6LECTURE29-30FIO-SEMICLASSICALFIOSTOscillatory half densities.Let d = dimZ-dimX be the fiber dimension.For anyk Z, we defineI(X, A), the space of compactly supported oscillatory half densities on X associatedwithA,tobe(X,A)= [μ=k-2*(a(z,h)eiT) / a E C(Z × R),(12)where T is a nowhere vanishing half-density on Z. (Obviously the space is indepen-dent of the choice of .)SimilarlywedefineIk(X,A),the space of oscillatoryhalfdensities on X associated with A, to be the set consists of those half densities μ sothat pμ E I(X,A) for all p E Co(X).Locally we may assume Z = X x S, where S is an open set in Rd. We maychoose ourfiber half-densitytobethe Euclidean one dsand choosetobeTodswith To a nowhere vanishing half-density on X. Then μ I(X, A) is of the form[a(r, s,h)etp(a,)dshk-2 Independence of generating function.We must show that the above definition is also independent of the choices ofgeneratingfunctions.Letπ:Z,→X,i=l,2betwo fibrations,and ibeagenerating function of A with respect to Ti.It is enough to do this locally.Recall that the two generating functions Pi andP2 are related by(a)Replaceby+c.(b) For a diffeomorphism g : Z -→ Z, replace by g* and by g*p.(c) Replace Z by Z = Z × Rd and by (z) +(Az,z), where A is a non-degenerate d x d matrix.We have already get rid of type (a)by requiring Ato satisfy the Bohr-Sommerfeldcondition (10) and fixing the constant in the generating function via the normal-ization condition (11). If two densities are related by a type (b) change, then by achange of variableargument it is not hard to prove(2)+(ae2g1)=(1)+(gae表P1T1)so the spaces defined via i and via P2 are the same.Now suppose 1 and p2 are related by a type (c) change. Without loss ofgenerality,we may assume Z2=Zi × S, where S is an open subset of Rm, andLLSAS,(P2(z, s) =(P1(z) +2where A is a symmetric non-degenerate m × m matrix. Let d be the fiber dimensionof Zi→X,then the fiber dimension of Z2 X is d +m.Let Ti be a nowhere
6 LECTURE 29-30 FIO – SEMICLASSICAL FIOS ¶ Oscillatory half densities. Let d = dim Z − dim X be the fiber dimension. For any k ∈ Z, we define I k 0 (X,Λ), the space of compactly supported oscillatory half densities on X associated with Λ, to be (12) I k 0 (X,Λ) = {µ = ~ k− d 2 π∗(a(z, ~)e i ϕ(z) ~ τ ) | a ∈ C ∞ 0 (Z × R)}, where τ is a nowhere vanishing half-density on Z. (Obviously the space is independent of the choice of τ .) Similarly we define I k (X,Λ), the space of oscillatory half densities on X associated with Λ, to be the set consists of those half densities µ so that ρµ ∈ I k 0 (X,Λ) for all ρ ∈ C ∞ 0 (X). Locally we may assume Z = X × S, where S is an open set in R d . We may choose our fiber half-density to be the Euclidean one ds 1 2 and choose τ to be τ0⊗ds 1 2 with τ0 a nowhere vanishing half-density on X. Then µ ∈ I k 0 (X,Λ) is of the form ~ k− d 2 Z S a(x, s, ~)e i ~ ϕ(x,s) ds τ0. ¶ Independence of generating function. We must show that the above definition is also independent of the choices of generating functions. Let π : Zi → X, i = 1, 2 be two fibrations, and ϕi be a generating function of Λ with respect to πi . It is enough to do this locally. Recall that the two generating functions ϕ1 and ϕ2 are related by (a) Replace ϕ by ϕ + c. (b) For a diffeomorphism g : Z → Z˜, replace π by g ∗π and ϕ by g ∗ϕ. (c) Replace Z by Z = Z × R d and ϕ by ϕ(z) + 1 2 hAz, zi, where A is a nondegenerate d × d matrix. We have already get rid of type (a) by requiring Λ to satisfy the Bohr-Sommerfeld condition (10) and fixing the constant in the generating function via the normalization condition (11). If two densities are related by a type (b) change, then by a change of variable argument it is not hard to prove (π2)∗(ae i ~ ϕ2 g∗τ1) = (π1)∗(g ∗ ae i ~ ϕ1 τ1) so the spaces defined via ϕ1 and via ϕ2 are the same. Now suppose ϕ1 and ϕ2 are related by a type (c) change. Without loss of generality, we may assume Z2 = Z1 × S, where S is an open subset of R m, and ϕ2(z, s) = ϕ1(z) + 1 2 s TAs, where A is a symmetric non-degenerate m×m matrix. Let d be the fiber dimension of Z1 → X, then the fiber dimension of Z2 → X is d + m. Let τ1 be a nowhere
LECTURE29-30FIO-SEMICLASSICALFIOS7vanishing half density on Zi, then Ti ds is a nowhere vanishing half density onZ2. Using the generating function p2 we get the expressionshk-(2)a2(z, s, h)e2(2,s)T1 @ ds2.Let T2.1 : Z2 → Zi be the projection on to the first factor so that (π2)= (πi) o(π2,1)*. Then by definition, (π2,1) acts as/a2(z,s,h)esTAsds)DekpiTi(2,1)(a2(z, s,)ep2(2,8)1 d) =Now the conclusion follows from the lemma of stationary phase (with quadraticphase).In conclusion, we provedTheorem 2.2. The space I(X,A) (and thus I*(X,A) is intrinsically defined (pro-vided A is eract and we fir a choice of px on A).3.SEMICLASSICALFOURIERINTEGRALOPERATORSTThe definition.Now suppose Xi,X, are smanifolds. We will denote M, = T*Xi, i = 1,2.Suppose F C Mi × M2 is an eract canonical relation. ThenA=0201is an exact Lagrangian submanifold of T*X, whereX=X×X2.Associated with Awe have the space of compactly supported oscillatory half densities I(X,A).If wefix a nowhere vanishing one density dci on Xi and a nowhere vanishing one densitydr2 on X2, then a typical element in I(X, A) is of the formμ=hk-号[a(r1, 2, s, h)et(1,m2,)ds) dredrWith someabuse of notion welet L?(X)be theHilbert spaceof L?half densitieson X,. Then associated to each μ = u(r1, 2,h)de dr e Ib(X,A) we can define anintegral operator Fμ : L?(Xi) -→ L?(X2) viaFu(fdat) = (/ f(ai)u(ri,r2, h)dai) das(13)Definition 3.1. Such operators are called compactly supported semi-classical Fouri-er integral operators of order m = k + , where n2 = dim X2. The space of theseoperators is denoted by F"(T).Remark. We could loose the conditions on u by requiring only u(ri, 2,h)dr eL?(Xi), or more generally, with distributional coefficients. In this case we drop thesubscript 0
LECTURE 29-30 FIO – SEMICLASSICAL FIOS 7 vanishing half density on Z1, then τ1 ⊗ ds 1 2 is a nowhere vanishing half density on Z2. Using the generating function ϕ2 we get the expressions ~ k− d+m 2 (π2)∗a2(z, s, ~)e i ~ ϕ2(z,s) τ1 ⊗ ds 1 2 . Let π2,1 : Z2 → Z1 be the projection on to the first factor so that (π2)∗ = (π1)∗ ◦ (π2,1)∗. Then by definition, (π2,1)∗ acts as (π2,1)∗(a2(z, s, ~)e i ~ ϕ2(z,s) τ1 ⊗ ds 1 2 ) = Z a2(z, s, ~)e i 2~ s T Asds e i ~ ϕ1 τ1. Now the conclusion follows from the lemma of stationary phase (with quadratic phase). In conclusion, we proved Theorem 2.2. The space I k 0 (X,Λ) (and thus I k (X,Λ)) is intrinsically defined (provided Λ is exact and we fix a choice of ϕΛ on Λ). 3. Semiclassical Fourier integral operators ¶ The definition. Now suppose X1, X2 are smanifolds. We will denote Mi = T ∗Xi , i = 1, 2. Suppose Γ ⊂ M1 × M− 2 is an exact canonical relation. Then Λ = σ2 ◦ Γ is an exact Lagrangian submanifold of T ∗X, where X = X1×X2. Associated with Λ we have the space of compactly supported oscillatory half densities I k 0 (X,Λ). If we fix a nowhere vanishing one density dx1 on X1 and a nowhere vanishing one density dx2 on X2, then a typical element in I k 0 (X,Λ) is of the form µ = ~ k− d 2 Z S a(x1, x2, s, ~)e i ~ ϕ(x1,x2,s) ds dx 1 2 1 dx 1 2 2 With some abuse of notion we let L 2 (Xi) be the Hilbert space of L 2 half densities on Xi . Then associated to each µ = u(x1, x2, ~)dx 1 2 1 dx 1 2 2 ∈ I k 0 (X,Λ) we can define an integral operator Fµ : L 2 (X1) → L 2 (X2) via (13) Fµ(f dx 1 2 1 ) = Z f(x1)u(x1, x2, ~)dx1 dx 1 2 2 Definition 3.1. Such operators are called compactly supported semi-classical Fourier integral operators of order m = k + n2 2 , where n2 = dim X2. The space of these operators is denoted by F m 0 (Γ). Remark. We could loose the conditions on u by requiring only u(x1, x2, ~)dx 1 2 1 ∈ L 2 (X1), or more generally, with distributional coefficients. In this case we drop the subscript 0