LECTURE17:EGOROV'STHEOREM1.THEPROPAGATORThepropagator.Let m ≥ 1 be an order function, and let q e S(m) be a real-valued symbolfunction.Assume q is “almost elliptic" in the sense that there exists constantsC,c > 0 such thatC +q ≥ cm.Since q is real-valued, we havela+=V+1≥(+1)>21162+2C=2+2CIn other words, q +i is elliptic in S(m). According to Corollary 2.2 in Lecture 16,for h small enough, the operatorQ = qW : Hr(m) C L2(Rn) -→ L?(Rn)is a densely-defined self-adjoint operator on L?(Rn). So by the Stone's theorem thatwe cited in Lecture 8,U(t) = e-itQ/h(1)is a strongly continuous one-parameter family of unitary operators on L?(R") solvingthe equationhDtU(t)+QU(t)=0, tER(2)U(0) = I.The family e-itQ/n is called the propagators generated by Q.Note that q + C is a positive-valued elliptic symbol in S(m). It follows that forany u E S(m),II(Q + C)e-itQ/hul/2 = le-itQ/h(Q + C)ullL2 = Il(Q + C)ull/2.As a consequence, e-itQ/h is a bounded linear operator on Hh(m). [We will NOTsay that le-tp/llc(H(m) equals one, since usually we wlluse another equivalentnorm on Hr(m).)We remark that in general the propagator e-itP/h is NOT a semiclassical pseu-dodifferential operator, because it violates Beals's theorem (Check this!). It is aFourier integral operator that we will study in more detail later
LECTURE 17: EGOROV’S THEOREM 1. The propagator ¶ The propagator. Let m ≥ 1 be an order function, and let q ∈ S(m) be a real-valued symbol function. Assume q is “almost elliptic” in the sense that there exists constants C, c > 0 such that C + q ≥ cm. Since q is real-valued, we have |q + i| = p q 2 + 1 ≥ 1 2 (q + 1) > q + C 2 + 2C ≥ c 2 + 2C m. In other words, q + i is elliptic in S(m). According to Corollary 2.2 in Lecture 16, for ~ small enough, the operator Q = qb W : H~(m) ⊂ L 2 (R n ) → L 2 (R n ) is a densely-defined self-adjoint operator on L 2 (R n ). So by the Stone’s theorem that we cited in Lecture 8, (1) U(t) = e −itQ/~ is a strongly continuous one-parameter family of unitary operators on L 2 (R n ) solving the equation (2) ~DtU(t) + QU(t) = 0, t ∈ R U(0) = I. The family e −itQ/~ is called the propagators generated by Q. Note that q + C is a positive-valued elliptic symbol in S(m). It follows that for any u ∈ S(m), k(Q + C)e −itQ/~ukL2 = ke −itQ/~ (Q + C)ukL2 = k(Q + C)ukL2 . As a consequence, e −itQ/~ is a bounded linear operator on H~(m). [We will NOT say that ke −itP/~kL(H~(m)) equals one, since usually we will use another equivalent norm on H~(m).] We remark that in general the propagator e −itP/~ is NOT a semiclassical pseudodifferential operator, because it violates Beals’s theorem (Check this!). It is a Fourier integral operator that we will study in more detail later. 1
2LECTURE 17:EGOROV'STHEOREMThe“propagator"generated by t-dependent symbols.For later purpose weextend thepropagatordefined above to t-dependent sym-bols. Now let (qt)ter c S(1) be a family of real-valued symbols that depend s-moothly on t. DenoteQ(t) = qW.Let the operator F(t) be the solution (which exists at least locally) to the systemhDtF(t)+Q(t)F(t)=0,tER(3) F(0) = I.We extend the properties of the propagator e-itQ/h to this time-dependent setting:Theorem 1.1. The solution F(t) to the system (3) erists for all t E R and is aunitary operator on L?(Rn). Moreover, for any order function m, F(t) is a boundedlinear map from Hr(m)to Hr(m) for anyt.Proof. Since qt E S(1), the operator Q(t) is bounded on L? (with uniform operatornorm bound on closed intervals since g(t) depends smoothly in t) and thus thesolution (in the Banach space of bounded linear operators)exists for all t.To prove F(t) is unitary, we need to find out F(t)*. Since qt is real-valued, Q(t)is formally self-adjoint, i.e. Q(t)*=Q(t). Take the adjoint of the system (3), we gethDtF(t)*-F(t)*Q(t)=0, tERF(O)* = I.It followshD:[F(t)*F(t)] = [D,F(t)*]F(t) + F(t)*[DtF(t)]=F(t)*Q(t)F(t) -F(t)*Q(t)F(t)= 0.So F(t)*F(t) = I since F(O)*F(0) = I. On the other hand,hDt[F(t)F(t)*-I] = -Q(t)F(t)F(t)*+ F(t)F(t)*Q(t)=[F(t)F(t)*-I,Q(t)]This is a homogeneous ODE on F(t)F(t)* - I, with initial conditionF(0)F(0)*- I = 0.It follows F(t)F(t)*= I. So F(t) is unitary on L?For the second half of the theorem, without loss of generality we assume m ES(m).Since m is elliptic in S(m),forh small enough the operator (mW)-1 isapseudo-differential operator with symbol in S(). Consider the operatorFm(t) =mWF(t)(mW)-1.Then to prove the C(Hr(m)) boundedness of F(t) is equivalent to prove the theC(L?) boundedness of Fm(t). Then Fm(t) satisfies the equationhDtFm(t) = -mWQ(t)F(t)(mW)-1 = -Qm(t)Fm(t)
2 LECTURE 17: EGOROV’S THEOREM ¶ The “propagator” generated by t-dependent symbols. For later purpose we extend the propagator defined above to t-dependent symbols. Now let {qt}t∈R ⊂ S(1) be a family of real-valued symbols that depend smoothly on t. Denote Q(t) = qbt W . Let the operator F(t) be the solution (which exists at least locally) to the system (3) ~DtF(t) + Q(t)F(t) = 0, t ∈ R F(0) = I. We extend the properties of the propagator e −itQ/~ to this time-dependent setting: Theorem 1.1. The solution F(t) to the system (3) exists for all t ∈ R and is a unitary operator on L 2 (R n ). Moreover, for any order function m, F(t) is a bounded linear map from H~(m) to H~(m) for any t. Proof. Since qt ∈ S(1), the operator Q(t) is bounded on L 2 (with uniform operator norm bound on closed intervals since q(t) depends smoothly in t) and thus the solution (in the Banach space of bounded linear operators) exists for all t. To prove F(t) is unitary, we need to find out F(t) ∗ . Since qt is real-valued, Q(t) is formally self-adjoint, i.e. Q(t) ∗ = Q(t). Take the adjoint of the system (3), we get ~DtF(t) ∗ − F(t) ∗Q(t) = 0, t ∈ R F(0)∗ = I. It follows ~Dt [F(t) ∗F(t)] = [~DtF(t) ∗ ]F(t) + F(t) ∗ [~DtF(t)] = F(t) ∗Q(t)F(t) − F(t) ∗Q(t)F(t) = 0. So F(t) ∗F(t) = I since F(0)∗F(0) = I. On the other hand, ~Dt [F(t)F(t) ∗ − I] = −Q(t)F(t)F(t) ∗ + F(t)F(t) ∗Q(t) = [F(t)F(t) ∗ − I, Q(t)]. This is a homogeneous ODE on F(t)F(t) ∗ − I, with initial condition F(0)F(0)∗ − I = 0. It follows F(t)F(t) ∗ = I. So F(t) is unitary on L 2 . For the second half of the theorem, without loss of generality we assume m ∈ S(m). Since m is elliptic in S(m), for ~ small enough the operator (mb W ) −1 is a pseudo-differential operator with symbol in S( 1 m ). Consider the operator Fm(t) = mb W F(t)(mb W ) −1 . Then to prove the L(H~(m)) boundedness of F(t) is equivalent to prove the the L(L 2 ) boundedness of Fm(t). Then Fm(t) satisfies the equation ~DtFm(t) = −mb W Q(t)F(t)(mb W ) −1 = −Qm(t)Fm(t)
3LECTURE17:EGOROV'STHEOREMforQm(t)=mWQ(t)(mW)-1,withinitialconditionFm(0) = I.Since qt E S(1), the Weyl symbol of Qm(t) = mWQ(t)(mW)-1 is also in S(1). SoQm(t)is abounded linearoperator onL?.HencebyGronwall's inequality,IIF(+)l(Ha(m) = IFm()l(L2) ≤e 1m()(2)ds o.Moreover, assume the symbolic estimate holds uniformlyfor t-derivatives:ot,eqtl ≤ Ck,am.Then the system (3) has a local solution which is invertible and maps Hr(mk) intoHn(mk)for any integer k.For details, c.f.Zworski, g10.1.32.EGOROV'STHEOREMHamiltonianflowRecall from Lecture 2 that associated to any smooth function q(r, ) defined onR2n one has a Hamiltonian flow (ptJter,that is, a family of diffeomorphismsPt = et=a : R2n → IR2nthat sends a point z0 = (ro, So) to the point zt = pt(zo) = 2o(t), where = zo(t) isthe unique integral curve of 三g starting at zo:(0) = 20, (t) =三g((t),Here, 三q is the Hamiltonian vector field generated by q,oga=OEkOCkOkEkAs we have seen in Lecture 2, for any smooth symbol a E Co(R2n), if we denotebt=pa,thenbt = (q,br]
LECTURE 17: EGOROV’S THEOREM 3 for Qm(t) = mb W Q(t)(mb W ) −1 , with initial condition Fm(0) = I. Since qt ∈ S(1), the Weyl symbol of Qm(t) = mb W Q(t)(mb W ) −1 is also in S(1). So Qm(t) is a bounded linear operator on L 2 . Hence by Gronwall’s inequality, kF(t)kL(H~(m)) = kFm(t)kL(L2) ≤ e R t 0 kQm(s)kL(L2) ds 0. Moreover, assume the symbolic estimate holds uniformly for t-derivatives: |∂ k t ∂ α x,ξqt | ≤ Ck,αm. Then the system (3) has a local solution which is invertible and maps H~(mk ) into H~(mk ) for any integer k. For details, c.f. Zworski, §10.1.3. 2. Egorov’s theorem ¶ Hamiltonian flow. Recall from Lecture 2 that associated to any smooth function q(x, ξ) defined on R 2n one has a Hamiltonian flow {ρt}t∈R, that is, a family of diffeomorphisms ρt = e tΞq : R 2n → R 2n that sends a point z0 = (x0, ξ0) to the point zt = ρt(z0) = γz0 (t), where γ = γz0 (t) is the unique integral curve of Ξq starting at z0: γ(0) = z0, γ˙(t) = Ξq(γ(t)). Here, Ξq is the Hamiltonian vector field generated by q, Ξq = X k ∂q ∂ξk ∂ ∂xk − ∂q ∂xk ∂ ∂ξk . As we have seen in Lecture 2, for any smooth symbol a ∈ C ∞(R 2n ), if we denote bt = ρ ∗ t a, then ˙bt = {q, bt}.
4LECTURE17:EGOROV'STHEOREMT A weak Egorov theorem.Now let a e S(m) be a symbol, and let bt = pta be the “"classical flow-out" ofthe symbol a(r, s) along the flow pt. A natural question is:Question:What is the Weyl quantization of bt?Without any control on theflow thefunction bt =pta could be very bad.Forsimplicity we assume p is identity outsidea compact set, this amounts to requirethatqis compactly supported (From theassumption onp we seethat qis a constantoutsidea compact set.Since subtracting a constant from a Hamiltonian functionwill not changetheflow,we may assume that constant is zero.)and in particularq E C S(1).The assumptionalsoimpliesbtES(m)sincebt=a outsideacompactset.NowwemaystatetheEgorovtheoremthatwementionedinLecture1 in more precise form:Theorem 2.1 (Weak Egorov's' theorem). Let Q = qW. Under the previous as-sumptions, namely g is compactly supported and a e S(m),we haveeitQ/naWe-itQ/h=btw+O(h2),where the estimate is uniform for O<t ≤T, where T is any fired timeRemark.Weneed to explain ournotation.Fora h-dependent bounded linear oper-ator An E C(H1, H2), when we write An = O(h), we mean [Allc(H,Ha) = O(t). Forthis theorem, we can take Hi = Hn(m) and H2 = L?.Proof. Since eitQ/h is bounded on each Hr(m), it is enough to proveaw- e-itQ/hb"etQ/h= O(h).Sincea = bo,we only need to prove(e-itQ/bt"Wet/t) = (n2),dti.e.*e-itQ/hb"-[o,6w eitQ/h = O(h2).一Conjugating by the bounded operator eitQ/h again, it is enough to prove= 0(h),(q*bt-br*)1This is proved by Y.V. Egorov (1939-2008), a Russian-Soviet mathematician who specializes indifferential equations. Another well-known Egorov theorem (in real analysis) was proven by D.F.Egorov (1869-1931), a Russian and Soviet mathematician known for significant contributions tothe areas of differential geometry and mathematical analysis
4 LECTURE 17: EGOROV’S THEOREM ¶ A weak Egorov theorem. Now let a ∈ S(m) be a symbol, and let bt = ρ ∗ t a be the “classical flow-out” of the symbol a(x, ξ) along the flow ρt . A natural question is: Question: What is the Weyl quantization of bt? Without any control on the flow the function bt = ρ ∗ t a could be very bad. For simplicity we assume ρ is identity outside a compact set, this amounts to require that q is compactly supported (From the assumption on ρ we see that q is a constant outside a compact set. Since subtracting a constant from a Hamiltonian function will not change the flow, we may assume that constant is zero.) and in particular q ∈ S ⊂ S(1). The assumption also implies bt ∈ S(m) since bt = a outside a compact set. Now we may state the Egorov theorem that we mentioned in Lecture 1 in more precise form: Theorem 2.1 (Weak Egorov’s1 theorem). Let Q = qb W . Under the previous assumptions, namely q is compactly supported and a ∈ S(m), we have e itQ/~ba W e −itQ/~ = bbt W + O(~ 2 ), where the estimate is uniform for 0 ≤ t ≤ T, where T is any fixed time. Remark. We need to explain our notation. For a ~-dependent bounded linear operator A~ ∈ L(H1, H2), when we write A~ = O(~), we mean kA~kL(H1,H2) = O(~). For this theorem, we can take H1 = H~(m) and H2 = L 2 . Proof. Since e itQ/~ is bounded on each H~(m), it is enough to prove a W − e −itQ/~ bbt W e itQ/~ = O(~ 2 ). Since a = b0, we only need to prove d dt e −itQ/~ bbt W e itQ/~ = O(~ 2 ), i.e. e −itQ/~ b˙bt W − i ~ h Q, bbt W i e itQ/~ = O(~ 2 ). Conjugating by the bounded operator e itQ/~ again, it is enough to prove b ˙ t − i \~ (q ? bt − bt ? q) W = O(~ 2 ), 1This is proved by Y.V. Egorov (1939-2008), a Russian-Soviet mathematician who specializes in differential equations. Another well-known Egorov theorem (in real analysis) was proven by D.F. Egorov (1869-1931), a Russian and Soviet mathematician known for significant contributions to the areas of differential geometry and mathematical analysis
5LECTURE17:EGOROV'S THEOREMSince bt= {g,bt) and since (in Lecture 9-10)方=(g,bt) +O(h3),q*bt-bt*q =口the conclusion follows.The weak version of Egorov's theorem claims that when conjugated by the prop-agator, a semiclassical pseudodifferential operator aw gets converted to an operatorwhich is semiclassically close to the semiclassical pseudodifferential operator whosesymbol is the flow-out of a. So roughly speaking, “conjugation by the propagator"isthe quantum analogue of “fow out by the classical fow".But we also want to know:what is the operator eitQ/hawe-itQ/h itself? Is it still a semiclassical pseudodiffer-ential operator.The answer is yes.[NOTE: as we have mentioned, the propagatoreitQ/hisNOTa semiclassical pseudodifferential operator.] The Time-dependent flow.We will prove the theorem in more general setting, namely, we will consider theflow generated by a family of symbols qt instead of generated by a single symbolq.We can repeat the construction of the Hamiltonian flow above:starting witha smooth family of symbols qt, which, as explained above, will be assumed to becompactly supported in a fixed compact set. Consider the associated time-dependentHamiltonian vector fieldsOqtaOqtaEqt=AaorkOrkOEkUnder the“compactly-supported"assumption,for any zo =(ro,So),Eq.admits aunique integral curve =zo(t) starting at z0,(0) =20,(t)=三g((t))This again gives us a family of diffeomorphisms (which need not satisfy the grouplawasintheprevioussetting)Pt=et=H:R2n→R2nthat sends a point zo to the point zt = pt(zo) := zo(t). Again for any smoothsymbol a e Co(R2n), if we denote at = pta, thenat = (qt, at].Egorov's theorem.Under the assumption that Iqtl vanish outside a fixed compact set, we haveq(t) e S(1) and thus we can apply Theorem 1.1 to Q(t) = qw to conclude that theoperatorequationhDtF(t)+Q(t)F(t)=0, 0≤t≤TF(0) = I
LECTURE 17: EGOROV’S THEOREM 5 Since ˙bt = {q, bt} and since (in Lecture 9-10) q ? bt − bt ? q = ~ i {q, bt} + O(~ 3 ), the conclusion follows. The weak version of Egorov’s theorem claims that when conjugated by the propagator, a semiclassical pseudodifferential operator ba W gets converted to an operator which is semiclassically close to the semiclassical pseudodifferential operator whose symbol is the flow-out of a. So roughly speaking, “conjugation by the propagator” is the quantum analogue of “flow out by the classical flow”. But we also want to know: what is the operator e itQ/~ba W e −itQ/~ itself? Is it still a semiclassical pseudodifferential operator. The answer is yes. [NOTE: as we have mentioned, the propagator e itQ/~ is NOT a semiclassical pseudodifferential operator.] ¶ The Time-dependent flow. We will prove the theorem in more general setting, namely, we will consider the flow generated by a family of symbols qt instead of generated by a single symbol q. We can repeat the construction of the Hamiltonian flow above: starting with a smooth family of symbols qt , which, as explained above, will be assumed to be compactly supported in a fixed compact set. Consider the associated time-dependent Hamiltonian vector fields Ξqt = X k ∂qt ∂ξk ∂ ∂xk − ∂qt ∂xk ∂ ∂ξk . Under the “compactly-supported” assumption, for any z0 = (x0, ξ0), Ξqt admits a unique integral curve γ = γz0 (t) starting at z0, γ(0) = z0, γ˙(t) = Ξqt (γ(t)) This again gives us a family of diffeomorphisms (which need not satisfy the group law as in the previous setting) ρt = e tΞH : R 2n → R 2n that sends a point z0 to the point zt = ρt(z0) := γz0 (t). Again for any smooth symbol a ∈ C ∞(R 2n ), if we denote at = ρ ∗ t a, then a˙t = {qt , at}. ¶ Egorov’s theorem. Under the assumption that {qt} vanish outside a fixed compact set, we have q(t) ∈ S(1) and thus we can apply Theorem 1.1 to Q(t) = qb W t to conclude that the operator equation ~DtF(t) + Q(t)F(t) = 0, 0 ≤ t ≤ T F(0) = I.
6LECTURE17:EGOROV'STHEOREMhas a unique solution F which is unitary on L? and is a bounded linear map fromHr(m) to Hr(m). In particular, by taking adjoint one getshD[F(t)-1] - F(t)-1Q(t) = 0.Now we can state the main theorem which tells us that the time-evolution of asemiclassical pseudodifferential operator aw is again a semiclassical pseudodifferen-tial operator whose symbol, to the leading order, is the time-evolution of a:Theorem 2.2 (Egorov's theorem).Suppose qt (t E[o,Tl) is a smooth family offunctions supported in a ficed compact set in R2n. Let m be an order function andsuppose a E S(m). Let F be as described above. Then for O ≤t ≤T, we haveF(t)-"aWF(t) = 6w(4)for somesymbol btES(m)of theformbt=pa+O(h)Ideaoftheproof:WewanttoshowthatB(t) := F(t)-laWF(t)is a semiclassical pseudodifferential operator. Of course we need to apply Beals'stheorem.But B(t)itself is not easy to handle, since F(t) is not a semiclassicalpseudodifferential operator. The idea is to approximate B(t) to arbitrary orderO(h)bypseudodifferential operators.First extend the weak Egorov theorem to the time-dependent setting, so thatB(t)ish-closetothepseudodifferential operatorBo(t)=ptaIterativelyfind pseudodifferential operator Br(t)that is hk+1-close to B(t)ApplyBealsstheoremtoboth Br(t) and B(t)-B(t)Proof.Step 1:Extend the weak Egorov theorem to the time-dependent setting.Denote A = aw, Bo(t) = pta" and defineB(t) = F(t)-1AF(t)We want to show B(t) - Bo(t) = O(h)We notice that by definition,B(t) - Bo(t) = F(t)-1(A- F(t)Bo(t)F(t)-1)F(t)So we first study F(t)-1Bo(t)F(t). Since Otpfa = [qt, pta), we haveWha克(5)hD,Bo(t) =(qt, Pta] "= [Q(t), Bo(t)] + Eo(t),Otpta"where Eo(t) = eo.tW for some eo,t = O(t2). eo,t is compactly supported since it canbe expressed as sums of products of derivatives of qt and pa. So eo,t e h?, and
6 LECTURE 17: EGOROV’S THEOREM has a unique solution F which is unitary on L 2 and is a bounded linear map from H~(m) to H~(m). In particular, by taking adjoint one gets ~Dt [F(t) −1 ] − F(t) −1Q(t) = 0. Now we can state the main theorem which tells us that the time-evolution of a semiclassical pseudodifferential operator ba W is again a semiclassical pseudodifferential operator whose symbol, to the leading order, is the time-evolution of a: Theorem 2.2 (Egorov’s theorem). Suppose qt (t ∈ [0, T]) is a smooth family of functions supported in a fixed compact set in R 2n . Let m be an order function and suppose a ∈ S(m). Let F be as described above. Then for 0 ≤ t ≤ T, we have (4) F(t) −1ba W F(t) = bbt W for some symbol bt ∈ S(m) of the form bt = ρ ∗ t a + O(~). Idea of the proof: We want to show that B(t) := F(t) −1ba W F(t) is a semiclassical pseudodifferential operator. Of course we need to apply Beals’s theorem. But B(t) itself is not easy to handle, since F(t) is not a semiclassical pseudodifferential operator. The idea is to approximate B(t) to arbitrary order O(~ k ) by pseudodifferential operators. • First extend the weak Egorov theorem to the time-dependent setting, so that B(t) is ~-close to the pseudodifferential operator B0(t) = ρc∗ t a W . • Iteratively find pseudodifferential operator Bk(t) that is ~ k+1-close to B(t). • Apply Beals’s theorem to both Bk(t) and Bk(t) − B(t). Proof. Step 1: Extend the weak Egorov theorem to the time-dependent setting. Denote A = ba W , B0(t) = ρc∗ t a W and define B(t) = F(t) −1AF(t). We want to show B(t) − B0(t) = O(~). We notice that by definition, B(t) − B0(t) = F(t) −1 (A − F(t)B0(t)F(t) −1 )F(t). So we first study F(t) −1B0(t)F(t). Since ∂tρ ∗ t a = {qt , ρ∗ t a}, we have (5) ~DtB0(t) = ~ i ∂[tρ ∗ t a W = ~ i {q\t , ρ∗ t a} W = [Q(t), B0(t)] + E0(t), where E0(t) = ec0,t W for some e0,t = O(~ 2 ). e0,t is compactly supported since it can be expressed as sums of products of derivatives of qt and ρ ∗ t a. So e0,t ∈ ~ 2S , and
LECTURE17:EGOROV'STHEOREM7Eo(t) is a bounded operator on L? with operator norm O(h2). It followshDt[F(t)Bo(t)F(t)-1|=[DtF(t)|Bo(t)F(t)-1 +F(t)[hDtBo(t)|F(t)-1 + F(t)Bo(t)hDt[F(t)-1]= -Q(t)F(t)Bo(t)F(t) + F(t)([Q(t), Bo(t)+ Eo(t)F(t)-1 + F(t)Bo(t)F(t)-1Q(t)= F(t)-1Eo(t)F(t)is bounded on L? with operator norm O(2). SoF(t)Bo(t)F(t)-1 = A + ~ /F(s)Eo(s)F(s)-1ds = A+O(h))九JSo we concludeB(t) -Bo(t)= O(t),Step 2:Construct h-PsDO Bk(t) such that B(t) - Bk(t) = O(hk).We just constructed Bo(t) satisfying[ hD,Bo(t) = [Q(t), Bo(t)l +Eo(t) Bo(0) = A.such that B(t)-Bo(t)=O(h). To improve the error, we will construct a sequenceof semiclassical pseudodifferential operators Br(t) so that for each k,(hDtBr(t) =[Q(t), Br(t) +Er(t)(6)Bk(0) = A.withEk(t) = ex, wfor some ek.t E hk+2 . Once this is done, the same computation giveshD,[F(t)Bk(t)F(t)-11 = F(t)E(t)F(t)-1 E O(hk+2)Again after integrating in t we will getB(t) = B(t) -_F(t)-1 [ / F(s)Es(s)F(s)-'ds|F(t),(7)which implies(8)B(t) = B(t) + O(hk+1).Wesolve (6)byinduction.Assumewehavefinishedk.ThenwetrytofindBk+of the form Bk+1 = Bk-Ck+1 which solves (6) for k+1, where Ck+1(t) := Ck+1,tW eO(hk+1).For this purpose we first find the equation that Ck+1 should satisfy.It isnot hard to write down it:hD,Ck+1(t) = [Q(t), Ck+1(t) + Ex(t) + O(tk+3).It is thus very clear that to the leading order, the symbol ck+1 of Ck+1 should satisfytheequationhDtck+1 = [qt, Ch+1) + ek(t)
LECTURE 17: EGOROV’S THEOREM 7 E0(t) is a bounded operator on L 2 with operator norm O(~ 2 ). It follows ~Dt [F(t)B0(t)F(t) −1 ] = [~DtF(t)]B0(t)F(t) −1 + F(t)[~DtB0(t)]F(t) −1 + F(t)B0(t)~Dt [F(t) −1 ] = −Q(t)F(t)B0(t)F(t) + F(t)([Q(t), B0(t)] + E0(t))F(t) −1 + F(t)B0(t)F(t) −1Q(t) = F(t) −1E0(t)F(t) is bounded on L 2 with operator norm O(~ 2 ). So F(t)B0(t)F(t) −1 = A + i ~ Z t 0 F(s)E0(s)F(s) −1 ds = A + O(~). So we conclude B(t) − B0(t) = O(~). Step 2: Construct ~-PsDO Bk(t) such that B(t) − Bk(t) = O(~ k ). We just constructed B0(t) satisfying ~DtB0(t) = [Q(t), B0(t)] + E0(t) B0(0) = A. such that B(t) − B0(t) = O(~). To improve the error, we will construct a sequence of semiclassical pseudodifferential operators Bk(t) so that for each k, (6) ~DtBk(t) = [Q(t), Bk(t)] + Ek(t) Bk(0) = A. with Ek(t) = eck,t W for some ek,t ∈ ~ k+2S . Once this is done, the same computation gives ~Dt [F(t)Bk(t)F(t) −1 ] = F(t)Ek(t)F(t) −1 ∈ O(~ k+2). Again after integrating in t we will get (7) B(t) = Bk(t) − i ~ F(t) −1 Z t 0 F(s)Ek(s)F(s) −1 ds F(t), which implies (8) B(t) = Bk(t) + O(~ k+1). We solve (6) by induction. Assume we have finished k. Then we try to find Bk+1 of the form Bk+1 = Bk −Ck+1 which solves (6) for k + 1, where Ck+1(t) := c[k+1,t W ∈ O(~ k+1). For this purpose we first find the equation that Ck+1 should satisfy. It is not hard to write down it: hDtCk+1(t) = [Q(t), Ck+1(t)] + Ek(t) + O(~ k+3). It is thus very clear that to the leading order, the symbol ck+1 of Ck+1 should satisfy the equation ~Dtck+1 = {qt , ck+1} + ek(t)
8LECTURE17:EGOROV'STHEOREMThis is an inhomogeneous evolution equation whose solution is given by Duhamel'sprinciple,(p-")'ek,sds.Ck+1,t 5So we take ck+1, as above, and define Ck+1(t) := Ck+1,W O(tk+1).. It is nothard to computehD,Ch+1(t) =(rpt / (ps1)*en,eds)Ot (qt, C+1,t) + ek,t=[Q(t),Ck+1(t)]+Er(t)-Ek+1(t)where Ex+1(t) = ek+1,tW for some ek+1,t E hk+3g. HenceBk+1(t) := B(t) - Ck+1(t)iswhatwewant.Step 3:Prove B(t) itself is a h-PsDO with B(t) - Bo(t) = O(h).According to Beals's theorem, it is enough to show that for any linear functionsli,..,lm,.. adw(B(t) - Bo(t) = O(hM+1).Wewill provethat foranykw... adgw(B(t) - B(t) = O(hk+1),(9)ad-so that for any M,adw ..adw(B(t) -Bo(t))=adow..adw(B(t) -Bm(t)) + adw .adw(Bm(t) -Bo(t)=O(hM+1).To prove (9), we notice that for any linear function l, the equation (7) implies2adhw(B(t) - B(t)) = -adw [F(t)-1F(s)Ek(s)F(s)-"F(t)] dshSince Ek(t) =ek,tW for ek,t E hk+2g, we haveAadwEx(s)=O(hk+1)杭Similar expression holds for more adw's. So it remains to proveadw..ad-wF(t) =O(1)
8 LECTURE 17: EGOROV’S THEOREM This is an inhomogeneous evolution equation whose solution is given by Duhamel’s principle, ck+1,t = i ~ ρ ∗ t Z t 0 (ρ −1 s ) ∗ ek,sds. So we take ck+1,t as above, and define Ck+1(t) := c[k+1,t W ∈ O(~ k+1). It is not hard to compute ~DtCk+1(t) = ~ \ i ∂t i ~ ρ ∗ t Z t 0 (ρ −1 s ) ∗ek,sds W = ~ \i {qt , ck+1,t} + ek,t W = [Q(t), Ck+1(t)] + Ek(t) − Ek+1(t) where Ek+1(t) = e[k+1,t W for some ek+1,t ∈ ~ k+3S . Hence Bk+1(t) := Bk(t) − Ck+1(t) is what we want. Step 3: Prove B(t) itself is a ~-PsDO with B(t) − B0(t) = O(~). According to Beals’s theorem, it is enough to show that for any linear functions l1, · · · , lM, adlb1 W · · · ad lcM W (B(t) − B0(t)) = O(~ M+1). We will prove that for any k (9) adlb1 W · · · adlbk W (B(t) − Bk(t)) = O(~ k+1), so that for any M, adlb1 W · · · ad lcM W (B(t) − B0(t)) =adlb1 W · · · ad lcM W (B(t) − BM(t)) + adlb1 W · · · ad lcM W (BM(t) − B0(t)) =O(~ M+1). To prove (9), we notice that for any linear function l, the equation (7) implies adblW (B(t) − Bk(t)) = − i ~ Z t 0 adblW F(t) −1F(s)Ek(s)F(s) −1F(t) ds. Since Ek(t) = eck,t W for ek,t ∈ ~ k+2S , we have i ~ adblW Ek(s) = O(~ k+1). Similar expression holds for more adlbj W ’s. So it remains to prove adlb1 W · · · adlbk W F(t) = O(1).
LECTURE17:EGOROV'S THEOREM9RecalladA(BC) = (adAB)C +B(adAC)andadAB = -B(adAB-1)B.Applying adaw to the equationhDtF(t) +Q(t)F(t) =0we gethDt(adhwF(t) + (adhwQ(t)Q(t) +Q(t)adhwF(t) = 0SohDt(F(t)-ladhwF(t) = F(t)-1Q(t)adwF(t) - F(t)-I[(adwQ(t)F(t) +Q(t)adwF(t)= -F(t)-[adwQ(t)]F(t),which moves the operator adaw from F(t) to Q(t)! Since Q(t) is a pseudodifferentialoperator,wehaveadwQ(t)=hQiforQi=O(h).HencehD(F(s)-ladwF(s)dsadwF(t) = F(t) F(t)F(s)-1Qi(s)F(s)ds = O(1)Apply the method iteratively, we get the same estimate for more adw's.Thiscompletes the proof.Remark.For the proof we can see that the estimate can be done only for t e[o, T],where T is a fixed interval. For example, if instead fixing T we take t =h-l, thenwe will get extra h-1 factor in the estimates and the proof will fail. It turns outthat Egorov theorem can be extend to longer time, namely,t E (0, -Clogh)The time-Clogh is called Ehrenfest time.For details about the Egorov theoremup to Ehrenfest time, c.f.Zworski, s11.4
LECTURE 17: EGOROV’S THEOREM 9 Recall adA(BC) = (adAB)C + B(adAC) and adAB = −B(adAB −1 )B. Applying adblW to the equation ~DtF(t) + Q(t)F(t) = 0 we get ~Dt(adblW F(t)) + (adblW Q(t))Q(t) + Q(t)adblW F(t) = 0 So ~Dt(F(t) −1 adblW F(t)) = F(t) −1Q(t)adblW F(t) − F(t) −1 [(adblW Q(t))F(t) + Q(t)adblW F(t)] = −F(t) −1 [adblW Q(t)]F(t), which moves the operator adblW from F(t) to Q(t)! Since Q(t) is a pseudodifferential operator, we have adblW Q(t) = ~Q1 for Q1 = O(~). Hence adblW F(t) = F(t) i ~ Z t 0 ~Ds(F(s) −1 adblW F(s))ds = i Z t 0 F(t)F(s) −1Q1(s)F(s)ds = O(1). Apply the method iteratively, we get the same estimate for more adblW ’s. This completes the proof. Remark. For the proof we can see that the estimate can be done only for t ∈ [0, T], where T is a fixed interval. For example, if instead fixing T we take t = ~ −1 , then we will get extra ~ −1 factor in the estimates and the proof will fail. It turns out that Egorov theorem can be extend to longer time, namely, t ∈ (0, −C log ~). The time −C log ~ is called Ehrenfest time. For details about the Egorov theorem up to Ehrenfest time, c.f. Zworski, §11.4.