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Mathematics of Chaos in 2019

Steve Zelditch

The purpose of this survey is to introduce the reader to the thor [Z17, Z18]. One of the leaders in the field, Nalini problems of mathematical QC (quantum chaos), moving Anantharaman, has just given a plenary address on quan- rapidly from the origins of the subject to some of the most tum chaos at the 2018 ICM, and interested readers may recent advances. Quantum chaos is now a rather mature consult her ICM Proceedings article [A18] as well as her field, and there exists a stream of expository articles over earlier expository article [A14]. Although this article be- the last twenty years, including several by the present au- gins at the beginning with the origins of quantum mechan- ics and its relation to classical , we aim to get Steve Zelditch is the Wayne and Elizabeth Jones Professor of Mathemat- to recent results and to avoid repeating material already ics at Northwestern University. His email address is zelditch@math nicely covered in the prior expositions.1 The new results .northwestern.edu. we discuss pertain to: Communicated by Notices Associate Editor Chikako Mese. • Applications of the FUP ( uncertainty prin- For permission to reprint this article, please contact: [email protected]. 1The number of references is limited to twenty; most are survey articles where DOI: https://doi.org/10.1090/noti1958 the reader can find precise references.

1412 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 9 ciple) to weak* limits of semiclassical measures When 휓0(푥) = 휓ℏ,푗 is an eigenfunction (2), then

(section “ and the Dyatlov- −푖푡퐸푗(ℏ) Jin Full Support Theorem”), 휓(푡, 푥) = 푒 ℏ 휓ℏ,푗(푥). (3) • 퐿푝 퐿푝 norms of eigenfunctions (subsection “ In general, the solution is given by applying the propagator norms”), and 푖푡 − 퐻ℏ̂ • Small-scale quantum and its applica- 푈ℏ(푡) ∶= 푒 ℏ tions to counting numbers of nodal domains, and to the initial state, where proving equidistribution of 퐿2 mass and of nodal 휕 sets on shrinking balls (subsection “Lower bounds 푖ℏ 푈ℏ(푡) = 퐻ℏ̂ 푈ℏ(푡), 푈ℏ(0) = 퐼 on numbers of nodal domains”). 휕푡 is a 1-parameter group of unitary operators on ℋ. He also Origins proposed that all physically relevant quantities should be The most familiar and historically the first quantum me- matrix elements (expectation values) of a bounded opera- chanical problem concerns the hydrogen atom, an elec- tor 퐴, tron moving around a nucleus. One might picture it as 휌ℏ,푗(퐴) = ⟨퐴휓ℏ,푗, 휓ℏ,푗⟩. (4) a classical 2-body problem, such as a planet going around the sun. But this picture cannot be correct, because an ac- In the case of an eigenfunction, celerating charge radiates energy and would spiral into the 휌ℏ,푗(푈ℏ(−푡)퐴푈ℏ(푡)) = 휌ℏ,푗(퐴), (5)

nucleus almost immediately. How can one reconcile this 푖푡퐸푗(ℏ) classical picture with the obvious fact that atoms do not since the two factors of 푒 ℏ cancel out; the probabilities immediately collapse? and the matrix elements are independent of 푡. For instance, 2 proposed the “,” which |휓ℏ,푗(푡, 푥)| 푑푥 is the probability density of finding the “quantizes” special periodic orbits to produce “stationary particle of energy 퐸푗(ℏ) at the point 푥. states” and “energy levels” of the hydrogen atom. These In effect, Schrödinger replaced the “Bohr-Sommerfeld” quantization conditions are still of of the 2-body problem by linear algebra. By passing from considerable interest today and are the precursor to the classical mechanics to linear algebra, Schrödinger resolved field of semiclassical connecting the problem of how an electron moving around a nucleus quantum and classical mechanics [Zw]. But the quantiza- can be moving and stationary at the same time. But the tion conditions do not apply to more complicated systems price one pays is that the intuitions of classical mechanics such as the helium atom. are lost in favor of linear algebra and functional analysis, Schrödinger then proposed his famous equation to de- about which most people have little intuition. fine “stationary states.” Instead of representing the elec- What puts classical mechanical intuition back into tron as a point particle, he represented it by a vector 휓 ∈ quantum mechanics is the behavior of the eigenfunctions ℋ in a . In the case of the hydrogen atom, and eigenvalues as the ℏ → 0. By classical 2 3 ℋ = 퐿 (ℝ , 푑푥) (푑푥 is Lebesgue measure). Schrödinger mechanics is meant the (푥푡, 휉푡) of points ∗ 푛 then introduced the operators of his name, (푥, 휉) in the 푇 ℝ (the cotangent bundle of 푛 2 ℝ ) under the classical Hamiltonian system 퐻ℏ̂ ∶= −ℏ Δ + 푉, (1) 푑 휕퐻 where ℏ is a very small constant (Planck’s constant). Here, ⎪⎧ 푥 = (푥 , 휉 ), 푑푡 푡 휕휉 푡 푡 Δ is the Laplace and 푉 is the potential, i.e., the (6) ⎪⎨ 푑 휕퐻 multiplication operator by a function 푉 on 푀. In the case 휉푡 = − (푥푡, 휉푡), 3 휕2 ⎩푑푡 휕푥 of the hydrogen atom, Δ푔 = ∑푗=1 2 is the standard 휕푥푗 generated by the classical Hamiltonian Euclidean Laplacian and 푉(푥) = − 1 . Schrödinger pro- |푥| 퐻(푥, 휉) = |휉|2 + 푉(푥) ∶ 푇∗ℝ푛 → ℝ (7) posed that a particle with a fixed energy 퐸푗(ℏ) be repre- sented by an 퐿2-normalized eigenvector of (1), i.e., associated to (1). Quantum mechanical objects tend to classical mechanical ones in the semiclassical limit ℏ → 0. 퐻̂ 휓 = 퐸 (ℏ)휓 . (2) ℏ ℏ,푗 푗 ℏ,푗 But which objects do they tend to? And how? In most He called 휓ℏ,푗 a : in some sense, it does problems, it is crucial to understand the joint behavior of not change with time. More precisely, the evolution of a the semiclassical limit ℏ → 0 and the long-time evolution state 휓 in quantum mechanics is governed by the time- 푡 → ∞. This is the subject of semiclassical quantum me- dependent Schrödinger equation, chanics [Zw]. 휕 The Hamiltonian flow (6) of any Hamiltonian preserves 푖ℏ 휓(푡, 푥) = 퐻̂ 휓(푡, 푥), 휓(0, 푥) = 휓 (푥). ∗ 푛 휕푡 ℏ 0 level sets (energy surfaces) Σ퐸 = {(푥, 휉) ∈ 푇 ℝ ∶

OCTOBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1413 퐻(푥, 휉) = 퐸}, and classical dynamics refers to the Hamil- Notation and Background tonian flow restricted to Σ퐸. Quantum chaos refers to the For the rest of the article, we specialize to the setting where special case where the classical Hamiltonian system is 푉 = 0 in (1), but Δ푔 is the Laplacian on a compact Rie- “chaotic” on an energy surface. The term “chaotic” is sug- mannian manifold (푀푛, 푔) of dimension 푛, locally given gestive rather than precise and can be taken to mean er- by godic, , hyperbolic, Bernoulli, or some more spe- 푛 1 휕 푖푗 휕 cific type of chaotic or unpredictable dynamical behavior. Δ푔 = ∑ 푔 √|푔| , (8) √|푔| 휕푥푖 휕푥푗 What impact does the chaotic behavior of the classical flow 푖,푗=1 have on the eigenvalues/eigenfunctions or the propagator 휕 휕 푖푗 where 푔푖푗 = 푔( 휕푥 , 휕푥 ), [푔 ] is the inverse matrix to of the quantum problem? If one holds ℏ fixed, then the an- 푖 푗 [푔푖푗], and |푔| = det[푔푖푗]. When 푀 is compact, there is swer is none. One merely has the spectral decomposition an orthonormal basis {휙푗} of eigenfunctions, of ℋ into eigenspaces of (1) and the simple evolution (3) of the eigenvectors. But if one considers the behavior of 2 Δ푔휙푗 = −휆푗휙푗, ∫ 휙푖휙푗푑푉푔 = 훿푖푗 (9) the matrix elements (4) as ℏ → 0 (or as the eigenvalue 푀

퐸푗(ℏ) → 퐸), or if one considers the joint asymptotics of with 0 = 휆0 < 휆1 ≤ 휆2 ≤ ⋯ ↑ ∞ repeated according to 푈ℏ(푡) as ℏ → 0, 푡 → ∞, then the effects of the chaotic their multiplicities. When 푀 has a nonempty boundary classical dynamics are felt. 휕푀, one imposes boundary conditions such as Dirichlet Although quantum chaos refers to quantizations of 퐵푢 = 푢|휕푀 = 0 or Neumann 퐵푢 = 휕휈푢|휕푀 = 0. −1 chaotic classical Hamiltonian systems, it should not be If one sets ℏ푗 = 휆푗 , then (9) takes the form thought that chaos is the only interesting type of system. 2 It is definitely of special interest, but the opposite extreme ℏ푗Δ푔휙푗 = −휙푗, (10) of “completely ” is equally of interest. It which is (2) with 푉 = 0 and with 퐸푗(ℏ) = −1. The is a familiar observation that the easiest dynamical systems limit ℏ푗 → 0 is thus the same as 휆푗 → ∞. It is customary to understand are the most chaotic ones or the most pre- in semiclassical analysis to use the form (10) and speak of dictable (integrable) ones. One of the most interesting the “semiclassical limit,” but also customary in PDE to use questions is to determine the semiclassical behavior of (2) and speak of the “high frequency limit.” When 푉 = 0 eigenfunctions/eigenvalues in mixed systems, which are the limits are the same. partly integrable, partly chaotic (see [Go18]). The Weyl eigenvalue counting function is For which potentials 푉 does the Hamiltonian (7) in Ki- 푛 푛−1 netic + Potential form generate chaotic dynamics on some 푁(휆) = #{푗 ∶ 휆푗 ≤ 휆} = 퐶푛Vol(푀, 푔)휆 + 푂푔(휆 ). Σ ⊂ 푇∗ℝ푛 energy surface 퐸 ? To our knowledge, it is un- Here and hereafter, a constant 퐶푛 (resp., 퐶푔) is understood known! G. Paternain and M. Paternain have proved that to depend only on the dimension (resp., metric). such a Newtonian Hamiltonian flow cannot be highly The classical Hamiltonian underlying (8) is the metric chaotic (namely, Anosov or hyperbolic). Singular poten- norm-square of a covector tials generating chaotic flows do exist, and an important 2 푖푗 ∗ example is provided by a hydrogen atom in a strong mag- |휉|푔 = ∑ 푔 (푥)휉푖휉푗 ∶ 푇 푀 → ℝ+. netic field.2 But there are many examples if we enlarge the 푖,푗 2 2 setting to Riemannian manifolds, where ℋ∶=퐿 (푀, 푑푉푔) In the language of pseudo-differential operators, |휉|푔 is 2 are the 퐿 functions on a Riemannian manifold (푀, 푔) the symbol of −Δ푔. We take square roots to get the first- with respect to its volume form 푑푉푔. Model examples in- order operator √−Δ푔 with symbol |휉|푔. The motivation clude the geodesic flow on the unit tangent bundle ofa is that √−Δ푔 generates the half-wave group compact surface of negative curvature, and one of the most popular examples of quantum chaos is generated by the 푈(푡) = exp 푖푡√−Δ푔; (11) Δ Laplacian 푔 on a surface of constant negative curvature. i.e., we can work with the half-wave propagator (11) rather This is in part due to its relevance to automorphic forms than with the Schrödinger flow. The Hamiltonian flow (6) and arithmetic quantum chaos (see [S95, S11]). In this generated by 퐻(푥, 휉) = |휉|푔 is the homogenous geodesic article we focus primarily on methods of PDE and semi- ∗ flow. We usually restrict it to an energy surface Σ1 = 푆 푀 classical analysis rather than on L-functions and arithmetic (the unit cosphere bundle {|휉|푔 = 1}) to obtain the key methods. dynamical flow 퐺푡 ∶ 푆∗푀 → 푆∗푀.

2Thanks to Andreas Knauf for the references; he has obtained other singular It preserves Liouville measure 푑휇퐿, the natural surface mea- ∗ examples. sure on 푆 푀.

1414 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 9 ∗ What is “quantization”? Quantization is a procedure for on the cotangent ball bundle 퐵휖 푀. For instance, if 푀 = 1 1 converting nice functions (classical ) 푎(푥, 휉) 푆 (the unit circle), then 푀ℂ ≃ 푆 × ℝ, 휙푗(푥) = sin 푗푥, 푇∗푀 Op (푎) 퐿2(푀) ℂ ℂ on phase space into operators ℏ on and 휙푗 (푧) = sin 푗푧. As this example indicates, 휙푗 is ex- in such a way that commutators [Opℏ(푎), Opℏ(푏)] = ponentially growing (or decaying) as 푗 → ∞. Therefore ℏ {푎, 푏}+푂(ℏ2), where {푎, 푏} = ∑ ( 휕푎 휕푏 − 휕푎 휕푏 ) is 2 ∗ 푖 푗 휕푥푗 휕휉푗 휕휉푗 휕푥푗 we 퐿 -normalize it on each “sphere bundle” 휕푀휖 ≃ 푆휖 푀, 휙ℂ the Poisson bracket. With this (and similar requirements), { 푗 } obtaining the sequence ‖휙ℂ‖ on phase space. quantum mechanics tends to classical mechanics as ℏ → 0. 푗 퐿2(휕푀휖) 푗≥1 Schrödinger was the first to quantize Hamiltonians of the Quantization, observables, and expectation values. As form |휉|2 + 푉(푥) as the operators (1), and H. Weyl soon mentioned above, the quantities in quantum mechanics are the “expectation values” (4) after defined the quantization Opℏ(푎) of a general classi- 푎 cal observable . For instance, the quantization of the lin- 휌푗(퐴) ∶= ⟨퐴휙푗, 휙푗⟩ (12) ℏ 휕 ear function 휉푗 is 푃푗 = 푖 휕푥 , and the quantization of 푥푗 푗 of a bounded self-adjoint operator 퐴 relative to an energy is 푄푗 = 푀푥푗 (multiplication by 푥푗) with the commuta- eigenstate (i.e., eigenfunction) 휙푗. Expectation values are [푃 , 푄 ] = ℏ 훿 tion relations 푗 푘 푖 푗푘. The quantization also ex- used to probe the behavior of eigenfunctions in the semi- tends to the groups generated by these operators, i.e., to : that is, we study the asymptotics of (12) 푖푡푄 푖푡푃 푖푡퐻̂ their exponentials 푒 , 푒 , 푒 ℎ . It is easy to see that for general “test operators” 퐴 as 푗 → ∞ to study the semi- 푖푡푄푗 푖푡푥푗 푖푡푃푘 푒 푓(푥) = 푒 푓(푥), 푒 푓(푥) = 푓(푥 + 푡푒푘), where classical asymptotics of eigenfunctions. As this statement 푒 푘 ℝ푛 푘 is the th standard basis element of . These opera- suggests, the main problems involve sequences {휙푗푘 } of tors are unitary and are the simplest examples of Fourier eigenfunctions rather than individual ones and their pos- integral operators which quantize general canonical (i.e., sible limit behavior. symplectic) transformations on cotangent bundles. There will not exist good asymptotics for (12) for a gen- Actually, quantization is not unique, since one can con- eral bounded operator 퐴. To obtain useful asymptotics, jugate any quantization by a unitary operator to obtain an- one needs to choose special types of bounded operators 퐴. other. Although the Schrödinger quantization is the most The most popular and standard choice in the Schrödinger common one in PDE, this article will often employ the representation is that 퐴 ∈ Ψ0(푀), i.e., 퐴 is a pseudo- holomorphic quantization on Bargmann– of differential operator of order zero on 퐿2(푀), either ho- 푛 ∗ 푛 entire holomorphic functions on ℂ ≃ 푇 ℝ which are mogeneous or semiclassical (see [Zw] for background). In- 2 in 퐿 with respect to Gaussian measure. The Bargmann deed, pseudo-differential operators were virtually invented 2 푛 transform takes 퐿 (ℝ ) to the Bargmann–Fock space for purposes such as this, to possess nice semiclassical be- 2 퐻2(ℂ푛, 푒−|푧| 푑푧), where 푑푧 is the Lebesgue measure on havior. A key property of 퐴 ∈ Ψ0(푀) is that it possesses 푛 ∞ ∗ ℂ . Bargmann–Fock space comes up naturally when one a principal symbol 휎퐴 ∈ 퐶 (푇 푀), which is homoge- works with creation/annihilation operators 푎 = 푃 + 푖푄, neous of degree zero when 퐴 is a homogeneous pseudo- 푎∗ = 푃 − 푖푄. It is somewhat simpler in that classical differential operator. It can then be identified with a mechanics and quantum mechanics both take place on smooth function on the unit cotangent bundle 푆∗푀, and ∗ 푛 푛 the phase space 푇 ℝ ≃ ℂ ; quantum mechanics is just one thinks of 퐴 as the quantization of 휎퐴. In the holo- the restriction of classical mechanics to holomorphic func- morphic representation one uses Toeplitz operators Π휎Π tions. where Π is the orthogonal projection to the holomorphic This procedure of equipping a cotangent bundle with a functions and 휎 is a symbol. complex structure makes sense on any real analytic man- The first thing to know about the expectation values 3 ifold. We can complexify 푀 to a complex manifold 푀ℂ, (or matrix elements) (12) is that they induce distributions ∗ ∗ which is canonically equivalent to 푇 푀 (or rather to a ball 푑Φ푗 on 푇 푀. In fact, with a proper choice of quantiza- ∗ ∗ ∗ bundle 퐵휖 푀) and identity 푇 푀 with 푀ℂ using the imag- tion, they induce probability measures on 푆 푀 by the rule inary time exponential map 퐸(푥, 휉) = exp푥 푖휉. Using ∗ 퐸 we endow 푇 푀 with a complex structure, adapted to ⟨Op(푎)휙푗, 휙푗⟩ = ∫ 푎(푥, 휉) 푑Φ푗. a real analytic metric 푔. More precisely, it lives on a ball 푆∗푀 ∗ ∞ ∗ bundle 퐵휖0 푀 of some radius 휖0, defined by |휉|푔 ≤ 휖0, In other words, the linear functional 푎 ∈ 퐶 (푆 푀) → 0 known as a Grauert tube (due to L. Lempert–R. Szoke and Op(푎) ∈ Ψ (푀) → 휌푗(Op(푎)) is a positive linear func- V. Guillemin–M. Stenzel; see [Z18]). tional of 푎 and therefore defines a measure 푑Φ푗. In various Each eigenfunction 휙푗 admits a holomorphic extension contexts, it has been called a “microlocal lift,” a “quantum ℂ ℂ 휙푗 to 푀휖. Thus, 휙푗 (퐸(푥, 휉)) is a holomorphic function lift,” a “Wigner distribution,” or a “microlocal defect mea- sure.” The key property is that 푑Φ푗 lives in phase space ∗ ∗ 3On general 퐶∞ manifolds one uses “almost analytic continuation.” 푇 푀; in the homogeneous setting, it lives on 푆 푀.

OCTOBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1415 ∞ In the holomorphic quantization, Op(푎) = Π푎Π, Definition 2. A subsequence {휙푗푘 }푘=1 of an orthonormal 2 2 where Π is the orthogonal projection to 퐻 (푀휖), the 퐿 basis is called QE (quantum ergodic) if 푑Φ푗푘 → 푑휇퐿 in the holomorphic functions on the Grauert tube. Then weak* sense.

The Laplacian Δ푔 is called QUE if one orthonormal 휌푗(Π푎Π) = ∫ 푎(푧) 푑Φ푗(푧), 휕푀휖 basis of eigenfunctions is QE. To the author’s knowledge, the only setting where QUE is known to occur is for Hecke with ℂ 2 eigenfunctions of arithmetic hyperbolic manifolds and re- 휙푗 (푧) lated settings (E. Lindenstrauss; see [S11]). 푑Φ푗(푧) ∶= | ℂ | 푑휇휖, (13) 1 2 ‖휙푗 ‖퐿 (휕푀휖) The simplest example is that of the circle 푀 = 푆 . The complex eigenfunctions are 푒푖푛푥, 푛 ∈ ℤ. We always as- where 푑휇휖 is the (Liouville) volume form on 휕푀휖 induced ∗ sume the eigenfunctions are real-valued and therefore by the symplectic volume form on 푇 푀. Note that 푑Φ푗 consider 1 cos 푛푥, 1 sin 푛푥. It is an immediate con- is manifestly a positive, smooth probability measure on √2 √2 ∗ 휕푀휖 ≃ 푆휖 푀. We opt to use the holomorphic representa- sequence of the Riemann–Lebesgue lemma that, for any 1 tion because (13) is most elementary in this setting. The 푓 ∈ 퐶(푆 ), measures (13) are called “Husimi distributions” in the ⨍ 푓(sin 푛푥)2푑푥 → ⨍ 푓푑푥. literature. 푆1 푆1 Weak* limit problem. We define Here, and henceforth, ⨍ denotes the average; i.e., one nor- 풬 ∶= weak*-limits of the sequence {푑Φ푗}. (14) malizes a measure to have unit mass. On the other hand, this calculation fails if we use the A probability measure 푑휈 is a weak* limit of a subse- 푒±푖푛푥 ∞ eigenfunctions . The failure is the simplest illustra- {푑Φ } ∫ ∗ 푓푑Φ → ∫ ∗ 푓푑휈 quence 푗푘 푘=1 if 푆 푀 푗푘 푆 푀 for every tion of quantum ergodicity versus quantum integrability. ∗ 푓 ∈ 퐶(푆 푀). In the holomorphic setting, one has the In phase space terms, 푒푖푛푥 corresponds to the Lagrangian rather explicit formulae (13) for the 푑Φ푗. ∗ 1 1 submanifold 휉 = 푛 in 푇 푆 = 푆 × ℝ휉. The unit cotan- The weak* limit problem is to determine all of the gent bundle |휉| = 1 has two connected components, so weak* limits. The only simple characterization of the lim- that the geodesic flow 퐺푡(푥, 휉) = (푥 + 푡휉, 휉) is not er- its is: godic. But the real eigenfunctions correspond to the quo- tient of 푇∗ℝ under 휉 → −휉, gluing together the two com- Proposition 1. If 푀 is a compact manifold, then 풬 ⊂ ℳ퐼, 푡 ponents, and the geodesic flow is ergodic on the quotient. where ℳ퐼 is the compact convex set of 퐺 -invariant probability ∗ There is a profound difference between studying the measures on 푆 푀 for the geodesic flow. The limits are time- ∗ reversal invariant if the eigenfunctions are real-valued. weak* limit problem in phase space 푇 푀 and in configu- ration (physical) space 푀, where the testing operators are Any weak* limit 휇 of {휌푗} is an invariant measure for just multiplications. The latter is best thought of as testing 퐺푡; i.e., 휇(퐸) = 휇(퐺푡퐸) for all Borel subsets 퐸 ⊂ 푆∗푀. for “flatness,” i.e., uniform distribution on the base man- This is because, by (5), 휌푗 is an invariant state for the auto- ifold 푀 rather than on the unit cotangent bundle. On a morphism 퐴→푈(푡)퐴푈(−푡). The proposition follows by flat torus, for instance, Egorov’s theorem, which says that 휌푗(푈(푡) Op(푎)푈(−푡)) 푖⟨푥,휆⟩ 푖⟨푥,휆⟩ 푡 Op(푎)푒 = 푎(푥, 휆)푒 . = 휌푗(Op(푎 ∘ 퐺 )) + 표(1) as 푗 → ∞ (see [Zw]). There are many invariant probability measures, and it is Hence difficult to characterize those that arise as quantum limits. 휆 ⟨Op(푎)푒푖⟨푥,휆⟩, 푒푖⟨푥,휆⟩⟩ = ∫ 푎 (푥, ) 푑푥. Some examples of invariant measures are: 푇푛 |휆| (1) Normalized Liouville measure 푑휇퐿. Thus, eigenfunctions concentrate on the invariant tori 휉 = (2) A periodic measure 휇훾 defined by ∫ 휎푑휇훾 = 휆/|휆|, although they have modulus 1 on the torus. 1 ∫ 휎푑푠, where 퐿훾 is the length of 훾. The cor- 퐿훾 훾 It is natural to wonder if a weak* limit measure 휇 might responding sequence of eigenfunctions is some- have some a priori regularity properties. Of course, in- times said to “scar” along 훾. variance under 퐺푡 implies smoothness in the direction of (3) A delta-function along an invariant Lagrangian geodesics, so the question is regularity in the transverse manifold Λ ⊂ 푆∗푀. The associated eigenfunc- direction. The short answer is no. For instance, Jakobson– tions are viewed as localizing along Λ. Zelditch proved that any invariant measure for the geo- (4) A more general measure that is singular with re- desic flow on the standard sphere 푆푛 is a weak* limit of spect to 푑휇퐿. There are many examples in the neg- some sequence of eigenfunctions; there are many general- atively curved case. izations of this result [Z17]. There is a profound result, due

1416 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 9 to Anantharaman and Anantharaman-Nonnenmacher Variance estimates. The fact that 푉퐴(휆) → 0 raises the (see [A14,A18]) giving lower bounds on the dynamical en- question of the “rate of quantum ergodicity.” The best −1 tropy of the limit measures; this is essentially a regularity that can be expected is that 푉퐴(휆) ≤ 퐶휆 . More gener- 푝 result. ally, one can study averages 푉퐴 (휆) of 푝th power variances 푡 2푝 Quantum ergodicity of eigenfunctions when 퐺 is er- |⟨퐴휙푗, 휙푗⟩−휔(퐴)| . Quantitative estimates have been godic. In this section, we assume that the geodesic flow studied by several authors, giving the following results: (푀, 푔) of is ergodic. Let • For negatively curved manifolds, 푉퐴(휆) ≤ −1 푝 −푝 퐶(log 휆) ; more generally 푉퐴 (휆)≤퐶(log 휆) 휔(퐴) ∶= ⨍ 휎퐴푑휇퐿. (15) 푆∗푀 (Zelditch, 1994; R. Schubert (2006, 2008)). There are generalizations to exponential functions The following is due to A. I. Schnirelman, S. Zelditch, Y. of |⟨퐴휙푗, 휙푗⟩ − 휔(퐴)| (Anantharaman–Rivière, Colin de Verdière, S. Zelditch–M. Zworski, P. Gerard–E. Le- 2012). ichtnam (see [A14, A18, Z17, Zw,So14] for background). • In the arithmetic setting such as the quotient of 2 Theorem 3. Let (푀, 푔) be a compact Riemannian manifold the upper-half-plane ℍ /SL(2, ℤ), 푉퐴(휆) ∼ −1 (possibly with boundary), and let {휆푗, 휙푗} be the spectral data 퐶퐴휆 , where 퐶퐴 is an explicit constant (Luo– 푡 ∗ of √−Δ푔. Then, if the geodesic flow 퐺 is ergodic on (푆 푀, Sarnak (2004), P. Zhao (2010), Zhao–Sarnak ∗ 푑휇퐿), there exists a subsequence 풮 of density one (퐷 (풮) = (2019)). See [Z17] for references. 1) such that The proofs involve estimates on decay of correlations for 푡 0 the geodesic flow, i.e., the rate of decay of ∫푆∗푀 푓(퐺 (휁))× lim ⟨퐴휙푗, 휙푗⟩ → 휔(퐴) for all 퐴 ∈ Ψ (푀). ∞ ∗ 푗→∞,푗∈풮 푓(휁)푑휇퐿(휁) when 푓 ∈ 퐶 (푆 푀) has mean value zero. The last result is amazingly accurate. It uses arithmetic 1 Density one means that 푁(휆) #{푗 ∈ 풮 ∶ 휆푗 ≤ 휆} → 1 techniques that have no analogue for general hyperbolic as 휆 → ∞. surfaces or other manifolds. Logarithmic estimates in the The proof splits into two parts: (i) a variance estimate negatively curved case reflect the exponential growth of the and (ii) extraction of a density one subsequence by a diag- geodesic flow and are now ubiquitous in quantum chaos. onal argument. The key quantities in (i) are the quantum It is a fundamental problem to break this exponential bar- variances rier—if, indeed, it is even possible.

1 2 QUE in terms of time and space averages. In the ergodic 푉퐴(휆) ∶= ∑ |⟨퐴휙푗, 휙푗⟩ − 휔(퐴)| . case, one knows that 휇퐿 ∈ 풬 (14) and that it is the limit 푁(휆) 푗∶휆 ≤휆 푗 of “almost all” eigenfunctions of an orthonormal basis. The general result relating variances and classical me- What is known about possible exceptional subsequences chanics is the following theorem, an improvement of The- of density zero? orem 3 due to Zelditch and T. Sunada (see [Z17] for back- This is known as the QUE problem. It may be put in a ground). functional analytic context as follows: Let ⟨퐴⟩ ∶= 푇 lim 1 ∫ 푈(푡)퐴푈(−푡) 푑푡. This is the diagonal part Theorem 4. Let (푀, 푔) be a compact Riemannian manifold 푇→∞ 2푇 −푇 퐴 {휙 } 퐴 ∈ Ψ0(푀) (possibly with boundary), and let {휆푗, 휙푗} be the spectral data of with respect to the basis 푗 . For , 푡 ∗ let 휔(퐴) be as defined in (15). QE implies that ⟨퐴⟩ = of √−Δ푔. Then the geodesic flow 퐺 is ergodic on (푆 푀, 0 휔(퐴)퐼+ 퐾, where all matrix elements of the diagonal op- 푑휇퐿) if and only if, for every 퐴 ∈ Ψ (푀), lim휆→∞ 푉퐴(휆) = erator 퐾 tend to zero along a density one subsequence. 0, and moreover Problem 5. Suppose the geodesic flow 퐺푡 of (푀, 푔) is er- 1 2 (∀휖)(∃훿) lim ∑ |(퐴휙푗, 휙푘)| < 휖. 푆∗푀 퐾 휆→∞ 푁(휆) godic on . Is a compact operator? Compactness 푗≠푘∶휆푗,휆푘≤휆 |휆푗−휆푘|<훿 of 퐾 implies that ⟨퐾휙푗, 휙푗⟩ → 0; hence ⟨퐴휙푗, 휙푗⟩ → 휔(퐴) along the entire sequence. Equivalently, is 풬 = The fact that 푉퐴(휆) → 0 implies that, for each 퐴, there {푑휇 } ∞ 퐿 ? exists a subsequence {푗푘}푘=1 ⊂ 퐍 of density one for which √−Δ 휌푗푘 (퐴) → 0. This is because all terms are positive. The fact In this case, is said to be QUE (quantum uniquely that the density one subsequence can be chosen indepen- ergodic). Rudnick–Sarnak conjectured that √−Δ of nega- dently of 퐴 requires a diagonal argument using a count- tively curved manifolds are QUE, i.e., that for any orthonor- able dense subset of symbols. This diagonal argument fails mal basis of eigenfunctions, the Liouville measure is the if one tries to prove QE for observables like multiplication only quantum limit. by the indicator function ퟏ퐸 of a general Borel set 퐸, since There do exist ergodic situations in which quantum lim- 퐿∞ is not separable. it measures other than Liouville occur. It was proved by A.

OCTOBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1417 Hassell [Ha16] that the well-known Bunimovich stadium ergodicity to the complex setting of Grauert tubes for the is an example. Numerical results of the physicist E. J. Heller Husimi measures (13) (see [Z18] for references). had strongly suggested this result, but it is difficult to prove. QER: Quantum ergodic restriction theorems. Let 퐻 ⊂ Mixed systems and converse QE. The weak* limit prob- 푀 be a hypersurface in a manifold with ergodic geodesic ∗ lem has mainly been studied in the case of completely in- flow. Then the set 푆퐻푀 of unit-covectors with footpoint tegrable systems (such as flat tori or surfaces of revolution) on 퐻 is a “cross section” of the geodesic flow in 푆∗푀. The and ergodic systems (such as manifolds of negative cur- quantum analogue is the space of Cauchy data vature). Most Hamiltonian systems are “mixed” in that (휙푗|퐻, 휕휈휙푗|퐻) of eigenfunctions on the hypersurface. they are neither integrable nor ergodic. It is hard to ana- The first return map of geodesics to the cross section isan lyze such systems because the “integrable component” can ergodic symplectic map, suggesting that the Cauchy data be Cantor-type sets whose indicator functions cannot be should be ergodic in 퐿2(퐻) (QER). Such QER phenom- quantized in an obvious way. For KAM systems, this Can- ena are useful in nodal set problems, especially in the case tor set is a kind of Cantor foliation by invariant tori, and where dim 푀 = 2, dim 퐻 = 1, because one can relate it seems very plausible that a positive proportion of eigen- nodal sets of 휙푗 on 푀 and zeros of 휙푗 on 퐻, which are functions concentrate on these tori and are not QE. Results much easier to study. of this kind have been recently established by S. Gomes QER has been proved by H. Christianson, J. A. Toth, and [Go18] and by Gomes–Hassell. the author following a proof in the case where 퐻 = 휕푀 Problem 6. Do there exist (푀, 푔) with nonergodic geo- due to Hassell and the author (see [Z17]). We state the re- sult in a special setting for later applications to nodal sets, desic flow for which √−Δ푔 is quantum ergodic (i.e., is Theorem 3 valid)? that of negatively curved surfaces possessing an isometric involution 휎 ∶ 푀 → 푀 with nonempty fixed point set An example due to B. Gutkin is a race-course stadium. Fix(σ), a finite union of closed geodesics, which divides There is a vague conjecture that “localization of eigenfunc- 푀 = 푀+ ∪ 푀− into two components. Such a Riemann tions” should be rare and that they should be “diffuse in surface is called a “real Riemann surface”: it is the complex- phase space” in quite general settings. The results of 2 ification of Fix(σ). The isometry 휎 acts on 퐿 (푀, 푑퐴푔), Gomes–Hassell indicate that QE is not a generic property. 2 2 and we define 퐿even(푀), respectively, 퐿odd(푀), to denote Length scales of QE. Another natural question is what is the subspace of even functions 푓(휎푥) = 푓(푥), respec- 푟(휆 ) the smallest length scale 푗 on which QE takes place; tively, odd elements 푓(휎푥) = −푓(푥). The even/odd eigen- i.e., whether Theorem 3 holds when the pseudo-differen- functions with respect to 휎 satisfy Dirichlet/Neumann tial operator is replaced by a sequence 퐴 = 퐴푗 whose prin- boundary conditions on 푀±. cipal symbols 휎퐴푗 are supported on balls 퐵(푝, 푟(휆푗)) of radii 푟(휆푗) → 0 in phase space or in configuration space. Theorem 7. Let (푀, 푔) be a real Riemann surface with iso- The natural length scales in are: metric involution 휎 as above, and let 훾 = Fix(휎) divide −1 푀 = 푀+ ∪ 푀− into two components. More generally, let (i) the wavelength scale ℏ푗 = 휆푗 , −훾 푀 be any compact surface with boundary 훾 = 휕푀 with er- (ii) logarithmic length scales (log 휆푗) (where 훾 > 훾 = 휕푀 0), godic geodesic flow, and let . Then, for a subsequence (iii) the mean level spacing between eigenvalues. of even (Neumann) eigenfunctions of density one, The Heisenberg (and other) uncertainty principles limit 2 4 ∫ 푓휙푗 푑푠 → ∫ 푓(푠) 푑푠, the smallness of the scale on which one can relate classical 훾 2휋Area(푀) 훾 and quantum mechanics. The spacing in (iii) is too small for any 푓 ∈ 퐶(훾). Similarly for normal derivatives of Dirichlet for current techniques to approach, although it is much (odd) eigenfunctions. discussed in the physics literature. On the wave length scale, eigenfunctions only oscillate a fixed number (e.g., In fact, there is a complementary general result: for a 푛 1, 000) of times, and it is hard to conceive of a semiclas- generic hypersurface 퐻 of any (푀 , 푔), there exists a den- ∞ sical limit on this scale, but one might imagine limit the- sity one subsequence such that {휙푗|퐻}푗=1 is QE on 퐻. −1 훾 orems on the scale 휆푗 (log 휆푗) . There is a sequence of The generic condition is known as “asymmetry” (see [TZ17, recent results on small-scale QE, due to X. Han [Han18], H. Z18]). However, the curves in Theorem 7 are symmetric, Hezari–G. Rivière [HR16], P. Humphries, and others. The and all odd eigenfunctions vanish on 훾. first three authors prove QE in a strong sense on log-scale Random wave/ONB models. A general heuristic (M. V. shrinking balls. Humphries [Hum18] has rather surpris- Berry) is that the ONB (orthonormal basis) of eigenfunc- ing counterexamples to small scale QE on balls of the scale tions behaves like a sequence of independent Gaussian −1 훾 휆푗 (log 휆푗) . Robert Chang and the author adapted the “random waves” of fixed frequency, such as random spher- proofs of Hezari–Rivière and Han of log-scale quantum ical harmonics on the sphere with fixed degree 푁. Since

1418 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 9 the eigenfunctions form an ONB, the author prefers a ran- ‖푓‖ = 1 such that ‖푃퐴푓‖ = 훼, ‖푄퐵푓‖ = 훽 if and only if −1 −1 −1 dom ONB model in which one chooses a random ONB cos 훼 + cos 훽 ≥ cos √휆1. of spherical harmonics of each degree 푁. Random waves/ The FUP in the sense of Bourgain, Dyatlov, Jin, and Zahl ONBs have many properties that are unknown and pos- is a kind of generalization where one replaces intervals by sibly false for ONBs of eigenfunctions, but the heuristic regular porous fractal sets. Some definitions: has inspired many interesting results on random nodal • For any set 푋 let 푋(푠) = 푋 + [−푠, 푠]. domains (Nazarov–Sodin), random topology (Canzani– • Given 휈 ∈ (0, 1) and 0 < 훼0 < 훼1, say that Ω ⊂ Sarnak–Wiegmann, Gayet–Welschinger), and QUE of ran- ℝ is 휈-porous on scales 훼0 to 훼1 if for each inter- dom ONBs (Zelditch, Chatterjee–Galkowski, Bourgade– val 퐼 of size |퐼| ∈ [훼0, 훼1] there exists a subinter- Yau, R. Chang (see [Z17] for some references)). val 퐽 ⊂ 퐼 with |퐽| = 휈|퐼| such that 퐽 ∩ Ω = ∅. Uncertainty Principle and the Dyatlov-Jin Full • Ahlfors–David regular: Let 훿 ∈ [0, 1], 퐶푅 ≥ Support Theorem 1, and 0 < 훼0 < 훼1. Say that 푋 ⊂ ℝ is 훿- regular with constant 퐶 on scales 훼 to 훼 if A recent breakthrough on the QUE problem for hyperbolic 푅 0 1 there exists a Borel measure 휇 on ℝ supported surfaces is the following result due to Dyatlov–Jin [DJ18]. 푋 −1 in 푋 such that (i) for any interval 퐼 with |퐼| ∈ We use semiclassical notation where ℏ푗 = 휆푗 and denote 훿 [훼0, 훼1], 휇푋(퐼) ≤ 퐶푅|퐼| ; (ii) if 퐼 is centered at eigenfunctions by 푢ℏ푗 or more simply 푢ℏ as in (10). −1 훿 푎 ∈ 푋, then 휇푋(퐼) ≥ 퐶푅 |퐼| . Theorem 8. Let (푀, 푔) be a compact hyperbolic surface. Let • Porous sets are embedded into Ahlfors–David reg- ∞ ∗ 푎 ∈ 퐶0 (푇 푀) with 푎|푆∗푀 not identically zero. Let 푢ℏ be ular sets of some dimension 훿 < 1. −2 an eigenfunction of eigenvalue ℏ and let ‖푢ℏ‖퐿2 = 1. Then The FUP for 훿-regular sets states the following. there exist constants ℏ0(푎) and 퐶푎 independent of ℏ so that, Proposition 10. Let 퐵(ℎ) be a semiclassical FIO on 퐿2(ℝ) for ℏ ≤ ℏ0(푎), of the form ‖ Opℏ(푎)푢ℏ‖퐿2 ≥ 퐶푎. − 1 푖Φ(푥,푦)/ℎ Op (푎) 퐵(ℎ)푓(푥) = ℎ 2 ∫ 푒 푏(푥, 푦)푓(푦)푑푦 Here, ℏ is the semiclassical pseudo-differential ℝ operator with symbol 푎. If 푎(푥, 휉) = 푉(푥) is a multi- with 푏 ∈ 퐶∞(푈) and 휕2 Φ ≠ 0 on 푈. Suppose that 푋, 푌 ⊂ plication operator, one gets that ∫ |푢 |2푑푉 ≥ 퐶 > 0, 0 푥푦 퐵 ℏ 퐵 ℝ are Ahlfors–David 훿-regular. Then there exists 훽 > 0 so that i.e., a uniform lower bound (in ℏ) of the 퐿2 mass on all 훽 balls. ‖ퟏ푋(ℎ)퐵(ℎ)ퟏ푌(ℎ)‖퐿2(ℝ)→퐿2(ℝ) ≤ 퐶ℎ . Corollary 9. All quantum limits of sequences of eigenfunctions Dyatlov–Jin apply this result to a porous set arising in on compact hyperbolic surfaces have full support in 푆∗푀, i.e., the dynamics of the geodesic flow on a compact hyper- charge every open set. bolic surface. Such a flow is hyperbolic, and it preserves two foliations transverse to the flow direction: one foli- The corollary does not imply that every quantum limit ation consists of stable leaves (horocycles) contracted by is Liouville (see Definition 2). For instance, a quantum the flow, the other of unstable leaves (antihorocycles) ex- limit could be a convergent sum of delta-functions along panded by the flow. Roughly speaking, these two folia- a dense set of closed geodesics. tions play the role of the position, respectively, momen- One of the main ingredients in the proof of Theorem tum, axes in the case of the . If there is an 8 is the so-called FUP (fractal uncertainty principle). It is open “hole” 퐴 in the support of the quantum limit mea- related to a classical problem of Landau–Slepian–Pollack 푡 sure, then the flowout ⋃푡∈ℝ 퐺 (퐴) of the hole is also a related to the uncertainty principle of quantum mechanics: hole, and the quantum limit measure must be supported 푓 ∈ 퐿2 Can there exist a function that is concentrated on in its complement. This complement is a regular porous 퐴 ℱ푓 an interval such that its Fourier transform is concen- set and is very “sparse” in the Liouville-measure sense. But 퐵 푃 = ퟏ trated on an interval ? To make this precise, let 퐴 퐴 until Anantharaman’s result (see [A18]), it could 푄 = ℱ∗ퟏ ℱ 푓 휖 and 퐵 퐵 . If there exists that is -concentrated not be proved that eigenfunctions do not concentrate on a 퐴 |푓|2 ≤ 휖2 |푓|2 on in the sense that ∫푋\퐴 ∫푋 and such that closed geodesic, much less a low measure set. The FUP is ℱ푓 is 훿-concentrated on 퐵, then 1 − 휖 − 훿 ≤ ‖푃퐴푄퐵‖. the new tool that shows that the support is full, i.e., dense, The operator 푄퐵푃퐴푄퐵 is self-adjoint and trace-class, even though the limit measure could still be a convergent and 2 sum of delta-functions on a dense set of closed geodesics. 휆1 = ‖푃퐴푄퐵‖ .

Thus, √휆1 is the cosine of the angle between the ranges of Applications of QE 푃퐴 and 푄퐵. Moreover, 휆1 < 1. In fact, if 0 ≤ 훼, 훽 ≤ In view of the amount of effort expended to prove QE and 1 and (훼, 훽) ≠ (1, 0), (0, 1), then there exists 푓 with QUE in various settings and on various length scales, it is

OCTOBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1419 natural to ask, what applications does QE have to the har- ments on the universal bound were proved by A. Selberg monic analysis of eigenfunctions? What features of eigen- (in the constant curvature case) and by P. Berard (in the functions should be sensitive to ergodicity? Here are some variable curvature case). The improvement in Theorem 11 intuitions. is that it makes no curvature assumptions, only ergodic- • Ergodicity (or chaos) causes eigenfunctions to ity of the geodesic flow. A precursor to Theorem 11 was a rapidly oscillate everywhere and in all directions. proof of the same statement in the special case of real an- Therefore, their nodal (zero) sets should be uni- alytic surfaces by Sogge–Toth–Zelditch, following general formly distributed on 푀 and (in some suitable results of Sogge–Zelditch (see [Z17]). Aside from 퐿∞ norms, the most attention has been de- sense) in phase space. Moreover, ergodic eigen- 4 4 functions should have a lot of nodal domains; i.e., voted to 퐿 norms, where it is conjectured that the 퐿 norm the connected components of 푀 where 휙푗 is pos- is uniformly bounded. The best evidence for this pertains 2 itive or negative should be rather small and there to (non-퐿 ) Eisenstein series on arithmetic quotients 2 should exist many of them. [Hum18] and to dihedral 퐿 Maass forms (Humphries– • The oscillations are on the wavelength scale ℏ푗 = Khan). −1 휆푗 . Therefore, if one “blows up” a small ball of Lower bounds on numbers of nodal domains. The nodal radius 푟푗(ℏ) ≫ ℏ푗 so that the number of wave- set of 휙푗 is the hypersurface lengths is still growing across the ball, one should 풩휙푗 = {푥 ∈ 푀 ∶ 휙휆(푥) = 0}. still see quantum ergodicity, i.e., oscillation in all 휙 directions and at all points. The nodal domains of 푗 are the connected components 푀\풩 푁(휙 ) • Ergodicity should prohibit exceptional concentra- of 휙푗 . Let 푗 be the number of its nodal do- tion in any region of 푀; therefore 퐿푝 norms of er- mains. The Courant upper bound states that the num- 푁(휙 ) 휙 godic eigenfunctions should be rather small com- ber 푗 of nodal domains of an eigenfunction 푗 is 푗 pared to extremal cases. Moreover, their 퐿2 norms bounded above by . The question arises whether a (pos- (푀, 푔) on any fixed ℏ-independent ball 퐵 ⊂ 푀 should sibly generic) possesses any sequence of eigenfunc- tions for which 푁(휙푗 ) → ∞. The first result on counting have a uniform lower bound in ℏ푗 (depending on 푘 퐵). nodal domains by counting intersections with a curve was proved by Ghosh–Reznikov–Sarnak for 푀 = ℍ2/SL(2, ℤ) • Cauchy data (휙푗|퐻, 휕휈휙푗|퐻) of eigenfunctions on hypersurfaces 퐻 ⊂ 푀 should be quantum er- [GRS13]. Junehyuk Jung and the author have proved a general result showing that the number of nodal domains godic along 퐻; 휕휈 is the normal derivative. For grows with the eigenvalue in certain ergodic cases, and QE generic 퐻, the restrictions 훾퐻휙푗 ∶= 휙푗|퐻 should have quantum ergodic behavior similar to that on plays an essential role in the proof. The setting is the same the global manifold. This is because the classical as that in Theorem 7. The following was proved in [JZ16] analogue of Cauchy data of eigenfunctions is the and then improved to the quantitative log estimate in a ∗ later article (see [Z18]). (See the discussion preceding The- cross section 푆퐻푀 (the set of unit covectors to 푀 with footpoint on 퐻) of 퐺푡 ∶ 푆∗푀 → 푆∗푀. The orem 7 for notation.) ∗ ∗ corresponding first return map Φ∶ 푆퐻푀 → 푆퐻푀 Theorem 12. Let (푀, 퐽, 휎) be a compact real Riemann sur- ∗ is ergodic on 푆퐻푀. face with Fix(휎) ≠ ∅ and dividing. Let 푔 be any 휎-invariant These heuristics are intentionally vague and debatable. In Riemannian metric. Then for any orthonormal eigenbasis {휙푗} 2 2 the next section we describe some rigorous results in this of 퐿even(푀), respectively, {휓푗} of 퐿odd(푀), one can find a direction. density 1 subset 퐴 of ℕ such that 푝 퐿 norms. As mentioned above, quantum ergodic lim 푁(휙푗) = ∞, lim 푁(휓푗) = ∞. 푗→∞ 푗→∞ sequences in the sense of Definition 2 should have rela- 푗∈퐴 푗∈퐴 tively small sup-norms. The following theorem is a special case of [Gal19, Theorem 1]: In fact, 퐾 1 ∞ 푁(휓푗) ≥ 퐶푔 (log 휆푗) for all 퐾 < 6 . Theorem 11. Let {휙푗푘 }푘=1 be a QE sequence of eigenfunc- 푛−1 2 The last statement uses the log-scale QE results of Han tions. Then ‖휙푗 ‖퐿∞ = 표(휆 ). 푘 푗푘 and Hezari–Rivière [HR16]. For generic negatively curved Galkowski’s result applies more generally to sequences invariant metrics, the eigenvalues are simple (multiplicity whose quantum limit measure is diffuse at 푥; we refer to one), and therefore all eigenfunctions are either even or [Gal19] for the definition. Y. Canzani–J. Galkowski have odd. proved further results in this direction. In the case of neg- The same result holds for any nonpositively curved sur- atively curved compact manifolds, logarithmic improve- face with concave boundary (J. Jung–Zelditch, 2016), al-

1420 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 9 though the quantitative lower bound is still open in that Item (1) is sometimes called a Kuznecov sum formula es- case. Hezari generalized the result to any domain with timate. It holds on any curve on any surface for a subse- ergodic billiards [H18]. Jung and Seung uk Jang proved quence of density 1 of the eigenfunctions. Item (2) is the the result in QUE cases for the entire orthonormal basis QER theorem. The proof shows that one may let 푓 = ퟏ훽, of eigenfunctions. Jakobson–Naud proved a related result the indicator function of any arc. Item (3) is the B´erard- on infinite area hyperbolic surfaces ([Z17, Z18]). Selberg sup norm estimate. All of these results pertain to dimension 2. The picture To prove that the restricted eigenfunctions 휙푗|훾 have in dimensions ≥ 3 seems to be quite different. There are an unbounded number of sign changes as 푗 → ∞, one many topological types of 3-manifolds, so we restrict atten- combines (2)–(3) to give tion to unit tangent bundles 푆푀of surfaces 푀. There ex- ‖휙 ‖2 ist natural “Kaluza–Klein” or bundle-metrics 푔퐾퐾 on 푆푀, 푗 퐿2(훽) −1/2 1/2 푆푀 → 푀 ∫ |휙푗| 푑푠 ≥ ≥ 퐶휆푗 (log 휆푗) , corresponding to a connection on and a met- 훽 ‖휙푗‖퐿∞ ric 푔푀 on 푀. Namely, the connection defines horizon- tal spaces. The horizontal part of 푔퐾퐾 is isometric to 푔푀 and this contradicts (1) if 휙푗 ≥ 0 on 훽. under the natural projection, the vertical spaces (tangent Intuitively, (1) shows that 휙푗 either is small on 훾 or os- to the fibers) are orthogonal to the horizontal spaces, and cillates a lot on 훾 so that its integral is small; (2) shows that 2 the fibers are geodesics of a fixed length. Jung–Zelditch 휙푗 is not small anywhere on 훾. There is quite a difference recently proved that for generic Kaluza–Klein metrics 푔퐾퐾 2 between 휙푗 being large everywhere and 휙푗 being large ev- (i.e., generic choices of 푔푀 or connections on 푆푀), the erywhere; (3) is used to soften this difference. Hence, (1) nodal set is regular (no self-intersections) and the number must be due to oscillations and in particular to zeros. of nodal domains for every eigenfunction is 2. Thus, for It is doubtful that the log estimates are sharp. It is a rea- a large class of topological 3-manifolds and metrics, the sonable conjecture that in the negatively curved case, the number of nodal domains is opposite to that of Theorem number of zeros of 휙푗|훾 is often roughly comparable with 12. 휆푗. So far, no techniques in nonarithmetic cases approach Sketch of the proof of Theorem 12. The proof may be this number. instructive, because it combines almost all of the known Log-scale equidistribution of complex zeros. The results results and tools of QE. Let 푀 be a negatively curved sur- in this section pertain to zeros of the Husimi measures face with involution 휎 and let 훾 be the fixed point set (13). It turns out that phase space zeros in 퐵∗푀 are sim- of 휎. We assume 훾 divides 푀 = 푀+ ∪ 푀− into two pler to study than physical space nodal sets in 푀 but have pieces interchanged by 휎, so that each piece is a surface basically the same interpretation: the least likely places for with boundary. Even/odd eigenfunctions correspond to the particle to be. R. Chang and the author proved log- Neumann/Dirichlet on the pieces. scale equidistribution results for phase space nodal sets (A) The first step is to show that the number 푁(휙푗) both in the setting of Grauert tubes and in the setting of 1 line bundles over Kähler manifolds (holomorphic auto- of nodal domains is ≥ 2 푁(휙푗|훾), the number of zeros of 휙푗 on 훾. This is purely topological and morphic forms). The results were partly motivated by a essentially uses that 훾 is a boundary. prior result of Lester–Matomaki–Radziwill for the special (B) The key is then to prove that Neumann eigenfunc- case of Hecke holomorphic forms for SL(2, ℤ). The equi- tions have a lot of zeros on 훾 = 휕푀+; respec- distribution law is quite different in the Grauert tube set- tively, Dirichlet eigenfunctions have many zeros ting and in the Kähler setting, although the two are analo- of 휕휈휙푗 = 0 on 훾. This is where QER (Theorem gous (see [Z18] for references to the parallel results). ℂ 7) is used. The proof is to show that The complex zero set of 휙푗 is the complex hypersurface

∗ ℂ 풵푗 ∶= {휁 ∈ 퐵휏0 푀 ∶ 휙푗 (휁) = 0}, ∫ 휙푗 푑푠 ≪ ∫ |휙푗| 푑푠 (16) 훽 훽 ℂ where 휙푗 is the analytic continuation of 휙푗 to the co-ball on any arc 훽 ⊂ 훾. To get a log lower bound, one ∗ bundle 퐵휏 푀. The zero sets define currents [풵푗] of inte- |훽| ≤ (log 휆 )−1 0 proves (16) for 푗 . gration in the sense that for every smooth (푛 − 1, 푛 − 1) 푛−1,푛−1 ∗ The proof of (16) uses the following estimates: test form 휂 ∈ 풟 (퐵휏0 푀), we have that the pairing −1/2 (1) For any arc 훽⊂훾, | ∫ 휙 푑푠|≤퐶휆 (log 휆 )1/4. 훽 푗 푗 푗 푖 ̄ ℂ 2 2 ⟨[풵푗], 휂⟩ ∶= ∫ 휂 = ∫ 휕휕 log|휙푗 | ∧ 휂 (2) For any 푓 ∈ 퐶(훾), ∫ 푓휙 푑푠 ≥ 1 (Theorem 7). ∗ 훾 푗 ℤ푗 퐵휏0 푀 2휋 1/2 휆푗 (3) ‖휙푗‖퐿∞ ≤ 퐶 . √log 휆푗 is a well-defined closed current. In the special case 휂 =

OCTOBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1421 푛−1 푓휔 , the zero set defines a positive measure |풵푗| by [JZ16] Jung J, Zelditch S. Number of nodal domains and singular points of eigenfunctions of negatively curved sur- 푛−1 ∗ faces with an isometric involution. J. Differential Geom. 102 ⟨|풵푗|, 푓⟩ ∶= ∫ 푓휔 , 푓 ∈ 퐶(퐵휏0 푀). 풵푗 (2016), no. 1, 37–66. MR3447086 [S95] Sarnak P. Arithmetic quantum chaos. The Schur lec- In a prior article, the author proved that the “currents of tures (1992) (Tel Aviv), 183–236, Israel Math. Conf. Proc., integration” of QE eigenfunctions have a weak* limit law: 8, Bar-Ilan Univ., Ramat Gan, 1995. MR1321639 [S11] Sarnak P. Recent progress on the quantum unique er- 1 푖 ̄ ∗ [풵푗 ] ⇀ 휕휕|휉|푔 weakly as currents on 퐵휏 푀 godicity conjecture. Bull. Amer. Math. Soc. (N.S.) 48 (2011), 휆 푘 휋 0 푗푘 no. 2, 211–228. MR2774090 [So14] Sogge CD. Hangzhou lectures on eigenfunctions of the along a density one subsequence of eigenvalues 휆푗푘 . It is further proved that a similar convergence theorem holds Laplacian. Ann. of Math. Stud., 188. Princeton University on balls in 푀 \푀 with logarithmically shrinking radii of Press, Princeton, NJ, 2014. MR3186367 휏0 [TZ17] Toth JA, Zelditch S. Nodal intersections and geomet- size (log 휆)−훾 for some explicit dimensional constant (in- ric control, J. Diff. Geom. (to appear) arXiv:1708.05754, dependent of the frequency 휆푗) 훾 > 0. 2017. In the real domain, for (푀, 푔) with ergodic geodesic [Z17] Zelditch S. Eigenfunctions of the Laplacian on a Rie- flow, it is conjectured that nodal sets become uniformly mannian manifold. CBMS Regional Conference Series in distributed on 푀. Han has proved that the limit distribu- Mathematics, 125. Amer. Math. Soc., Providence, RI, 2017. tion is at least absolutely continuous with respect to 푑푉푔 MR3729409 if small scale QE holds [Han18]. [Z18] Zelditch S. Local and global analysis of nodal sets. Sur- veys in differential geometry 2017. Celebrating the 50th an- References niversary of the Journal of Differential Geometry, 365–406, [A14] Anantharaman N. Le th´eorèmed’ergodicit´equantique. Surv. Differ. Geom., 22, Int. Press, Somerville, MA, 2018. (French) [The quantum ergodicity theorem] Chaos en MR3838125 m´ecaniquequantique, 101–146, Ed. Èc. Polytech., Palaiseau, [Zw] Zworski M. Semi-classical analysis, Graduate Studies in 2014. MR3362293 Mathematics, 138. Amer. Math. Soc., Providence, RI, 2012. [A18] Anantharaman N. Delocalization of Schrödinger MR2952218 eigenfunctions, Proceedings ICM 2018, Vol. 1, 341–376, M. Viana (ed.), World Scientific, 2019. [D19] Dyatlov S. Notes on the fractal uncertainty principle, arXiv:1903.02599, 2019. [DJ18] Dyatlov S and Jin L. Semiclassical measures on hyper- bolic surfaces have full support. Acta Math. 220 (2018), no. 2, 297–339. MR3849286 [Gal19] Galkowski J. Defect measures of eigenfunctions with maximal 퐿∞ growth, Ann. Institut Fourier (to appear), arXiv:1704.01452, 2017. [GRS13] Ghosh A, Reznikov A, Sarnak P. Nodal domains of Maass forms I. Geom. Funct. Anal. 23(5):1515–1568, 2013. Steve Zelditch MR3102912 Credits [Go18] Gomes S. KAM Hamiltonians are not quantum er- Opening figure is courtesy of Alexander Strohmaier. godic, arXiv:1811.07718, 2018. [Han18] Han X. Distribution of the nodal sets of eigenfunc- Photo of author is courtesy of Northwestern University. tions on analytic manifolds. J. Spectr. Theory 8 (2018), no. 4, 1281–1293. MR3870068 [Ha16] Hassell A. Ergodic billiards that are not quantum unique ergodic. With an appendix by the author and Luc Hillairet. Ann. of Math. (2) 171 (2010), no. 1, 605–619. MR2630052 [H18] Hezari H. Applications of small scale quantum ergod- icity in nodal sets. Anal. PDE 11 (2018), no. 4, 855–871. MR3749369 [HR16] Hezari H, Rivière G. 퐿푝 norms, nodal sets, and quantum ergodicity. Adv. Math. 290 (2016), 938–966. MR3451943 [Hum18] Humphries P. Equidistribution in shrinking sets and L4-norm bounds for automorphic forms. Math. Ann. 371, no. 3–4, (2018), 1497–1543. MR3831279

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