctions with Dirichlet boundary condition. In polar coordinates , the real-valued Dirichlet eigenfunctions are
Here, is the -th Bessel function and , where is the -th nonnegative zero of , and . So all the Dirichlet eigenvalues are the squares of zeros of Bessel functions. In particular, the eigenvalues are distinct for different values of and , (c.f. [W, Section 15.23]) and the multiplicity of is two with eigenspace spanned by
The nodal set (in the interior of the disc) of the eigenfunction with eigenvalue is a collection of radials (i.e. diameters) and concentric circles with radii , . Our main theorem states that
In the disc , let be a Dirichlet eigenfunction with eigenvalue . Then
in which the limit is achieved by the eigenfunctions as ; and
in which the limit is achieved by the eigenfunctions as .
A simple dilation gives
In the disc , let be a Dirichlet eigenfunction with eigenvalue . Then
where, as before, is the first zero of the Bessel function . On a smooth Riemannian surface , Savo [S, Theorem 13] proved that
So our calculation of in Corollary 2 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 or of as . Here, we make the observation that the nodal set of
is a collection of diameters that pass through the origin. Hence,
In the disc ,
is saturated by a sequence of eigenfunctions whose nodal curves in the interior are geodesics, i.e. pieces of straight lines.
Recall that on proved by Gichev [G], is saturated by the Gaussian beams, whose nodal sets are totally geodesic. Based on these evidence, we propose the following problem.
In what manifold , one has that
for a sequence of eigenfunctions with eigenvalues such that the nodal sets of are totally geodesic in the interior of ?
The answer to Problem 4 is positive on the spheres by Gichev [G] and in the disc by Corollary 3. In the irrational rectangles (see §2.1), the nodal curves of all eigenfunctions are geodesics so the answer to Problem 4 is trivially positive. It would be interesting to see if Problem 4 holds in other rectangles (or tori). On a general manifold, the answer to Problem 4 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 . So a natural question follows as
In what manifold , ?
In the case when , there is a unique limit of as 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 [Z1, Corollary 1.2], Zelditch showed that on an analytic manifold with ergodic geodesic flow, there is a full density subsequence of eigenfunctions for which has a unique limit as . Here, denotes the complex extension of the eigenfunctions and denotes the complex hypersurface measure of the nodal set of . 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 5 might be positive.
2. Proof of Theorem 1
2.1. The irrational rectangles and tori
Before proving Theorem 1, we discuss the proof of (1.3) in an irrational rectangle , where is irrational. (This is observed in [BG].) We then make some remarks about finding and in more general rectangles and tori.
The Dirichlet eigenfunctions in have the form
with the eigenvalues
Notice that if is irrational, then all the eigenvalues are simple. Indeed, if for another pair , , then
which forces since is irrational. The nodal set (in the interior of ) consists of line segments of length and line segments of length . So the nodal length of is
One then sees from for that
in which the limit is achieved by as and by as , and
in which the limit is achieved by such that as .
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 , there are eigenvalues with multiplicity of the order as . In the case of high multiplicity, one has to estimate the precise nodal lengths of linear combinations of eigenfunctions of the form (2.1). These linear combinations have complex nodal portraits and the problem of finding and becomes challenging.
In the disc, the eigenvalues have multiplicity two. Therefore, in the following subsection, we can use explicit formulae to deduce and .
If we identify the two opposing sides of the rectangle and define the torus (i.e. without boundary), then the real-valued eigenfunctions are spanned by
Given that is irrational, the eigenvalues are not simple (except when ) but their multiplicity is uniformly bounded by . A similar argument as in the corresponding irrational rectangle shows that the same results of and in (2.2) and (2.3) hold on the torus. However, and remain unknown on other tori, for the same reason as described in the above remark.
2.2. Proof of Theorem 1
Now we prove Theorem 1. Recall that the Dirichlet eigenfunction with eigenvalue has the form
Since the nodal length is independent of here, we assume without loss of generality. The nodal set of is a collection of diameters and concentric circles with radii , . In particular, consists of circles only and consists of diameters only.
Here, we provide the graphs of the nodal curves of some eigenfunctions with different nodal portrait.
From left to right: , , and . Their eigenvalues are approximately and one can check that , which is a reflection of Theorem 1.
we pick any subsequence of such that
and divide into three cases.
Case 1: tends to infinity and is bounded;
Case 2: tends to infinity and is bounded;
Case 3: and both tend to infinity.
Case 1. As , is bounded so . First set . Then the nodal set of is the union of radials, that is,
Here, we use the fact that from [AS, Equation 9.5.14, pp. 371],
This argument works for all the subsequences of for which is bounded and . Indeed, If is bounded by , then
So by squeezing,
Here, we need to use a formula [AS, Equation 9.5.22, pp. 371] that if is bounded, then
Case 2. As , is bounded so . First set . Then the nodal set of is the union of concentric circles with radii , , that is,
Here, we use the fact that from [AS, Equation 9.5.12, pp. 371]: If , then
This argument works for all the subsequences of for which is bounded and . Indeed, If is bounded by , then contains circles and at most diameters. Hence,
So by squeezing, using (2.5) again, we have that
Case 3. As , . Suppose that in such a subsequence
Our goal is then to prove that .
We need the uniform bounds of the zeros of Bessel functions as . By [Bre, Equation (1) in Theorem 1], we have that
Here, is the -th negative zero of the Airy function. By [AS, Equations 10.4.94 and 10.4.105], we have that as ,
We now proceed to prove the upper bound that . Using (2.6),
We then prove the lower bound that . By [Bre, Equation (2) in Theorem 1], we have that
Notice that , , are fractions which are distributed in the interval . If , then by the asymptotic formula (2.5), these fractions are rather equidistributed. So the above inequality is natural in this case. If , then by (2.4) and (2.7), one sees that and therefore for all . 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 and .
Now by (2.7) again,
Hence, the lower bound is proved.
2.3. Neumann eigenfunctions
The Neumann eigenfunctions in can be written as
Here, is the -th Bessel function and , where is the -th nonnegative zero of . 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 . However, notice that Neumann eigenfunctions in extends to and in fact defines a Dirichlet eigenfunction in a slightly larger disc. So we can estimate the nodal set of Neumann eigenfunctions by Corollary 2 for Dirichlet eigenfunctions.
Indeed, the zeros and interlace according to
See [AS, Equation 9.5.2]. Using this relation, we see that extends from to as a Dirichlet eigenfunction with
Now the nodal set of in and the nodal set of in differ by the radials in . That is,
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 4; XH also wants to thank Zeév Rudnick for informing him the results in Gichev [G] and Werner Horn for his translation of Brüning-Gromes [BG].
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