PROBLEM SET 4 SEMICLASSICAL MICROLOCAL ANALYSIS DUE: DEC. 28, 2020 (1) [Adjoint in KN quantization] (a) Suppose m is an order function, a ∈ S(m). Find the Kohn-Nirenberg symbol of the adjoint (ba KN ) ∗ . (b) Suppose k is a real number and a ∈ S k is an invariant symbol. Show that (ba KN ) ∗ is also in Ψk . (2) [PsDO on manifolds] Prove Proposition 2.2 in Lecture 20. (3) [Dyadic P.O.U.] Prove Lemma 2.6 in Lecture 18. [ Ref: Zworski, Semiclassical Analysis. Lemma 7.14] (4) [Elliptic Estimate] (a) Suppose a ∈ S(1) and p ∈ S(m), where m ≥ 1 is an order function. Moreover, suppose p is elliptic on supp(a) in the following sense: |p(x, ξ)| ≥ cm(x, ξ), ∀(x, ξ) ∈ supp a(·, ~), ∀~. Prove: (i) There exits q1, q2 ∈ S(1/m) with supp(q1),supp(q2) ∈ supp(a) such that a = q1 ? p + ~ ∞S(1), a = p ? q2 + ~ ∞S(1). (ii) There exists C > 0 such that for all u ∈ L 2 (R n ), kba W ukL2 ≤ Ckpb W ukH~(1/m) + O(~ ∞)kukL2 . (b) Suppose M is compact, a ∈ S 0 (T ∗M), P ∈ Ψk (M), and P is elliptic on supp(a) in the following sense: |σ(P)| ≥ chξi k , ∀(x, ξ) ∈ suppa(·, ~), ∀~. (i) There exits Q1, Q2 ∈ Ψ−k (M) such that Op(a) = Q1 ? P + ~ ∞Ψ −∞, Op(a) = P ? Q2 + ~ ∞Ψ −∞. (ii) There exists C > 0 such that for all u ∈ L 2 (M), kOp(a)ukL2 ≤ CkP ub kH −k ~ (M) + O(~ ∞)kukL2 . 1
2 PROBLEM SET 4 SEMICLASSICAL MICROLOCAL ANALYSIS DUE: DEC. 28, 2020 (5) [A weak Egorov theorem on manifolds] Let (M, g) be a compact Riemannian manifold. Suppose a ∈ S −∞(T ∗M). Let p(x, ξ) = kξk 2 g + V (x) and P = −~ 2∆ + V (x), where V ∈ C∞(M). Let ϕt : T ∗M → T ∗M be the Hamilton flow generated by p, and let at = ϕ ∗ t a. Prove: at ∈ S −∞(T ∗M) and ke itP/~Op(a)e −itP/~ − Op(at)kL(L2(M)) = O(~), where the estimate is uniform for t ∈ [0, T]. [ Ref: Zworski, Semiclassical Analysis. §15.2] (6) [Sub-principal symbol] At the end of Lecture 19, we wrote a definition of sub-principal symbol for differential operators. Let P be a differential operator of order k. Prove σsub(P) is well-defined by checking (a) Q = (P − (−1)kP ∗ )/2 is a differential operator of order k − 1. (b) σsub(P) = σk−1(Q). [Ref: Guillemin-Sternberg, Semi-classical Analysis, §1.3.4]