Dipartimento di Matematica
Corso di Laurea Magistrale in Matematica
Tesi di Laurea Magistrale
A Semiclassical proof of the Weyl law
Candidato: Relatore:
Viviana Grasselli
Prof. Nicola Visciglia
Controrelatore:
Prof. Giovanni Alberti
Contents
Introduction iii
1 Notations 1
1.1 General setting . . . 1
1.2 Fourier transform on the Schwartz space . . . 3
1.3 Hilbert-Schmidt operators . . . 4
1.4 Trace class operators . . . 5
2 The Weyl Law 9 2.1 Statement of the result . . . 9
2.2 Roadmap of the proof . . . 9
2.2.1 Rewriting N as trace of an operator . . . 10
2.2.2 Proof of Weyl law by approximation with smooth functions . 10 2.2.3 Performing functional calculus on −h2∆g . . . 12
2.2.4 Finding an approximate inverse of −h2∆g− z . . . 12
2.2.5 Computing the trace . . . 13
3 Semiclassical operators 15 3.1 Space of symbols . . . 15
3.2 Definition . . . 18
3.3 Properties of symbolic calculus . . . 19
4 Helffer-Sj¨ostrand functional calculus 23 4.1 Almost analytic extension . . . 23
4.2 Integrals of almost analytic extension . . . 24
4.3 Helffer-Sj¨ostrand formula . . . 29
5 Parametrix 32 5.1 Parametrix of an elliptic operator on Rn . . . 32
5.2 Parametrix of a local elliptic operator . . . 38
5.3 Approximating the resolvent of −h2∆g . . . 43
5.4 Applying Helffer-Sj¨ostrand formula . . . 47
CONTENTS ii
6 Computing the trace 49
6.1 Trace of semiclassical operators . . . 49 6.2 Computing the trace of FN(f, h) . . . 53
Introduction
In 1911 for the first time a result from Hermann Weyl on the asymptotic behaviour of eigenvalues of the Laplacian was presented ([10]). Weyl then proceeded to publish several other papers on the subject, giving the start to a thread of many more related works aimed at improving the estimates on remainder terms and generalizing the domains in which the problem was set.
Although Weyl’s initial interest in the topic sparked from some conjectures stated by Sommerfeld and Lorentz in 1910, other works concerning the asymptotics of eigenvalues date much earlier and are related to acoustics.
Already in 1877, Lord Rayleigh made computations to count the number of normal tones of a cubic room in a given range of frequency. In order to do so, he reduced himself to a three dimensional lattice-point problem, finding that the number of overtones in a high enough range of frequencies, [ν, ν + dν], grows as V· ν3, where V is the volume of the cube.
Years later, the question of acoustics of rooms was again taken into consideration by Sommerfeld when justifying the importance of a rigorous and general study of the asymptotic behaviour of the Laplacian eigenvalues.
At the same time, the relevance of such a problem was also stated by Lorentz, albeit for different reasons. As noticed in 1900 by Lord Rayleigh, his results on the overtones of a room can be applied to a phisically different problem: that of the heat radiating from a black body. It is then not surprising that in 1910, while discussing the problem of heat radiation in a series of lectures in G¨ottingen, Lorentz brought attention to the problem of proving that the number of high overtones (for ν → ∞) is independent of the shape of the body and only depends on the volume, in accordance to what was previously proved by Lord Rayleigh.
It is exactly this remark from Lorentz that brought Weyl to focus his interests on such a topic.
Analytic expressions of the eigenvalues are known only in the few cases of very simple shapes of the domain, hence it is natural to study only their asymptotic distribution.
INTRODUCTION iv In his first work on the subject, Weyl considered the operator −∆ on a bounded domain Ω ⊂ R2 with eigenvalues (λj)j.
He used Fredholm-Hilbert theory on integral equations to show that lim j→∞ j λj = |Ω| 4π; which translates to N (λ) = |Ω| 4πλ + o(λ) as λ → ∞,
where N (λ) is the counting function defined by N (λ) := #{λj|λj ≤ λ}.
In following papers, Weyl also conjectured the existence of a second asymptotic term, depending on the measure of ∂Ω, both for Dirichlet and Neumann conditions. The statement was proved only many years later and under assumptions for the domain Ω ([6], [7]).
Weyl’s asymptotic studies are closely related to acoustics, as we said at the be-ginning and in particular they are of interest in a very famous issue.
The Laplacian with Dirichlet condition models the vibration of a membrane, where the pure tones produced by it are linked to the eigenvalues. From this point of view, interpreting Weyl law as an inverse problem, we can see how it is linked to the famous question “Can one hear the shape of a drum?”. In its full generality, the answer to this query is still unknown, but from Weyl law we have a relation between the frequencies and the size of the membrane, meaning one can at least “hear the area” of a drum.
As the eigenvalues of the Laplacian appear in many different models of physics phenomena, Weyl law proved to be a very versatile result, finding applications in several fields. For example in models describing oscillatory phenomena or the black body radiation problem, as mentioned before, but also in the fluctuation of the gravitational field or in the study of the Scr¨odinger equation describing the wave function of a quantum system.
The thesis is divided into six chapters.
In Chapter 1 we give preliminary notions to fix the notations of the objects that will be used throughout the work.
In Chapter 2 we state the Weyl law and give a sketch of the main steps of the proof presented in [2]. In particular, it is underlined how the proof of the Weyl law reduces to demonstrating a result on smooth functions of the Laplacian, that will then become the main goal of the thesis.
In Chapter 3 we give the definition of semiclassical operators and the main properties that will be used in the sequel.
their integration that are then used to prove Helffer-Sj¨ostrand formula.
In Chapter 5 we construct a paramterix first for an elliptic operator on Rn and then for a local one, finally we apply this result to the Laplace-Beltrami operator. In Chapter 6 we compute the traces that eventually allow us to complete the proof of the result needed to demonstrate the Weyl law.
Chapter 1
Notations
1.1
General setting
The general setting we will consider is that of a compact Riemannian manifold M of dimension n, that is a smooth manifold with a volume density dvolg and the
Laplace-Beltrami operator ∆g.
Since we are assuming our manifold to be compact we can consider a finite collection of charts (Ui, Vi, ki)i∈F with associated partition of unity (θi)i such that supp θi ⊂ Ui
and P
iθi = 1
To fix the notations we state now all the properties satisfied by the objects we defined.
• Pullback and pushforward Consider the coordinate chart (Ui, Vi, ki) with open
sets Ui ⊂ M and Vi ⊂ Rn. We will use the pushforward and pullback maps
denoted respectively by
ki∗: C∞(Ui) → C∞(Vi) ki∗: C ∞
(Vi) → C∞(Ui)
u 7→ u ◦ k−1i v 7→ v ◦ ki.
• Volume density For any chart (U, V, k) there exists a matrix (gjk(x))j,k with
smooth coefficients such that, setting
|g(x)| := det(gjk(x))−
1 2,
we can write the integral of a function φ ∈ C0(U ) as an integral on Rn by
pushing forward the function φ Z
φ dvolg =
Z
k(U )
(φ ◦ k−1)(x)|g(x)|dx. (1.1) In short, we will write k∗dvolg= |g(x)|dx.
Having defined the integral of a function on the manifold we can consider the space Lp(M ) = Lp(M, dvolg) as the closure of C∞(M ) with respect to the norm
R |φ|pdvol g
1p .
In the case p = 2 it is an Hilbert space with the usual scalar product (φ, ψ)L2(M ) =
Z
M
φψdvolg
for φ, ψ ∈ C(M ).
• Laplace-Beltrami operator As stated before we will consider ∆g, which is a
linear map from C∞(M ) into itself.
• Local coordinates representation For every coordinate chart (Ui, Vi, ki) there
exists a symmetric matrix
(gijk(x))nj,k=1 with gijk ∈ C∞(Vi, R) (1.2)
and such that (gjki (x)) is positive semidefinite at every fixed point x ∈ Vi.
These matrices give us the volume density
ki∗dvolg = |gi(x)|dx (1.3)
and allow us to write ∆g in local coordinates.
In operator form we have the following differential operator defined on func-tions belonging to C0∞(Vi) ki∗∆gk∗i : = n X j,k=1 gijk(x) ∂ 2 ∂xj∂xk + |gi(x)|−1 ∂ ∂xj |gi(x)|gijk(x) ∂ ∂xk . (1.4)
Hence, given a function defined on the manifold φ ∈ C0∞(Ui) we can apply the
previous operator to ki∗φ ∈ C0∞(Vi).
• Selfadjointness The operator ∆g is selfadjoint on C∞(M ) with respect to the volume density defined in (1.3).
• Diagonalization There exists an orthonormal basis (ej)J of L2(M ) consisting
of eigenfunctions of −∆g, that is
(ej)j∈N⊂ L2(M ) such that − ∆gej = λjej, with (λj)j∈N ⊂ R. (1.5)
By positivity of the operator one also gets λj = (−∆gej, ej)L2(M ) ≥ 0.
CHAPTER 1. NOTATIONS 3 • Functional calculus Consider the representation of u ∈ L2(M ) on the
orthonor-mal basis of eigenfunctions
u =X
j
ujej,
where uj = (u, ej)L2(M ).
For any bounded Borel function f ∈ B(R) we will define the functional calculus on the operator ∆g considering the morphism of algebras
B(R) → L(L2(M )) (1.6) f 7→ f (−∆g), defined as f (−∆g)u = X j f (λj)ujej. (1.7)
Analogoulsy, functional calculus can be defined for any selfadjoint operator with a basis of eigenfunctions.
1.2
Fourier transform on the Schwartz space
We state here the definition of Fourier transform in order to fix the notations that will be used throughout.
In the following we will use the japanese bracket defined for x ∈ Rn as
hxi := (1 + |x|2)12. (1.8)
The Schwartz space S = S(Rn) is defined as
S = {u ∈ C∞(Rn)| ∀α ∈ Nn, ∀N ≥ 0 |∂α
xu(x)| ≤ CαNhxi−N}, (1.9)
equivalently one can require that for all α, β ∈ Nn sup
x∈Rn
|xβ∂xαu(x)| ≤ Cαβ. (1.10)
We will consider the set of seminorms on S NNS(u) := max
|α|≤NsupRn
hxiN|∂xαu(x)|, (1.11) for N ≥ 0. The whole family of seminorms induces the natural topology considered on the Schwartz space, hence it is a locally convex topological vector space, and allows us to define the convergent sequences in S: given u ∈ S and (uj)j ⊂ S
Definition 1.1. For u ∈ S the Fourier transform of u is a function ˆu : Rn → C defined by ˆ u(ξ) = Z Rn e−iy·ξu(y)dy. (1.12) The map u 7→ ˆu is linear and continuous from S to S, meaning that for all N ≥ 0 there exist C > 0, M > 0 such that for any u ∈ S
NNS(ˆu) ≤ CNMS(u).
By Fourier inversion formula for all u ∈ S and x ∈ Rn we have u(x) = 1
(2π)n
Z
eix·ξu(ξ)dξˆ (1.13) from which we see that, up to a factor i, taking the derivative with respect to x corresponds to multiplying by ξ. Indeed, having defined
Dj :=
1
i∂j and D := (D1, . . . , Dn) (1.14) if we compute the derivative of u differentiating under the integral sign in (1.13) we get
∂ju(x) = (2π)−n
Z
eix·ξiξju(ξ)dξ,ˆ
so by (1.14)
Dju(x) = (2π)−n
Z
eix·ξξju(ξ)dξˆ
and more in general, taking α ∈ Nn Dαu(x) = (2π)−n
Z
eix·ξξαu(ξ)dξ.ˆ (1.15) Sobolev spaces Through Fourier transform we can also define the L2 based Sobolev spaces.
For s ∈ R, a temperate distribution u is in Hs(Rn) if its Fourier transform ˆu is in L2locand it is such that
kuk2
Hs := (2π)−n
Z
hξi2s|ˆu(ξ)|2dξ < ∞.
1.3
Hilbert-Schmidt operators
Let H and K separable Hilbert spaces over C and (L(H, K), || · ||L(H,K)) the Banach
CHAPTER 1. NOTATIONS 5 Definition 1.2. An operator A ∈ L(H, K) is Hilbert-Schmidt if
||A||HS := X j ||Aej||2K 1/2 < ∞, (1.16)
for some orthonormal basis (ej)j of H.
The space of Hilbert-Schmidt operators (S2(H, K), || · ||HS) is a Banach space.
Remark 1.1. The definition in (1.16) does not depend on the choice of the orthonor-mal basis. Indeed, given any orthonororthonor-mal basis (fj)j of H and (gj)j of K we have
X
j
||Afj||2K=X
j
||A∗gj||2H. (1.17)
Then, applying equality (1.17) first considering (ej)j as basis of H and then
consid-ering (˜ej)j we obtain the desired invariance
X j ||Aej||2K=X j ||A∗fj||2H= X j ||A˜ej||2K.
Hilbert-Schmidt operators are compact, since they are limit of finite rank op-erators, and they also satisfy the following properties.
Proposition 1.1.
i) Let A ∈ S2(H, K), then A∗ ∈ S2(K, H) and ||A||HS = ||A∗||HS.
ii) Let A ∈ S2(H, K), then we have ||A||L(H,K)≤ ||A||HS.
iii) If H1 and K1 are Hilbert spaces and A ∈ S2(H, K), B ∈ L(H1, H),
C ∈ L(K1, K) then CAB ∈ S2(H1, K1) and
||CAB||HS ≤ ||C||L(K1,K)||A||HS||B||L(H1,H).
1.4
Trace class operators
Let H a separable Hilbert space and T : H → H a linear operator. T is trace class if there exist N ∈ N and
i) K1, . . . KN separable Hilbert spaces
such that T = N X j=1 A∗jBj. (1.18)
By this definition we get that T ∈ S2(H), thanks to the fact that S2(H) is a vector
space and by item i) and iii) of Proposition 1.1. We define ||T ||tr := inf N X j=1 ||Aj||HS||Bj||HS , (1.19)
taking the infimum over all N, (Kj)j, (Aj)j, (Bj)j such that the definition in (1.18)
is satisfied.
The set of trace operators (S1(H), || · ||tr) is a normed vector space. Similarly to
Hilbert-Schmidt operators we have the following properties. Proposition 1.2. Let T ∈ S1(H), then the following holds.
i) ||T ||L(H,H)≤ ||T ||tr,
ii) T∗ ∈ S1(H) and ||T∗||tr = ||T ||tr,
iii) if K is a separable Hilbert space and A, B ∈ L(K, H) then A∗T B ∈ S1(K) and
||A∗T B||tr ≤ ||A||L(K,H)||T ||tr||B||L(K,H).
Definition 1.3. Let T ∈ S1(H), then the trace of T is the complex number
tr(T ) :=X
j
(ej, T ej)H, (1.20)
where (ej)j is any orthonormal basis of H.
Remark 1.2. The quantity in (1.20) is well defined since it does not depend on the choice of the orthonormal basis. Indeed, one can prove that the seriesP
j(ej, T ej)H
is absolutely convergent and the sum does not depend on the choice of the basis. Remark 1.3. If we assume that T is a selfadjoint operator on the Hilbert space H we can easily check that this definition of trace simply results in the number of non zero eigenvalues counted with multiplicity.
Indeed, if T is selfadjoint there exists an orthonormal basis of H made of eigenfunc-tions (ej)j, then tr(T ) =X j (ej, T ej)H= X j λj(ej, ej)H= X j λj.
In the case of a selfadjoint operator on a Hilbert space algebraic and geometric multiplicity of eigenvalues are always the same, thus whenever an eigenvalue λ has
CHAPTER 1. NOTATIONS 7 algebraic multiplicity m > 1 it also has m corresponding eigenvectors in the or-thonormal basis. This means that the sum above takes into account λ as many times as needed.
Moreover, if we consider f (T ) ∈ L(H) by functional calculus we know that f (T )ej = f (λj)ej, then in the same way as before we get tr(f (T )) =Pjf (λj).
Proposition 1.3. i) the map
tr(·) : (S1(H), || · ||tr) → (C, | · |)
is linear and continuous. ii) We have ciclicity of the trace:
trH(A∗B) = trK(BA∗) for A, B ∈ S2(H, K),
trH(A∗T C) = trK(T CA∗) for T ∈ S1(K), A, C ∈ L(H, K).
iii) Let A, B ∈ L(L2(Rn)) integral operators with kernels K1, K2 ∈ L2(R2n)
re-spectively. Then
trL2(Rn)(AB) =
Z Z
K1(x, y)K2(y, x)dxdy. (1.21)
Proof. We will just give proof of item iii).
First of all, we should prove that A and B are Hilbert-Schmidt operators, which implies that AB is trace class and hence its trace is well defined.
In order to do so, we consider an orthonormal basis (ej)j of L2(Rn) and the
corre-sponding orthonormal basis of L2(R2n)
(ek⊗ ej)(j,k)∈N2, where ek⊗ ej(x, y) := ek(x)ej(y).
Now evaluating the Hilbert-Schmidt norm of A we get kAk2HS =X j kAejk2L2(Rn) = X j X k |(ek, Aej)L2(Rn)|2 =X j X k Z ek(x)Aej(x)dx 2 =X j X k Z ek(x) Z K1(x, y)ej(y)dydx 2 =X j X k |(ek⊗ ej, K1)L2(R2n)|2= kK1k2L2(R2n) < ∞,
which implies A ∈ S2(L2(Rn)). In the same way we obtain that also B is
Now observe that A∗ is also an integral operator with kernel K1∗(x, y) = K1(y, x).
Indeed, A∗ is an operator that must satisfy the relation (A∗v, u)L2(Rn) = (v, Au)L2(Rn)
and expanding both terms we get Z A∗v(x)u(x)dx = Z Z v(x)K1(x, y)u(y)dydx = Z Z v(x)K1(x, y)dx u(y)dy, which implies A∗v(x) = Z
v(y)K1(y, x)dy.
Now, coming to evaluate the trace of AB by definition we have trL2(Rn)(AB) = X j (ej, ABej)L2(Rn)= X j (A∗ej, Bej)L2(Rn) =X j X k (ek, A∗ej)L2(Rn)(ek, Bej)L2(Rn) =X j X k (ek⊗ ej, K1∗)L2(R2n)(ek⊗ ej, K2)L2(R2n) =(K1∗, K2)L2(R2n) = Z Z
K1(x, y)K2(y, x)dxdy,
Chapter 2
The Weyl Law
In this chapter we will state the main result and give a summary of the steps needed to prove it. The role of semiclassic calculus in the general idea of the proof will be underlined here, while the mathematical aspects will be developed in the following chapters.
2.1
Statement of the result
The Weyl law allows one to study the asymptotic behaviour of a counting function that keeps trace of how many eigenvalue of the operator −∆g are smaller than a
given threshold.
We define the counting function as
N (λ) := #{j ∈ N|λj ≤ λ}, (2.1)
such function is finite for any fixed λ since the eigenvalues λj go to ∞.
In addition we notice that N counts the eigenvalues with multiplicity, indeed we will rewrite N as the trace of a finite rank, selfadjoint operator which takes into account the multiplicity of eigenvalues because it counts the dimensions of the eigenpsaces. The result that we will consider is the following.
Theorem 2.1 (Weyl law). Let N (λ) the counting function defined in (2.1), then lim
λ→+∞λ −n
2N (λ) = (2π)−nωnvolg(M ), (2.2)
where dimM = n, ωn is the volume of the unit ball of Rn and volg(M ) =
R
Mdvolg.
2.2
Roadmap of the proof
In this section we will break down the main steps performed to prove the Weyl law. Then in the following chapters we will concentrate on the role played by semiclassical
calculus and the Hellfer-Sj¨ostrand formula that will allow us to write explicitly the functional calculus on ∆g at least for smooth functions.
2.2.1 Rewriting N as trace of an operator
Let 1[0,1] be the characteristic function which is bounded and Borel, hence we can
perform functional calculus and evaluate 1[0,1] −∆g λ
.
From the definition in (1.7) f (∆g) has eigenvalues f (λj), then, given that the trace
of a finite rank selfadjoint operator is the sum of its eigenvalues counted with mul-tiplicity, we get tr 1[0,1] −∆g λ =X j 1[0,1] λj λ = X λj∈[0,λ] 1 = N (λ).
Remark 2.1. Thanks to Remark 1.3, the previous equality holds even in the case that the trace is defined according to (1.20).
By rewriting N in this form, we resort to studying the trace of operators which are functions of ∆g.
We will see that is enough to study smooth functions of ∆g, for which we have a
more useful representation of f (∆g)u than the one given in (1.7). We can then get
the result for 1[0,1] by approximation.
2.2.2 Proof of Weyl law by approximation with smooth functions We will reduce ourselves to proving the following result.
Theorem 2.2. Let f ∈ C0∞(R), then lim h→0h ntr f −h2∆ g = (2π)−nvolg(M ) Z Rn f (|η|2)dη. If we consider h := √1 λ, the previous statement reads as
lim λ→∞λ −n 2 tr f −∆g λ = (2π)−nvolg(M ) Z Rn f (|η|2)dη (2.3) for any f ∈ C0∞(R). Once we got this, we can obtain the asymptotic behaviour of the trace even for non smooth functions of the Laplacian, as the one used in N (λ) = tr1[0,1] −∆ g λ .
CHAPTER 2. THE WEYL LAW 11 Indeed we can find f−, f+∈ C0∞(R) such that
f−≤1[0,1] ≤ f+.
Now ˜f := f+−1[0,1] ≥ 0, and from the definition (1.7) it is easily seen that also
˜ f −∆ g λ
is a non negative operator, meaning that for any φ ∈ L2(M ) we have ˜ f −∆g λ φ, φ L2(M ) = X j ˜ f λj λ ejφj, X k ekφk L2(M ) =X j,k ˜ f λj λ φjφk(ej, ek)L2(M ) =X j ˜ f λj λ φ2j ≥ 0.
Now the eigenvalues of ˜f−∆g
λ
will also be non negative because they are given by the scalar products ( ˜f
−∆ g λ ej, ej)L2(M ), hence tr ˜ f −∆g λ = tr f+−1[0,1] −∆g λ ≥ 0, same goes for
tr 1[0,1]− f− −∆g λ ≥ 0.
Considering both inequalities and by linearity of the trace we get tr f− −∆g λ ≤ N (λ) ≤ tr f+ −∆g λ . (2.4)
In general we don’t know if the limit of λ−n2 N (λ) exists and we can only say that
lim inf λ→∞ λ −n 2 N (λ) ≤ lim sup λ→∞ λ−n2 N (λ).
Instead, by Theorem 2.2 we know the value of the limit of the terms on the left and on the right of inequality (2.4) and we get
CM Z Rn f−(|η|2)dη ≤ lim inf λ→∞ λ −n 2 N (λ) ≤ lim sup λ→∞ λ−n2 N (λ) ≤ CM Z Rn f+(|η|2)dη, (2.5) setting CM = (2π)−nvolg(M ).
Now choosing as f−, ε, f+, ε∈ C0∞([−1, 2]) such that they assume values in [0, 1] and
are defined by f−, ε(x) = ( 1 if x ∈ (ε, 1 − ε), 0 if x ∈ (0, 1)c ; f+, ε(x) = ( 1 if x ∈ (−ε, 1 + ε), 0 if x ∈ (−2ε, 1 + 2ε)c
we get f−, ε≤1[0,1]≤ f+, εand the following convergence as ε goes to 0 f−, ε L1(R) −−−→1(0,1), f+, ε L1(R) −−−→1[0,1].
At this point we can repeat the same reasoning that led to inequality (2.5) with f−,ε, f+,ε and then pass to the limit for ε going to 0 in the integrals to the far left
and far right, which happen to have the same limit. In this way we obtain lim λ→∞λ −n 2N (λ) =CM Z Rn 1(0,1)(|η|2)dη = CM Z {|η|<1} dη =(2π)−nvolg(M )ωn,
which gives us exactly the statement of the Weyl law as in (2.2). 2.2.3 Performing functional calculus on −h2∆
g
Thanks to the previous section we reduced the proof of the Weyl law to Theorem 2.2, which involves tr(f (−h2∆g)).
First of all, we apply the following result to compute the functional calculus on −∆g, where the almost analytic extension of a function will be defined in Chapter
4, while the integral in the statement is defined as a limit in (4.2).
Theorem 2.3 (Helffer-Sj¨ostrand formula). Let f ∈ C0∞(R) and ˜f an almost analytic extension of f . Given a selfadjoint operator A on a Hilbert space H, then
f (A) = 1 2π Z C ∂ ˜f (z)(A − z)−1L(dz), (2.6) where ∂ = ∂x+ i∂y.
Remark 2.2. Functional calculus on a selfadjoint operator might be defined in a very general setting (i.e. for any Borel bounded f ), but the definition of f (A), analogous to the one given in (1.7) for −∆g, is quite abstract and useless for concrete
computations.
In our case, thanks to the fact that Theorem 2.2 is stated only for smooth functions, we are able to apply this formula, taking A = −h2∆g and compute explicitly a
function of ∆g, provided we have an expression for the resolvent (−h2∆g− z)−1.
2.2.4 Finding an approximate inverse of −h2∆ g− z
It is at this point that we resort to semiclassical calculus, since it allows us to find a good enough approximation of the resolvent thanks to the following result. Lemma 2.4. Take z ∈ C \ [0, +∞) and N > n, then there exist QN(h, z) and
RN(h, z) operators on M such that
CHAPTER 2. THE WEYL LAW 13 as operators on L2(M ). Moreover QN(h, z) is sum of semiclassical operators such
that the integral
FN(f, h) :=
1 2π
Z
∂ ˜f (z)QN(h, z)L(dz)
is a trace class operator with trace depending on volg(M ), as in (2.11). On the other
hand, RN(h, z) is a trace class operator itself such that the integral
TN(f, h) :=
1 2π
Z
∂ ˜f (z)(−h2∆g− z)−1RN(h, z)L(dz)
has a trace with an upper bound given by O(h−n).
QN(h, z) and RN(h, z) are built as sum of operators supported on each
coordi-nate patch. Indeed, they are elements of L(L2(M )) of the form QN(h, z) :=X i∈F ki∗Qi,N(h, z)ki∗, RN(h, z) := X i∈F k∗iRi,N(h, z)ki∗,
where (ki, Ui, Vi)i∈F is an atlas for M and Qi,N(h, z), Ri,N(h, z) are semiclassical
operators defined on Vi ⊂ Rn. Here h will be a semiclassical parameter in (0, 1] and
z a spectral parameter in C \ [0, +∞).
Moreover, we will find bounds on kRN(h, z)kL(L2(M )) assuring us of the fact that
QN(h, z) is a good approximation of the resolvent.
Using the Helffer-Sj¨ostrand formula and the results stated above we can write f (−h2∆g) = 1 2π Z ∂ ˜f (z)(−h2∆g− z)−1L(dz) = 1 2π Z ∂ ˜f (z)QN(h, z)L(dz) − hN 1 2π Z ∂ ˜f (z)(−h2∆g− z)−1RN(h, z)L(dz) =FN(f, h) − hNTN(f, h). (2.8)
2.2.5 Computing the trace
Once we get to this point, we compute the trace of f (−h2∆g) considering the two
terms FN(f, h) and TN(f, h) separately.
To deal with FN(f, h) the main result we need is the following, about the trace
of a semiclassical operator Oph(a), which are defined in Chapter 3 together with the
spaces Sµ,m.
Theorem 2.5. Let ρ > n, if a ∈ S−ρ,−ρ then Oph(a) is trace class on L2(Rn) and
there exist C > 0 and N ∈ N such that kOph(a)ktr ≤ C
1 hn N
S−ρ,−ρ
Moreover the following identity for the trace holds tr(Oph(a)) = 1 (2πh)n Z Z a(x, ξ)dxdξ. (2.10) Note that FN(f, h) = 1 2π Z ∂ ˜f (z)QN(h, z) = 1 2π X i∈F Z ∂ ˜f (z)ki∗Qi,N(h, z)ki∗L(dz)
is sum of integrals whose integrand functions are semiclassical operators. We will prove actually that FN(f, h) itself can be written as a sum of semiclassical operators
to which we will apply the previous Theorem obtaining tr(FN(f, h)) =
1
(2πh)n volg(M )
Z
f (|η|2)dη + O(h1−n). (2.11) Next, coming to the second term TN(f, h) we will write explicitely the structrure
of RN(h, z) as a trace operator, which is
RN(h, z) =X
i∈F
Ai,N(h, z)∗Bi,N(h) (2.12)
for Ai,N(h, z) and Bi,N(h, z) Hilbert-Schmidt operators defined on suitable spaces.
We will then take advantage of this expression to estimate the trace norm of TN(f, h)
and the bound we will find will ensure us that the term hNtr(TN(f, h)), that arises
when taking the trace in (2.8), vanishes as h goes to 0.
Given this fact, together with the expression found for the trace of FN(f, h) we can
compute the required limit of the quantity hnf (−h2∆g) and finally prove Theorem
Chapter 3
Semiclassical operators:
definition and properties
We will now introduce semiclassical pseudodifferential operators, which will be the main tool to solve the core of our problem: finding an expression for the resolvent of the Laplacian, in order to apply formula (2.6) and computing the traces needed to prove Theorem 2.2. Some properties about the operations that can be performed with these operators will also be presented.
3.1
Space of symbols
Consider a differential operator
P = X
|α|≤m
aα(x)Dα
with associated symbol p(x, ξ) :=P
|α|≤maα(x)ξα. Thanks to the expression of the
derivative found in (1.15), applying operator P to u ∈ S we obtain P u(x) = X |α|≤m (2π)−n Z eix·ξaα(x)ξαu(ξ)dξˆ = (2π)−n Z eix·ξp(x, ξ)ˆu(ξ)dξ. (3.1) In the case of a differential operator the symbol p is a polynomial in ξ with non constant coefficients, semiclassical pseudodifferential operators are a generalization since the symbol need not to be a polynomial with respect to ξ. In particular, we will require the symbols to be in the following space of functions.
Definition 3.1. For m, µ ∈ R the space of functions Sµ,m= Sµ,m(Rn× Rn)
is defined as
Sµ,m = {a ∈ C∞(R2n) | ∀α, β ∈ Nn |∂xα∂ξβa(x, ξ)| ≤ Cαβhxiµhξim−|β| }. (3.2)
In the case µ = 0 we will just write Sm:= S0,m. In Sµ,m we consider the family of seminorms for N ∈ N
NNSµ,m(a) = max
|α+β|≤Nx,ξ∈Rsupn
hxi−µhξi−m+|β||∂xα∂ξβa(x, ξ)|. (3.3) Remark 3.1. Clearly, any polynomial of the form
p(x, ξ) := X
|α|≤m
aα(x)ξα
is in Smif and only if the functions a
αare bounded together with all their derivatives.
Lemma 3.1. For real numbers m1, m2, µ1, µ2 such that m1 ≤ m2 and µ1 ≤ µ2 we
have the continuous embedding
Sµ1,m1 ,→ Sµ2,m2,
meaning that for all N2∈ N there exist C > 0 and N1 ∈ N such that
NNSµ2,m2
2 (a) ≤ CN
Sµ1,m1
N1 (a) (3.4)
for all a ∈ Sµ1,m1.
From this fact we also get continuity of the product and the derivative. Lemma 3.2. Let m1, m2, µ1, µ2∈ R.
i) The map
Sµ1,m1 × Sµ2,m2 → Sµ1+µ2,m1+m2
(a, b) 7→ ab
is continuous, that is for all N ∈ N there exist C > 0 and N1, N2 ∈ N such
that
NNSµ1+µ2,m1+m2(ab) ≤ CNN 1Sµ1,m1(a)NN 2Sµ2,m2(b) for all a ∈ Sµ1,m1 and b ∈ Sµ2,m2.
ii) For fixed α, β ∈ Nn the map
Sµ1,m1 → Sµ1,m1−|β|
a 7→ ∂xα∂ξβa is continuous.
CHAPTER 3. SEMICLASSICAL OPERATORS 17 We give now a useful property for the special case of elliptic symbols.
Lemma 3.3. Let a ∈ Sm an elliptic symbol, namely there exists c > 0 such that
|a(x, ξ)| ≥ chξim. Then 1
a ∈ S
−m.
Proof. The proof relies on the fact that given any f ∈ C∞(Rd) non vanishing func-tion, then computing directly the derivative ∂γ 1
f
we can write it as a linear combination of terms of the form
∂γ1f . . . ∂γjf
f1+j ,
for all the j ∈ N such that 1 ≤ j ≤ |γ| and
γ1, . . . , γj 6= 0 such that γ1+ . . . + γj = γ.
To see that 1 a is in S
−mwe need to check that all the derivatives satisfy the inequality
in (3.2). Thanks to the expression given above, we obtain
∂(x,ξ)(α,β) 1 a = X j≤|α| k≤|β| cjk ∂β1 ξ a . . . ∂ βj ξ a ∂xα1a . . . ∂xαka a1+j+k , (3.5)
with β1+ . . . βk= β and α1+ . . . αk= α and cjk real constants.
Now, by assumption a is elliptic and in Sm, hence |∂βi
ξ a| ≤ Cβihξi
m−|βi| and |∂αi
x a| ≤ Cαihξi
m,
thanks to these inequalities we can estimate the derivative of 1/a and prove the statement ∂(x,ξ)(α,β) 1 a ≤X j,k |cjk| hξijm−(|β1|+...+|βj|) hξikm c1+j+khξim+jm+km = hξi −|β| hξim X j,k |cjk| c1+j+k ≤ Cαβhξi−m−|β|.
3.2
Definition
Let a ∈ Sµ,m, u ∈ S and h ∈ (0, 1], we define the semiclassical operator of symbol a as
Oph(a)u(x) := (2π)−n
Z
Rn
eix·ξa(x, hξ)ˆu(ξ)dξ, (3.6) which will also be denoted by Oph(a) := a(x, hD).
The special case of h = 1 gives us the so called pseudodifferential operator of symbol a
a(x, D)u(x) := (2π)−n Z
Rn
eix·ξa(x, ξ)ˆu(ξ)dξ, (3.7) also denoted as Op(a) := a(x, D).
Therefore, setting ah(x, ξ) := a(x, hξ) we have the relation Oph(a) = Op(ah).
Notice that from Fourier inversion formula (1.13) Oph(1) = I.
Remark 3.2. Pseudodifferential operators are a generalization of classic differential operators, because from Remark 3.1 any differential operator whose coefficients are bounded and with all derivatives bounded is an operator with symbol in Sm, hence it is also pseudodifferential.
Remark 3.3. As we stated before, one of the reasons we introduce semiclassical operators is to find inverses of elliptic operators. We can see that in the simple case of an operator with constant coefficients it is quite easy to find an exact inverse using pseudodifferential operators. Here we give as an example the resolvent of the Laplacian on Rn.
Take z ∈ C \ [0, ∞) and
az(ξ) =
1 |ξ|2− z.
If we set dz(ξ) = |ξ|2− z then dz ∈ S2 and it is an elliptic symbol. Thanks to the
previous Lemma we get az ∈ S−2 and we can directly check that
(−∆ − z)Op(az) = I.
Indeed, by the definition given in (3.7) we have Op(az)u(x) = (2π)−n
Z
eix·ξ 1
|ξ|2− zu(ξ)dξ,ˆ
now taking into account that −∆(eix·ξ) = |ξ|2eix·ξand the Fourier inversion formula we get
(−∆−z)Op(az)u(x) = (2π)−n
Z
−∆(eix·ξ) u(ξ)ˆ |ξ|2− zdξ−
Z
zeix·ξ u(ξ)ˆ
|ξ|2− zu(ξ)dξˆ
CHAPTER 3. SEMICLASSICAL OPERATORS 19 = (2π)−n Z u(ξ)ˆ |ξ|2− z(|ξ| 2− z)eix·ξdξ = u(x)
What happens here is thanks to the fact that the symbol of the operator ∆ − z is a polynomial in ξ with coefficients that don’t depend on x. In general, as stated in Section 2.2.3, if we compute the composition Op(p)Op(1/p) we get the identity only up to a remainder term Op(r). Instead, if the symbols do not depend on x like in this case, the symbol of the composition is simply the product of symbols, with no remainder.
For this reason we will instead use semiclassical operators: the introduction of the semiclassical parameter h solves the problem of the remainder because we will see that we can find explicitly the inverse of an elliptic operator up to a corrective term which will be a remainder of the form hNOph(r). Applying this fact to our
case with h that goes to 0 will allow us to find a good approximation of the inverse.
3.3
Properties of symbolic calculus
We now give some results about symbolic calculus with semiclassical operators that will be used later on, once we will see −h2∆g− z as sum of semiclassical operators.
The following statements hold in particular in the case of pseudodifferential opera-tors, obtained by h = 1.
First of all, we have continuity of the action of semiclassical operators on the Schwartz and Sobolev spaces.
Proposition 3.4. For fixed h ∈ (0, 1] the bilinear map Sµ,m×S → S
(a, u) 7→ Oph(a)u
is continuous.
Proposition 3.5. Let m, s ∈ R and a ∈ Sm, then the map Hs(Rn) → Hs−m(Rn)
u 7→ Oph(a)u
is continuous.
Thanks to the fact that Oph(a) maps S into itself we can compose two
Theorem 3.6 (Composition). Let a ∈ Sµ1,m1, b ∈ Sµ2,m2 and h ∈ (0, 1], then Oph(a)Oph(b) = Oph((a#b)(h)) (3.8) with (a#b)(h) =X j<J hj(a#b)j+ hJrJ#(a, b, h),
for any J ∈ N, where
(a#b)j = X |α|=j 1 α!∂ α ξaDαxb
and r#J is a remainder such that the map
Sµ1,m1 × Sµ2,m2 → Sµ1+µ2,m1+m2−J
(a, b) 7→ rJ#(a, b, h)
is bilinear and equicontinuous, that is for every N ∈ N there exist C > 0, not dependent on h, and N1, N2 ∈ N such that
NNSµ1+µ2,m1+m2−J(r#J(a, b, h)) ≤ CNN 1Sµ1,m1(a)NN 2Sµ2,m2(b) (3.9) for all a, b, h.
Remark 3.4. Thanks to the continuity of the product and of the derivative, stated in 3.2, we have that the bilinear map
Sµ1,m1 × Sµ2,m2 → Sµ1+µ2,m1+m2−j
(a, b) 7→ (a#b)j
is continuous, while from the continuity of the embedding in Lemma 3.1 it follows that the map
Sµ1,m1 × Sµ2,m2 → Sµ1+µ2,m1+m2
(a, b) 7→ (a#b)(h) is equicontinuous with respect to h.
In some cases a semiclassical operator can be extended by a continuous linear map to act on the whole space L2(Rn) thanks to the following result.
Theorem 3.7. There exist C > 0 and N ≥ 0 such that kOph(a)ukL2(Rn) ≤ CNS
0
N (a)kukL2(Rn)
CHAPTER 3. SEMICLASSICAL OPERATORS 21 We record also the following formula that will be used when computing the trace of semiclassical operators.
Theorem 3.8. Let ρ > n2, h ∈ (0, 1] and a ∈ S−ρ,−ρ, then Oph(a) is Hilbert-Schmidt
on L2(Rn) with norm kOph(a)kHS = 1 2πh n2 kakL2(R2n). (3.10)
Proof. We consider first a semiclassical operator with symbol a ∈ S, expanding the definition of Oph(a) we get
Oph(a) =(2π)−n
Z
eix·ξa(x, hξ)ˆu(ξ)dξ =(2π)−n
Z Z
ei(x−y)·ξa(x, hξ)u(y)dydξ =(2πh)−n
Z Z
ei(x−y)h ·ηa(x, η)u(y)dydη
=(2πh)−n Z ˆ a x,y − x h u(y)dy, hence Oph(a) is an integral operator with kernel
K(x, y) = (2πh)−naˆ x,y − x h ,
where we are considering the Fourier transform of a with respect to the second variable.
As we found in the proof of Proposition 1.3 the Hilbert-Schmidt norm of an integral operator is the L2 norm of its kernel, then
kOph(a)kHS = (2πh)−n ˆ a x,y − x h L2(R2n) . (3.11)
Computing the L2 norm gives us ˆ a x,y − x h 2 L2(R2n) = Z Z ˆ a x,y − x h 2 dxdy = hn Z Z |ˆa(x, z)|2dxdz =(2πh)n Z Z |a(x, ξ)|2dxdξ = (2πh)nkak2L2(R2n),
from which the statement follows, since kOph(a)kHS = (2πh)−n+ n 2kakL2(R2n)= 1 2πh n 2 kakL2(R2n).
Next we can prove the result for a symbol in S−ρ,−ρ by density. Indeed, multiplying by suitable cutoff functions, we can approximate a with a sequence (aj)j ⊂ C0∞(R2n)
such that
aj → a in S−ρ
0,−ρ0
with ρ > ρ0 > n 2. By the previous step we have
kOph(aj)kHS = (2πh)
−n 2 kajk
L2(R2n). (3.12)
Moreover, since ρ0 > n2, S−ρ0,−ρ0 is continuously embedded in L2(R2n) and this fact, together with the embedding of S−ρ0,−ρ0 in S0,0, gives us
aj → a in L2(R2n) and aj → a in S0,0.
From equality (3.12) we obtain that (Oph(aj))j is a Cauchy sequence in S2(L2(Rn)),
implying also that Oph(a) is a bounded operator on (L2(Rn).
On the other hand, thanks to Proposition 3.4, the convergence in S0,0 implies that Oph(aj)u → Oph(a)u for any u ∈ S
and, applying Theorem 3.7, this pointwise convergence can be extended to L2(Rn). Now, by definition of Hilbert-Schmidt norm, the limit of Oph(aj) in S2(L2(Rn)) must
be Oph(a), which is then Hilbert-Schmidt, since S2(L2(Rn)) is a Banach space.
As a result of these observations, passing to the limit for j → ∞ in (3.12) we get the statement.
Chapter 4
Helffer-Sj¨
ostrand functional
calculus
We recall the fact that our aim is to prove the following result about the trace of an operator: given f ∈ C0∞(R) then
lim λ→∞h ntr(f (−h2∆ g)) = (2π)−nvolg(M ) Z Rn f (|η|2)dη.
For this reason, we display here the proof of the Helffer-Sj¨ostrand formula that will allow us to perform the first step: computing functional calculus on the operator −h2∆
g.
4.1
Almost analytic extension
Given a function of real variable we introduce its almost analytic extension, which essentially is an extension to the whole complex plane whose derivative with respect to the complex variable ¯z, ∂ = ∂x+ i∂y, tends to zero when approaching the real
line.
Definition 4.1. Let f ∈ C0∞(R), its almost analytic extension is a function ˜
f ∈ C0∞(R2) such that
i) ˜f (x, 0) = f (x): the restriction of ˜f to the real line coincides with f ; ii) for all N ≥ 0, ∂ ˜f (x, y) = O(|y|N) as y → 0 for any x ∈ R.
Remark 4.1. Item ii) in the definition of ˜f actually means that ∂ ˜f vanishes on the real line with order N for any N .
Indeed, it implies that there exists a smooth function ω(y) bounded in a neighbor-hood of zero and such that ∂ ˜f (x, y) = ω(y)|y|N for any N ≥ 0, it follows that for
k < N we have ∂yk(∂ ˜f )|y=0= k X j=0 k j ω(k−j)(y)N (N − 1) . . . (N − j + 1)|y|N −j y=0 = 0.
Taking the N -th derivative instead
∂yN(∂ ˜f )|y=0 = ω(0)N ! = CNN !.
So, the Taylor expansion with respect to y of ∂ ˜f around a point of the real line (x, 0) reduces to the remainder of order N: letting χ ∈ C0∞(R2) a cutoff function that is constantly one near the support of ˜f we can write
∂ ˜f (x, y) = CNyNχ(x, y) for any N ≥ 0,
since the polynomial part in the Taylor expansion vanishes.
Remark 4.2. The almost analytic extension must be a function with complex values. Indeed, if we assume that ˜f (x, y) takes values in R, properties i) and ii) in Definition 4.1 combined give
∂ ˜f (x, 0) = f0(x) + i∂yf (x, 0) = 0,˜ (4.1)
that would imply that the real part f0(x) must be zero, hence f is a constant. Instead, if ˜f assumes complex values, the real part in (4.1) is not only f0(x) anymore and we can’t reason as before.
The following result gives an example of how to construct an almost analytic extension and we can see that, indeed, it is complex valued.
Proposition 4.1. Let f ∈ C0∞(R) and χ1, χ2 ∈ C0∞(R) cutoff functions such that
χ1 ≡ 1 near supp f, χ2≡ 1 near 0.
Then an almost analytic extension of f is given by ˜ f (x, y) = χ1(x)χ2(y) 1 2π Z ei(x+iy)ξχ2(yξ) ˆf (ξ)dξ.
4.2
Integrals of almost analytic extension
Let B be a function with values in a Banach space B, B : C \ R → B, we define the following integrals, whose integrand is an element of the Banach space.
Z |Im(z)|≥ε ∂ ˜f (z)B(z)L(dz) := Z |y|≥ε Z R ∂ ˜f (x, y)B(x, y)dx dy Z C ∂ ˜f (z)B(z)L(dz) := lim ε→0 Z |Im(z)|≥ε ∂ ˜f (z)B(z)L(dz). (4.2) Under suitable assumptions, these quantities are well defined as shown below.
CHAPTER 4. HELFFER-SJ ¨OSTRAND FUNCTIONAL CALCULUS 25 Proposition 4.2. Let f ∈ C0∞(R) and ˜f an almost analytic extension, supported in a rectangle W := [a, b] + i[c, d].
Let B : W → B continuous and such that there exist C, M ≥ 0 that verify ||B(z)||B ≤ C|Im(z)|−M, for all z ∈ W \ R,
then we have that the integral in (4.2) is well defined and such that Z C ∂ ˜f (z)B(z)L(dz) B ≤ ˜C sup W \R ||Im(z)MB(z)||B. (4.3)
Remark 4.3. Since M > 0, the assumption on kB(z)kB tells us that the more we
move far away from the real line the more B(z) is small in norm, while getting closer to the real line the function can be arbitrarily large.
Proof. Consider the map
F : [c, d] \ {0} −→ B y 7→
Z b
a
∂ ˜f (x, y)B(x + iy)dx,
which is well defined since the integrand in the definition of F (y) is Bochner inte-grable because Z b a k∂ ˜f (x, y)B(x + iy)kBdx ≤ Z b a |∂ ˜f (x, y)|kB(x + iy)kBdx ≤(b − a) sup x∈[a,b] kB(x + iy)kBk∂ ˜f (·, y)k∞ < ∞.
Since the quantity ∂ ˜f (x, y)B(x + iy) is integrable with respect to x we also have continuity of F by the properties of the Bochner integral:
kF (y)kB ≤|y| (b − a) sup x
kyMB(x + iy)kBsup x
|y−M −1∂ ˜f (x, y)| (4.4) ≤|y|(b − a)C sup
x,y
|y−M −1∂ ˜f (x, y)|, where the supremum in the last line is finite by Remark 4.1.
Now, to prove that the integral in (4.2) is well defined we must show that the following limit in B exists
lim ε→0 Z −ε c F (y)dy + Z d ε F (y)dy .
This is true because, being that F (y) is continuous, the map l 7→RclF (y)dy is also continuous from R to B.
It remains to prove the bound on the integral, thanks to what we found before in (4.4) we get immediately Z C ∂ ˜f (z)B(z)L(dz) B ≤ lim sup ε→0 Z |y|≥ε, y∈[c,d] Z R ∂ ˜f (x, y)B(x + iy)dx B dy ≤ lim sup ε→0 Z |y|≥ε, y∈[c,d] kF (y)kBdy ≤ ˜C sup (x,y)∈W \R kyMB(x + iy)kB
where the constant is ˜C = (b − a)(c − d)ky−M∂ ˜f k∞.
The integral defined in (4.2) is an element of the Banach space B, hence in the case that B is an operator space the integral is an operator itself, whose action is described by the following.
Lemma 4.3. Let H, K Hilbert spaces. Take f ∈ C0∞(R) and B with values in the Banach space L(H, K) such that they satisfy the assumptions of Proposition 4.2. If u ∈ H, then Z C ∂ ˜f (z)B(z)L(dz) u = Z C ∂ ˜f (z)B(z)[u] L(dz). (4.5) Through almost analytic extension we can give a sort of generalization of the Cauchy integral formula for non analytic functions.
Proposition 4.4. Let f ∈ C0∞(R) and ˜f ∈ C0∞(R2) its almost analytic extension. Then for all j ∈ N and λ ∈ R
1 2π Z C ∂ ˜f (z)(λ − z)−1−jL(dz) = (−1) j j! f (j)(λ).
Proof. First, we show that is enough to prove the statement for j = 0. Set
g(x, y) := ∂xjf (x, y).˜
By definition of almost analytic extension ˜f (x, 0) and f (x) agree on the whole real line, then also g(x, 0) = f(j)(x) for any x in R. Moreover, g is an almost analytic
extension of f(j) since, as stated in Remark 4.1, condition ii) in Definition 4.1 is equivalent to the fact that ∂yk(∂g(x, 0)) = 0 for any k and this is true because we can swap derivatives:
CHAPTER 4. HELFFER-SJ ¨OSTRAND FUNCTIONAL CALCULUS 27 Now if we integrate by parts in x j times, the boundary terms vanish since ∂ ˜f has compact support and we obtain
Z |Im(z)|≥ε ∂ ˜f (z)(λ − z)−1−jL(dz) = Z |y|≥ε Z R ∂ ˜f (x, y) dx (λ − x − iy)1+jdy = (−1) j j! Z |y|≥ε Z R ∂xj∂ ˜f (x, y) dx (λ − x − iy)dy = (−1) j j! Z |Im(z)|≥ε ∂xj∂ ˜f (z)(λ − z)−1L(dz) = (−1) j j! Z |Im(z)|≥ε ∂g(z)(λ − z)−1L(dz). If the statement is true for j = 0, thanks to the fact that g is an almost analytic extension of f(j) we can substitute in the above equality and get the statement for generic j.
It remains to prove the statement for j = 0. Recalling that (λ − z)−1 is an holomo-prhic function, hence ∂(λ − z)−1 = 0, and ˜f has compact support, we integrate by parts with respect to ∂
Z |Im(z)|≥ε ∂ ˜f (z)(λ − z)−1L(dz) = Z |y|≥ε 0 − Z R ˜ f (x, y)∂x 1 λ − x − iy dx dy + i Z R ˜ f (x, y) λ − x − iy y=∞ y=ε + ˜ f (x, y) λ − x − iy y=−ε y=−∞dx − i Z R Z |y|≥ε ˜ f (x, y)∂y 1 λ − x − iy dy dx = i Z R ˜ f (x, −ε) λ − x + iε− ˜ f (x, ε) λ − x − iε dx.
Now we consider the Taylor expansion of ˜f with respect to y in a neighborhood of zero, evaluating at ±ε we get
˜
f (x, ±ε) = f (x) ± ε∂yf (x, 0) + ε˜ 2∂y2f (x, η)˜
with η ∈ R in a neighborhood of zero.
Substituting in the previous expression we obtain Z |Im(z)|≥ε ∂ ˜f (z)(λ − z)−1L(dz) = i Z R f (x) − ε∂yf (x, 0) + ε˜ 2∂y2f (x, η)˜ (λ − x)2+ ε2 (λ − x − iε)dx − i Z R f (x) + ε∂yf (x, 0) + ε˜ 2∂y2f (x, η)˜ (λ − x)2+ ε2 (λ − x + iε)dx =2ε Z R f (x) (λ − x)2+ ε2dx + 2iε Z R ∂yf (x, 0)(x − λ)˜ (λ − x)2+ ε2 dx
+ 2ε3 Z R ∂2 yf (x, η)˜ (λ − x)2+ ε2dx =:I1+ I2+ I3 (4.6)
We consider each term in (4.6) separately.
In I1 we perform the change of variables x−λε = t, which results in
I1= 2 Z R f (x) x−λ ε 2 + 1ε dx = 2 Z R f (λ + εt) 1 + t2 dt,
now f is smooth with compact support, hence f ∈ L1(R) and we can pass to the limit for ε → 0 in I1 thanks to dominated convergence, obtaining
I1 = 2
Z
R
f (λ + εt)
1 + t2 dt −−−ε→0→2πf (λ). (4.7)
Next, we can see that I3 vanishes as ε approaches zero thanks to the fact that ˜f is
integrable together with all its derivatives, again by smoothness and compactness of the support: |I3| ≤ 2ε3 Z R |∂2 yf (x, η)|˜ (λ − x)2+ ε2dx ≤ 2εk∂ 2 yf (·, η)k˜ L1(R)−−−→ ε→0 0. (4.8)
Last, we perform the change of variables x − λ = s in I2 and then integrate by parts
getting I2=iε Z R ∂yf (λ + s, 0)˜ 2s s2+ ε2ds =iε∂yf (λ + s, 0) ln(s˜ 2+ ε2) ∞ −∞− iε Z R ∂xy2 f (λ + s, 0) ln(s˜ 2+ ε2)ds. (4.9) Notice that the boundary term vanishes. Since ˜f ∈ C0∞(R2) ⊂ S(R2) we have growth assumptions on the derivatives of ˜f which give us
|∂yf (λ + s, 0)| ≤ Ch(λ + s, 0)i˜ −N = C(1 + (λ + s)2)−N/2 for any N ≥ 0, so
|∂yf (λ + s, 0) ln(s˜ 2+ ε2)| ≤ C | ln(s
2+ ε2)|
(1 + (λ + s)2)N/2 −−−−→s→±∞ 0.
Being that ˜f is compactly supported, the integral remaining in (4.9) is actually an integral over a compact set. Let K compact such that supp f ⊂ K, then ∂xy2 f (λ + s, 0) ln(s˜ 2+ ε2) ∈ L1(K), since it is a product of functions in L2(K). By dominated convergence, passing to the limit in (4.9) we have
I2= −iε
Z
R
∂xy2 f (λ + s, 0) ln(s˜ 2+ ε2)ds −−−→
ε→0 0. (4.10)
Going back to (4.6), thanks to (4.7), (4.8) and (4.10), as ε goes to zero we get the statement for j = 0.
CHAPTER 4. HELFFER-SJ ¨OSTRAND FUNCTIONAL CALCULUS 29
4.3
Helffer-Sj¨
ostrand formula
The main result proven here will give us an easy expression to compute the functional calculus on −h2∆
g through Helffer-Sj¨ostrand formula obtaining
f (−h2∆g) = 1 2π Z C ∂ ˜f (z)(−h2∆g− z)−1L(dz).
Let A a selfadjoint operator on a Hilbert space H, z ∈ C \ R, which is the resolvent set of A, and fz(λ) = (λ − z)−1 with λ ∈ R.
We define the resolvent of A the operator
R(z) := (A − z)−1 = fz(A). (4.11)
Let (fj)j ⊂ H the orthonormal basis of eigenfunctions with corresponding
eigenval-ues (µj)j ⊂ R. By definition of functional calculus for u ∈ H we have
R(z)u =X
j
uj
µj− z
fj. (4.12)
In general, in any case when f (A) is well defined, we have the following
kf (A)kL(H) = sup u kf (A)ukH kukH = sup u P j |f (µj)|2|uj|2 P j |uj|2 1/2 ≤ sup R |f |.
In the specific case of the resolvent, for z ∈ C \ R we get kR(z)kL(H) =k(A − z)−1kL(H)≤ sup λ∈R |fz(λ)| = sup λ∈R 1 p(λ − Re(z))2+ Im(z)2 = 1 |Im(z)|. (4.13) Remark 4.4. In the following we will apply the result taking A = −h2∆g, hence
considering (−h2∆g−z)−1, which is a sort of semiclassical resolvent of the Laplacian.
In this case the operator is also non negative, thus the resolvent set is actually bigger, being it C \ [0, +∞).
If we take z ∈ C \ [0, +∞), then we need to consider fz only for λ ∈ [0, +∞) for the
functional calculus to be well defined.
We prove now that the resolvent map is continuous, which will allow us to consider the integral that appears in the Helffer-Sj¨ostrand formula.
Lemma 4.5. The resolvent map
C \ [0, +∞) → L(L2(M )) z 7→ R(z) is continuous.
Proof. We show first that R(z) − R(ζ) = (z − ζ)R(z)R(ζ). Since the series defining functional calculus in (4.12) is absolutely convergent we can write
(R(z) − R(ζ))[u] = fz(A)[u] − fζ(A)[u] =
=X j ujfj µj− z −X j ujfj µj − ζ = = (z − ζ)X j ujfj (µj− z)(µj− ζ) = (z − ζ)X j fz(µj)fζ(µj)ujfj = (z − ζ)R(z)R(ζ)[u],
where the last equality follows from the fact that functional calculus is a morphism of algebras, hence it is such that f1(−∆g)f2(−∆g) = (f1f2)(−∆g). Now continuity
follows easily from the following estimate
kR(z) − R(ζ)kL(H) = |z − ζ|kR(z)R(ζ)kL(H) ≤ |z − ζ| sup λ∈R+ 1 |λ − z||λ − ζ| = |z − ζ| 1 |z||ζ|, with |z|, |ζ| 6= 0.
We are now able to prove the formula already cited in Chapter 2.
Theorem 4.6 (Helffer-Sj¨ostrand formula). Let f ∈ C0∞(R) and ˜f its almost analytic extension. Given A a selfadjoint operator on the Hilbert space H, the following identity holds f (A) = 1 2π Z C ∂ ˜f (z)(A − z)−1L(dz). (4.14) Proof. First of all, notice that the integral on the right hand side of the statement is well defined since the assumptions of Proposition 4.2 are satisfied as we remarked before in (4.13).
On both sides we have bounded operators on H so we can reduce ourselves to check that they coincide on a dense subset of H.
We know that there exists the orthonormal basis of eigenfunctions (fj)j, so we
consider the dense subspace of elements of the form u := P
j≤J(fj, u)Hfj. Thanks
to Lemma 4.3 applied in the Banach space L(H) we have 1 2π Z C ∂ ˜f (z)(A − z)−1L(dz) u = 1 2π Z C ∂ ˜f (z)(A − z)−1uL(dz)
CHAPTER 4. HELFFER-SJ ¨OSTRAND FUNCTIONAL CALCULUS 31 =X j≤J (fj, u)H 1 2π Z C ∂ ˜f (z)(A − z)−1fjL(dz) =X j≤J (fj, u)H 1 2π Z C ∂ ˜f (z) fj µj − z L(dz) (4.15) =X j≤J (fj, u)Hf (µj)ej =f (A)u,
Parametrix: construction via
semiclassical operators
Thanks to the results in the previous chapter we can obtain an explicit formula for f (−h2∆g) which involves the resolvent of −h2∆g, so now we will see how to
construct an approximation of the resolvent of an operator and then apply it to our case, namely to approximate (−h2∆g− z)−1.
5.1
Parametrix of an elliptic operator on R
nWe consider a differential operator defined on Rnsplitting it into homogeneous parts P = p2(x, D) + p1(x, D) + p0(x) (5.1) = − n X j,k=1 ajk(x) ∂2 ∂xj∂xk + n X j=1 bj(x) ∂ ∂xj + p0(x).
Hence the symbols of P are polynomials which are homogeneous in ξ of the form p2(x, ξ) = n X j,k=1 ajk(x)ξjξk, p1(x, ξ) = n X j=1 bj(x)iξj,
where we assume ajk = akj and the following regularity assumptions on the symbols
p2−j ∈ S2−j for j = 0, 1, 2,
which, thanks to Remark 3.1, is equivalent to the fact that the coefficients ajk, bj, p0
are smooth, bounded functions on Rn with derivatives of any order bounded. We also assume the operator to be elliptic, which means we request the principal symbol p2 to satisfy the lower bound
|p2(x, ξ)| ≥ c|ξ|2 32
CHAPTER 5. PARAMETRIX 33 for some c > 0 and for any x, ξ ∈ Rn.
We want to apply the following results to find a parametrix of the operator −h2∆g,
so we consider the generic operator P multiplied by h2, by doing so we are also able to write h2P in terms of semiclassical operators.
By the homogeneity of p2 and p1 we have p2(x, hξ) = h2p2(x, ξ) and
p1(x, hξ) = hp1(x, ξ) so, taking z ∈ C \ [0, +∞) as in the previuos chapter, we get
h2P − z = p2(x, hD) − z + hp1(x, hD) + h2p0(x)
= Oph(p2− z) + hOph(p1) + h2Oph(p0). (5.2)
By the ellipticity assumption, the function 1 p2− z
is well defined and thanks to Lemma 3.3, belongs to S−2.
We can actually give more precise bounds on the derivatives of 1 p2− z
, that we will use to estimate the seminorms NS−N(·) of the remainder of the paramterix. Proposition 5.1. The map
C \ [0, +∞) → S−2 z 7→ 1
p2− z
is continuous and for all α, β ∈ Nn there exists a constant Cαβ such that
∂xα∂ξβ 1 p2(x, ξ) − z ≤ Cαβ hzi d(z, R+) |α|+|β|+1 hξi−2−|β|. (5.3) Proof. First, we prove inequality (5.3) for α = β = 0 by considering
1 p2− z = p2+ 1 p2− z · 1 p2+ 1 = 1 + z + 1 p2− z 1 p2+ 1 · (5.4)
Now, by the ellpiticity of p2
1 p2+ 1 ≤ 1 c|ξ|2+ 1 ≤ C1hξi −2,
while using the fact that |p2− z| ≥ d(z, R+) and d(z, R+) ≤ |z| we can estimate the
remaining factor in (5.4) 1 + z + 1 p2− z ≤1 + |z| + 1 d(z, R+) ≤ 2|z| + 1 d(z, R+) ≤C2 hzi d(z, R+)
choosing C2 suitable real constant. Using these two bounds to estimate (5.4) we get
the statement in the case α = β = 0.
Next, we consider generic α and β in Nn. As we said in the proof of Lemma 3.3 the derivative ∂xα∂ξβ
1 p2− z
is a linear combination of terms of the form
∂α1 x ∂ β1 ξ p2. . . ∂ αj x ∂ βj ξ p2 (p2− z)j+1 (5.5) with α1+ . . . + αj = α, β1+ . . . + βj = β and j ≤ |α| + |β|. By the fact that p2 ∈ S2
we have |∂αi x ∂ βi ξ p2| ≤ Cαiβihξi 2−|βi|
for every i = 1, . . . , j and from the case α = β = 0 we get 1 (p2− z)j+1 ≤ Cj+1 hzi j+1 d(z, R+)j+1hξi −2j−2≤ Cj+1 hzi d(z, R+) 1+|α|+|β| hξi−2j−2 since hzi
d(z, R+) ≥ 1 and j ≤ |α| + |β|. We can now estimate the terms in (5.5)
∂α1 x ∂ β1 ξ p2. . . ∂ αj x ∂ βj ξ p2 (p2− z)j+1 ≤ Cαβhξi2j−|β1|−...−|βj| hzi d(z, R+) 1+|α|+|β| hξi−2j−2 = Cαβhξi−2−|β| hzi d(z, R+) 1+|α|+|β| . It remains to prove the continuity of the map, fixing z0 we can write
1 p2− z − 1 p2− z0 = (z − z0) 1 p2− z · 1 p2− z0 . From Lemma 3.2 we have continuity of the map
S−2× S0→ S−2 (a, b) 7→ ab, considering 1
p2− z0
∈ S0 and recalling that we have continuous embedding
S−2 ,→ S0 there exist N, N1, N2 such that
NNS−2 1 p2− z − 1 p2− z0 =|z − z0| NS −2 N 1 p2− z · 1 p2− z0 ≤ C1|z − z0| NS −2 N 1 1 p2− z NN 2S0 1 p2− z0
CHAPTER 5. PARAMETRIX 35 ≤ C2|z − z0| NS −2 N 1 1 p2− z NN 2S−2 1 p2− z0 ≤ C3|z − z0| hzi d(z, R+) M
for some suitable M , where in the last inequality we used estimate (5.3).
The existence of an approximate inverse of (h2P − z) is given by the following theorem.
Theorem 5.2 (Global parametrix). Consider P elliptic operator defined as in (5.1) and take z ∈ C \ [0, +∞), N ∈ N and h ∈ (0, 1].
There exist symbols djk ∈ S2j−k, which are polynomials in ξ independent from z and
h, such that if we define
Qz(h) := N −1 X k=0 hkOph(qz,−2−k), Rz(h) := Oph(rz,−N(h)) (5.6) with rz,−N(h) ∈ S−N and qz,−2 := 1 p2− z , qz,−2−k:= 2k X j=1 djk (p2− z)j+1 , (5.7)
operators Qz(h) and Rz(h) satisfy the following equality
(h2P − z)Qz(h) = I + hNRz(h). (5.8)
Moreover, for the remainder the following estimate holds |∂α x∂ β ξrz,−N(x, ξ, h)| ≤ Cαβhξi −N −|β| hzi d(z, R+) Mαβ (5.9) for all z ∈ C \ [0, +∞), h ∈ (0, 1], x, ξ ∈ Rn and more precisely the following maps
C \ [0, +∞) → S−2−k C \ [0, +∞) → S−N z 7→ qz,−2−k z 7→ rz,−N(h)
are continuous.
Remark 5.1. In the construction of the symbols qz,−2−k it can be found that for
k ≥ 1 they satisfy the following recursive relation qz,−2−k= −1 p2− z X j+l+m=k l<k (p2−j#qz,−2−l)m.
Heuristic construction for N = 2 In the previous theorem we would need to prove the following identity
(h2P − z) 1 X k=0 hkOph(qz,−2−k) ! = I + h2Oph(rz,−2(h)), (5.10)
where h2P − z is a sum of three terms, so the left hand side of (5.10) is Oph(p2− z) + hOph(p1) + h2Oph(p0) Oph(qz,−2) + hOph(qz,−3) . (5.11) By symbolic calculus, precisely Theorem 3.6, for any M, P, Q ∈ N we have
Oph(p2− z)Oph(qz,−2−k) = M −1 X j=0 hjOph(((p2− z)#qz,−2−k)j) + hMOph(r#M,2,k(h)), (5.12) Oph(p1)Oph(qz,−2−k) = P −1 X j=0 hjOph((p1#qz,−2−k)j) + hPOph(r#P,1,k(h)), (5.13) Oph(p0)Oph(qz,−2−k) = Q−1 X j=0 hjOph((p0#qz,−2−k)j) + hQOph(rQ,0,k# (h)), (5.14)
where we used the simplified notations
rM,2,k# (h) := r#M(p2− z, qz,−2−k, h),
rP,1,k# (h) := r#P(p1, qz,−2−k, h), (5.15)
rQ,0,k# (h) := rQ#(p0, qz,−2−k, h).
We want a remainder of order 2, hence each term of the form (5.12), (5.13) or (5.14) needs to be expanded up to suitable M, P or Q.
For example the terms Oph(p2− z)Oph(qz,−2−k), that are multiplied by the power
hk, will be expanded according to (5.12) with M = 2 − k and k = 0, 1, resulting in
Oph(p2− z)Oph(qz,−2) = Oph((p2− z)qz,−2) + hOph((p2#qz,−2)1)
+ h2Oph(r2,2,0# (h)),
for k = 0 and
Oph(p2− z)Oph(q−z,−3) = Oph((p2− z)qz,−3) + hOph(r#1,2,1(h))
CHAPTER 5. PARAMETRIX 37 The terms Oph(p1)Oph(qz,−2) and Oph(p1)Oph(qz,−3) instead are multiplied by
h and h2 respectively, hence we expand the composition in (5.13) once with P = 1 and once with P = 0, meaning in this last case that the sum is empty and (5.13) is reduced to the remainder only.
Oph(p1)Oph(q−z,−2) =Oph(p1qz,−2) + hOph(r#1,1,0(h)),
Oph(p1)Oph(q−z,−3) =Oph(r#0,1,1(h)).
For the last two terms Oph(p0)Oph(q−z,−2) and Oph(p0)Oph(q−z,−3) we expand
the composition with Q = 0 in both cases, obtaining
Oph(p0)Oph(q−z,−2) =Oph(r0,0,0# (h)),
Oph(p0)Oph(q−z,−3) =Oph(r0,0,1# (h)).
Now, if we replace all these expressions in (5.11) and regroup the terms with the same power of h we get
Oph((p2− z)qz,−2) + h Oph((p2#qz,−2)1) + Oph((p2− z)qz,−3) + Oph(p1qz,−2) + h2 Oph(r2,2,0# (h)) + Oph(r#1,2,1(h)) + Oph(r1,1,0# (h)) + Oph(r#0,1,1(h)) + Oph(r#0,0,0(h)) + hOph(r#0,0,1(h)) .
We recall that, by (5.10), this term must equal I + h2Oph(rz,−2(h)) therefore we
need first of all
Oph((p2− z)qz,−2) = I
and recalling that Oph(1) = I we can determine qz,−2 from
qz,−2=
1 p2− z
.
Once we have qz,−2 we can get qz,−3 by requiring the coefficient of h to be zero
qz,−3 =
−1 p2− z
(p1qz,−2+ (p2#qz,−2)1)
and it is easily seen that it satisfies the relation in Remark 5.1.
The continuous dependence on z follows from Proposition 5.1 and the properties about composition in Theorem 3.6.
Also by Theorem 3.6, the norms of the remainder terms have bounds depending on the norms of p2, p1, p0 and qz,−2, qz,−3.
Thanks to the fact that qz,−2 and qz,−3 involve p2− z and its derivatives, we can
Remark 5.2. Thanks to Theorem 3.7 and inequality (5.9) we have kRz(h)kL(L2(Rn))≤ C1 NS 0 (rz,−N(h)) ≤ C2 NS −N (rz,−N(h)) ≤ C3 hzi d(z, R+) M
for some M ≥ 0, where we used the fact that the embedding S−N ,→ S0 is
continu-ous. It follows that, for fixed z, hNRz(h) is small in operator norm when also h is
small, meaning that (h2P − z)Qz(h) is close to the identity in operator norm.
Remark 5.3. If we consider the symbol of Qz(h) as in S0 we see that it can be
extended to an operator on L2(Rn) thanks to Theorem 3.7. In the same way, also Rz(h) can be extended to act on L2(Rn).
Moreover, by Proposition 3.5 applied to Qz(h), whose symbols are in S−2, we have
that Qz(h) maps continuously Hs(Rn) in Hs+2(Rn).
5.2
Parametrix of a local elliptic operator
In the previous section we have proved that given P elliptic differential operator on Rn there exist Qz(h) and Rz(h) pseudodifferential operators such that
(h2P − z)Qz(h) = I + hNRz(h),
so if we define u := Qz(h)f ∈ H2 for some f ∈ L2(Rn) we have an “almost solution”
to the problem (h2P − z)v = f since
(h2P − z)u = f + hNRz(h)f. (5.16)
Now, we come to the case of a differential operator PV defined on an open set V of
Rn of the form PV := − n X j,k=1 gjk(x) ∂ 2 ∂xj∂xk + n X j=1 cj(x) ∂ ∂xj + c0(x) (5.17)
with smooth coefficients such that for any x ∈ V (gjk(x))j,k is a positive definite
matrix.
In particular the operator ∆g in local coordinates, as in (1.4), is of this form.
This time we want to find an “almost solution” to the problem (h2PV − z)˜v = ˜f
with ˜v and ˜f functions defined on V and with compact supports.
CHAPTER 5. PARAMETRIX 39 form (5.1) such that P |V = PV, namely we can consider PV as a restriction to V of
an operator defined on all of Rn.
As we said before, thanks to the previous section we have a function that is almost a solution to the global problem on Rn, we are now looking for a solution to the problem restricted on the open set V . It is then natural to consider the restriction to V of the global “almost solution”.
We define a localized datum ˜
f := χf, with χ ∈ C0∞(V0), V0⊂ V (5.18)
and consider
u := Qz(h) ˜f
that satisfies the relation in (5.16) with datum ˜f . Defining the localized function ˜
u := χ1u, with χ1 ∈ C0∞(V0), χ1 ≡ 1 near supp χ, (5.19)
we apply PV to the localized approximate solution ˜u obtaining
(h2PV − z)˜u =h2PV(χ1u) − χ1zu =
=χ1(h2PVu) − χ1(h2PVu) + h2PV(χ1u) − χ1zu
=χ1(h2PV − z)u + h2(PV(χ1u) − χ1PVu)
=χ1χf + hNχ1Rz(h)χf + [h2PV, χ1]u (5.20)
=χf + hNχ1Rz(h)χf + [h2PV, χ1](Qz(h)χf ) (5.21)
where the commutator in (5.20) is defined as
[h2PV, χ1] = h2(PVχ1− χ1PV). (5.22)
We remark that to get (5.20) we used the fact that u satisfies a relation of the form (5.16), while to obtain (5.21) we used the support properties required on χ1 that
imply
χ1(x)χ(x) = χ(x), for any x ∈ V0. (5.23)
We concentrate now on the last term of the expression in (5.21).
Consider other cutoff functions ˜χ, χ2∈ C0∞(V0) with the following support properties
˜
χ ≡ 1 near supp χ, hence ˜χχ ≡ χ, (5.24) χ1≡ 1 near supp ˜χ, hence χ1χ ≡ ˜˜ χ, (5.25)
the need of all these additional functions will be clarified later on (see Remarks 5.4, 5.5 and 5.6).
We want to prove the following identity
[h2PV, χ1](Qz(h)χ) = χ2([h2PV, χ1]Qz(h) ˜χ)χ, (5.27)
to do so, we expand the right hand side first, where we are composing on the right with the multiplication operator by χ and on the left with the multiplication by χ2,
χ2([h2PV, χ1]Qz(h) ˜χ)χ =χ2([h2PV, χ1]Qz(h)χ) (5.28)
=χ2(h2PV(χ1Qz(h)χ) − h2χ1PV(Qz(h)χ))
=χ2h2PV(χ1Qz(h)χ) − h2χ1PV(Qz(h)χ), (5.29)
where in (5.28) and (5.29) we applied properties (5.24) and (5.26) respectively. By the definition of differential operator it follows that for any u ∈ L2(Rn)
supp PV(χ1Qz(h)χu) ⊆ supp (χ1Qz(h)χ)u ⊆ supp χ1,
which means that whenever PV(χ1Qz(h)χ) is non zero, we also have χ2 ≡ 1 by
(5.26), hence continuing from (5.29) we get
χ2([h2PV, χ1]Qz(h) ˜χ)χ =χ2h2PV(χ1Qz(h)χ) − h2χ1PV(Qz(h)χ)
=h2PV(χ1Qz(h)χ) − h2χ1PV(Qz(h)χ)
=[h2PV, χ1](Qz(h)χ)
that is exactly (5.27).
Now, [h2PV, χ1]Qz(h) ˜χ is a composition between the differential operator [h2PV, χ1]
and the semiclassical operator Qz(h) ˜χ, it follows that the result is still a
semiclas-sical operator.
Given that PVχ1− χ1PV is a classic differential operator, its symbol will be a
poly-nomial in ξ and we notice that its coefficients vanish whenever χ1 ≡ 1, namely on
the support of ˜χ. We conclude that in [h2PV, χ1](Qz(h) ˜χ) one of the two operators
always has vanishing symbol, hence all the terms of the expansion given by symbolic calculus vanish and the composition reduces to the remainder only, that is
[h2PV, χ1](Qz(h) ˜χ) = hNOph(rN(h)) (5.30)
for some rN(h) ∈ S−N. The regularity of rN(h) follows from symbolic calculus
recalling that the symbol of Qz(h) ˜χ is in S−2 and the symbol of PV is in S2. Using
(5.30) in (5.27) we finally get
