Notes available on my website, under Downloadable Lecture Notes 8. Seminar talks and AMS talks See also 4. Spectral theory 7. Quantum mechanics connections
Basic quantization: a function on phase space is taken to an operator on a Hilbert space H. Euclidean case. H = L 2 (R n ). Position: x j Q j, Q j f = x j f. Momentum: p j P j, P j f = (1/i) f / x j. Laplace operator: p 2. Quantization of motion in a force field + V (x). Quantization of free motion on a Riemannian manifold M. H = L 2 (M). Use Laplace-Beltrami operator, i.e., ) f = g 1/2 j (g 1/2 g jk k f.
Classical free motion on M: geodesic flow dx j dt = E ξ j, dξ j dt = E x j E(x, ξ) = g jk ξ j ξ k = symbol of. Get flow on cotangent bundle T M, preserving level sets of E(x, ξ). Hence get flow. G t : S M S M The flow G t preserves a natural Liouville measure ds on S M. We normalize this so that S M ds = 1.
Basic problem: relate dynamical properties of the geodesic flow G t to spectral properties of. Assume M is a compact Riemannian manifold. L 2 (M) has an orthonormal basis {φ k : k N} of eigenfunctions of : φ k = λ 2 k φ k, λ k +. Weyl law: λ k (Ck) 1/n, as k, where n = dim M, and C = Γ(n/2 + 1)(4π) n/2 /Vol M. Here and below, normalize the metric on M so that Vol M = 1. Mean equidistribution of eigenfunctions: 1 N N φ k (x) 2 1, k=1 uniformly in x M, as N. One tool for these results: heat kernel asymptotics.
Question: when can lots of eigenfunctions concentrate on some subset of M? Example: Unit sphere S n in R n+1. Theorem (Shnirelman, 1974) Assume the flow G t on S M is ergodic. Then there is a subset N N, of density 0, such that for all b C(M), lim b(x) φ k (x) 2 dv (x) = b(x) dv (x). k,k / N M M
Further phase space localization brings in a quantization of C (S M), Quantizations include op : C (S M) OPS 0 (M) L(L 2 (M)). Kohn-Nirenberg quantization, op KN, Weyl quantization, op W, Friedrichs quantization, op F. These differ by maps from C (S M) to OPS 1 (M), a space of compact operators on L 2 (M). Special property of op F : a C (S M), a 0 = op F (a) 0. Constructions involve oscillatory integrals. See, e.g., [T1].
Kohn-Nirenberg quantization on R n a(x, D)u(x) = (2π) n/2 a(x, ξ)û(ξ)e ix ξ dξ, R n û(ξ) = (2π) n/2 u(y)e iyξ dy. R n
Theorem (Colin de Verdière, 1985) Assume the flow G t on S M is ergodic. Then there is a subset N N, of density 0, such that for all a C (S M), lim (Aφ k, φ k ) L 2 = a, k,k / N where A = op(a), a = S M a ds. Ingredients in the proof. Weyl law lim N 1 N N (Aφ k, φ k ) L 2 = a. k=1 (Does not require ergodicity of G t.)
Egorov theorem op(a G t ) U t AU t OPS 1 (M), where U t = e it. Mean ergodic theorem a T Pa in L 2 (S M, ds), as T, given a L 2 (S M), a T = 1 T T 0 a G t dt, P = orthogonal projection of L 2 (S M) onto G t -invariant elements.
Theorem (Schrader-Taylor 1989, Taylor 2015) Do not assume the flow G t on S M is ergodic. There is a subset N N, of density 0, such that if a C(S M), A = op F (a), then More generally, Pa = a = Pa C(S M) lim (Aφ k, φ k ) L 2 = a. k,k / N = lim k,k / N (Aφ k, φ k ) L 2 (op F (Pa)φ k, φ k ) L 2 = 0. First part proved in [ST], for a C (S M). Rest done in [T2]. Example Take M = T n, flat torus (so G t is integrable). Then Pa = a for a = a(x), i.e., for Af (x) = a(x)f (x).
This theorem uses the fact that, thanks to positivity a 0 = op F (a) 0, op F has a unique continuous extension from C (S M) to still satisfying such positivity. op F : C(S M) L(L 2 (M)), Example M is an inner tube, a non-flat torus of revolution in R 3. As shown in [T2], P : C(S M) C(S M), but P does not map C (S M) to C (S M), or even to the space of Hölder continuous functions on S M.
Quantization of discontinuous symbols ([T3]) The map op F has a unique extension from C(S M) to satisfying op F : L (S M) L(L 2 (M)), a ν a weak in L (S M) = op F (a ν ) op F (a) in the weak operator topology. Special case: op F : R(S M) L(L 2 (M)), where R(S M) is the space of bounded functions on S M that are Riemann integrable.
Theorem ([T3]) Assume the flow G t on S M is ergodic. Then there is a subset N N of density 0 such that lim (Aφ k, φ k ) L 2 = a ds, k,k / N S M whenever a R(S M), A = op F (a). Elementary special case: b(x) φ k (x) 2 dv (x) = lim k,k / N M M b(x) dv (x), provided b R(M).
Theorem ([T3]) Do not assume the flow G t on S M is ergodic. There is a subset N N, of density 0, such that lim (Aφ k, φ k ) L 2 (A p φ k, φ k ) L 2 = 0, k,k / N whenever A = op F (a), A p = op F (Pa), with a C(S M), Pa R(S M).
Local (in phase space) equidistribution result. (See also [Riv], [Gal].) Theorem ([T3]) Assume G t acts ergodically on an open set U S M. Then there is a subset N N, of density 0, such that if a, b R(S M) are supported on a compact subset of U, then U a ds = U b ds = lim k,k / N (Aφ k, φ k ) L 2 (Bφ k, φ k ) L 2 = 0, for A = op F (a), B = op F (b).
Concentration of eigenfunctions on S 2 G t periodic of period 2π. Take Λ = + 1 4 1 4, Spec Λ = {k Z : k 0}. Rotation about x 3 -axis, R(t) = e itx. k-eigenspace V k, dim V k = 2k + 1. Eigenfunctions φ kl, l k. Λφ kl = kφ kl, X φ kl = lφ kl.
Pa(x, ξ) = 1 2π a(g t (x, ξ)) dt. 2π 0 Take Au(x) = a(x)u(x), a C (S 2 ), rotationally symmetric. Π(A) = 1 2π e itλ Ae itλ dt, 2π 0 commutes with Λ and X. (Λ, X ) simple spectrum Π(A) = F (Λ, X ). Task: analyze F.
Egorov theorem Π(A) op F (Pa) OPS 1 (S 2 ). Proposition. (Via [T1], Chapter 12.) for op F (Pa) = F 0 (Λ, X ) mod OPS 1 (S 2 ), F 0 (1, λ) = Pa(x 0, (λ, 1 λ 2 )) = g(λ), x 0 on equator of S 2. Hence Π(A) F 0 (Λ, X ) OPS 1 (S 2 ). Corollary. For rotationally symmetric a C (S 2 ), S 2 a(x) φ kl (x) 2 ds(x) = (Aφ kl, φ kl ) = (Π(A)φ kl, φ kl ) ( l ) = g + O(k 1 ). k
Apply the corollary to a vanishing on x 3 β, with β (0, 1). Conclusion. The orthonormal eigenfunctions φ kl concentrate on the strip x 3 β, for l k 1 β 2.
References [CV] Y. Colin de Verdière, Ergodicité et fonctions propre du laplacian, Comm. Math. Phys. 102 (1985), 497 502. [Gal] J. Galkowski, Quantum ergodicity for a class of mixed systems, J. of Spectral Theory 4 (2014), 65 85. [Riv] G. Riviere, Remarks on quantum ergodicity, J. Mod. Dyn. 7 (2013), 119 133. [ST] R. Schrader and M. Taylor, Semiclassical asymptotics, gauge fields, and quantum chaos, J. Funct. Anal. 83 (1989), 258 316. [Sn] A. Shnirelman, Ergodic properties of eigenfunctions, Usp. Mat. Nauk. 29 (1974), 181 182.
[T1] M. Taylor, Pseudodifferential Operators, Princeton Univ. Press, 1981. [T2] M. Taylor, Variations on quantum ergodic theorems, Potential Anal. 43 (2015), 625 651. [T3] M. Taylor, Quantization of discontinuous symbols and quantum ergodic theorems, Preprint, 2018. [Ze] S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987), 919 941. [ZZ] S. Zelditch and M. Zworski, Ergodicity of eigenfunctions for ergodic billiards, Comm. Math. Phys. 175 (1996), 673 682.