PROBLEMSET 2SEMICLASSICALMICROLOCAL ANALYSISDUE:NOV.16. 2020(1)[The Wigner Transform]Given f, g E (R"), define the Fourier-Wigner transform of f, g to be the function1(et(q-Q+p-P) f, g)V(f, g)(p,q) :=(2元h)nand definetheir Wigner transform to be the functionW(f,9)(r,E) = (Fn)(p,g)(r,s)V(f,g)(a) Prove:1etvaf(y+)g(y-P)dyV(f, g)(p,q) =(2元h)n(b) Prove:e-p f(r+)g(a-W(f,g)(r,s) =)d(c) Prove: For any a ES,1(aW f,9) =a(r,s)W(f,g)(r,s) drde.(2元h)nJRn2(d) Prove: [Moyal's identity](W(fi,91),W(f2,92))= (2元h)"(f1,f2)《g1,92)(e) Prove:[W(f,g)"](r) = (p,g)f(r).In particular, W(f,f)"p is the projection of onto f[Ref:Folland, Harmonic analysis on phase space](2)[Symplectic invariance of the Weyl quantization]We mentioned three special cases of the symplectic invariance of the Weyl quantiza-tion, proved the case (A) and a special case of (B) (with C-Id). (Lec 7, p. 4)(a) Prove case (B) for general symmetric matrix C.(b) Prove case (C).(c) Prove: for any a,b e Rn and ce R, we have[(ar+b.+c)m]"=(aQ+b.P+cId)m.(Hint: First prove the identity with c = 0.)[Ref:Folland,Harmonicanalysisonphasespace.]1
PROBLEM SET 2 SEMICLASSICAL MICROLOCAL ANALYSIS DUE: NOV. 16, 2020 (1) [The Wigner Transform] Given f, g ∈ S (R n ), define the Fourier-Wigner transform of f, g to be the function V (f, g)(p, q) := 1 (2π~) n he i ~ (q·Q+p·P) f, gi and define their Wigner transform to be the function W(f, g)(x, ξ) = (F~)(p,q)→(x,ξ)V (f, g). (a) Prove: V (f, g)(p, q) = 1 (2π~) n Z Rn e i ~ y·q f(y + p 2 )g(y − p 2 ) dy. (b) Prove: W(f, g)(x, ξ) = Z Rn e − i ~ p·ξ f(x + p 2 )g(x − p 2 ) dp. (c) Prove: For any a ∈ S , hba W f, gi = 1 (2π~) n Z Rn Z Rn a(x, ξ)W(f, g)(x, ξ) dxdξ. (d) Prove: [Moyal’s identity] hW(f1, g1), W(f2, g2)i = (2π~) n hf1, f2ihg1, g2i. (e) Prove: [W\(f, g) W ϕ](x) = hϕ, gif(x). In particular, W\(f, f) W ϕ is the projection of ϕ onto f. [Ref: Folland, Harmonic analysis on phase space] (2) [Symplectic invariance of the Weyl quantization] We mentioned three special cases of the symplectic invariance of the Weyl quantization, proved the case (A) and a special case of (B) (with C=Id). (Lec 7, p. 4) (a) Prove case (B) for general symmetric matrix C. (b) Prove case (C). (c) Prove: for any a, b ∈ R n and c ∈ R, we have [(a · x +\b · ξ + c)m] W = (a · Q + b · P + c Id)m. (Hint: First prove the identity with c = 0.) [ Ref: Folland, Harmonic analysis on phase space.] 1
2PROBLEMSET2SEMICLASSICALMICROLOCALANALYSISDUE:NOV:16,2020(3)[AncounterexampletouncertaintyandBCH]Consider the Hilbert space H = L?([-1, 1). Note that1wn(r) :nEZV2form an orthonormal basis of H.DefineQ:H-→H, $-pandhdpP : Dom(P) C H →H,6idwhere Dom(P) = [ E Ci([-1, 1]) [ (1) = (-1))(a) Prove: Q is self-adjoint on H, and P is essentially self-adjoint on Dom(P).(b) Prove: The uncertainty (c.f.PSet1-5)3n(Q)5n(P)=0.Explain why this doesnot violate PSet1-5-(b).(c)ForanyaER, defineSa:H→H by(Sap)(r):=(r+a-2mra),where mr,a is the unique integer such that + a - 2mr,a E [-1,1). Prove: forany t e R, eitB/h - St.(d) Prove:Q,P violate the Baker-Campbell-Hausdorff formula. In other words,e-[lee[Q,[Q, P] = [P,[Q, P] = 0 but e"(f[Ref:Hall,Quantum theoryformathematicians,12.2and Example14.5.](4)[Mehler formula]For a symbol a(r,s), although we proved the following equality for several specialcases (e.g.a is linear or a(,)= ip($). where p is a real-valued polynomial), ingeneral (where we assume eta(r,t) is also a nice symbol function)etaw+etawIn this problem let's explore the case a(r,5) = -H(r,) = -, where for sim-plity we assumed n=1. Then etaw =e-th, where =-+ be the 1-dimharmonic oscillator.We want to find a function qt(r,S,t) so thate-th=eaw(a) Prove:et=-H*e%.(b) For the non-semiclassical setting (namely, there is no h), I saw from literature aformula1e-(r2+$2)tanheqt=coshPlease write down a semiclassical version of this formula, and check it via (a).(c) Write down the Schwartz kernel of the operator e-th
2 PROBLEM SET 2 SEMICLASSICAL MICROLOCAL ANALYSIS DUE: NOV. 16, 2020 (3) [An counterexample to uncertainty and BCH] Consider the Hilbert space H = L 2 ([−1, 1]). Note that ψn(x) := 1 √ 2 e inx, n ∈ Z form an orthonormal basis of H. Define Q : H → H, ϕ 7→ xϕ and P : Dom(P) ⊂ H → H, ϕ 7→ ~ i dϕ dx , where Dom(P) = {ϕ ∈ C 1 ([−1, 1]) | ϕ(1) = ϕ(−1)}. (a) Prove: Q is self-adjoint on H, and P is essentially self-adjoint on Dom(P). (b) Prove: The uncertainty (c.f. PSet1-5) δ ψn (Q)δ ψn (P) = 0. Explain why this does not violate PSet1-5-(b). (c) For any a ∈ R, define Sa : H → H by (Saϕ)(x) := ϕ(x + a − 2mx,a), where mx,a is the unique integer such that x + a − 2mx,a ∈ [−1, 1). Prove: for any t ∈ R, e itB/~ = St . (d) Prove: Q, P violate the Baker-Campbell-Hausdorff formula. In other words, [Q, [Q, P]] = [P, [Q, P]] = 0 but e it(Q+P ) ~ 6= e −[ itQ ~ , itP ~ ] e itQ ~ e itP ~ . [Ref: Hall, Quantum theory for mathematicians, §12.2 and Example 14.5.] (4) [Mehler formula] For a symbol a(x, ξ), although we proved the following equality for several special cases (e.g. a is linear or a(x, ξ) = ip(ξ), where p is a real-valued polynomial), in general (where we assume e ta(x,ξ) is also a nice symbol function) e tbaW 6= ecta W . In this problem let’s explore the case a(x, ξ) = −H(x, ξ) = − x 2+ξ 2 2 , where for simplicity we assumed n = 1. Then e tbaW = e −tHb , where Hb = − ~ 2 2 d 2 dx2 + x 2 2 be the 1-dim harmonic oscillator. We want to find a function qt(x, ξ, t) so that e −tHb = ecqt W . (a) Prove: ∂qt ∂t e qt = −H ? eqt . (b) For the non-semiclassical setting (namely, there is no ~), I saw from literature a formula e qt = 1 cosh t 2 e −(x 2+ξ 2 ) tanh t 2 . Please write down a semiclassical version of this formula, and check it via (a). (c) Write down the Schwartz kernel of the operator e −tHb
3PROBLEMSET2SEMICLASSICAL MICROLOCAL ANALYSISDUE:NOV.16,2020(5)[t-quantizations]Prove the following properties of semiclassical t-quantizations(a) Prove the formulas (10) and (11) in Lecture 9, page 9.(b)ProveTheorem3.2inLecture9.(c) Prove: For a E S(IR2n), if we let b(r,E) = eihDaDea(r,E), then (aW)*=BW[Ref: Zworski, Semiclassical Analysis, 84.3.3 and 84.3.4](6)[Exact quantization condition]Prove Proposition 2.2 and Proposition 2.3 in Lecture 9 (page 7)Hint:Usethecommutativerelation [A,BC=[A,B]C+B[A,]for operators[Ref:Hall,Quantumtheoryformathematicians,Proposition13.11.](7) [Oscillatory testing for standard quantization]Suppose m is an order function, and a Ss(m). Prove:a(r,S)=e-r-s aKN(etr-s)So one can easily recover the symbol a(r, s) from its Kohn-Nirenberg quantization[Semiclassical differential operators]A semiclassical differential operator is an operator of the formP= aia(r)(hD)°,=0alkwhere m, k,'s are positive integers. Prove: A semiclassical pseudodifferential operatoris a semiclassical differential operator if and only if it is the Weyl quantization of afunction which is a polynomial in both and h.Ref:Folland,Harmonicanalysisonphasespace,Prop.2.11.(9)[Dirichlet-to-Neumann map as pseudodifferential operator]Consider the upper half space R+1 := R+ × Rn. We use t as the coordinate on R+and = (ri,.., &n) as the coordinates on Rn. The Dirichlet-to-Neumann operatorA : (Rn) → Co(Rn) is defined as follows: Given any f E (Rn) = (oRn+1), letu = u(t, r) E Co(Rn+l) be the solution to the equation【 (?2+?)u(t,) =0, u(0,a) = f(r).which decays rapidly as t → +oo. Define A(f) to be the exterior normal derivativeduA(f) =-hXIt turns out that A is a semiclassical pseudodifferential operator with symbol /sl:(a) Prove: i(t, E) = Ce-tisl/hf($), where i(t, ) =[(Fh)r→u](t, s).(b) Prove: A = F-l o|slo Fh
PROBLEM SET 2 SEMICLASSICAL MICROLOCAL ANALYSIS DUE: NOV. 16, 2020 3 (5) [t-quantizations] Prove the following properties of semiclassical t-quantizations: (a) Prove the formulas (10) and (11) in Lecture 9, page 9. (b) Prove Theorem 3.2 in Lecture 9. (c) Prove: For a ∈ S (R 2n ), if we let b(x, ξ) = e i~Dx·Dξ a(x, ξ), then (baW ) ∗ = bbW . [Ref: Zworski, Semiclassical Analysis, §4.3.3 and §4.3.4] (6) [Exact quantization condition] Prove Proposition 2.2 and Proposition 2.3 in Lecture 9 (page 7). Hint: Use the commutative relation [A, BC] = [A, B]C + B[A, C] for operators. [Ref: Hall, Quantum theory for mathematicians, Proposition 13.11.] (7) [Oscillatory testing for standard quantization] Suppose m is an order function, and a ∈ Sδ(m). Prove: a(x, ξ) = e − i ~ x·ξ ba KN (e i ~ x·ξ ). So one can easily recover the symbol a(x, ξ) from its Kohn-Nirenberg quantization! [Ref: Zworski, Semiclassical Analysis, Theorem 4.19] (8) [Semiclassical differential operators] A semiclassical differential operator is an operator of the form P = Xm i=0 ~ i X |α|≤ki ai,α(x)(~D) α , where m, ki ’s are positive integers. Prove: A semiclassical pseudodifferential operator is a semiclassical differential operator if and only if it is the Weyl quantization of a function which is a polynomial in both ξ and ~. [ Ref: Folland, Harmonic analysis on phase space, Prop. 2.11.] (9) [Dirichlet-to-Neumann map as pseudodifferential operator] Consider the upper half space R n+1 + := R+ × R n . We use t as the coordinate on R+ and x = (x1, · · · , xn) as the coordinates on R n . The Dirichlet-to-Neumann operator Λ : S (R n ) → C∞(R n ) is defined as follows: Given any f ∈ S (R n ) = S (∂R n+1 + ), let u = u(t, x) ∈ C∞(R n+1 + ) be the solution to the equation (~ 2∂ 2 t + ~ 2∆)u(t, x) = 0, u(0, x) = f(x). which decays rapidly as t → +∞. Define Λ(f) to be the exterior normal derivative Λ(f) = −~ ∂u ∂t t=0 . It turns out that Λ is a semiclassical pseudodifferential operator with symbol |ξ|: (a) Prove: ˆu(t, ξ) = Ce−t|ξ|/~ ˆf(ξ), where ˆu(t, ξ) = [(F~)x→ξu](t, ξ). (b) Prove: Λ = F −1 ~ ◦ |ξ| ◦ F~