LECTURE5—09/30/2020THEMETHODOFSTATIONARYPHASE1.THE METHOD OF STATIONARY PHASE: SIMPLE MODELS Asymptotic series.When talking about the asymptotic behavioras h→o',we will use the notations“big O" and “"small o" as in mathematical analysis, namely for functions f, g whichare depending on h,. f = O(g) means “3 constant C > 0 such that Ifl ≤ Cgl for all small t"..f=o(g)means“ash→0,the quotient f/g→0"We will use the following conceptions/notations from asymptotic analysis:Definition 1.1. Let f=f(h)(1) We say f ~=oa,h if for each non-negative integer N,f-a=O(hN+1),h→0.(2) We say f = O(h) if f ~ 0, ie. f = O(hN) for all N.Remark. When we write f ~ E=o axhk, we don't require the series E=o axhk toconverge! Moreover, even if the series converges at a point, it need not converge tothe value of f at that point. For example, for any smooth function f = f(h), itsTaylor series is an asymptotic series,((0)f(h) ~k!However, the series converges for all small h to the value f(h) only if f is analyticat 0.EmpleHereisamoreconcree exmple Conside the function (h) -/,h<0.which is widely used in building cut-off functions.Then f is smooth everywherebut it is not analytic at 0. Moreover, by induction one can easily prove fk(O) = 0for all k. It follows that f ~ O, i.e. f = O(h).IIn this course, when we say as h -→ 0", we always means "as h -→ O+".1
LECTURE 5 — 09/30/2020 THE METHOD OF STATIONARY PHASE 1. The method of stationary phase: simple models ¶ Asymptotic series. When talking about the asymptotic behavior as ~ → 0 1 , we will use the notations “big O” and “small o” as in mathematical analysis, namely for functions f, g which are depending on ~, • f = O(g) means “∃ constant C > 0 such that |f| ≤ C|g| for all small ~”. • f = o(g) means “as ~ → 0, the quotient f/g → 0”. We will use the following conceptions/notations from asymptotic analysis: Definition 1.1. Let f = f(~). (1) We say f ∼ P∞ k=0 ak~ k if for each non-negative integer N, f − ❳ N k=0 ak~ k = O(~ N+1), ~ → 0. (2) We say f = O(~ ∞) if f ∼ 0, i.e. f = O(~ N ) for all N. Remark. When we write f ∼ P∞ k=0 ak~ k , we don’t require the series P∞ k=0 ak~ k to converge! Moreover, even if the series converges at a point, it need not converge to the value of f at that point. For example, for any smooth function f = f(~), its Taylor series is an asymptotic series, f(~) ∼ ❳∞ k=0 f (k) (0) k! ~ k . However, the series converges for all small ~ to the value f(~) only if f is analytic at 0. Example. Here is a more concrete example: Consider the function f(~) = ➝ e −1/~ , ~ > 0 0, ~ ≤ 0. which is widely used in building cut-off functions. Then f is smooth everywhere, but it is not analytic at 0. Moreover, by induction one can easily prove f k (0) = 0 for all k. It follows that f ∼ 0, i.e. f = O(~ ∞). 1 In this course, when we say “as ~ → 0”, we always means “as ~ → 0+”. 1
2LECTURE5-09/30/2020 THEMETHODOF STATIONARYPHASEWe can perform standard operations on asymptotic series. For example, iff(h)~a,hl,g(h)~b,hjthen wewill havef(h)±g(h) ~E(a,±b,)nand f(h)g(h) ~c,f),where c, = Ei-o abj-1. Similarly one can calculate the quotient of two asymptoticseries: If bo 0, thenf(h)/g(h)~d;hi,where d,'s are defined iteratively via do = ao/bo and d, = b-'(aj - Ei-o dibj-1).↑ Oscillatory integrals.Very often in semiclassical analysis we will need to evaluate the asymptoticbehavior of the oscillatory integrals of theform( eia(r)dr,(1)Ih=where Co(R", R) is called the phase, and a E C(R", C) is called the amplitude.The method of stationary phase is the correct tool for this purpose.?To illustrate, let's start with two extremal cases:: Suppose p(r) = c is a constant. ThenIn=eic/hAwhich is fast oscillating as h → O, where A = Jrna(r)dr is a constantindependent of h..Supposen =1 and (r)=.Thenby definition In=F(a)(-)Since ais compactly supported, F(a) is Schwartz. It follows I=O(hN) for any Ni.e.In = O(h).Intuitively,in the second case theexponential exp(i)oscillates fast in anyintervalof for small h, so that many cancellations takeplace and thus weget a functionrapidlydecreasing inh.Thisis infactthecaseatanypointrwhichisnotacriticalpoint of g. On the other hand, near a critical point of r the phase function doesn'tchange much, i.e.“looks like"a constant, so that we are in case l.According tothese intuitive observation, we expect that the main contributions to In should arisefrom the critical points of the phase function p.2Here we assume the phase function p is real-valued. In the case the phase function is complex,there is a similar way to evaluate the asymptotic of the integral: the method of steepest descent
2 LECTURE 5 — 09/30/2020 THE METHOD OF STATIONARY PHASE We can perform standard operations on asymptotic series. For example, if f(~) ∼ ❳aj~ j , g(~) ∼ ❳bj~ j , then we will have f(~) ± g(~) ∼ ❳(aj ± bj )~ j and f(~)g(~) ∼ ❳cj~ j , where cj = Pj l=0 albj−l . Similarly one can calculate the quotient of two asymptotic series: If b0 6= 0, then f(~)/g(~) ∼ ❳dj~ j , where dj ’s are defined iteratively via d0 = a0/b0 and dj = b −1 0 (aj − Pj−1 l=0 dlbj−l). ¶ Oscillatory integrals. Very often in semiclassical analysis we will need to evaluate the asymptotic behavior of the oscillatory integrals of the form (1) I~ = ❩ Rn e i ϕ(x) ~ a(x)dx, where ϕ ∈ C ∞(R n , R) is called the phase, and a ∈ C ∞ c (R n , C) is called the amplitude. The method of stationary phase is the correct tool for this purpose.2 To illustrate, let’s start with two extremal cases: • Suppose ϕ(x) = c is a constant. Then I~ = e ic/~A which is fast oscillating as ~ → 0, where A = ❘ Rn a(x)dx is a constant independent of ~. • Suppose n = 1 and ϕ(x) = x. Then by definition I~ = F(a)(− 1 ~ ). Since a is compactly supported, F(a) is Schwartz. It follows I~ = O(~ N ) for any N, i.e. I~ = O(~ ∞). Intuitively, in the second case the exponential exp(i x ~ ) oscillates fast in any interval of x for small ~, so that many cancellations take place and thus we get a function rapidly decreasing in ~. This is in fact the case at any point x which is not a critical point of ϕ. On the other hand, near a critical point of x the phase function ϕ doesn’t change much, i.e. “looks like” a constant, so that we are in case 1. According to these intuitive observation, we expect that the main contributions to I~ should arise from the critical points of the phase function ϕ. 2Here we assume the phase function ϕ is real-valued. In the case the phase function is complex, there is a similar way to evaluate the asymptotic of the integral: the method of steepest descent
LECTURE5—09/30/2020THEMETHODOFSTATIONARYPHASE3 Non-stationary phase.Proposition 1.2. Suppose the phase function p has no critical point in a neighbor.hood of the support of a. Then(2)In = O(h~).Proof. Let x be a smooth cut-off function such that: x is identically one on the support of a.has nocritical point onthesupportof xThn)()() dmohmtioThe trick (histandard in semiclassical microlocal analysis)is to introduce an operator L given byLf(a)= Z)(0,(a)/V0()27Then it is easy to checkiveireX00L(ei)() =ite/V012九2It follows [, L(ei)a(r)da= ei(L*a)(r)drIn=Rwhere L* is the adjoint of L, explicitly given byrro-- a(Mp/a)Repeating this N times, we getL e(L*)Na(r)da =O(hN).In=口Thestationaryphaseformulafor quadraticphase.Nowwe consider phasefunctions o with very simple critical points,in which caseone can imagine that the main contribution to the oscillatory integral comes fromthe critical point. We start with a model, i.e. (r) = rTQr for some non-singularreal symmetric n × n matrix Q.Theorem 1.3 (Stationary phase for non-singular quadratic phase). Let Q be a real,symmetric, non-singular n × n matrir. For any a E Co(Rn), one has, eQra(r)d ~ (2h)/2 eg(0)Idet Q((-Po-(D) a(0),(3)where for a matrir A= (ai), we denote pA(D)=auD,D
LECTURE 5 — 09/30/2020 THE METHOD OF STATIONARY PHASE 3 ¶ Non-stationary phase. Proposition 1.2. Suppose the phase function ϕ has no critical point in a neighborhood of the support of a. Then (2) I~ = O(~ ∞). Proof. Let χ be a smooth cut-off function such that • χ is identically one on the support of a • ϕ has no critical point on the support of χ. Then a(x) = χ(x)a(x), and χ(x) |∇ϕ(x)| 2 is a smooth function. The trick (which is standard in semiclassical microlocal analysis) is to introduce an operator L given by Lf(x) = ❳ j χ(x)∂jϕ(x) |∇ϕ(x)| 2 ∂jf(x). Then it is easy to check L(e i ϕ(x) ~ )(x) = χ |∇ϕ| 2 ❳ j ∂jϕ i∂jϕ ~ e i ϕ(x) ~ = i ~ χei ϕ(x) ~ . It follows I~ = ~ i ❩ Rn L(e i ϕ(x) ~ )a(x)dx = ~ i ❩ Rn e i ϕ(x) ~ (L ∗ a)(x)dx, where L ∗ is the adjoint of L, explicitly given by L ∗ f(x) = − ❳ j ∂j ❶ χ(x)∂jϕ |∇ϕ(x)| 2 f(x) ➀ . Repeating this N times, we get I~ = ❶ ~ i ➀N ❩ Rn e i ϕ(x) ~ (L ∗ ) N a(x)dx = O(~ N ). ¶ The stationary phase formula for quadratic phase. Now we consider phase functions ϕ with very simple critical points, in which case one can imagine that the main contribution to the oscillatory integral comes from the critical point. We start with a model, i.e. ϕ(x) = 1 2 x TQx for some non-singular real symmetric n × n matrix Q. Theorem 1.3 (Stationary phase for non-singular quadratic phase). Let Q be a real, symmetric, non-singular n × n matrix. For any a ∈ C ∞ c (R n ), one has (3) ❩ Rn e i 2~ x T Qxa(x)dx ∼ (2π~) n/2 e i π 4 sgn(Q) | det Q| 1/2 ❳ k ~ k 1 k! ✒ − i 2 pQ−1 (D) ✓k a(0), where for a matrix A = (aij ), we denote pA(D) = P aklDkDl .
4LECTURE5-09/30/2020 THEMETHODOFSTATIONARYPHASEProof. Recall Theorem 2.3 in Lecture 4: If (r) = er Qr, then(2元)n/2eisgn(Q)Fp(s) = Idet(0)/2-e-0-sNow we apply the multiplication formula/F()(s)d= /(r)Fb(r)dato getFa(S)es(-h-1)d= (2m)n/2eign(--)eaTQra(r)drIdet(-hQ-1)1/2i.e.hn/2eiisgn(Q)erTQa(r)dre-Q-'Fa($)ds(2)n/2 /|det Q|1/2Using the Taylor's expansion formula for the exponential function, we see that forany non-negative integer N, the differenceZ(e-TQ-"Fa(E)de-)()(TQ-)Fa(E)dk!is bounded by (a multiple of)1hN+1I($TQ-1e)*Fa($)dE2N+1(N + 1)! JSotheconclusionfollowsfromLemma 1.4. For any a E and any polynomial pF(p(D)a)() = p($)Fa($)Proof. This is just a consequence of the fact F(Da) = saFa and口the linearity of F.which implies(sTQ-1s)*Fa($) = F(pQ-1(D)*a)($),togetherwith the Fourier inversion formula, which impliesPo-(D)a(0)=2) /.(Q-1),Fa(2)d,口This completes the proof.Remark.By calculating more carefully, one can prove that for any N, the remaindern e+Qra(r)dr -(2h)/2 e(@)E（-含(-2Po-(D)a(0)RN :=detQ|1/2is controlled byan explicitupper bound10°al.R≤Csup[a|≤2N+n+
4 LECTURE 5 — 09/30/2020 THE METHOD OF STATIONARY PHASE Proof. Recall Theorem 2.3 in Lecture 4: If ϕ(x) = e i 2 x T Qx, then Fϕ(ξ) = (2π) n/2 e i π 4 sgn(Q) | det(Q)| 1/2 e − i 2 ξ T Q−1 ξ . Now we apply the multiplication formula ❩ Fϕ(ξ)ψ(ξ)dξ = ❩ ϕ(x)Fψ(x)dx to get ❩ Rn Fa(ξ)e i 2 ξ T (−~Q−1 )ξ dξ = (2π) n/2 e i π 4 sgn(−~Q−1 ) | det(−~Q−1 )| 1/2 ❩ Rn e i 2~ x T Qxa(x)dx, i.e. ❩ Rn e i 2~ x T Qxa(x)dx = ~ n/2 (2π) n/2 e i π 4 sgn(Q) | det Q| 1/2 ❩ Rn e − i~ 2 ξ T Q−1 ξFa(ξ)dξ. Using the Taylor’s expansion formula for the exponential function, we see that for any non-negative integer N, the difference ❩ Rn e − i~ 2 ξ T Q−1 ξFa(ξ)dξ − ❳ N k=0 1 k! (− i~ 2 ) k ❩ Rn (ξ TQ −1 ξ) kFa(ξ)dξ is bounded by (a multiple of) ~ N+1 1 2 N+1(N + 1)! ❩ Rn |(ξ TQ −1 ξ) kFa(ξ)|dξ. So the conclusion follows from Lemma 1.4. For any a ∈ S and any polynomial p, F(p(D)a)(ξ) = p(ξ)Fa(ξ). Proof. This is just a consequence of the fact F(Dαa) = ξ αFa and the linearity of F. which implies (ξ TQ −1 ξ) kFa(ξ) = F(pQ−1 (D) k a)(ξ), together with the Fourier inversion formula, which implies pQ−1 (D) k a(0) = 1 (2π) n ❩ Rn (ξ TQ −1 ξ) kFa(ξ)dξ. This completes the proof. Remark. By calculating more carefully, one can prove that for any N, the remainder RN := ❩ Rn e i 2~ x T Qxa(x)dx − (2π~) n/2 e i π 4 sgn(Q) | det Q| 1/2 N ❳−1 k=0 ~ k 1 k! ✒ − i 2 pQ−1 (D) ✓k a(0) is controlled by an explicit upper bound RN ≤ CN sup |α|≤2N+n+1 |∂ α a|.
LECTURE5—09/30/2020THEMETHODOFSTATIONARYPHASE52.THEMETHODOFSTATIONARYPHASE:GENERALCASE Morse lemma.We first introduce some standard conceptions in global analysissDefinition 2.1. Let be a smooth function.(1) A point p is called a critical point of ifVp(p) = 0.(2) A critical point p of is called non-degenerate if the Hessian matrix d(p)isnon-degenerate,i.e.a0±0det d'(p) = detOron(3) A smooth function is called a Morse function if all of its critical points arenon-degenerate.3To study the stationary phase expansion for more general phase functions, onefirst convert the general (non-degenerate)phase function to a quadratic one by usingthe Morse lemma:Theorem 2.2 (Morse Lemma, version 1).Let E C(Rn).Suppose p is a non-degenerate critical point of p. Then there erists a neighborhood U of O, a neighbor-hood V of p and a diffeomorphism p:U-→V so that p(O) =p and(4)ps(a) = (p) + (a + + -+1-..-),where r is the number of positive eigenvalues of the Hessian matrir d'p(p). The stationary phase formula for general phase.As we have seen, only the critical points of the phase function give an essentialcontribution to the oscillatory integraleira(r)daIh:In whatfollows we will assume that ECo(Rn)admits only non-degenerate criticalpoints in a neighborhood of the support of a. Since non-degenerate critical pointsmust be discrete, has only finitely many critical points in the support of a. Thusone can find apartition of unity[U;/1≤ i≤ N+1] of the support of a sothateach Ui, 1≤i≤N, contains exactly one critical point p of ,and Un+i contains3Note that non-degenerate critical points must be discrete. In more subtle examples where thecritical points are not necessary discrete, but still nice enough, say, form smooth manifolds, onehas an analogous conception, namely the Morse-Bott function. Many results for Morse functionscan begeneralized to Morse-Bott functions
LECTURE 5 — 09/30/2020 THE METHOD OF STATIONARY PHASE 5 2. The method of stationary phase: general case ¶ Morse lemma. We first introduce some standard conceptions in global analysis: Definition 2.1. Let ϕ be a smooth function. (1) A point p is called a critical point of ϕ if ∇ϕ(p) = 0. (2) A critical point p of ϕ is called non-degenerate if the Hessian matrix d 2ϕ(p) is non-degenerate, i.e. det d 2ϕ(p) = det ➊ ∂ 2ϕ ∂xi∂xj (p) ➍ 6= 0. (3) A smooth function is called a Morse function if all of its critical points are non-degenerate.3 To study the stationary phase expansion for more general phase functions, one first convert the general (non-degenerate) phase function to a quadratic one by using the Morse lemma: Theorem 2.2 (Morse Lemma, version 1). Let ϕ ∈ C ∞(R n ). Suppose p is a nondegenerate critical point of ϕ. Then there exists a neighborhood U of 0, a neighborhood V of p and a diffeomorphism ρ : U → V so that ρ(0) = p and (4) ρ ∗ϕ(x) = ϕ(p) + 1 2 (x 2 1 + · · · + x 2 r − x 2 r+1 − · · · − x 2 n ), where r is the number of positive eigenvalues of the Hessian matrix d 2ϕ(p). ¶ The stationary phase formula for general phase. As we have seen, only the critical points of the phase function ϕ give an essential contribution to the oscillatory integral I~ = ❩ Rn e i ϕ(x) ~ a(x)dx. In what follows we will assume that ϕ ∈ C ∞(R n ) admits only non-degenerate critical points in a neighborhood of the support of a. Since non-degenerate critical points must be discrete, ϕ has only finitely many critical points in the support of a. Thus one can find a partition of unity {Ui | 1 ≤ i ≤ N + 1} of the support of a so that each Ui , 1 ≤ i ≤ N, contains exactly one critical point pi of ϕ, and UN+1 contains 3Note that non-degenerate critical points must be discrete. In more subtle examples where the critical points are not necessary discrete, but still nice enough, say, form smooth manifolds, one has an analogous conception, namely the Morse-Bott function. Many results for Morse functions can be generalized to Morse-Bott functions