Mathematics of Quantum 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 mechanics, 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 (fractal 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 “Uncertainty Principle 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 ergodicity 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 Niels Bohr proposed the “old quantum theory,” 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 classical mechanics “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 quantum mechanics 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 Hilbert space. In the case of the hydrogen atom, and eigenvalues as the Planck constant ℏ → 0. By classical 2 3 ℋ = 퐿 (ℝ , 푑푥) (푑푥 is Lebesgue measure). Schrödinger mechanics is meant the time evolution (푥푡, 휉푡) of points ∗ 푛 then introduced the operators of his name, (푥, 휉) in the phase space 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 operator 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 stationary state: 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, mixing, 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 integrable system” 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).