Semi classical analysis of the complex Airy operator (γ = 1)174

The goal of this subsection is the proof of Theorem 5.3.1.

We introduce two model-operators, that have well known spectral and pseu- dospectral behavior. Let A⁽−R,+∞) and A⁽−∞,R) be the Dirichlet realizations of the operator−dy^d22+iyon the intervals (−R,+∞) and (−∞, R) respectively.

We are going to approximate the resolvent ofA⁽−R,R)by the one of A⁽−R,+∞)

orA⁽−∞,R)depending on where we are, respectively close to−Ror close to +R.

Let us remark that, if

TR:u(x)7→u(x+R) and UR:u(x)7→u(R−x) (5.3.7) then

T_R⁻¹(A⁽−R,+∞)−λ)TR=A^(0,+∞)−(λ+iR), (5.3.8) U_R⁻¹(A⁽−∞,R)−λ)UR=A^∗(0,+∞)−(λ−iR), (5.3.9) thus

inf Reσ

A(−R,∞)

= inf Reσ

A(−∞,R)

=|µ1|

2 , (5.3.10)

because inf Reσ

A(0,+∞)

=|µ1|/2, see [7].

Step 1 : We prove lim

R→+∞

inf Reσ

A⁽−R,R)

>|µ1|

2 (5.3.11)

and (5.3.2).

Letε >0. We searchRε>0 such that

∀R≥Rε, σ

A⁽−R,R)

∩(]− ∞,|µ1|/2−ε] +iR) =∅. (5.3.12)

5.3. NONOBSERVABILITY ON A VERTICAL STRIP 175

We recall that, by [75], there existsCε>0 such that sup

γ≤ |µ1|/2−ε, ν∈R

A(0,+∞)−(γ+iν)−1

L(L²(0,+∞)) 6Cε, (5.3.13) sup

γ≤ |µ1|/2−ε, ν∈R

A^∗(0,+∞)−(γ+iν)−1

L(L²(0,+∞)) 6Cε. (5.3.14) Let

λ=γ+iν , γ≤ |µ1|/2−ε], ν∈R, (5.3.15) andh+, h₋∈ C^∞(R; [0,1]) be such that

Supp (h₋)⊂(−∞,1/2), h₋≡1 on (−∞,−1/2], Supp (h+)⊂(−1/2,+∞), h+≡1 on [1/2,+∞),

h²₋+h²₊≡1 on (−∞,+∞). ForR >0, we define

η^±_R(x) =h_±x R

1(−R,R)(x) (5.3.16)

and

R^R(λ) =η⁻_R

A⁽−R,+∞)−λ−1

η⁻_R+η_R⁺

A⁽−∞,R)−λ−1

η⁺_R. (5.3.17) R^R(λ) will be used as an approximation of the resolvent ofA⁽−R,R). We have

A⁽−R,R)−λ

R^R(λ) =I + [A⁽−R,R), η⁻_R]

A⁽−R,+∞)−λ−1

η_R⁻ + [A⁽−R,R), η⁺_R]

A⁽−∞,R)−λ−1

η⁺_R(5.3.18) as an equality between operators onL²(−R, R).

We estimate the second term on the right hand side. In what follows, the esti- mates are uniform with respect toν= Imλ. We have

[A⁽−R,R), η_R⁻]

A⁽−R,+∞)−λ−1

η⁻_R=

−(η⁻_R)^′′−2(η_R⁻)^′ d

dy A⁽−R,+∞)−λ−1

η_R⁻, (5.3.19) Usingk(η_R⁻)^′k^L∞(−R,R)=O(R⁻¹) andk(η⁻_R)^′′k^L∞(−R,R)=O(R⁻²), we get, by (5.3.8) and (5.3.13),

(η_R⁻)^′′

A(−R,+∞)−λ−1

η⁻_R

L(L²(−R,R)) =O 1

R²

. (5.3.20)

Moreover, for everyv∈L²(−R,+∞),

^dyd

A⁽−R,+∞)−λ−1

v _L2(

−R,+∞)

≤

A⁽−R,+∞)−λ−1

1/2

+√γ

A⁽−R,+∞)−λ−1

kvk^L2(−R,+∞). (5.3.21)

176 CHAPITRE 5. PARABOLIC OPERATORS WITH A GCC Indeed, letw:= (A(−R,+∞)−λ)⁻¹v, i.e.

−w^′′(y) +iyw(y)−λw(y) =v(y), y∈(−R,+∞), w(−R) =w(+∞) = 0.

We have

kw^′k²L²(−R,+∞) =−Re

+R∞

−R

w(y)w^′′(y)dy

!

= Re

+R∞

−R

w[iyw+λw+v]

!

=γ

+R∞

−R |w|²+ Re

+R∞

−R

wv

!

6γkwk²L²(−R,+∞)+kwkL²(−R,+∞)kvkL²(−R,+∞). By taking the square root of this inequality, we get

kw^′k^L2(−R,+∞)6√γkwk^L2(−R,+∞)+kwk^1/2L²(−R,+∞)kvk^1/2L²(−R,+∞), which proves (5.3.21). By applying (5.3.21) tov=η⁻_Ru,u∈L²(R), we get

(η_R⁻)^′ d

dy

A(−R,+∞)−λ−1

η_R⁻

L(L²(−R,R))=O 1

R

, (5.3.22)

which gives, with (5.3.19) and (5.3.20),

[A(−R,R), η⁻_R]

A(−R,+∞)−λ−1

η⁻_R

L(L²(−R,R)) =O 1

R

. (5.3.23) In the same way, we verify that

[A⁽−R,R), η_R⁺]

A⁽−∞,R)−λ−1

η_R⁺

L(L²(−R,R))=O 1

R

. (5.3.24) Equality (5.3.18) can be written

(A⁽−R,R)−λ)R^R(λ) =I+E^R(λ),

withkE^R(λ)kL(L²(−R,R)) =O(R⁻¹), uniformly with respect toλ∈]−∞,|µ1|/2− ε] +iR. We deduce the existence of Rε > 0 such that, for every R ≥ Rε, (A(−R,R)−λ) is invertible, with inverse

A⁽−R,R)−λ−1

=R^R(λ)

I+E^R(λ)−1

.

We have proved (5.3.12). Moreover, according to the definition (5.3.17) ofR^R(λ), (5.3.8), (5.3.9), (5.3.13) and (5.3.14) yield the estimate (5.3.2).

Step 2 : We prove that

R→lim+∞

inf Reσ

A⁽−R,R)

6 |µ1|

2 . (5.3.25)

5.3. NONOBSERVABILITY ON A VERTICAL STRIP 177 First, we reduce the study to the complex Airy operator A(0,R) on the in- terval (0, R) . Indeed, applying the translationTR:u(x)7→u(x+R), we get

T_R⁻¹(A(−R,R)−λ)TR=A(0,2R)−(λ+iR),

thus Reσ(A(−R,R)) = Reσ(A(0,2R)) . Therefore, in order to prove (5.3.25), we are going to prove that

R→lim+∞ inf Reσ A(0,R)

6 |µ1|

2 . (5.3.26)

Letθ1,θ2∈ C^∞(R; [0,1]) be such that

Supp (θ1)⊂(−∞,2/3), θ1≡1 on (−∞,1/2), Supp (θ2)⊂(1/2,+∞), θ2≡1 on (2/3,+∞),

θ₁²+θ₂²≡1 onR. Forj= 1,2 andR >0 , we define

χ^j_R(x) =θj

x R

1(0,R)(x). (5.3.27)

We want to prove that 1(0,R)

A^(0,R)+ 1−1

1(0,R)_R−→

→+∞

A^(0,+∞)+ 1−1

inL(L²(R⁺)). (5.3.28) Let us remark that

σ

1(0,R)

A^(0,R)+ 1−1

1(0,R)

=σ

A^(0,R)+ 1−1

with non vanishing eigenvalues that have the same multiplicity for both opera- tors.

Step 2.a : We prove that 1(0,R)

A^(0,R)+ 1−1

1(0,R)−χ¹_R

A^(0,+∞)+ 1−1

χ¹_RR−→

→+∞0 in L(L²(R⁺)). For this, we use the following approximations of the resolvent of (A^(0,R)+ 1),

R˜^R=χ¹_R

A^(0,+∞)+ 1−1

χ¹_R+χ²_R

A^(0,R)+ 1−1

χ²_R. Then, we have

(A^(0,R)+ 1) ˜R^R=I + [A^(0,R)+ 1, χ¹_R]

A^(0,+∞)+ 1−1

χ¹_R + [A^(0,R)+ 1, χ²_R]

A^(0,2R)+ 1−1

χ²_R, thus, by composing on the left by1(0,R)

A(0,R)+ 1−1

1(0,R), we get 1(0,R)

A^(0,R)+ 1−1

1(0,R)−χ¹_R

A^(0,+∞)+ 1−1

χ¹_R=χ²_R

A^(0,R)+ 1−1

χ²_R

−1(0,R)

A^(0,R)+ 1−1

1(0,R)[A^(0,R)+ 1, χ¹_R]

A^(0,+∞)+ 1−1

χ¹_R

−1(0,R)

A^(0,R)+ 1−1

1(0,R)[A^(0,R)+ 1, χ²_R]

A^(0,R)+ 1−1

χ²_R. (5.3.29)

178 CHAPITRE 5. PARABOLIC OPERATORS WITH A GCC Now, we control the different terms on the right hand side. The terms involving commutators can be estimated as in Step 1, thanks to (5.3.2), and we get

1(0,R)

A(0,R)+1−1

1(0,R)[A(0,R)+1, χ¹_R]

A(0,+∞)+1−1

χ¹_R

L(L²(R+))=O 1

R

, (5.3.30)

1_(0,R)

A(0,R)+1−1

1_(0,R)[A(0,R)+1, χ²_R]

A(0,R)+1−1

χ²_R|L(L²(R+))=O 1

R

. (5.3.31) Moreover, foru∈L²((0, R),C), we have

Im

(A(0,R)+ 1)u , u

=hyu , ui (5.3.32) whereh., .idenotes theL²((0, R),C)-hermitian product.

This relation, applied to u=χ²_R

A^(0,R)+ 1−1

χ²_Rf,f ∈L²(0,+∞), which is supported in (R/2, R), gives

Im

(A(0,R)+ 1)u , u> R 2kuk². Moreover,

(A(0,R)+ 1)u= (χ²_R)²f+ [A(0,R)+ 1, χ²_R]

A(0,R)+ 1−1

χ²_Rf . Thus, estimating the commutator as in Step 1, we get

Im

(A^(0,R)+ 1)u, u6C

1 + 1 R

kfkkuk. Therefore,

R

2kuk²6C

1 + 1 R

kfkkuk. We have proved that

χ²_R

A(0,R)+ 1−1

χ²_R

L(L²(0,+∞)) =O 1

R

. (5.3.33)

By (5.3.29), (5.3.30), (5.3.31) and (5.3.33), we have

1(0,R)

A^(0,R)+ 1−1

1(0,R)−χ¹_R

A^(0,+∞)+ 1−1

χ¹_R

L(L²(0,+∞))=O 1

R (5.3.34) which ends Step 2.a.

Step 2.b : We verify that χ¹_R

A(0,+∞)+ 1−1

χ¹_R −→

R→+∞

A(0,+∞)+ 1−1

inL(L²(0,+∞)), (5.3.35) which ends the proof of (5.3.28).

To simplify notation, let us introduce

A⁺=A^(0,+∞)+ 1.

5.3. NONOBSERVABILITY ON A VERTICAL STRIP 179 First, we write

χ¹_RA⁻+¹χ¹_RA⁺= (χ¹_R)²−χ¹_RA⁻+¹[A⁺, χ¹_R],

then, composing on the right byA⁻+¹ and using that (χ¹_R)²= 1−(χ²_R)², A⁻+¹−χ¹_RA⁻+¹χ¹_R= (χ²_R)²A⁻+¹+χ¹_RA⁻+¹[A⁺, χ¹_R]A⁻+¹. (5.3.36) The term involving a commutator can be estimated as in Step 1,

χ¹_RA⁻+¹[A⁺, χ¹_R]A⁻+¹

L(L²(R⁺))=O 1

R

. (5.3.37)

Forf ∈L²(0,+∞), we have R

2k(χ²_R)²A⁻+¹fk² 6ky^1/2(χ²_R)²A⁻+¹fk² (because Supp (χ²_R)⊂(R/2, R))

= ImhA⁺(χ²_R)²A⁻+¹f ,(χ²_R)²A⁻+¹fi 6kA⁺(χ²_R)²A⁻+¹fkk(χ²_R)²A⁻+¹fk 6

k(χ²_R)²fk+k[A⁺,(χ²_R)²]A⁻+¹fk

k(χ²_R)²A⁻+¹fk, whereh., .idenotes theL²((0,+∞),C)-hermitian product andk.kis the associ- ated norm. Estimating the term with a commutator as in Step 1, we get

Rk(χ²_R)²A⁻+¹fk^L2(0,+∞)6C

1 + 1 R

kfk^L2(0,+∞). Thus (χ²_R)²A⁻+¹

L(L²(0,+∞)) =O 1

R

. (5.3.38)

Finally, (5.3.36), (5.3.37) and (5.3.38) imply (5.3.35).

Step 2.c : Conclusion.

Step 2.a and Step 2.b prove (5.3.28). The eigenvalues ofA⁻+¹are isolated, thus we can apply [96, Section IV, §3.5]. For any subsequence Rj → +∞ and any eigenvalueλ∈σ(A⁻+¹)\ {0}, there exists a sequence (λj) such that, for everyj large enough

λj ∈σ

1(0,Rj)

A^(0,Rj)+ 1−1

1(0,Rj)

\ {0}=σ

A^(0,Rj)+ 1−1

\ {0} andλj→λwhenj→+∞.

In particular, with λ = 1/(˜λ+ 1), where ˜λ = e^iπ/3|µ1| ∈ σ(A(0,+∞)) is the eigenvalue ofA(0,+∞) with smallest real part (see [7]), we get a sequence ˜λj = 1/λj−1∈σ(A^(0,Rj)) such that Re ˜λj →Re ˜λ=|µ1|/2, from which we deduce (5.3.25).

5.3.4 Semi classical analysis of the Davies operator ( γ = 2)

The goal of this section is the proof of Theorem 5.3.2, which is similar to the one of Theorem 5.3.1.

180 CHAPITRE 5. PARABOLIC OPERATORS WITH A GCC Step 1 : Let ε >0. We search Rε>0 such that

∀R≥Rε, σ H⁽−R,R)

∩

(−∞,√

2/2−ε) +iR

=∅ (5.3.39)

and we prove (5.3.4).

Letα∈(0,1/3) andζ_R¹, ζ_R², ζ_R³ ∈ C^∞(R; [0,1]) be such that

Suppζ_R¹ ⊂(−∞,−R+R^α), ζ_R¹ ≡1 on (−∞,−R+R^α/2), Suppζ_R² ⊂(−R+R^α/2, R−R^α/2), ζ_R² ≡1 on (−R+R^α, R−R^α),

Suppζ_R³ ⊂(R−R^α,+∞), ζ_R³ ≡1 on (R−R^α/2,+∞), (ζ_R¹)²+ (ζ_R²)²+ (ζ_R³)²≡1 onR,

k(ζ_R^j)^′kL^∞(R)= O

R→+∞(R^−α), k(ζ_R^j)^′′kL^∞(R)= O

R→+∞(R^−2α), (5.3.40) Close toy=−R, we have

y²=−2R(y+R) +R²+o(|y+R|).

Thus, we are going to approximate H(−R,R), close toy =−R, by the complex Airy type operator on (−R,+∞)

A⁻R :=− d²

dy²−2iR(y+R) +iR².

In the same way, we will approximateH(−R,R) close toy= +Rby the complex Airy type operator on (−∞,+R)

A⁺R:=− d²

dy² −2iR(R−y) +iR².

Then, we remark that, ifTR andUR are defined by (5.3.7), then we have A⁻R=TRA˜^∗2RT_R⁻¹+iR² and A⁺R=URA˜^∗2RU_R⁻¹+iR²,

where ˜A^R is the Dirichlet realization of the complex Airy operator−dy^d22+iRy on (0,+∞).

Following [75], we deduce that inf Reσ A⁺R

= inf Reσ A⁻R

= (2R)^2/3|µ1|

2 , (5.3.41)

and, for everyε >0, there existsCε>0 such that sup

γ∈[0, R2/3|µ1|/2−ε], ν∈R

A^±R−(γ+iν)−16 Cε

R^2/3. (5.3.42) We call H⁰ the complex harmonic oscillator −dy^d22 +iy² onR, that will serve to approximate H(−R,R) on the support of ζ_R². We recall that inf Reσ(H⁰) = cosπ/4 =√

2/2 (see [41]) and sup

γ≤√2/2−ε, ν∈R

H⁰−(γ+iν)−16C_ε^′, (5.3.43)

5.3. NONOBSERVABILITY ON A VERTICAL STRIP 181 for someC_ε^′ >0, see for instance [116].

Now, we takeλ=γ+iν∈(0,√

2/2−ε) +iRand we set Q^R(λ) =ζ_R¹

A⁻R−λ−1

ζ_R¹ +ζ_R²

H⁰−λ−1

ζ_R² +ζ_R³

A⁺R−λ−1

ζ_R³. (5.3.44) Then, we have

(H⁽−R,R)−λ)Q^R(λ) =I+ [H⁽−R,R), ζ_R¹]

A⁻R−λ−1

ζ_R¹ +[H⁽−R,R), ζ_R²]

H⁰−λ−1

ζ_R² + [H⁽−R,R), ζ_R³]

A⁺R−λ−1

ζ_R³ +ζ_R¹(H(−R,R)− A⁻R)

A⁻R−λ−1

ζ_R¹ +ζ_R³(H(−R,R)− A⁺R)

A⁺R−λ−1

ζ_R³, as equality between operators onL²(−R, R) . The terms involving commutators can be estimated as in Step 1 of the previous section, by using (5.3.40), (5.3.42), (5.3.43) and we get

[H⁽−R,R), ζ_R¹]

A⁻R−λ−1

ζ_R¹

L(L²(−R,R))+[H⁽−R,R), ζ_R²]

H⁰−λ−1

ζ_R²

L(L²(−R,R))

+[H⁽−R,R), ζ_R³]

A⁺R−λ−1

ζ_R³

L(L²(−R,R))=O(R^−α). Moreover, we have, by definition ofA⁻R,

(H⁽−R,R)− A⁻R)u(y) =i(y+R)²u(y),

and on the support ofζ_R¹, we havey+R≤R^α. Therefore, by (5.3.42)

ζ_R¹(H⁽−R,R)− A⁻R)

A⁻R−λ−1

ζ_R¹

L(L²(−R,R)) ≤ R^2α

A⁻R−λ−1

L(L²(−R,+∞))

≤ CεR^2(α−1/3). In the same way, we verify

ζ_R³(H(−R,R)− A⁺R)

A⁺R−λ−1

ζ_R³

L(L²(−R,R)) ≤CεR^2(α−1/3). Thus, we have proved that

(H(−R,R)−λ)Q^R(λ) =I+ ˜E^R(λ),

with kE˜^R(λ)k → 0 as R → +∞, uniformly with respect to λ in the interval (0,√

2/2−ε) +iR. Thus, there exists Rε > 0 such that, for every R ≥ Rε, (H(−R,R)−λ) is invertible, with

H(−R,R)−λ−1

=Q^R(λ)

I+ ˜E^R(λ)−1

. (5.3.45)

This proves the existence ofRǫ >0 such that (5.3.39) holds. The resolvent es- timate (5.3.4) follows from (5.3.42), (5.3.43) and (5.3.44).

182 CHAPITRE 5. PARABOLIC OPERATORS WITH A GCC Step 2 : We prove

R→lim+∞inf Reσ H⁽−R,R)6

√2

2 . (5.3.46)

Letϕ¹_R, ϕ²_R∈C^∞(R,[0,1]) be such that

Supp (ϕ¹_R)⊂(−∞,−R/2)∪(R/2,+∞), ϕ¹_R≡1 on (−∞,−2R/3)∪(2R/3,+∞), Supp (ϕ²_R)⊂(−2R/3,2R/3), ϕ²_R≡1 on (−R/2, R/2),

(ϕ¹_R)²+ (ϕ²_R)²≡1 on R, k(ϕ^j_R)^′kL^∞(R)=O R⁻¹

, k(ϕ^j_R)^′′kL^∞(R)=O R⁻² . We recall thatH⁰denotes the operator−dx^d22 +ix² defined onR, and we set

Q˜^R=ϕ²_R

H⁰+ 1−1

ϕ²_R+ϕ¹_R

H(−R,R)+ 1−1

ϕ¹_R. Thus, we have

H⁽−R,R)+ 1Q˜^R=I+P^R, where

P^R= [H(−R,R), ϕ²_R]

H⁰+ 1−1

ϕ²_R+ [H(−R,R), ϕ¹_R]

H(−R,R)+ 1−1

ϕ¹_R, and

kP^RkL(L²(−R,R))=O(R⁻¹). (5.3.47) By composing on the left with (H⁽−R,R)+ 1)⁻¹, we get

H⁽−R,R)+1−1

−ϕ²_R

H⁰+1−1

ϕ²_R=ϕ¹_R

H⁽−R,R)+1−1

ϕ¹_R−

H⁽−R,R)+1−1

P^R. (5.3.48) By going back over the proof of (5.3.33) and replacing (5.3.32) by

Im

H(−R,R)u, u

=hx²u, ui, (5.3.49) we get

ϕ¹_R

H(−R,R)+ 1−1

ϕ¹_R

L(L²(−R,R))=O 1

R

.

By (5.3.48), the previous relation, together with (5.3.47) and (5.3.4) imply

H⁽−R,R)+ 1−1

−ϕ²_R

H⁰+ 1−1

ϕ²_R

L(L²(−R,R))=O 1

R

. (5.3.50) Then, we prove that the operator ϕ²_R(H⁰+ 1)⁻¹ϕ²_R converges to (H⁰+ 1)⁻¹ in L(L²(R)), when R → +∞, with the same arguments as in Step 2.b of the previous section. Thus, (5.3.46) is proved, with the same arguments as in Step 2.c of the previous section, and this ends the proof of Theorem 5.3.2.

No documento Ecole Doctorale de Math´ ematiques de la R´ egion Paris-Sud (páginas 174-183)