Quantum unique ergodicity and the number of nodal domains of eigenfunctions

Abstract

We prove that the Hecke-Maass eigenforms for a compact arithmetic triangle group have a growing number of nodal domains as the eigenvalue tends to $+\infty$. More generally the same is proved for eigenfunctions on negatively curved surfaces that are even or odd with respect to a geodesic symmetry and for which quantum unique ergodicity holds.

1. Introduction

1.1. Nodal domains of eigenfunctions on a surface

Let $(M,g)$ be a smooth compact Riemannian surface without boundary, and let $\{u_n\}$ be an orthonormal Laplacian eigenbasis ordered by the eigenvalue, i.e.,

$$\begin{align*} -\Delta _g u_n &= \lambda _n^2 u_n,\\ \langle u_n,u_m \rangle _M&=\delta _{nm},\\ 0=\lambda _0 &<\lambda _1 \leq \lambda _2 \leq \ldots , \end{align*}$$

where $\Delta _g$ is the Laplace-Beltrami operator on $M$. Here $\langle f,h \rangle _M=\int _M f\bar{h}dV_g$, where $dV_g$ is the volume form of the metric $g$. We assume throughout the paper that every eigenfunction is real valued. We denote by $Z_{u_n}$ the nodal set $\{x \in M\nobreakspace :\nobreakspace u_n(x)=0\}$ of $u_n$ and by $\mathcal{N}(u_n)$ the number of nodal domains of $u_n$, where nodal domains are the connected components of $M \backslash Z_{u_n}$.

The purpose of this paper is to understand the growth of $\mathcal{N}(u_n)$ as $n$ tends to $+\infty$. Note that Courant’s nodal domain theorem CH53 and Weyl law imply that $\mathcal{N}(u_n)= O(\lambda _n^2)$. However, it is not true in general that the number of nodal domains necessarily grows with the eigenvalue. For instance, when $M=S^2$ (the standard sphere) or $M=T^2$ (the flat torus), there exists a sequence of eigenfunctions $\{u_{n_k}\}$ with $\lambda _{n_k}\to \infty$ that satisfy $\mathcal{N}(\phi _{n_k})\leq 3$ Ste25Lew77JN99.

We first state the main result of the paper.

Theorem 1.1.

Let $\phi$ be a Hecke-Maass eigenform for an arithmetic triangle group with eigenvalue $\lambda$. Then we have $\lim _{\lambda \to +\infty } \mathcal{N}(\phi )=+\infty$.

Note that there are $76$ arithmetic triangle groups Tak77a which are divided into $18$ commensurable classes Tak77b.

Remark 1.2.

This result in the stronger form of a lower bound of $\gg _{\epsilon } \lambda ^{\frac{2}{27}-\epsilon }$ for the number of nodal domains is obtained in GRS15, however, assuming the generalized Lindelöf hypothesis for a certain family of $L$-functions.

Remark 1.3.

We refer the readers to Section $6$ of GRS15 for further examples; Section $2.5$ of GRS15 might be required in order to apply Theorem 1.6 to those examples.

Theorem 1.1 is a consequence of Theorem 1.6 given below which considers the number of nodal domains when we have quantum unique ergodicity (QUE). Note that the arithmetic quantum unique ergodicity theorem by Lindenstrauss Lin06 asserts that QUE holds for Maass-Hecke eigenforms on these triangles. In order to state Theorem 1.6, we first fix $a \mapsto a(x,hD)$, a quantization of a symbol $a(x,\xi ) \in C^\infty \left(T^*M\right)$, to a pseudo-differential operator. (We refer the readers to Zwo12 for detailed discussion on the subject.) We say QUE holds for the sequence of eigenfunctions $\{u_n\}_{n\geq 1}$ if we have

$$\begin{equation} \lim _{n \to \infty }\left\langle a\left(x,\lambda _{n}^{-1}D\right) u_{n},u_{n}\right\rangle _M =\int _{S^*M} a(x,\xi ) d\mu {\tag{1.1}} \end{equation}$$

for any fixed symbol $a \in C^\infty \left(T^*M\right)$ of finite order. Here $d\mu$ is a normalized Liouville measure on the unit cotangent bundle $S^*M$. We often write $\mathrm{Op}(a)$ for an operator that acts on an eigenfunction $u$ with the eigenvalue $\lambda$ as $a(x,\lambda ^{-1}D)$.

Remark 1.4.

The classical notions of equidistribution of these “Wigner measures” Šni74CdV85Zel87 are concerned with 1.1 for degree zero homogeneous symbols. One can prove that if 1.1 holds with degree zero homogeneous symbols, then 1.1 holds with finite order symbols, by observing that any weak limit of these measures is supported in $S^*M \subset T^*M$.

Remark 1.5.

For a compact smooth negatively curved Riemannian manifold, it is conjectured by Rudnick and Sarnak RS94 that QUE holds for any given orthonormal eigenbasis $\{u_{n}\}$.

Theorem 1.6.

Let $M$ be a smooth compact Riemannian surface without boundary. Assume that there exists an orientation-reversing isometric involution $\tau :M \to M$ such that $\mathrm{Fix}(\tau )$ is separating. Let $\{u_n\}$ be an orthonormal basis of $L^2(M)$ such that each $u_n$ is a joint eigenfunction of the Laplacian and $\tau$. Assume that QUE holds for the sequence $\{u_n\}$. Then

$$\begin{equation*} \lim _{n\to \infty } \mathcal{N}(u_{n}) = +\infty . \end{equation*}$$

We say a function $f$ on $M$ is even (resp. odd) if $\tau f=f$ (resp. $\tau f=-f$). In order to prove Theorem 1.6, we first use a topological argument to bound the number of nodal domains of an even (resp. odd) eigenfunction from below by the number of sign changes (resp. the number of singular points) of the eigenfunction along $\mathrm{Fix}(\tau )$. Such an argument is first developed in GRS13, and we review in Section 4.1 in terms of the nodal graphs and Euler’s inequality as in JZ16. We then use Bochner’s theorem and a Rellich type identity to deduce from QUE that even (resp. odd) eigenfunctions $\{u_n\}$ have a growing number of sign changes (resp. singular points) along $\mathrm{Fix}(\tau )$ as $n$ tends to $+\infty$. This is the main contribution of the paper, and we sketch the argument in the following section.

Remark 1.7.

In JZ16, the same assertion has been obtained when $M$ is a negatively curved surface, but for a density one subsequence of $\{u_n\}$. The argument of JZ16 to detect a sign change of an eigenfunction $u_n$ on a curve $\beta$ is to compare

$$\begin{equation*} \left|\int _\beta u_n(s) ds\right| \quad \text{and}\quad \int _\beta |u_n(s)| ds. \end{equation*}$$

(See GRS13Jun16BR15JZ16Mag15GRS15, where such an idea is used to prove a lower bound for the number of sign changes in various contexts.) In order to bound $\|u_n\|_{L^1(\beta )}$ from below using Hölder’s inequality, the authors use the quantum ergodic restriction (QER) theorem TZ13DZ13 for the lower bound of $\|u_n\|_{L^2(\beta )}$ and the point-wise Weyl law with an improved error term Bér77 for the upper bound of $\|u_n\|_{L^\infty (\beta )}$. For the upper bound of the integral of $u_n$ over $\beta$, the authors use the Kuznecov sum formulas Zel92. Note that the result of Bér77 requires a global assumption on the geometry of $M$ that it does not have conjugate points, which is satisfied if $M$ is negatively curved. Also note that in order to bound such quantities using QER theorem and Kuznecov sum formulas, it is necessary to remove a density $0$ subsequence.

1.2. Sketch of the proof: sign changes of even eigenfunctions

The main step in the proof of Theorem 1.6 is to show that all but finitely many $u_{n}$ have at least one sign change on any given fixed segment $\beta$ of $\mathrm{Fix}(\tau )$.

To simplify the discussion, let $\{\psi _n\}$ be a sequence of functions in $C_0^\infty ([0,1])$. Assume that for any fixed integer $m \geq 0$ we have

$$\begin{equation} \lim _{n \to \infty } \int _0^1 \left|\frac{\partial ^m \psi _n}{\partial s^m}(s)\right|^2 ds = a_{2m}, {\tag{1.2}} \end{equation}$$

for some positive real number $a_{2m}$. Let $h_n(\xi )=|g_n(\xi )|^2/\|g_n\|_2^2$, where $g_n(\xi )$ is the Fourier transform of $\psi _n$,

$$\begin{equation*} g_n(\xi ) = (2\pi )^{-\frac{1}{2}} \int _0^1 e^{i\xi s} \psi _n(s) ds. \end{equation*}$$

Assume that there exists a unique probability measure $d\mu (\xi )$ whose $2m$th moment is $a_{2m}/a_0$ and whose $(2m+1)$th moment is zero for any $m\geq 0$. Then 1.2 implies that a sequence of probability measures $h_n(\xi ) d\xi$ converges to $d\mu (\xi )$ in moments.

We claim that all but finitely many $\psi _n$ have at least one sign change on $(0,1)$ under the assumption that $d\mu (\xi )$ is not positive-definite, i.e., not a Fourier transform of a positive measure (Lemma 4.6). Assume for contradiction that there exists a subsequence $\{\psi _{n_k}\}$ of $\{\psi _n\}$ such that $\psi _{n_k}$ does not change sign on $(0,1)$ for all $k$. Then by Bochner’s theorem, $\{h_{n_k}(\xi )\}$ is a sequence of positive-definite functions, and it cannot converge in moments to a measure that is not positive-definite, contradicting the assumption that $d\mu (\xi )$ is not positive-definite.

Now let $f \in C_0^\infty (\beta )$ be a non-negative function. Our aim is to apply the above argument to $\psi _n(s)=f(s)u_{n}|_{\beta }(s)$, when QUE holds for the sequence of eigenfunctions $\{u_{n}\}$. Note that it is not known whether the limit

$$\begin{equation*} \lim _{n \to \infty }\int _{\beta }|\psi _n(s)|^2 ds \end{equation*}$$

should exist. However, under the assumption that QUE holds for $\{u_{n}\}$, we may instead compute the limit (Theorem 3.1)

$$\begin{equation} \lim _{n \to \infty } \int _\beta \left|\psi _n(s)\right|^2 - \left|\frac{1}{\lambda _{n}^m}\frac{\partial ^m \psi _n}{\partial s^m}(s)\right|^2 ds = 2(1- b_{2m}) \int _{\beta } f^2(s) ds {\tag{1.3}} \end{equation}$$

for each fixed $m \geq 0$ with an explicit constant $0<b_{2m}\leq 1$ using the Rellich identity, as in the proof of the quantum uniquely ergodic restriction theorem of CTZ13.

We first deduce from 1.3 that (Corollary 3.2)

$$\begin{equation*} \liminf _{n \to \infty } \int _\beta \left|\psi _n(s)\right|^2 ds \geq 2 \int _\beta f^2(s) ds, \end{equation*}$$

and so

$$\begin{equation*} \limsup _{n \to \infty } 2 \int _\beta f^2(s) ds\left(\int _\beta \left|\psi _n(s)\right|^2 ds \right)^{-1}\leq 1. \end{equation*}$$

Assume for simplicity that, for some $0 \leq a\leq 1$, we have

$$\begin{equation*} \lim _{n \to \infty } 2 \int _\beta f^2(s) ds\left(\int _\beta \left|\psi _n(s)\right|^2 ds \right)^{-1} = a. \end{equation*}$$

Then 1.3 implies that

$$\begin{equation*} \lim _{n \to \infty } \int _\beta \left|\frac{1}{\lambda _{n}^m}\frac{\partial ^m \psi _n}{\partial s^m}(s)\right|^2 ds \left(\int _\beta \left|\psi _n(s)\right|^2 ds \right)^{-1}= (1-a) + a b_{2m}, \end{equation*}$$

and we may apply the argument to

$$\begin{equation*} h_n(\xi ) = \lambda _{n} \left|\widehat{\psi _n}(\lambda _{n} \xi )\right|^2\left(\int _\beta \left|\psi _n(s)\right|^2 ds \right)^{-1} \end{equation*}$$

to conclude that all but finitely many $u_{n}$ have at least one sign change on $\beta$, by verifying that the unique measure having $(1-a)+a b_{2m}$ as the $2m$th moment and $0$ as the $(2m+1)$th moment is not positive-definite for any given $0 \leq a \leq 1$. This implies that the number of sign changes of $u_{n}$ along $\mathrm{Fix}(\sigma )$ tends to $+\infty$ as $n \to \infty$ (Theorem 4.3).

2. $L^p$ estimates for the restriction to a curve of derivatives of eigenfunctions

Let $u$ be a Laplacian eigenfunction with the eigenvalue $\lambda$. Let $L$ be a degree $m$ linear differential operator on $M$; i.e., for any coordinate patch $(U,p)$ there exist smooth functions $a_\alpha \in C^\infty (\mathbb{R}^n)$ (in which $a_\alpha \not \equiv 0$ for some $\alpha$ with $|\alpha |=m$) such that for any $\phi ,\psi \in C_0^\infty (U)$ and for each $f \in C^\infty (M)$,

$$\begin{equation*} \phi L(\psi f)=\phi p^* \sum _{|\alpha |\leq m} a_\alpha (x)\partial ^\alpha \left(p^{-1}\right)^* (\psi f). \end{equation*}$$

Recall that

$$\begin{equation} \sup _{x \in M}\left|Lu(x)\right| =O\left(\lambda ^{m+\frac{1}{2}}\right), {\tag{2.1}} \end{equation}$$

which is a consequence of the generalization of remainder estimate for spectral function by Avakumovic-Levitan-Hörmander to that for the derivatives of spectral function Bin04. Denoting by $\langle f, g\rangle _\beta = \int _\beta f(s)\overline{g(s)}ds$, 2.1 implies that

$$\begin{equation} |\langle L u, u\rangle _{\beta }| \ll _\beta \sup _{x \in M}\left|Lu(x)\right|\sup _{x \in M}\left|u(x)\right|= O\left(\lambda ^{m+1}\right). {\tag{2.2}} \end{equation}$$

In the proof of Theorem 3.1, we need an improvement over 2.2, and we achieve an improvement by combining the $L^2$ eigenfunction restrictions estimates along curves due to Burq, Gérard, and Tzvetkov BGT07 and 2.1.

Lemma 2.1.

For any fixed degree $m$ differential operator $L$, we have

$$\begin{equation*} |\langle L u, u\rangle _{\beta }| =O\left(\lambda ^{m+\frac{3}{4}}\right). \end{equation*}$$

Proof.

By Hölder’s inequality,

$$\begin{equation*} |\langle L u, u\rangle _{\beta }| \leq \sup _{x \in M}\left|Lu(x)\right| \|u\|_{L^1(\beta )}. \end{equation*}$$

From BGT07, we have $\|u\|_{L^2(\beta )}=O\left(\lambda ^{\frac{1}{4}}\right)$; hence

$$\begin{equation*} \|u\|_{L^1(\beta )}\leq l(\beta )^{\frac{1}{2}}\|u\|_{L^2(\beta )} = O_{\beta } \left(\lambda ^{\frac{1}{4}}\right). \end{equation*}$$

Therefore by 2.1, we conclude

$$\begin{equation*} |\langle L u, u\rangle _{\beta }|= O\left(\lambda ^{m+\frac{3}{4}}\right). \end{equation*}$$■

Since we only need any power saving over $O\left(\lambda ^{m+1}\right)$ in 2.2 in our proof, it is unnecessary to optimize our bound in Lemma 2.1. The optimal upper bound for $\|Lu\|_{L^2(\beta )}$ is $O\left(\lambda ^{m+\frac{1}{4}}\right)$ which is sharp when $L=1$ and $M$ is the standard sphere $S^2$. Note that when $L$ corresponds to a normal derivative along $\beta$, the bound can be improved to $O(1)$ using second-microlocalization techniques, due to CHT15.

3. Rellich type analysis when QUE holds: even eigenfunctions

In this section, we prove 1.3 with explicit constants $\{b_{2m}\}$. The main idea is to follow the computation involving the Rellich identity in CTZ13, with a specific choice of symbols.

Theorem 3.1.

Assume that QUE holds for the sequence of even eigenfunctions $\{u_{n}\}$. Fix a segment $\beta \subset \mathrm{Fix}(\tau )$. For any fixed real valued function $f \in C_0^\infty (\beta )$ and for any fixed non-negative integer $m$, we have

$$\begin{align*} &\quad \lim _{n \to \infty }\int _\beta |f(t)u_{n}(t)|^2 dt - \lambda _{n}^{-2m}\int _{\beta } \left|\partial _t^{m}(f(t)u_{n}(t))\right|^2 dt\\ &=2\left(1-\frac{1}{\pi }\int _{-1}^1 \xi ^{2m} \frac{d\xi }{\sqrt {1-\xi ^2}}\right)\int _\beta f^2(t) dt. \end{align*}$$

Proof.

We drop the subscript $n$ in $u_n$ and $\lambda _n$ for simplicity.

Let $(t,n)$ be Fermi normal coordinates in a small tubular neighborhood $U_\epsilon$ of $\beta$ near a point $x_0 \in \beta$. Let $p:U_\epsilon \to \mathbb{R}^2$ be the coordinate chart. We may assume that

$$\begin{equation*} U=U_\epsilon = p^{-1}\left(\{(t,n)\nobreakspace |\nobreakspace t\in V,\nobreakspace |n|<\epsilon \}\right) \end{equation*}$$

in these coordinates, where $V \subset \mathbb{R}$ is a coordinate chart that contains $x_0$. Let $(t,n,\xi _t,\xi _n)$ be the local coordinates of $T^*(U)$ under the identification

$$\begin{align*} \mathbb{R}^2& \to T_{x=(t,n)}^*(U),\\ (\xi _t,\xi _n) &\mapsto \xi _t dt+\xi _n dn. \end{align*}$$

We consider the standard quantization in these coordinates; i.e., for any given symbol $a(t,n,\xi _t,\xi _n)$ of finite order, we let

$$\begin{multline*} \mathrm{Op}(a)u(t_0,n_0)\\ =\frac{\lambda ^n}{(2\pi )^n}\int _{p(U)\times \mathbb{R}^2} e^{\lambda i ((t_0-t)\xi _t+(n_0-n)\xi _n)}a(t_0,n_0,\xi _t,\xi _n)u(t,n) dtdnd\xi _t d\xi _n. \end{multline*}$$

For example, if $a(t,n,\xi _t,\xi _n)=\sum _{|\alpha |\leq N}a_\alpha (t,n)\xi _t^{\alpha _1}\xi _n^{\alpha _2}$, then

$$\begin{equation*} \mathrm{Op}(a)u=\sum _{|\alpha |\leq N} a_\alpha (t,n)\left(\frac{\partial _t}{i\lambda }\right)^{\alpha _1}\left(\frac{\partial _n}{i\lambda }\right)^{\alpha _2}u . \end{equation*}$$

Let $U_{-}\subset U$ be given by

$$\begin{equation*} U_{-}=\{(t,n) \in U\nobreakspace |\nobreakspace n<0\}. \end{equation*}$$

For any pseudo-differential operator $T$ on $M$, from Green’s formula, we have

$$\begin{equation} \langle \Delta _g T u,u\rangle _{U_-}-\langle T u, \Delta _g u\rangle _{U_-}=\langle \partial _n T u|_\beta ,u|_\beta \rangle _\beta -\langle T u|_\beta , \partial _n u|_\beta \rangle _\beta . {\tag{3.1}} \end{equation}$$

Since $u$ is an eigenfunction, $\langle T u, \Delta _g u\rangle _{U_-}=\langle T \Delta _g u, u\rangle _{U_-}$. Also since we are assuming that $u$ is even, $\langle T u|_\beta , \partial _n u|_\beta \rangle _\beta =0$. Therefore we have the Rellich identity,

$$\begin{equation} \frac{1}{\lambda }\langle \lbrack -\Delta _g, T\rbrack u,u\rangle _{U_-} = -\frac{1}{\lambda }\langle \partial _n T u|_\beta ,u|_\beta \rangle _\beta , {\tag{3.2}} \end{equation}$$

where $\lambda ^{-1}$ is the normalizing factor.

Now fix $\chi \in C_0^\infty (\mathbb{R})$ such that

$$\begin{equation*} \chi (x)= \left\{ \begin{array}{cl} 0 & \hspace{10mm}\text{if } |x|\geq 1,\\1 & \hspace{10mm} \text{if }|x|\leq \frac{1}{2}. \end{array} \right. \end{equation*}$$

We define a symbol supported near $\beta$ for $0<\delta <\epsilon$,

$$\begin{equation*} a_{\delta ,m}(x,\xi ) = \chi \left(\frac{n}{\delta }\right)f^2(t) i\xi _n \sum _{k=0}^{m-1}\xi _t^{2k}, \end{equation*}$$

and let $T=\mathrm{Op}(a_{\delta ,m})$. Observing that $-(\partial _n^2+\partial _t^2)u = \lambda ^2 u$ along $\beta$, we may rewrite the right-hand side (RHS) of 3.2 as

$$\begin{equation*} \left\langle f^2\left(1+(-1)^{m-1}\lambda ^{-2m}\partial _t^{2m}\right)u|_\beta , u|_\beta \right\rangle _{\beta }. \end{equation*}$$

We integrate by parts to further simplify the second term as follows:

$$\begin{align*} &\left\langle (-1)^{m-1}f^2\lambda ^{-2m}\partial _t^{2m}u|_\beta , u|_\beta \right\rangle _{\beta }+\lambda ^{-2m}\left\langle \partial _t^{m}\left(fu|_\beta \right), \partial _t^{m}\left(fu|_\beta \right) \right\rangle _{\beta }\\ =&\left\langle (-1)^{m-1}\lambda ^{-2m}f\lbrack f,\partial _t^{2m}\rbrack u|_\beta , u|_\beta \right\rangle _{\beta }\\ =&\,O_{m,f}(\lambda ^{-\frac{1}{4}}), \end{align*}$$

where we used Lemma 2.1 with $L=f\lbrack f,\partial _t^{2m}\rbrack$ in the last estimate. So we have

$$\begin{equation} -\frac{1}{\lambda ^2}\langle \partial _n T u|_\beta ,u|_\beta \rangle _\beta =\int _\beta |f(t)u(t)|^2 dt - \lambda ^{-2m}\int _{\beta } \left|\partial _t^{m}(f(t)u(t))\right|^2 dt+O_{m,f}\left(\lambda ^{-\frac{1}{4}}\right). {\tag{3.3}} \end{equation}$$

Now let $-\Delta _g=\sum _{|\alpha |=1,2}b_\alpha (n,t)\left(\frac{\partial _n}{i}\right)^{\alpha _1}\left(\frac{\partial _t}{i}\right)^{\alpha _2}$. Observe from 2.1 that if $\alpha _1+\alpha _2=1$,

$$\begin{equation*} \frac{1}{\lambda } \left\lbrack b_\alpha (n,t) \left(\frac{\partial _n}{i}\right)^{\alpha _1}\left(\frac{\partial _t}{i}\right)^{\alpha _2},\chi \left(\frac{n}{\delta }\right)f^2(t) \frac{(-1)^k \partial _n \partial _t^{2k}}{\lambda ^{2k+1}}\right\rbrack u= O_{\delta ,m,f}(\lambda ^{-\frac{1}{2}}), \end{equation*}$$

and that if $\alpha _1+\alpha _2=2$,

$$\begin{align*} &\frac{1}{\lambda } \left\lbrack b_\alpha (n,t) \left(\frac{\partial _n}{i}\right)^{\alpha _1}\left(\frac{\partial _t}{i}\right)^{\alpha _2},\chi \left(\frac{n}{\delta }\right)f^2(t) \frac{\partial _n (-1)^k\partial _t^{2k}}{\lambda ^{2k+1}}\right\rbrack u\\ =&\,\mathrm{Op}\left(\frac{\alpha _1}{\delta }b_\alpha (n,t)\chi '\left(\frac{n}{\delta }\right)f^2(t)\xi _n^{\alpha _1}\xi _t^{2k+\alpha _2}\right)u\\ +&\,\mathrm{Op}\left(\chi \left(\frac{n}{\delta }\right)R_{m,f,\alpha }(n,t,\xi _n,\xi _t)\right)u+O_{\delta ,m,f}(\lambda ^{-\frac{1}{2}}) \end{align*}$$

for some symbol $R_{m,f,\alpha }$ of finite order depending only on $m,f,\alpha$. Therefore we may reexpress the left-hand side (LHS) of 3.2 as

$$\begin{align*} &\left\langle \mathrm{Op}\left(\sum _{|\alpha |=2}\frac{\alpha _1\xi _n^{\alpha _1}\xi _t^{\alpha _2}}{\delta }b_\alpha (n,t)\chi '\left(\frac{n}{\delta }\right)f^2(t)\sum _{k=0}^{m-1}\xi _t^{2k}\right)u,u\right\rangle _{U_-}\\ +&\left\langle \mathrm{Op}\left(\chi \left(\frac{n}{\delta }\right)R_{m,f}(n,t,\xi _n,\xi _t)\right)u,u\right\rangle _{U_-}\\ +&\,O_{\delta ,m,f}(\lambda ^{-\frac{1}{2}}) \end{align*}$$

for some finite order symbol $R_{m,f}$.

We bound the second inner product using Cauchy-Schwartz inequality by

$$\begin{align*} &\left|\left\langle \mathrm{Op}\left(\chi \left(\frac{n}{\delta }\right)R_{m,f}(n,t,\xi _n,\xi _t)\right) u, u\right\rangle _{U_-}\right| \\ \leq &\left\|\mathrm{Op}\left(\chi \left(\frac{n}{\delta }\right)R_{m,f}(n,t,\xi _n,\xi _t)\right) u\right\|_{L^2(U_-)}^2\\ \leq &\left\|\mathrm{Op}\left(\chi \left(\frac{n}{\delta }\right)R_{m,f}(n,t,\xi _n,\xi _t)\right) u\right\|_{L^2(U)}^2, \end{align*}$$

and from the assumption that the QUE holds, we may estimate the last quantity as $O_{m,f}(\delta )+o_{\delta ,m,f}(1)$ as $\lambda$ tends to $+\infty$.

Now let $\chi _0\in C_0^\infty (\mathbb{R})$ be given by $\chi _0(x)=\chi '(x)$ if $x<0$, and $\chi _0(x)=0$ otherwise. We then have

$$\begin{align} &\left\langle \mathrm{Op}\left(\sum _{|\alpha |=2}\frac{\alpha _1\xi _n^{\alpha _1}\xi _t^{\alpha _2}}{\delta }b_\alpha (n,t)\chi '\left(\frac{n}{\delta }\right)f^2(t)\sum _{k=0}^{m-1}\xi _t^{2k}\right)u,u\right\rangle _{U_-}\\ =&\left\langle \mathrm{Op}\left(\sum _{|\alpha |=2}\frac{\alpha _1\xi _n^{\alpha _1}\xi _t^{\alpha _2}}{\delta }b_\alpha (n,t)\chi _0\left(\frac{n}{\delta }\right)f^2(t)\sum _{k=0}^{m-1}\xi _t^{2k}\right)u,u\right\rangle _{U}\\ =&\int _{S^*U} \sum _{|\alpha |=2}\frac{\alpha _1\xi _n^{\alpha _1}\xi _t^{\alpha _2}}{\delta }b_\alpha (n,t)\chi _0\left(\frac{n}{\delta }\right)f^2(t)\sum _{k=0}^{m-1}\xi _t^{2k}d\mu +o_{\delta ,m,f}(1) {\tag{3.4}} \end{align}$$

as $\lambda$ tends to $+\infty$ from the assumption that QUE holds.

We therefore conclude from 3.3 and 3.4 that

$$\begin{multline*} \int _\beta |f(t)u(t)|^2 dt - \lambda ^{-2m}\int _{\beta } \left|\partial _t^{m}(f(t)u(t))\right|^2 dt\\ =\int _{S^*U} \sum _{|\alpha |=2}\frac{\alpha _1\xi _n^{\alpha _1}\xi _t^{\alpha _2}}{\delta }b_\alpha (n,t)\chi _0\left(\frac{n}{\delta }\right)f^2(t)\sum _{k=0}^{m-1}\xi _t^{2k}d\mu \\ +O_{m,f}\left(\lambda ^{-\frac{1}{4}}\right)+O_{m,f}(\delta )+o_{\delta ,m,f}(1), \end{multline*}$$

and so

$$\begin{multline*} \lim _{\lambda \to \infty }\int _\beta |f(t)u(t)|^2 dt - \lambda ^{-2m}\int _{\beta } \left|\partial _t^{m}(f(t)u(t))\right|^2 dt\\ =\int _{S^*U} \sum _{|\alpha |=2}\frac{\alpha _1\xi _n^{\alpha _1}\xi _t^{\alpha _2}}{\delta }b_\alpha (n,t)\chi _0\left(\frac{n}{\delta }\right)f^2(t)\sum _{k=0}^{m-1}\xi _t^{2k}d\mu +O_{m,f}(\delta ). \end{multline*}$$

Note that no terms in the left-hand side depend on $\delta$. Also note that $b_{20}(0,t)=b_{02}(0,t)=1$ and $b_{11}(0,t)=0$ since we are taking the Fermi normal coordinate. Therefore by taking $\delta \to 0$, we have

$$\begin{align*} &\lim _{\delta \to 0}\int _{S^*U} \sum _{|\alpha |=2}\frac{\alpha _1\xi _n^{\alpha _1}\xi _t^{\alpha _2}}{\delta }b_\alpha (n,t)\chi _0\left(\frac{n}{\delta }\right)f^2(t)\sum _{k=0}^{m-1}\xi _t^{2k}d\mu +O_{m,f}(\delta )\\ =&\int _{S_\beta ^*U} \sum _{|\alpha |=2}\alpha _1 b_\alpha (0,t)f^2(t)\xi _n^{\alpha _1}\xi _t^{\alpha _2}\sum _{k=0}^{m-1}\xi _t^{2k}d\mu \\ =&\,\frac{1}{\pi } \int _\beta f^2(t) dt \int _{\xi _t^2+\xi _n^2=1}(1-\xi _t^{2m}) d\xi \\ =&\,2\left(1-\frac{1}{\pi }\int _{-1}^1 \xi ^{2m} \frac{d\xi }{\sqrt {1-\xi ^2}}\right)\int _\beta f^2(t) dt. \end{align*}$$

This implies that

$$\begin{multline*} \lim _{\lambda \to \infty }\int _\beta |f(t)u(t)|^2 dt - \lambda ^{-2m}\int _{\beta } \left|\partial _t^{m}(f(t)u(t))\right|^2 dt\\ =2\left(1-\frac{1}{\pi }\int _{-1}^1 \xi ^{2m} \frac{d\xi }{\sqrt {1-\xi ^2}}\right)\int _\beta f^2(t) dt+o_{m,f}(1) \end{multline*}$$

as $\delta \to 0$, and since $\delta$ can be chosen arbitrarily small, we conclude that

$$\begin{multline*} \lim _{\lambda \to \infty }\int _\beta |f(t)u(t)|^2 dt - \lambda ^{-2m}\int _{\beta } \left|\partial _t^{m}(f(t)u(t))\right|^2 dt\\ =2\left(1-\frac{1}{\pi }\int _{-1}^1 \xi ^{2m} \frac{d\xi }{\sqrt {1-\xi ^2}}\right)\int _\beta f^2(t) dt. \end{multline*}$$ ■

As an immediate application of Theorem 3.1, we give a sharp lower bound for the $L^2$ estimate of the restriction of eigenfunctions.

Corollary 3.2.

Assume that QUE holds for the sequence of even eigenfunctions $\{u_{n}\}$. Then for any fixed real valued function $f \in C_0^\infty (\beta )$, we have

$$\begin{equation*} \liminf _{n \to \infty } \int _{\beta } f^2(t)|u_{n}(t)|^2 dt \geq 2\int _\beta f^2(t) dt. \end{equation*}$$

Proof.

By the positivity of $\lambda ^{-2m}\int _{\beta } \left|\partial _t^{m}(f(t)u(t))\right|^2 dt$,

$$\begin{align*} \liminf _{n \to \infty } \int _{\beta } f^2(t)|u_{n}(t)|^2 dt&\geq 2\left(1-\frac{1}{\pi }\int _{-1}^1 \xi ^{2m} \frac{d\xi }{\sqrt {1-\xi ^2}}\right)\int _\beta f^2(t) dt\\ &=2\int _\beta f^2(t) dt +O(1/m^{1/2}). \end{align*}$$

Since the limit does not depend on $m$, we conclude that

$$\begin{equation*} \liminf _{n \to \infty } \int _{\beta } f^2 (t)|u_{n}(t)|^2 dt \geq 2\int _\beta f^2(t) dt. \end{equation*}$$■

Remark 3.3.

A constant lower bound for the $L^2$ norm of the restriction of an eigenfunction to a geodesic segment is first proven in GRS13, when the geodesic segment is sufficiently long, from the arithmetic QUE theorem Lin06Sou10.

Remark 3.4.

If the geodesic flow on $M$ is ergodic, it is known that there exists a density $1$ subsequence $\{u_{n}\}$ of even eigenfunctions that satisfies

$$\begin{equation} \lim _{n \to \infty } \|u_{n}\|_{L^2(\beta )}^2 = 2 l(\beta ); {\tag{3.5}} \end{equation}$$

hence the lower bound in Corollary 3.2 is sharp. The existence of such a subsequence is a consequence of results which are studied in Bur05TZ13DZ13CTZ13.

Remark 3.5.

If $\beta$ is not a part of $\mathrm{Fix}(\tau )$ and satisfies a certain asymmetry condition (see, for instance, TZ13, Definition 1), then

$$\begin{equation*} \lim _{k \to \infty } \|u_{n_k}\|_{L^2(\beta )}^2 = l(\beta ) \end{equation*}$$

along a density $1$ subsequence $\{u_{n_k}\}$ of $\{u_n\}$. When $\beta$ is a segment of $\mathrm{Fix}(\tau )$, then every odd eigenfunction vanishes identically on $\beta$, hence explaining why we expect the factor $2$ in 3.5.

4. The number of nodal domains of even eigenfunctions

4.1. Graph structure of the nodal set and Euler’s inequality

In this section we briefly review the topological argument in GRS13JZ16 on bounding the number of nodal domains from below by the number of zeros on $\mathrm{Fix}(\tau )$. We refer the readers to JZ16 for details.

First note that if there exists a segment of $\eta \subset \mathrm{Fix}(\tau )$ such that $\eta \subset Z_u$, then because the normal derivative of $u$ vanishes along $\mathrm{Fix}(\tau )$, any point on $\eta$ is a singular point, contradicting the upper bound on the number of singular points in Don92. Therefore together with the following lemma on the local structure of the nodal set, we conclude that $Z_u \cap \mathrm{Fix}(\tau )$ is a finite set of points.

Lemma 4.1 (Section 6.1, JZ16).

Assume that $u$ vanishes to order $N$ at $x_0$. Then there exists a small neighborhood $U$ of $x_0$ such that the nodal set in $U$ is $C^1$ equivalent to $2N$ equi-angular rays emanating from $x_0$.

From Lemma 4.1, we may view the nodal set as a graph (a nodal graph) embedded on a surface as follows:

(1)

For each connected component of $Z_u$ that is homeomorphic to a circle and that does not intersect $\mathrm{Fix}(\tau )$, we add a vertex.

(2)

Each singular point is a vertex.

(3)

Each intersection point in $\mathrm{Fix}(\tau ) \cap Z_u$ is a vertex.

(4)

Edges are the arcs of $Z_u \cup \mathrm{Fix}(\tau )$ that join the vertices listed above.

Let $V(u)$ and $E(u)$ be the finite set of vertices and the finite set of edges given above, respectively. This way, we obtain a nodal graph $V(u),E(u)$ of $u$ embedded into the surface $M$.

From the assumption that $\mathrm{Fix}(\tau )$ is separating, the nodal domains that intersect $\mathrm{Fix}(\tau )$ are cut in two by $\mathrm{Fix}(\tau )$. Therefore the number of faces divided by two bounds the number of nodal domains $\mathcal{N}(u)$ from below.

Observe from Lemma 4.1 that every vertex of a nodal graph has a degree at least $2$. Then by Euler’s inequality JZ16, (6.1),

$$\begin{equation*} |V(u)|-|E(u)|+|F(u)| -m(u) \geq 1- 2 \mathfrak{g}, \end{equation*}$$

we obtain a lower bound for the number of nodal domains by the number of zeros on $\mathrm{Fix}(\tau )$. Here $m(G)$ is the number of connected components of the nodal graph, and $\mathfrak{g}$ is the genus of the surface $M$.

Lemma 4.2 (Lemma 6.4, JZ16).

$$\begin{equation*} \mathcal{N}(u) \geq \frac{1}{2}\#\left(Z_{u} \cap \mathrm{Fix}(\tau ) \right)+1-\mathfrak{g}. \end{equation*}$$

Therefore in order to prove Theorem 1.6, it is sufficient to prove the following theorem.

Theorem 4.3.

Assume that QUE holds for the sequence of even eigenfunctions $\{u_{n}\}_{n \geq 1}$. Then

$$\begin{equation*} \lim _{n\to \infty }\#\left(Z_{u_{n}}\cap \mathrm{Fix}(\tau )\right) = +\infty . \end{equation*}$$

4.2. Lemmata from probability theory

In order to prove Theorem 4.3, we first recall some facts about probability measures. We assume that all random variables in this section are defined on the real line.

Lemma 4.4.

Suppose a random variable $X$ has moments $\mu _k=\mathbb{E}[X^k]$ that satisfies the condition

$$\begin{equation*} \limsup _{k\rightarrow \infty }\mu _{2k}^{\frac{1}{2k}}/2k=r<\infty . \end{equation*}$$

Then, $X$ has the unique distribution with moments $(\mu _k)_{k\geq 1}$.

Proof.

See Dur10, Theorem 3.3.11.

■

Lemma 4.5.

If $X_n$ converges to $X$ in moments and the distribution of $X$ is uniquely determined by its moments, then for each $t\in \mathbb{R}$, $\mathbb{E}[e^{itX_n}]$ converges to $\mathbb{E}[e^{itX}]$.

Proof.

Suppose we have a counterexample of this lemma. That is, we have a sequence $(X_n)$ of random variables and $X$ a random variable, such that $\mathbb{E}\lbrack X_n^m\rbrack \rightarrow \mathbb{E}\lbrack X^m\rbrack$ for all $m>0$, but $\mathbb{E}\lbrack \exp (it_0X_n)\rbrack \not \rightarrow \mathbb{E}\lbrack \exp (it_0 X)\rbrack$ for some $t_0\in \mathbb{R}$.

Let $F_n(x):=\Pr \lbrack X_n\leq x\rbrack$ be the cumulative distribution functions of random variable $X_n$. By Helly’s selection theorem Dur10, Theorem 3.2.6, together with the tightness of $(X_{n})$’s Dur10, Theorems 3.2.7 and 3.2.8 there exists a subsequence $(F_{n_k})$ that converges to a cumulative distribution function $G$ of some random variable $Y$ on the real line.

Dur10, Theorem 3.2.2 implies that, by appropriately settling the probability space $\Omega$ for $X_{n_k}$’s and $Y$, we can have $X_{n_k}(\omega )\rightarrow Y(\omega )$ almost surely for $\omega \in \Omega$. (For instance, we can set $\Omega =(0,1)$, $\Pr =($the Lebesgue measure), and $X_{n_k}(\omega )=\sup \{x\in \mathbb{R}\nobreakspace |\nobreakspace F_{n_k}(x)<\omega \}$, etc.) In particular, as $\exp (it_0X_{n_k})\rightarrow \exp (it_0Y)$ almost surely, together with $|\exp (it_0X_{n_k})|\leq 1$ for all $k$ implies that $\mathbb{E}\lbrack \exp (it_0X_{n_k})\rbrack \rightarrow \mathbb{E}\lbrack \exp (it_0Y)\rbrack$ by the bounded convergence theorem.

From the assumption that $X$ is the unique random variable with the sequence of the moments $\left(\mathbb{E}[X^m]\right)$, we claim $X=Y$ by showing that $\mathbb{E}\lbrack X^m\rbrack =\mathbb{E}\lbrack Y^m\rbrack$ for all $m>0$. Equivalently, $\mathbb{E}\lbrack X_{n_k}^m\rbrack \rightarrow \mathbb{E}\lbrack Y^m\rbrack$ as $k\rightarrow \infty$. Denote by $\chi _M$ the indicator function of $\lbrack -M,M\rbrack$ for $M>0$. We first estimate

$$\begin{align*} |\mathbb{E}\lbrack X_{n_k}^m\rbrack -\mathbb{E}\lbrack Y^m\rbrack |&\leq \mathbb{E}\lbrack |X_{n_k}|^m(1-\chi _M(X_{n_k}))\rbrack +\mathbb{E}\lbrack |Y|^m(1-\chi _M(Y))\rbrack \\ &\qquad \,+|\mathbb{E}\lbrack X_{n_k}^m\chi _M(X_{n_k})-Y^m\chi _M(Y)\rbrack |. \end{align*}$$

We bound the first term by Cauchy-Schwarz inequality and Markov inequality,

$$\begin{align*} \mathbb{E}\lbrack |X_{n_k}|^m(1-\chi _M(X_{n_k}))\rbrack &\leq \mathbb{E}\lbrack |X_{n_k}|^{2m}\rbrack ^{\frac{1}{2}}\mathbb{E}\lbrack (1-\chi _M(X_{n_k}))^2\rbrack ^{\frac{1}{2}} \\ &=\mathbb{E}\lbrack X_{n_k}^{2m}\rbrack ^{\frac{1}{2}}(\Pr \lbrack |X_{n_k}|>M\rbrack )^{\frac{1}{2}} \\ &\leq \mathbb{E}\lbrack X_{n_k}^{2m}\rbrack ^{\frac{1}{2}}\mathbb{E}\lbrack X_{n_k}^2\rbrack ^{\frac{1}{2}}M^{-1} \leq KM^{-1}, \end{align*}$$

where $K=\sup \{\mathbb{E}\lbrack X_{n_k}^{2m}\rbrack ,\mathbb{E}\lbrack X_{n_k}^2\rbrack\nobreakspace |\nobreakspace k\}<\infty$. We bound the second term by Fatou’s lemma,

$$\begin{equation*} \mathbb{E}\lbrack |Y|^m(1-\chi _M(Y))\rbrack \leq \liminf _{k\rightarrow \infty }\mathbb{E}\lbrack |X_{n_k}|^m(1-\chi _M(X_{n_k}))\rbrack \leq KM^{-1}, \end{equation*}$$

where we used the estimate of the first term in the last inequality. Finally, observe that the third term converges to $0$, i.e., $|\mathbb{E}\lbrack X_{n_k}^m\chi _M(X_{n_k})-Y^m\chi _M(Y)\rbrack |\rightarrow 0$ as $k\rightarrow \infty$, by the bounded convergence theorem.

Therefore

$$\begin{equation*} \limsup _{k \to \infty }|\mathbb{E}\lbrack X_{n_k}^m\rbrack -\mathbb{E}\lbrack Y^m\rbrack | =O(M^{-1}), \end{equation*}$$

and since $M$ can be chosen arbitrarily large, we conclude $\mathbb{E}\lbrack X_{n_k}^m\rbrack \rightarrow \mathbb{E}\lbrack Y^m\rbrack$ which implies that $X=Y$. Therefore $\mathbb{E}\lbrack \exp (it_0X_{n_k})\rbrack \rightarrow \mathbb{E}\lbrack \exp (it_0Y)\rbrack =\mathbb{E}\lbrack \exp (it_0X)\rbrack$, contradicting the initial assumption

$$\begin{equation*} \mathbb{E}\lbrack \exp (it_0X_{n_k})\rbrack \not \rightarrow \mathbb{E}\lbrack \exp (it_0X)\rbrack . \end{equation*}$$■

We now present a new method for detecting sign changes of functions using Lemma 4.4, Lemma 4.5, and Bochner’s theorem.

Lemma 4.6.

Let $\{f_n\}$ be a sequence of real valued functions in $C_0^\infty ([0,1])$, and let $\{a_n\}$ be a sequence of positive reals such that for each fixed non-negative integer $m$ we have

$$\begin{equation} \lim _{n \to \infty } a_n^{-2m}\int _0^1 \left|\partial _x^m f_n(x)\right|^2 dx = b_{2m} {\tag{4.1}} \end{equation}$$

for some positive real numbers $b_{2m}$. Assume that $d\mu (\xi )$ is the unique probability distribution whose $2m$th moment is $b_{2m}/b_0$ and whose $(2m+1)$th moment is zero for any $m \geq 0$. If $d\mu (\xi )$ is not positive-definite, then all but finitely many $f_n$ has at least one sign change on $(0,1)$.

Proof.

Assume for contradiction that there exists a subsequence $\{f_{n_k}\}$ such that $f_{n_k}$ does not change sign on $(0,1)$ for all $k \geq 1$. Let $h_k$ be given by

$$\begin{equation*} h_k(\xi )= \frac{a_{n_k}}{2\pi }\left|\int _0^1 f_{n_k}(x) e^{ia_{n_k} \xi x}dx\right|^2 \left(\int _0^1 |f_{n_k}(x)|^2 dx\right)^{-1}. \end{equation*}$$

Then from 4.1, we have for each $m \geq 0$,

$$\begin{equation*} \lim _{k \to \infty } \int _{-\infty }^\infty \xi ^{2m} h_k(\xi ) d\xi = b_{2m}/b_0, \end{equation*}$$

and since $h_k(\xi )$ is an even function in $\xi$, the sequence of probability distribution $\{h_k(\xi )d\xi \}$ converges in moments to $d\mu (\xi )$. We therefore conclude from Lemma 4.5 that the sequence of characteristic functions of $h_k(\xi )d\xi$ converges point-wise to the characteristic function of $d\mu (\xi )$.

Now observe that since $f_{n_k}$ does not change sign along $(0,1)$, $h_k(\xi )$ is a positive-definite function in $\xi$ for each $k$ by Bochner’s theorem. Therefore the characteristic function of $h_k(\xi )d\xi$ is a non-negative function for each $k$. However, since we assumed $d\mu (\xi )$ is not positive-definite, the characteristic function $\int _{-\infty }^\infty e^{it\xi }d\mu (\xi )$ is negative for some $t \in \mathbb{R}$, which contradicts the point-wise convergence of characteristic functions. We therefore conclude that all but finitely many $f_n$ has at least one sign change on $(0,1)$.

■

4.3. Sign changes of even eigenfunctions on fixed segments

Lemma 4.7.

Assume that QUE holds for the sequence of even eigenfunctions $\{u_{n}\}_{n \geq 1}$. For any fixed segment $\beta \subset \mathrm{Fix}(\tau )$, all but finitely many $u_{n}$ have at least one sign change on $\beta$.

Proof.

Assume for contradiction that there exists a subsequence of even eigenfunctions $\{u_{n_k}\}_{k \geq 1}$ such that $u_{n_k}$ does not change sign along $\beta$ for all $k \geq 1$. Fix a non-negative function $f \in C_0^\infty (\beta )$.

First, by Corollary 3.2, we can find a subsequence $\{u_{j_k}\}_{k \geq 1}\subset \{u_{n_k}\}_{k \geq 1}$ such that

$$\begin{equation*} \lim _{k \to \infty }2 \int _\beta f^2(t) dt||fu_{j_k}||_{L^2(\beta )}^{-2} = a \end{equation*}$$

for some $0\leq a\leq 1$. Then by Theorem 3.1, we have that

$$\begin{equation*} \lim _{k \to \infty }1 - \lambda _{j_k}^{-2m}\int _{\beta } \left|\partial _t^{m}(f(t)u_{j_k}(t))\right|^2 ||fu_{j_k}||_{L^2(\beta )}^{-2}dt=a-\frac{a}{\pi }\int _{-1}^1 \xi ^{2m} \frac{d\xi }{\sqrt {1-\xi ^2}}. \end{equation*}$$

We therefore have

$$\begin{align*} \lim _{k \to \infty }\lambda _{j_k}^{-2m}\int _{\beta } \left|\partial _t^{m}(f(t)u_{j_k}(t))\right|^2 ||fu_{j_k}||_{L^2(\beta )}^{-2}dt &= (1-a)+\frac{a}{\pi }\int _{-1}^1 \xi ^{2m} \frac{d\xi }{\sqrt {1-\xi ^2}}\\ &=\int _{-\infty }^\infty \xi ^{2m} d\mu _a(\xi )\\ &=b_{2m}, \end{align*}$$

where $d\mu _a(\xi )$ is the probability measure given by

$$\begin{equation*} d\mu _a(\xi )=\frac{(1-a)}{2}(\delta _{-1}(\xi )+ \delta _{1}(\xi ))d\xi +\frac{a}{\pi }I_{[-1,1]}(\xi )\frac{d\xi }{\sqrt {1-\xi ^2}}. \end{equation*}$$

Here $I_{[-1,1]}(\xi )$ is the indicator function of $[-1,1]$. Observe that

$$\begin{equation*} \int _{-\infty }^\infty \xi ^{2m} d\mu _a(\xi ) \leq (1-a)+\frac{a}{\pi }\int _{-1}^1 \frac{d\xi }{\sqrt {1-\xi ^2}} =1, \end{equation*}$$

and

$$\begin{equation*} \limsup _{m \to \infty }\frac{1}{2m} b_{2m}^{\frac{1}{2m}} = 0 < +\infty , \end{equation*}$$

so by Lemma 4.4, $d\mu _a(\xi )$ is the only probability measure on $\mathbb{R}$ whose $2m$th moment is $b_{2m}$ and whose $(2m+1)$th moment is zero for any $m \geq 0$.

Note that the characteristic function of $d\mu _a(\xi )$ is given by

$$\begin{equation*} \int _{-\infty }^\infty e^{it\xi } d\mu _a(\xi )= (1-a) \cos (t)+ a J_0(t), \end{equation*}$$

where $J_0(t)$ is the Bessel function of the first kind and that

$$\begin{equation*} (1-a) \cos (\pi )+ a J_0(\pi ) \leq J_0(\pi ) = -0.3042\ldots <0 \end{equation*}$$

which implies that $d\mu _a(\xi )$ is not positive-definite.

It now follows from Lemma 4.6 that $f(t)u_{j_k}$ has at least one sign change along $\beta$ for all but finitely many $k$, which contradicts the assumption that $u_{n_k}$ does not change sign on $\beta$ for all $k\geq 1$. We therefore conclude that all but finitely many $u_{n}$ have at least one sign change on $\beta$.

■

We complete the proof of Theorem 1.6 by proving Theorem 4.3.

Proof of Theorem 4.3.

Fix $N \in \mathbb{N}$. Let $\beta _1, \ldots , \beta _N \subset \mathrm{Fix}(\tau )$ be a set of disjoint segments. Then by Lemma 4.7, for all sufficiently large $k$, $u_{n}$ has at least one sign change on each curve $\beta _i$ for $i=1,\ldots , N$. Hence we have

$$\begin{equation*} \liminf _{k\to \infty }\#\left(Z_{u_{n}}\cap \mathrm{Fix}(\tau )\right) \geq N, \end{equation*}$$

and since $N$ can be chosen arbitrarily large, we conclude that

$$\begin{equation*} \lim _{k\to \infty }\#\left(Z_{u_{n}}\cap \mathrm{Fix}(\tau )\right) = +\infty . \end{equation*}$$■

5. Nodal domains of odd eigenfunctions

In this section we prove an analogy of Theorem 1.6 for sequence odd eigenfunctions assuming QUE. Recall from 3.1 that

$$\begin{equation*} \langle \Delta _g T u,u\rangle _{U_-}-\langle T u, \Delta _g u\rangle _{U_-}=\langle \partial _n T u|_\beta ,u|_\beta \rangle _\beta -\langle T u|_\beta , \partial _n u|_\beta \rangle _\beta . \end{equation*}$$

From the assumption that $u$ is an odd eigenfunction, we have the Rellich identity for odd eigenfunctions

$$\begin{equation} \frac{1}{\lambda }\langle \lbrack -\Delta _g, T\rbrack u,u\rangle _{U_-}=\frac{1}{\lambda }\langle T u|_\beta , \partial _n u|_\beta \rangle _\beta . {\tag{5.1}} \end{equation}$$

Let

$$\begin{equation*} a_{\delta ,m}(x,\xi )=\chi \left(\frac{n}{\delta }\right)f^2(t) \xi _t^{2m}\xi _n, \end{equation*}$$

and let $T= \mathrm{Op}(a_{\delta ,m})$. For simplicity, let $Nu=\lambda ^{-1}\partial _n u|_\beta$. Then the RHS of 5.1 is

$$\begin{multline*} (-1)^m\lambda ^{-2m}\int _{\beta } f^2(t)\left(\partial _t^{2m}Nu(t) \right) Nu(t)dt\\ =\lambda ^{-2m}\int _{\beta } \left|\partial _t^{m}\left(f(t)Nu(t) \right)\right|^2 dt+O_{m,f} \left(\lambda ^{-\frac{1}{4}}\right), \end{multline*}$$

and the LHS of 5.1 is

$$\begin{equation*} \frac{2}{\pi }\int _{-1}^1 \xi ^{2m}\sqrt {1-\xi ^2} d\xi \int _\beta f^2(t) dt+o_{\delta ,m,f}(1)+O_{m,f}(\delta ). \end{equation*}$$

Therefore Theorem 3.1 for odd eigenfunctions assuming QUE is

$$\begin{align*} \lim _{k \to \infty } \lambda _{n}^{-2m}\int _\beta |\partial _t^m \left(f(t)Nu_{n}(t)\right) |^2 dt &= \frac{2}{\pi } \int _\beta f^2(t)dt \int _{-1}^1 \xi ^{2m}\sqrt {1-\xi ^2}d\xi \\ &=b_{2m} \int _\beta f^2(t)dt . \end{align*}$$

Let $d\mu (\xi )=\pi ^{-1}\sqrt {1-\xi ^2}d\xi$. Observe that

$$\begin{equation*} b_{2m}= \frac{2}{\pi } \int _{-1}^1 \xi ^{2m}\sqrt {1-\xi ^2}d\xi \leq \frac{2}{\pi } \int _{-1}^1 \sqrt {1-\xi ^2}d\xi =1, \end{equation*}$$

and

$$\begin{equation*} \limsup _{m \to \infty } \frac{1}{2m}b_{2m}^{\frac{1}{2m}}=0<+\infty , \end{equation*}$$

so by Lemma 4.4, $d\mu (\xi )$ is the only probability measure on $\mathbb{R}$ whose $2m$th moment is $b_{2m}$ and whose $(2m+1)$th moment is zero for any $m\geq 0$. Now note that

$$\begin{equation*} \int e^{it\xi } d\mu (\xi ) = J_1(t)/t, \end{equation*}$$

and since $J_1(5)/5=-0.0655\ldots <0$, $d\mu (\xi )$ is not positive-definite, so we may apply Lemma 4.6 to conclude.

Lemma 5.1.

Assume that QUE holds for a sequence of odd eigenfunctions $\{u_{n}\}$. For any fixed segment $\beta \subset \mathrm{Fix}(\tau )$, all but finitely many $\partial _n u_{n}|_\beta$ has at least one sign change on $\beta$.

As in Theorem 4.3, Lemma 5.1 implies the following.

Theorem 5.2.

Assume that QUE holds for a sequence of odd eigenfunctions $\{u_{n}\}$. Then

$$\begin{equation*} \lim _{k \to \infty } \#\{x \in \mathrm{Fix}(\tau )\nobreakspace :\nobreakspace (\partial _n u_{n})(x)=0\} = +\infty . \end{equation*}$$

We now use the topological argument in GRS13JZ16 to conclude an analogy of Theorem 1.6 for odd eigenfunctions.

Theorem 5.3.

Assume that QUE holds for a sequence of odd eigenfunctions $\{u_{n}\}$. Then

$$\begin{equation*} \lim _{k \to \infty } \mathcal{N}(u_{n}) \to +\infty . \end{equation*}$$

6. Proof of Theorem 1.1

We now prove Theorem 1.1 using Theorem 1.6. Let $\Gamma$ be an arithmetic triangle group, and let $\mathbb{X}=\Gamma \backslash \mathbb{H}$. Let $\{\phi _j\}_j$ be the complete sequence of Hecke-Maass eigenforms on $\mathbb{X}$; i.e., it is a joint eigenfunction of $-\Delta _g$ and Hecke operators $\{T_n\}_{n\geq 1}$. It is shown in GRS15 that there exists an orientation-reversing isometric involution $\tau : \mathbb{X} \to \mathbb{X}$ such that $\mathrm{Fix} (\tau )$ is separating and that $\tau$ commutes with all $T_n$. From the multiplicity one theorem for Hecke eigenforms AL70, the sequence of Hecke eigenvalues $\{\lambda _\phi (n)\}_{n \geq 1}$ of $T_n$ (i.e., $T_n\phi =\lambda _\phi (n) \phi$) determines $\phi$ uniquely. Hence any Hecke-Maass eigenform $\phi _j$ on $\mathbb{X}$ is an eigenfunction of $\tau$ so that we have either $\tau \phi _j =\phi _j$ or $\tau \phi _j=-\phi _j$ for all $j$. Now from the arithmetic quantum unique ergodicity theorem by Lindenstrauss Lin06, QUE holds for $\{\phi _j\}_j$—hence we conclude that $\lim _{j \to +\infty } \mathcal{N}(\phi _j)=+\infty$ by Theorem 1.6.