The Spectral Position of Neumann Domains on the Torus

Abstract

Neumann domains of Laplacian eigenfunctions form a natural counterpart of nodal domains. The restriction of an eigenfunction to one of its nodal domains is the first Dirichlet eigenfunction of that domain. This simple observation is fundamental in many works on nodal domains. We consider a similar property for Neumann domains. Namely, given a Laplacian eigenfunction f and its Neumann domain \(\Omega \), what is the position of \(\left. f\right| _{\Omega }\) in the Neumann spectrum of \(\Omega \)? The current paper treats this spectral position problem on the two-dimensional torus. We fully solve it for separable eigenfunctions on the torus and complement our analytic solution with numerics for random waves on the torus. These results answer questions from (Band and Fajman in Ann Henri Poincaré, 17(9):2379–2407, 2016; Zelditch in Surv Differ Geom 18:237–308, 2013) and raise new ones.

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Notes

  1. 1.

    The reason for this is that the boundary of a general Neumann domain might include a cusp and in general we do not have an explicit expression of the cusp.

  2. 2.

    We thank Michael Levitin for suggesting this experiment and pointing out FEM++ for this purpose, [*, [19]].

  3. 3.

    To apply the theory in [7, 28] for our case, we take the group to be \(C_{2}\times C_{2}\) (the direct product of two copies of the cyclic group, \(C_{2}\)) with its regular representation.

  4. 4.

    We thank John Hannay for pointing out this interesting geometrical meaning to us.

  5. 5.

    As a matter of fact, \(y_{0}=-b\) also gives a Neumann line, but it is connected to a different saddle point.

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Acknowledgements

We are grateful to Michael Levitin for his encouraging comments and useful ideas for further investigations. We would like to thank Emanuel Milman for stimulating discussions and for pointing out helpful references. We thank Gregory Berkolaiko and Mark Dennis for interesting discussions in various stages of this ongoing work. We thank Luc Hillairet and Graham Cox for pointing out to us the second proof of Theorem 1.4,(1). Band and Egger were supported by ISF (Grant No. 494/14). Taylor was funded by the Leverhulme Trust Research Programme Grant No. RP2013-K-009.

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A. The Boundary of \(\Omega _{a,b}^{\mathrm {star}}\) and its Area

A. The Boundary of \(\Omega _{a,b}^{\mathrm {star}}\) and its Area

We consider a separable eigenfunction \(f_{a,b}\) on the torus, (8), and its star-like Neumann domain, \(\Omega _{a,b}^{\mathrm {star}}\). In this appendix, we derive the explicit expression for the boundary of \(\Omega _{a,b}^{\mathrm {star}}\) (Lemma 7.1) and show that it is of class C (Lemma 7.3). This boundary characterization is needed to justify the application of some Sobolev space analysis (done in Proposition 3.1). Furthermore, we perform here an asymptotic calculation of the \(\Omega _{a,b}^{\mathrm {star}}\) area (Lemma 7.4) which is used in the proofs of Theorem 1.4, (1) (first proof) and Lemma 5.2.

Lemma 7.1

We have

$$ \Omega _{a,b}^{\mathrm {star}}=\left\{ \left( x,y\right) ~:~\left| x\right|<a,~~\left| y\right| <\gamma _{a,b}(x)\right\} , $$

where

$$\begin{aligned} \gamma _{a,b}(x):=\frac{2b}{\pi }\arcsin \left( \left[ \cos \left( \frac{\pi }{2a}x\right) \right] ^{\left( \frac{a}{b}\right) ^{2}}\right) . \end{aligned}$$
(33)

Proof

To prove the lemma, we parameterize the Neumann line which connects the extremal point, (a, 0) to the saddle point, (0, b) (see Fig. 2,(ii)) and show that it is given by (33). The other four Neumann lines which form the boundary of \(\Omega _{a,b}^{\mathrm {star}}\) are obtained by noting that \(\Omega _{a,b}^{\mathrm {star}}\) is symmetric with respect to horizontal and vertical reflections (see Fig. 3). Plugging the expression of the eigenfunction (8) in the flow equations (2), we get

$$ \begin{pmatrix}\dot{x}\\ \dot{y} \end{pmatrix}=-\frac{\pi }{2}\begin{pmatrix}a^{-1}\cos \left( \frac{\pi }{2a}x\right) \cos \left( \frac{\pi }{2b}y\right) \\ b^{-1}\sin \left( \frac{\pi }{2a}x\right) \sin \left( \frac{\pi }{2b}y\right) \end{pmatrix}. $$

Hence, the tangent to any gradient flow line is

$$ \frac{\mathrm {d}y}{\mathrm {d}x}=\frac{a}{b}\tan \left( \frac{\pi }{2a}x\right) \tan \left( \frac{\pi }{2b}y\right) . $$

Integrating this, we obtain the gradient flow lines

$$\begin{aligned} y(x)=\frac{2b}{\pi }\arcsin \left( \sin \left( \frac{\pi }{2b}y_{0}\right) \left[ \cos \left( \frac{\pi }{2a}x\right) \right] ^{\left( \frac{a}{b}\right) {}^{2}}\right) , \end{aligned}$$
(34)

where \((0,y_{0})\) is a point through which the gradient flow line passes. Note that for \(-b{<}y_{0}{<}b\), each of the gradient flow lines in (34) is connected to the extremal point (a, 0), but only the one with \(y_{0}=b\) is connected to the saddle point (0, b) and hence it is the desired Neumann lineFootnote 5. \(\square \)

Remark 7.2

From the proof of Lemma 7.1, one may also obtain that there is no gradient flow line which connects two saddle points of the eigenfunction \(f_{a,b}\). From this, we conclude that \(f_{a,b}\) is a Morse–Smale function [5, Proposition A.7].

The next lemma shows that the boundary of \(\Omega _{a,b}^{\mathrm {star}}\) is regular enough for applying an appropriate Sobolev space analysis. The classification of the boundary in the lemma is based on [12, Definition 4.1].

Lemma 7.3

The boundary of the star-like domain, \(\partial \Omega _{a,b}^{\mathrm {star}}\), is of class C.

Namely, for any \(\varvec{p}\in \partial \Omega _{a,b}^{\mathrm {star}}\), there exists an open neighborhood \(U(\varvec{p})\subset \mathbb {R}^{2}\) and a continuous function \(h\in C(I)\) on an interval \(I\subset \mathbb {R}\) such that for suitable local Cartesian coordinates

$$\begin{aligned} \partial \Omega _{a,b}^{\mathrm {star}}\cap U(\varvec{p})=\left\{ (s,t):\ t=h(s),\ s\in I\right\} \end{aligned}$$
(35)

holds.

Proof

Since the boundary consists of gradient flow lines, the claim is obvious for every point \(\varvec{p}\) not being an end point of such a flow line (i.e., for every \(\varvec{p}\) which is not a critical point). At a saddle point, any two adjacent Neumann lines meet with an angle of \(\tfrac{\pi }{2}\), [27, Theorem 3.2.]. Hence, the boundary at a neighborhood of a saddle point is also a continuous function. At the extremal points \((\pm a,0)\), adjacent Neumann lines meet with an angle of 0 and form a cusp. We derive the asymptotics of \(\gamma _{a,b}(a-x)\), \(x\rightarrow 0^{+}\). Using

$$\begin{aligned} \cos \left( \frac{\pi }{2a}(a-x)\right)&=\sin \left( \tfrac{\pi }{2a}x\right) =\tfrac{\pi }{2a}x+\mathrm {O}\left( \left( \tfrac{\pi }{2a}x\right) {}^{3}\right) \\ (1+x)^{\beta }&=1+\mathrm {O}(x)\quad \quad \text {for }\beta >0\\ \arcsin (x)&=x+\mathrm {O}\left( x^{3}\right) , \end{aligned}$$

we get that for \(x\rightarrow 0^{+}\)

$$\begin{aligned} \gamma _{a,b}(a-x)=\frac{2b}{\pi }\arcsin \left( \left[ \cos \left( \frac{\pi }{2a}(a-x)\right) \right] ^{\left( \frac{a}{b}\right) ^{2}}\right) =\frac{2b}{\pi }\left( \frac{\pi }{2a}x\right) {}^{\left( \frac{a}{b}\right) {}^{2}}+\mathrm {O}\left( x^{3\left( \frac{a}{b}\right) {}^{2}}\right) . \end{aligned}$$
(36)

These asymptotics show that \(\gamma _{a,b}\) is strictly monotonically decreasing in a left neighborhood of (a, 0) and its inverse exists there. Hence, the condition (35) is satisfied in a neighborhood of (a, 0) by choosing

$$ h(s)={\left\{ \begin{array}{ll} \gamma _{a,b}^{-1}(s) &{} s>0\\ \gamma _{a,b}^{-1}(-s) &{} s<0 \end{array}\right. }. $$

\(\square \)

Finally, we use the expression of \(\gamma _{a,b}\) to bound the area of \(\Omega _{a,b}^{\mathrm {star}}\) which is needed in the proofs of Theorem 1.4,(1) (first proof) and Lemma 5.2 (see (28) in that proof).

Lemma 7.4

There exists \(c>1\) such if then

$$\begin{aligned} \frac{1}{ab}\left( \frac{b}{a}+\frac{a}{b}\right) \left| \Omega _{a,b}^{\mathrm {star}}\right| <\frac{2}{\pi }(j_{0,1})^{2} \end{aligned}$$
(37)

where \(j_{0,1}\approx 2.4048\) is the first zero of \(J_{0}\), the zeroth Bessel function.

Proof

Using Lemma 7.1 we have

(38)

We may use the Taylor expansion of \(\mathrm {ln}\left[ \cos \left( z\right) \right] \), which converges for \(\left| z\right| <\frac{\pi }{2}\) (see e.g., [1, 4.3.72] and [23, p. 27]) to obtain the bound

$$\begin{aligned} \forall z\in (0,\frac{\pi }{2}),\quad \left[ \cos \left( z\right) \right] ^{\left( \frac{a}{b}\right) ^{2}}<\exp \left[ -\frac{1}{2}\left( \frac{a}{b}\right) ^{2}z^{2}\right] . \end{aligned}$$
(39)
Fig. 9
figure9

The left-hand side of (37) plotted as a function of . The right-hand side of (37) is indicated together with the corresponding value

Fig. 10
figure10

The first eigenvalue of \(-\Delta ^{h}_{a,b}\), the first eigenvalue of \(-\Delta ^{v}_{a,b}\), and \(\lambda _{a,b}\) plotted as a function of (i.e., we chose \(a=1\))

Another bound which we use is

(40)

To validate (40), we may observe that both functions at the RHS and LHS coincide for \(w=0\) and \(w=1\) and further check that the difference does not vanish anywhere else in \(\left( 0,1\right) \) (for example, since the difference has only a single critical point in this interval).

Plugging the bounds (39), (40) in (38) and using also the monotonicity of \(\arcsin (w)\) for \(w\in (0,1)\) we get

(41)

where moving to the last line we used integration over (half) Gaussian.

From the above, we get \(\frac{1}{ab}\left( \frac{a}{b}+\frac{b}{a}\right) \left| \Omega _{a,b}^{\mathrm {star}}\right| \lessapprox 2.7014\cdot \left( 1+\left( \frac{b}{a}\right) ^{2}\right) \). Now, since \(\frac{2}{\pi }(j_{0,1})^{2}\approx 3.68\) we get that (37) holds if \(\frac{b}{a}\) is small enough. \(\square \)

Remark 7.5

From the proof, one easily gets that (37) holds for . Numerically, it seems that choosing \(c\approx 1.1407\) already guarantees this bound. This can be seen in Fig. 9 and shows that the methods in the proof of Theorem 1.4, (2) cannot reduce the constant in the theorem below \(c\approx 1.1407\). Yet, a numerical experiment (see Figure 10) shows that the statement of the theorem should be valid also for \(c=1\) (which is the optimal result).

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Band, R., Egger, S.K. & Taylor, A.J. The Spectral Position of Neumann Domains on the Torus. J Geom Anal (2020). https://doi.org/10.1007/s12220-020-00444-9

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Keywords

  • Neumann domains
  • Neumann lines
  • nodal domains
  • Laplacian eigenfunctions
  • Morse–Smale complexes

Mathematics Subject Classification

  • 58C40
  • 58J50
  • 35P05