# 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 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 be a smooth compact Riemannian surface without boundary, and let be an orthonormal Laplacian eigenbasis ordered by the eigenvalue, i.e.,

where is the Laplace-Beltrami operator on Here . where , is the volume form of the metric We assume throughout the paper that every eigenfunction is real valued. We denote by . the nodal set of and by the number of nodal domains of where nodal domains are the connected components of , .

The purpose of this paper is to understand the growth of as tends to Note that Courant’s nodal domain theorem .Reference CH53 and Weyl law imply that However, it is not true in general that the number of nodal domains necessarily grows with the eigenvalue. For instance, when . (the standard sphere) or (the flat torus), there exists a sequence of eigenfunctions with that satisfy Reference Ste25Reference Lew77Reference JN99.

We first state the main result of the paper.

Note that there are arithmetic triangle groups Reference Tak77a which are divided into commensurable classes Reference Tak77b.

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 Reference Lin06 asserts that QUE holds for Maass-Hecke eigenforms on these triangles. In order to state Theorem 1.6, we first fix a quantization of a symbol , to a pseudo-differential operator. (We refer the readers to ,Reference Zwo12 for detailed discussion on the subject.) We say QUE holds for the sequence of eigenfunctions if we have

for any fixed symbol of finite order. Here is a normalized Liouville measure on the unit cotangent bundle We often write . for an operator that acts on an eigenfunction with the eigenvalue as .

We say a function on is even (resp. odd) if (resp. 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 Such an argument is first developed in .Reference GRS13, and we review in Section 4.1 in terms of the nodal graphs and Euler’s inequality as in Reference JZ16. We then use Bochner’s theorem and a Rellich type identity to deduce from QUE that even (resp. odd) eigenfunctions have a growing number of sign changes (resp. singular points) along as tends to This is the main contribution of the paper, and we sketch the argument in the following section. .

### 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 have at least one sign change on any given fixed segment of .

To simplify the discussion, let be a sequence of functions in Assume that for any fixed integer . we have

for some positive real number

Assume that there exists a unique probability measure

We claim that all but finitely many

Now let

should exist. However, under the assumption that QUE holds for

for each fixed

We first deduce from Equation 1.3 that (Corollary 3.2)

and so

Assume for simplicity that, for some

Then Equation 1.3 implies that

and we may apply the argument to

to conclude that all but finitely many

## 2. estimates for the restriction to a curve of derivatives of eigenfunctions

Let

Recall that

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 Reference Bin04. Denoting by

In the proof of Theorem 3.1, we need an improvement over Equation 2.2, and we achieve an improvement by combining the

Since we only need any power saving over

## 3. Rellich type analysis when QUE holds: even eigenfunctions

In this section, we prove Equation 1.3 with explicit constants