Birkhoff normal form

that is the matrix ^λI₂ +Ahas no imaginary eigenvalues.

Applying the above conditions we can define a generical radial point⁶.

3.4 Birkhoff normal form

Next chapter we will give the normal form of our operators in section 1.3, it is worth to mention some knowledge of Birkhoff normal form, the reference is [1].

Write the hamiltonian asH = ¹₂(Ax, x), where x = (p₁,· · · , p_n;q₁,· · · , q_n) is a vector written in a symplectic basis andAis a symmetric linear operator. The canonical equations have the form

By eigenvalues of the hamiltonian we mean the eigenvalues of the linear infinitesimally symplectic operatorIA, and by a Jordan block we mean a Jordan block of the operatorIA.

The eigenvalues of the hamiltonian are of four types: real pairs (a,−a), purely imaginary pairs (√

−1b,−√

−1b), quadruples (±a ± √

−1b) and zero eigenvalues.

The Jordan blocks corresponding to the two members of a pair or four members of a quadruple always have the same structure.

The complete list of normal forms follows:

(1) For a pair of Jordan blocks of orderkwith eigenvalues±a, then hamiltonian is H =−a

(2) For a quadruple of Jordan blocks of orderk with eigenvalues±a±√

−1b the

(3) For a pair of Jordan blocks of orderkwith eigenvalues zero the hamiltonian is H =

(4) For a Jordan blocks of order2k with eigenvalues zero the hamiltonian is of one of the following two inequivalent types:

H =±1

(5) For a pair of Jordan blocks of order 2k + 1 with purely imaginary eigenvalues

±√

−1bthe hamiltonian is of one of the following two inequivalent types:

H =± 1

(6) For a pair of Jordan blocks of order2kwith eigenvalues±√

−1bthe hamiltonian is of one of the following two inequivalent types:

H =± 1

Theorem 3.4.1. (Williamson’s theorem) A real symplectic vector space with a given quadratic form H can be decomposed into a direct sum of pairwise skew orthogonal real symplectic subspaces so that the form H is represented as a sum of forms of the types indicated above on these subspaces.

For our case, we only need to consider all the Jordan blocks are of first order, while the individual hamiltonian in “general position” does not have multiple eigenvalues and reduces to a simple form. Moreover, in our case, we consider the principal symbol as the Hamiltonian of quadratic form in the variable(x⁰, ξ⁰).

Chapter 4

The normal form

In this chapter we are going to simplify the microdifferential operatorP near a generic radial point. Basically first we introduce the subprincipal symbol in section 4.1, after that we study analytic properties of radial points and clarify the generic conditions in section 4.2, then obtain the normal form of the original operatorP in section 4.3. We discuss the projected null bicharacteristics at the end this chapter.

4.1 Subprincipal symbol

LetXbe anndimensional real analytic manifold, letP be a microdifferential operator of orderm defined onS^∗X, and Take a local coordinate systemx = (x₁, x₂,· · · , x_n) ofX. By considering the adjoint operatorP^∗ofP ={P_k(x, ξ)}_k, we have

(P^∗)_m(x, ξ) = P_m(x,−ξ) = (−1)^mP_m(x, ξ), (P^∗)m−1(x, ξ) = Pm−1(x,−ξ)−X

j

∂²

∂x_j∂ξ_jPm(x,−ξ)

= (−1)^m

Pm−1(x, ξ)−X

j

∂²

∂x_j∂ξ_jP_m(x, ξ) . HenceP −(−1)^mP^∗ ∈ E_X(m−1).

Definition 4.1.1. We set

σ_sub^m−1(P) := 1

2σm−1(P −(−1)^mP^∗)

=Pm−1−1 2

X

j

∂²P_m

∂x_j∂ξ_j

(4.1)

and call it thesubprincipal symbol¹ofP.

1Notice that if we write a pseudo-differential operatorP(x, D) = P₀(x) +Pm

i=1P_i(x, D)onRⁿ withD= ^√1_−1∂x^∂ , the formula of its subprincipal symbol is likeσ_sub^m−1(P) :=Pm−1−₂^√1₋₁P

j

∂²Pm

∂xj∂ξj.

The mapσ_sub^m−1 : EX(m) → OT^∗X extends the principal symbol σm−1 : EX(m− 1) → OT^∗X. However, unlike the principal symbol, it depends on the choices of coordinate systems. SupposeP˜ = ( ˜P_k(˜x,ξ))˜ be the associated operator ofP in another coordinated system x˜ = (˜x1,· · · ,x˜n). One has d˜xP˜^∗dx˜⁻¹ = dxP^∗dx⁻¹, and let

˜

σm−1( ˜P)be the subprincipal symbol inx, one has˜

˜

σ_sub^m−1( ˜P) =σ_sub^m−1(P)−1 2

n

σ_m(P),logd(˜x) dx

o .

Then one can define a first order differential operatorL_P^(m−1) acting onOT^∗X ⊗Ω^−1/2_X by

L_P^(m−1)(a/√

dx) := H_σm(P)a+σ^m−1_sub (P)a /√

dx,

where Ω^−1/2_X = (Ω^1/2_X )^⊗−1, and Ω^1/2_X is an invertible sheaf such that(Ω^1/2_X )^⊗2 = Ω_X, and let √

dxdenote a section ofΩ^1/2_X such that(√

dx)^⊗2 =dx, heredx =dx₁∧ · · · ∧ dx_n ∈ Ω_X. By definition, (dx)¹²LP^(m−1)(dx)¹² is independent of the choice of local coordinated system.

Proposition 4.1.2. ForP ∈ E_X(m)andQ∈ E_X(n), one has (1) σ^m+n−1_sub (P Q) = σ_m(P)σ_subⁿ⁻¹(Q) +σ^m−1_sub (P)σ_n(Q) + ¹₂

σ_m(P), σ_n(Q) , (2) σ^m+n−2_sub ([P, Q]) = {σ_m(P), σⁿ⁻¹_sub (Q)}+{σ_sub^m−1(P), σ_n(Q)},

(3) LP Q^m+n−1 =P_mL⁽ⁿ⁻¹⁾_Q +Q_nLP^(m−1)+ ¹₂[P_m, Q_n], (4) L_[P,Q]^m+n−2 = [LP^(m−1),LQ⁽ⁿ⁻¹⁾].

4.2 Classification of radial points

LetXbe ann-dimensional manifold and let(x, ξ) = (x₁,· · · , x_n, ξ₁,· · · , ξ_n)be local coordinates ofT^∗X. LetP be anm-th order microdifferential operator defined onT^∗X with principal symbolp_m(x, ξ), thecharacteristic varietyChar(P)of the operatorP is defined by

Char(P) :={(x, ξ)∈T^∗X\0|p_m(x, ξ) = 0}, which is a closed subset ofT^∗X.

Definition 4.2.1. A pointν0 = (x0, ξ0)inChar(P)is said to be a radial point of the operator P if the Hamiltonian vector field H_pm associated with the principal symbol p_m of P is a (necessarily nonzero) multiple of the radial vector field R = ξ_∂ξ^∂, i.e., H_pm +γR = 0holds atν₀ for some γ ∈ R\0. Conversely, ifH_pm andR are linear independent atν⁰, then we say the operatorP isof principal typeatν⁰.

Remark 4.2.2. There is an equivalent definition of radial point that dp_m and the canonical 1-form α = ξdx are collinear at (x0, ξ0). A point (x0, ξ0) is radial if and

4.2 Classification of radial points only if it solves the system

( _∂p

m

∂_ξi = 0, i= 1,· · · , n, ξ_n^∂p_∂x^m

i =ξ_i^∂p_∂x^m

n, i= 1,· · · , n−1. (4.2) If (x₀, ξ₀) is a radial point, then all points(tx₀, tξ₀) (t 6= 0) in the direction are radial, eventually we can consider problem on cotangent sphere bundle S^∗X. Here we prefer to considerS^∗X instead ofP^∗X, because it is more convenient to study the singular spectrum.

Example 4.2.3. Consider a simple example of Euler operator P =xD_x−θ, x∈R

where_√ θ ∈ C. It has two radial points (0,±1)onS^∗R, and its subprincipal symbol is

−1 2 −θ.

A radial pointν₀ is isolated in microlocal sense if there is no other ray nearR+ν₀ consisting of radial points.

We will see later in next section that linearizing the contact vector fieldH_pm+γR gives a matrix ^γ₂I +A, where A is a symplectic matrix. According to section 3.3, suppose the eigenvalues of the symplectic mapping A at the radial point(x₀, ξ₀) are distinct, and we can write them as

λ1,−λ1, λ2,−λ2,· · · , λn,−λn.

Definition 4.2.4. (Non-resonant condition) A radial point (x₀, ξ₀) is generic if the equation

mγ 2 =

n

X

i=1

miReλi

has no integer solution(m, m1,· · · , mn)∈Zⁿ⁺¹ withm 6= 0.

Definition 4.2.5. A generic radial point ν₀ is called to be elliptic (hyperbolic, loxodromic or of mixed type, respectively) if the associated symplectic matrix A is elliptic (hyperbolic, loxodromic or of mixed type, respectively).

Near a generic radial point, the microlocal equivalence of operators is classified by three invariants [75]:

(i) The factorγ ∈ R\0, withH_pm +γR = 0at the radial pointν₀. One can seeγ is homogeneous inξ₀ of orderm−1, i.e.,

γ_P^m−1(tν₀) = t^m−1γ_P^m−1(ν₀), here we writeγasγ_P^m−1(ν0).

(ii) The subprincipal symbol ofP at(x0, ξ0)

The subprincipal symbol is an invariant in the sense as following [75]:

Lemma 4.2.6. LetP be an m-th order pseudo-differential operator andν₀be a radial point, and letU be an open neighborhood ofν₀inT^∗X\0. Let J be an elliptic Fourier integral operator with associated canonical transformationJ :U →J(U). Then

σ_sub^m−1(J P J⁻¹) J(ν₀)

=σ^m−1_sub (P)(ν₀)), andJ⁻¹is the parametrix ofJ.

(iii) Conformal symplectic map ^γ₂I+A.

From the definition, at radial point(x0, ξ0)we have (H_p+γR)|_(x0,ξ0)= 0,

whereH_p is symplectic vector field andγR is conformal symplectic vector field with constant conformal factor u. ExtendHp +γR to a vector fieldHpm +ϕ(x, ξ)R with (x₀, ξ₀)as its zero. Hereϕ : T^∗X \0 → Ris arbitrary real function homogeneous of degreem−1with value γ at (x₀, ξ₀). Let V_P⁰ be the linear part of the vector field at (x0, ξ0), then it defines a conformal symplectic linear map

V_P⁰ :T_(x0,ξ0)(T^∗X\0)→T_(x0,ξ0)(T^∗X\0).

where A is symplectic. Also we have tensor contraction hR, dp_mi = kdp_m and hHpm, dpmi= 0, which imply that the transpose ofV_P⁰

(V_p⁰)^t :T_(x^∗

0,ξ0)(T^∗X\0)→T_(x^∗

0,ξ0)(T^∗X\0)

maps dp_m onto a multiple of itself, and alternatively, V_P⁰ maps the codimension one subspace ofT_ν0(T^∗X\0)defined bydp_m(ν₀) = 0into itself.

Moreover, it introduces two invariant spaces, one is spanned by Hpm(x0, ξ0) (or R|_(x0,ξ0)), another is the kernel ofdp_m(x, ξ)(or α) quotient the subspace spanned by

4.3 Normal form

Im Dokument Analytic singularities near radial points (Seite 53-59)