Normal form

4.3 Normal form

Similar as Witt [75], we have

Theorem 4.3.1. LetM and M⁰ are two E_X-modules associated with twom-th order micro-differential operatorsP andP⁰, respectively, and let ν₀ andν₀⁰ be radial points ofP andP⁰, respectively. IfM nearν₀ andM⁰ nearν₀⁰ are microlocally equivalent, via quantized contact transformation, then one has three invariants:

(i) γ_P^m−1(ν₀) = γ_P^m−10 (ν₀⁰),

(ii) σ^(m−1)_sub (P)|_ν0 =σ^(m−1)_sub (P⁰)|_ν⁰

0,

(iii) There is a linear symplectic mapW :Eb_νˆ0 →Ebνˆ⁰0 such that A^m−1_P0 (ν₀⁰) =W A^m−1_P (ν₀)W⁻¹.

If ν₀ and ν₀⁰ are generic, then these conditions are also sufficient, that is, the three invariants imply thatM andM⁰ are microlocally equivalent.

Proof. Suppose that two E_X-modules M near ν₀ and M⁰ near ν₀⁰ are microlocally equivalent. That is, P⁰ is equivalent toP via conjugating invertible microdifferential operator of order zero. Letp= σ^m(P)andp⁰ =σ^m(P⁰)be the principal symbol ofP andP⁰ respectively.

(i) Since H_p + γ_P^m−1R = 0, and by assumption we know the principal symbols are same near the radial points by conjugating a zeroth order elliptic micro-differential operatorJ, i.e.,

σ^m(P⁰)(ν₀⁰) =σ^m(J⁻¹P J)(ν₀)

=σ⁰(J⁻¹)(ν₀)σ^m(P)(ν₀)σ⁰(J)(ν₀)

=σ^m(P)(ν₀), then (i) is true.

(ii) If A ∈ E_X(n) is a n-th order micro-differential operator, notice that from Proposition 4.1.2:

σ_sub^m+n−1(P A) = σ^m(P)σⁿ⁻¹_sub (A) +σ^m−1_sub (P)σⁿ(A) + 1 2i

σ^m(P), σⁿ(A) ,

σ_sub^m+n−1(AP) = σⁿ(A)σ^m−1_sub (P) +σⁿ⁻¹_sub (A)σ^m(P) + 1 2i

σⁿ(A), σ^m(P) . We know the principal symbol vanishes at radial point, then at the radial point we

have

σ_sub^m+n−1(P A) = σ^m−1_sub (P)σⁿ(A)

=σⁿ(A)σ^m−1_sub (P)

=σ^m+n−1_sub (AP).

SupposeAis elliptic, then at the radial point we have σ_sub^m−1(P⁰)(ν₀⁰) = σ^m−1_sub (A⁻¹P A)(ν0)

=σ^m−1_sub (AA⁻¹P)(ν₀)

=σ^m−1_sub (P)(ν₀).

(iii) Assume we obtain two matrices ^γ₂ +Aand ^γ₂ +A⁰ by linearizing the symplectic vector fieldsH_p+γRnearν₀andH_p⁰+γRnearν₀⁰, respectively. Since the zeroth order microdifferential operator provided a morphism mapping p to p⁰, which induced a symplectic transformation mapsH_ptoH_p⁰. Then the associated contact transformation maps V_p toV_p⁰. Moreover, we have two contact transformations linearize Vp and Vp⁰ respectively. Hence there exist a linear symplectic map V such thatA⁰ =V AV⁻¹.

The rest of the proof is similar to the proof of the following corollary.

Corollary 4.3.2. Let P ∈ E(2) be a second order micro-differential operator on a real analytic manifold M. Let M be theE-module associated withP, i.e., M ' _EP^E . Furthermore if ν₀ = (x₀, ξ₀) is a generic radial point of P, then near (x₀, ξ₀), via quantized contact transformation, one can transformM into the system

N ' E EP_N, andP_N is the normal form ofP of form

P_N =hAD_x⁰, D_x⁰i+hBx⁰, D_x⁰iD_xn +hCx⁰, x⁰iD²_xn+γx_nD_x²_n +θD_xn. HereA,B,Care constant(n−1)×(n−1)matrices,AandCare symmetric,D_xi =

√1

−1

∂

∂xi,x= (x⁰, x_n),γ ∈Randθ ∈C.

Proof. With slight modification of Guillemin and Schaeffer’s proof, the essential idea works equally well here. Roughly, the steps are:

STEP I) One can conjugate P into a micro-differential operator having the same principal symbol of the normal formP_N.

By direct computation, the normal form P_N has a radial point at ν₀⁰ = (x⁰₀, ξ₀⁰) wherex⁰₀ = 0,ξ⁰₀ = (0,· · · ,0,1).

For technical reasons it is much easier to deal with first order operators, we consider a micro-differential operator P ∈ E_X(1), the normal form P_N can be looked as a first order operatorP⁰ by the action of a micro-differential operator(Dxn)⁻¹ ∈ E_Rⁿ(−1), it

4.3 Normal form is elliptic near the radial pointν₀⁰, then

P⁰ = (D_xn)⁻¹(P_N)∈ E_Rⁿ(1). (4.3) It is easy to seeν₀⁰ is also the radial point ofP⁰.

Letp and p⁰ be the principal symbol of P and P⁰, which by assumption are both homogeneous of degree one inξ. Then HpandHp⁰ are homogeneous of degree zero on T^∗X andT^∗Rⁿ, respectively. We can assign the Hamiltonian vector fields the related contact vector fieldsV_pandV_p⁰ on onS^∗X andS^∗Rⁿ, respectively.

radial point ν₀ ν₀⁰

operator P P⁰

l (a) l

principal symbol p p⁰

l (b) l Hamiltonian vector field H_p H_p⁰ l (c) l Contact vector field Vp Vp⁰

l (d) l

Linear part V_p⁰ V_p⁰0

In the above table, (a) and (b) are self-evident, (c) is canonical projection and (d) is the linearization of contact vector fields that we studied in chapter 3.

(S^∗X, ν0) −−−→^ϕ (S^∗X, ν0)

Choose suitable coefficients ofP_N such that there is a contact transformationφsuch thatφ∗ can conjugateV_p⁰ to V_p⁰0, and one knows that the mapsϕandϕ⁰, related to the linearization ϕ_∗ and ϕ⁰_∗ of the contact vector fields V_p and V_p⁰, respectively, are also contact transformations. A combination of mappingsψ , ϕ◦φ◦ϕ⁰⁻¹ shows us, near generical radial points, there is a contact transformationsψ fromS^∗X to S^∗Rⁿ. The related symplectic transformationψ˜which mapsT^∗X\0toT^∗Rⁿ\0, i.e.,

ψ(x, ξ) = (y, η)↔ψ(x, tξ) = (y, tη),˜ ∀t∈∈R\0. (4.4) With the introduction of contact transformation and quantized contact transformation in (A.3), one knows the contact transformationψ : S^∗X → S^∗Rⁿ can be expressed by composition of two contact transformations with generating functions.

To coincide with former introduction, we still use ψ to denote the associated contact transformation fromP^∗X toP^∗Rⁿ.

Decomposeψ = π₂◦π⁻¹₁ with a generating functions Γ(x, r). Here we only need the existence of Γ and we do not care what Γ is. Let Λ ⊂ X ×Rⁿ be non-singular

hypersurface defined byΓ(x, r) = 0, that is,

∇_(x,r)Γ(x, r)6= 0 holds onΛ. Moreover, onΛ, one has

det

0 d_yΓ dxΓ dxdyΓ

6= 0.

And then

π₁ :P_Λ^∗(X×Rⁿ)→P^∗X and

π₂ :P_Λ^∗(X×Rⁿ)→P^∗Rⁿ.

Then we can lifting the contact transformationψ :P^∗X →P^∗Rⁿto the sheaves of germs of microdifferential operatorsE_X → E_Rⁿ, via quantized contact transformation.

From theorem A.3.7, one can find an isomorphism between the two sheaves of germs of micro-differential operatorsπ₂ ◦π₁⁻¹E_X andE_Rⁿ, as well asπ₂◦π₁⁻¹E_X(m) and E_Rⁿ(m), for any integerm. Moreover the isomorphism is unique up to an inner automorphism by an invertible microdifferential operator of order zero.

At least we can claim that we can find an invertible microdifferential operator of order zero, denote byF, such that

P⁰ =F⁻¹P F +W,

ord(W)≤1, σ¹(W) = 0 near ν₀. (4.5) STEP II) Deal with the lower order term.

Redefine the operatorP byF⁻¹P F, so far we have σ¹(P) = p=p⁰ =σ¹(P⁰)

We want to find an invertible micro-differential operatorQof order zero, such that

Q⁻¹P⁰Q'P (4.6)

That is the operatorQ⁻¹P⁰QandP are microlocally equivalent in the sense of Q⁻¹P⁰Q−P ∈ ∩m∈ZE_X(m).

Now we are going to figure outQ. Consider the principal symbol of the zero order operatorQ⁻¹P⁰Q−P nearν₀. We have

σ⁰(Q⁻¹P⁰Q−P) = σ⁰ Q⁻¹([P⁰, Q] +P⁰−P)

=σ⁰(Q⁻¹)σ⁰([P⁰, Q]) +σ⁰(P⁰)−σ⁰(P)

= {p, q₀}

q₀ +θ−s.

(4.7)

4.3 Normal form Here s is the zeroth order component of the total symbol of P, and q0 the principal symbolQ,θis the arbitrary constant in the first order term ofP_N.

Writeq₀(x, ξ) =exp a₀(x, ξ)

, and both ofq₀(x, ξ)anda₀(x, ξ)are homogeneous of order zero inξ. Let the above equation be zero then

Hpa0 =s−θ,b0 (4.8)

Suppose (4.8) can be solved, then chooseQ₀ to be any micro-differential operator of order zero with principal symbol q₀, Q⁻¹₀ P⁰Q₀ − P will be of order −1 in a neighborhood ofν₀.

Now again setP⁰ to beQ⁻¹₀ P⁰Q₀, thenP⁰−P ∈ E_Rⁿ(−1). Consider a zero order operatorQ₁ = I+A₁ withA₁ ∈ E_X(−1), then the principal symbol of the operator Q⁻¹₁ P⁰Q1−P is

σ⁻¹(Q⁻¹₁ P⁰Q₁−P) =σ⁻¹ (X

k∈N

(−A₁)^k)P⁰(I +A₁)−P

=σ⁻¹([P⁰, A₁]) +σ⁻¹(P⁰−P)

={p, a₁} −b₁,

(4.9)

wherea₁is the principal symbol ofA₁andb₁is the principal symbol ofP−P⁰. Suppose {p, a₁} −b₁ vanishes, thena₁satisfies the equation

H_pa₁ =b₁,

repeat the above steps, after conjugating by a finite sequence of Q⁰_is, the principal symbol ofQ⁻¹_k P⁰Qk−P is

{p, a_k} −b_k, (4.10) wherea_k is the principal symbol of A_k = Q_k −I ∈ E_X(−k)and b_k is the principal symbol ofP⁰−P nearν₀.

If the above equation vanishes, then

H_pa_k =b_k.

If it is satisfied, then we can constructQby combining allQ_i’s,i= 0,1,· · ·. Consider the above all those unsolved equations

Hpak =bk, k= 0,1,· · · .

If those above equations are solvable, then we can finish the proof.

Now we try to solve

(H_pa₀ =s−θ H_pa_k =b_k

Since we have a diffeomorphism

S^∗X×R+∼=T^∗X\0,

we can treat a contact vector fieldV_pas a vector field onT^∗X\0, V_p =H_p+f R,

wheref is a homogeneous function of order zero onT˙^∗X, andR is the radial vector field. At radial point,f(0) =λ. Letckbe the pull-back ofak via

S^∗X →T^∗X\0,

Sincea_k is homogenous of degree−k,Ra_k =−ka_k, then it is equivalent to solve the following equations.

(V_pc₀ =b₀ Vpck+kf ck =bk

(4.11) wheref(0) =−γ.

ReplayingS^∗X byR²ⁿ⁻¹ with the contact structure dz+ 1

2

n−1

X

i=1

(y_idx_i−x_idy_i),

by making a contact transformation and replayV_p by some linear contract vector field V_P⁰ onR²ⁿ⁻¹ of the form in section 3.3. Due to Guillemin and Schaeffer’s discussion in [22], we can solve it formally.

Roughly, let Sⁱ be the set of all polynomials in x, y and z which are linear combinations of polynomials of the form z^lh(x, y), where h is homogeneous in x, y of ordermand2n+m=i. So the vector fieldV induces a linear mapVⁱ :Sⁱ →Sⁱ. Lemma 4.3.3. To be able to solve the above equations formally (in power series about the origin), it is necessary and sufficient that−kunot be an eigenvalue ofVⁱ fori >0 and fork >0.

The lemma shows that (4.11) can be solved up to error terms whichvanish to order infinite² at the origin ofRⁿ.

However, we do not need the argument as the smooth case, which need to discuss the error terms vanishing to infinite order at the origin. In fact the real analytic setting ensure that once we can solve the equations (4.11) formally, the problems have been solved.

We could have simpler normal form with additional conditions.

2A functionu(x)vanishes to infinite order atx₀if for every positive integerN,lim_x→x0|x−x^u(x)

0|^N = 0 is true in a neighborhood ofx0.

4.3 Normal form Corollary 4.3.4. Let X be a n-dimensional real analytical manifold and M be a E_X-module associated with a first order micro-differential operator P ∈ E_X(1).

Let ν₀ = (x₀, ξ₀) be a generical radial point. If ν₀ is hyperbolic, then near ν₀ M is microlocally equivalent to the E_Rⁿ-module N ∼= _EP^E

N h via quantized contact transformation, where

P_{N h} =hBx˜ ⁰, D_x⁰i+γx_nD_xn+θ. (4.12) HereB˜is a constant(n−1)×(n−1)matrix.

Corollary 4.3.5. Let X be a n-dimensional real analytical manifold and M be a E_X-module associated with a second order micro-differential operatorP ∈ E_X(2). Let ν₀ = (x₀, ξ₀)be a generical radial point. Ifν₀is elliptic, then nearν₀M is microlocally equivalent to theE_Rⁿ-moduleN ∼= _EP^E

N e via quantized contact transformation, where P_{N e} =hAD˜ _x⁰, D_x⁰i+hCx˜ ⁰, x⁰iD_x²_n +γx_nD²_xn+θD_xn. (4.13) HereA˜is a(n−1)×(n−1)identity matrix,C˜ = (˜c_i)is a(n−1)×(n−1)diagonal matrix with all non-vanishing entries larger thanγ²/16.

Now we are going to prove the above corollaries. First we have:

Lemma 4.3.6. Let P_N be the normal form of the operator P with radial point ν₀ = (0; 0⁰,1), and p is the principal symbol of P_N. Then the linear part V_p⁰ of related with respect to the symplectic basis

∂

From the discussion in section 4.2, the spacedp = 0divided by the space spanned by H_p (i.e. Eb_ˆν0) can be identified with the space spanned by the _∂ξ^∂

Hence we obtain a homogeneous symplectic vector field Hp +γR. Denote byVp the

To prove the Corollary 4.3.5, if P has the normal formP_{N e}, according to Lemma 4.3.6, the matrixAis given by

_γ

The eigenvalues ofAis purely imaginary if and only if the determinant of the matrix

−γ² 4 + 4˜c_i

is greater than zero, orc˜_i > ^γ₁₆². Conversely, every elliptic case can be obtained by this way.

To prove the Corollary 4.3.4, ifP is microlocally equivalent to the normal form P_{N h} = X

near the radial point. According to Lemma 4.3.6, the matrixAis given by −b_ij +^γ₂ 0

0 b_ij − ^γ₂

Conversely, every hyperbolic case can be obtained by this way.

Go back to our normal formP_N in Corollary 4.3.2:

Remark 4.3.7. Notice that P_N is invariant invariant up to some constant κ under the coordinates transformation:

(x⁰, x_n)7→(κx⁰, κ²x_n), which will help to construct special form of solutions.

Im Dokument Analytic singularities near radial points (Seite 59-67)