NOTES ON RANDOM PERTURBATION OF NON-SELF-ADJOINT OPERATORS ZHONGKAI TAO Contents 1. Introduction 2 1.1. Motivation from differential equations 2 2. Random perturbation of Jordan blocks 3 2.1. Review of Spectral theory 4 2.2. Grushin problem 7 2.3. Review of Probability theory 14 2.4. Proof 16 3. Random perturbation of differential operators 19 3.1. Unbounded operators 19 3.2. Hager’s theorem 23 3.3. Semiclassical analysis 23 4. Higher oder generalizations 35 4.1. Basic constructions 36 5. WKB methods for analytic PDEs 41 References 43 1
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NOTES ON RANDOM PERTURBATION OF NON-SELF-ADJOINTOPERATORS
ZHONGKAI TAO
Contents
1. Introduction 2
1.1. Motivation from differential equations 2
2. Random perturbation of Jordan blocks 3
2.1. Review of Spectral theory 4
2.2. Grushin problem 7
2.3. Review of Probability theory 14
2.4. Proof 16
3. Random perturbation of differential operators 19
3.1. Unbounded operators 19
3.2. Hager’s theorem 23
3.3. Semiclassical analysis 23
4. Higher oder generalizations 35
4.1. Basic constructions 36
5. WKB methods for analytic PDEs 41
References 43
1
2 ZHONGKAI TAO
1. Introduction
This is the notes from Professor Maciej Zworski’s Spring 2021 topics course at Berke-
ley. The primary reference is Sjostrand’s book [Sj19].
1.1. Motivation from differential equations. One central problem of PDEs is the
stability of the equation under perturbation, in particular, the nonlinear perturbation.
Example 1. Consider the equation
∂tu = Au+ F (u), A ∈MN×N(C), F (u) = O(|u|ε).
Here F is considered as a small perturbation of the ODE. If F = 0, then as long as
σ(A) has negative real parts, the system is stable. However, let A = JN − 1/2 where
JN is the Jordan block matrix. Let
F (u) =
u2
1
0...
0
.
Then for initial value
u0 =
0
0...
ε
,
the system will blow up for ε ∼(
34
)N. [Lack of proof or reference here.]
Example 2. Here is a PDE version of our previous example. Consider the following
PDE
∂tu =1
ihPu+ au2
where P = hi∂x + ig(x) and g(x) is a real valued smooth function on T = R/2πZ. For
the linear problem, we can simply solve it and get the eigenvalues of P .
z = kh+ ig, k ∈ Z, g =1
2π
∫Tg(x)dx.
Suppose g < 0, then the linear problem is stable. However, for the nonlinear (quadratic)
perturbation, the system will blow up. Let us write the equation as
(∂t + ∂x)u =1
hg(x)u+ bu2
NOTES ON RANDOM PERTURBATION OF NON-SELF-ADJOINT OPERATORS 3
where b = − iah
. Let
G(a, b) =
∫ a
b
g(τ)dτ,
the solution is
u(t, x) =e
1hG(x,x−t)u0(x− t)
1− be 1hG(x,x−t)u0(x− t)
∫ t0e
1hG(x−s,x)ds
.
If g(x0) > 0 and bu0 is a bump function at x0, we find a blow up at time t ∼ tδ for
initial data u0 of size e−h−1+δ
.
Example 3. Here is another example by.
∂tu = (−∂2x + ∂x +
1
8)u+ u2.
[Lack of proof or reference here.]
The nonlinear instability is related to the linear stability as shown in the following
theorem by Hager.
Theorem 1. Let
Pδ = hDx + ig(x) + δQ
on T1 where Q is a random operator with
Qu =∑
|k|,|l|.h−1
αk,l(ω)〈u, ek〉el(x), el(x) = (2π)−1/2e2πilx
and αk,l are i.i.d standard Gaussian random variables. Let Γ b Ω = z ∈ C :
min Reg < Rez < max Reg, then for e−h−1+ε ≤ δ ≤ h4, we have
](σ(Pδ) ∩ Γ) ∼ 1
2πh|p−1(Γ)|
where p(x, ξ) = ξ + ig(x).
2. Random perturbation of Jordan blocks
In this section we are going to prove the following theorem.
Theorem 2. Let JN : CN → CN be the Jordan block matrix, and Q is a matrix with
entries i.i.d. standard Gaussian random variables. Let e−N1−ε ≤ δ ≤ N−4, then
1
N
∑λ∈σ(JN+δQ)
δλ →1
2πδS1 .
4 ZHONGKAI TAO
2.1. Review of Spectral theory. In this section we breifly review the basics of
spectral theory.
Definition 1. Let A be a bounded operator on a Banach space, then the resolvant set
is
ρ(A) = z ∈ C : A− z is inertible
and the spectrum of A is
σ(A) = C \ ρ(A).
Roughly speaking, spectral theory is the study of spectrum of linear operators. An
important observation from linear algebra suggests some operators behave better from
the spectral point of view.
Definition 2. An bounded operator A on a Hilbert space H is called
• a self-adjoint operator if A = A∗,
• a unitary operator if AA∗ = A∗A = I,
• a normal operator if [A,A∗] = AA∗ − A∗A = 0.
For self-adjoint operators, we have the following psectral theorem.
Theorem 3. If A is a self-adjoint operator on a Hilbert space H, then there exists a
projection-valued measure dE(λ) such that
A =
∫σ(A)
λdE(λ).
This is also true for normal operators, since we can always write any normal operator
A = ReA+ iImA, where
ReA =1
2(A+ A∗), ImA =
1
2i(A− A∗).
Also, self-adjoint operators only have real spectrum. If A is self-adjoint, then by
spectral theorem we obtain
‖(A− z)−1‖ =1
d(z, σ(A)).
But this is dramatically not true for non-self-adjoint operators, as the following exam-
ple shows.
Example 4. Let JN ∈MN×N(C) be the Jordan block matrix, then
(JN − z)−1 = −z−1
(1− JN
z
)−1
= −z−1
N−1∑k=0
JkNz−k
NOTES ON RANDOM PERTURBATION OF NON-SELF-ADJOINT OPERATORS 5
and
‖(JN − z)−1‖ ≥ |z|−N .
To study non-self-adjoint operators, there is a more ’stable’ version of spectrum:
pseudospectrum.
Definition 3. Let A be a bounded operator on a Hilbert space, define the ε-pseudospectrum
of A as
σε(A) =
z ∈ C : ‖(A− z)−1‖ > 1
ε
∪ σ(A).
We have the following direct properties.
Proposition 4. • σ(A) +D(0, ε) ⊂ σε(A)
• When A is a normal operator, σε(A) = σ(A) +D(0, ε).
Also, we have the following equivalent definitions.
Proposition 5. The following are equivalent.
• (a) z ∈ σε(A);
• (b) There exists u with ‖u‖ = 1 and ‖(A− z)u‖ < ε;
• (c) There exists an operator B with ‖B‖ < 1 such that z ∈ σ(A+ εB).
Proof. Only (b)⇒ (c) is not trivial. But taking
Bv = −(A− z)u
ε(v, u)u
would work.
We also have another property of pseudospectrum.
Proposition 6. If U is a bounded component of σε(A), then U ∩ σ(A) is nonempty.
Proof. We recall a function on Ω ⊂ C is called subharmonic if it is upper semi-
continuous and for any h harmonic in K b Ω, u ≤ h on ∂K implies u ≤ h in
K.
By writing
‖(A− z)−1‖ = sup‖u‖=‖v‖=1
Re〈(A− z)−1u, v〉
as the supremum of a family of harmonic functions, we obtain ‖(A − z)−1‖ is a sub-
harmonic function in z.
6 ZHONGKAI TAO
If U ⊂ ρ(A), then ‖(A − z)−1‖ is subharmonic in U . Since ‖(A − z)−1‖ = ε−1 on
∂U , by subharmonicity we have
‖(A− z)−1‖ ≤ 1
εon U
This is a contradiction.
2.1.1. Properties of matrix exponentials. There is a general theorem by Trefethen-
Embree relating matrix exponentials and its spectrum.
Theorem 4.
limt→+∞
t−1 log ‖etA‖ = α(a) := max Reσ(A). (2.1)
limt→0+
t−1 log ‖etA‖ = ω(a) := max σ(ReA). (2.2)
etα(A) ≤ ‖etA‖ ≤ etω(A), t ≥ 0. (2.3)
Proof. To prove (2.1), we write the Jordan normal form A = V JV −1, then
Now our lemma follows easily: First ∂αz ∂βz eWKB is of tempered growth, then by our
estimate of ∂αz ∂βz Π(z), e0 is also of tempered growth. Then since e0 − eWKB is small,
we get ∂αz ∂βz (e0 − eWKB) = O(e−
1Ch ) be interpolation
|f ′(0)| ≤ Cε(‖f‖12
L∞(−ε,ε)‖f′′‖
12
L∞(−ε,ε) + ‖f‖L∞(−ε,ε)).
Lemma 42.
Re∆F = 4Re∂zf =2
h
(1
1ip, p(ρ+)
− 11ip, p(ρ−)
)+O(1).
Proof. Recall f+ = (e0, ∂ze0) = (eWKB, ∂zeWKB)+O(e−1Ch ). A direct calculation shows
that
(eWKB, ∂zeWKB) = − ih∂zφ+(x+(z), z) +O(1)
=i
hξ+(z)∂zx+(z) +O(1).
So
Re∂zf+ = Rei
2h∂zx+(z) +O(1).
30 ZHONGKAI TAO
A similar computation for f− proves the lemma.
Corollary 43.
Re∆Fdy ∧ dx =1
h(dξ+ ∧ dx+ − dξ− ∧ dx−).
3.3.1. The Grushin problem. To prove Hager’s theorem, we set up the following Grushin
problem.
Pδ(z) =
(P − z + δQ R−
R+ 0
).
The following lemma is similar to the one we proved before.
Lemma 44.
‖Q‖HS ≤C
h
with probability ≥ 1−O(e−1
Ch2 ).
Now we know ‖P(z)‖ = O(h−1/2), so for ‖δQ‖ √h we have Pδ(z) is invertible.
A direct calculation shows that
Eδ = E +O(δ
h2
)Eδ
+ = E+ +O(δ
h32
)Eδ− = E− +O
(δ
h32
)Eδ−+ = E−+ − δE−QE+ +O
(δ2
h52
).
Lemma 45.
|eWKB(k)| = O((
h
|k|
)∞).
Proof. The crucail thing is
eWKB ≈ h−14e−
x2
h .
A direct calculation shows that
∂nxeWKB(x) . h−14
((xh
)n+ h−
n2
)e−
x2
h
NOTES ON RANDOM PERTURBATION OF NON-SELF-ADJOINT OPERATORS 31
and ∫eWKB(x)e−ikxdx =
1
kn
∫DnxeWKB(x)e−ikxdx
. h−14k−nh−
n2
. h−14h
n4 |k|−
n4 .
Corollary 46.
E−QE+ ∼ NC(0, 1−O(h∞)).
Proof. This is because
E−QE+ = 〈f0, Qe0〉
=∑
|k|,|j|≤Ch
αjk(ω)f0(j)e0(k)
∼ NC(0,∑
|k|,|j|≤Ch
|e0(k)|2|f0(j)|2).
Now we have
Proposition 47. For 0 < t 1, 0 < δ h32 , δt e−
1Ch , t δ
h52
, we have
• ”∀z ∈ Ω, |Eδ−+(z)| ≤ e−
1Ch + Cδ
h”, with probability ≥ 1−O(e−
1Ch ).
• ∀z ∈ Ω, ”|Eδ−+(z)| ≥ tδ
C, with probability ≥ 1−O(t2)−O(e−
1Ch )”.
Proof. This follows from
Eδ−+ = E−+ − δE−QE+ +O
(δ2
h52
).
3.3.2. Counting zeros of holomorphic functions. Now we can estimate the zeros of
Eδ−+(z) by the following lemma due to Hager-Sjostrand.
Theorem 13. Let Ω b Ω b C, ∂Ω is smooth. ϕ ∈ C2(Ω), z 7→ u(z, h) is a holomor-
phic function in Ω, 0 < ε 1. Suppose
• |u(z, h)| ≤ exp( 1h(ϕ(z) + ε)), for z ∈ nbhd(∂Ω).
• z1, z2, · · · , zn ∈ ∂Ω, zj = zj(h), N ∼ 1√ε, and ∂Ω ⊂ ∪jD(Zj,
√ε), such that
|u(zj, h)| ≥ exp(1
h(ϕ(z)− ε)).
32 ZHONGKAI TAO
Then
]u−1(0) ∩ Ω =1
2πh
∫Ω
∆ϕdm(z) +O(
√ε
h).
This theorem follow from the local version of Hadamard’s factorization theorem.
Theorem 14. Suppose f(z) is a holomorphic function in |z| ≤ 2R and |f(z)| ≤ M
for |z| ≤ 2R. Also, |f(0)| ≥M−1. Then there exists C > 0 idependent of R such that
f(z) = eiθeg(z)N∏j=1
(z − zj), |z| ≤ R,
where zj are zeros of f in |z| ≤ 3R2
, and
N ≤ C logM, |g(z)| ≤ C logM(1 + log〈R〉).
Proof. We will use three steps to prove this theorem.
Step 1: Jensen’s formula.
log |f(0)|+∫ r
0
N(t)
tdt =
1
2π
∫ 2π
0
log |f(reiθ)|dθ.
Suppose f(z) does not no zeros in |z| ≤ r, then it follows directly from the fact
that Re log f(z) is a harmonic function.
If f(z) has no zero on the circle |z| = r, then we can apply the formula to
f(z) =N∏j=1
r2 − zzjr(z − zj)
f(z)
and get the desired formula. Finally, the case when there are zeros on the circle
|z| = r follows by continuity.
The estiamte for the number of zeros N ≤ C logM follows directly from
Jensen’s formula. But to find a bound for g(z), we need a lower bound for the
polynomialN∏j=1
(z − zj), which is obtained by the following Cartan’s lemma.
Step 2: Cartan’s lemma.
Lemma 48. Let µ be a finite Radon measure on C and consider the logrithmic
potential of µ:
u(z) =
∫C
log |z − ζ|dµ(ζ).
Then for any 0 < η < 1, there exists a set of discs Cj of radii rj, s.t.
–∑j
rj < 5η
– For z /∈ ∪Cj, |u(z)| ≥ µ(C) log ηe.
For polynomials, the constant 5 can be replaced by 2.
NOTES ON RANDOM PERTURBATION OF NON-SELF-ADJOINT OPERATORS 33
Proof. We only prove for the polynomial case, since this is the case we will be
using. Let Z = zj with multiplicity, and set
C = D(z, λη
N) : ]Z ∩D(z, λ
η
N) = λ.
If we take discs near the boundary of the convex hall of Z, it is easy to see Cis not empty. Now let λ1 = maxλ : D(z, λ η
N) ∈ C. Then we observe
λ > λ1 ⇒ ]Z ∩D(z, λη
N) < λ.
Now let C1 be a disc of radius λ ηN
such that ]Z ∩ C1 = λ1 (we call the points
of rank λ1), and let Z1 = Z \C1. For this new Z1, we can repeat the procedure
and get smaller and smaller discs C2, C3, · · · , Ck, with λ1 ≥ λ2 ≥ · · · ≥ λk,∑λi = N . Now let Cj be the concentric discs with Cj with twice radii. We
have
z /∈p⋃1
Cj ⇒ D(z, λη
N)⋂ ⋃
λ≤λj
Cj = ∅
⇒ rank of points in D(z, λη
N) < λ
⇒ ]Z ∩D(z, λη
N) ≤ λ− 1.
Suppose
|z − z1| ≤ |z − z2| ≤ · · · ≤ |z − zN |,
then
]Z ∩D(z, λη
N) ≤ λ− 1⇒ |z − zj| ≥
jη
N.
Thus ∏j
|z − zj| ≥∏j
jη
N≥( ηN
)NN ! ≥
(ηe
)N.
Step 3: Borel-Caratheodory inequality.
For a holomorphic function g(z) in |z| ≤ R, and |z| = r < R, we have the
following Borel-Caratheodory inequality.
|g(z)| ≤ 2r
R− rmax|z|≤R
Reg(z) +R + r
R− r|g(0)|.
To prove the lemma, we can first assume g(0) = 0 without loss of generality,
then let
u(z) =g(z)
2 max|z|≤R
Reg(z)− g(z),
34 ZHONGKAI TAO
we have
u(0) = 0 and |u(z)|2 =|g(z)|2
(2 max|z|≤R
Reg(z)− Reg(z))2 + (Img(z))2≤ 1.
By Schwarz lemma we have
|u(z)| ≤ |z|R
and then
|g(z)| ≤ |z|R
∣∣∣∣2 max|z|≤R
Reg(z)− g(z)
∣∣∣∣⇒ |g(z)| ≤ 2r
R− rmax|z|≤R
Reg(z).
The final step to to apply the Borel-Caratheodory inequality to g(z) given by
the decomposition
f(z) = eg(z)∏j
(z − zj)
Since
Reg(z) ≤ log |f(z)| − log |∏j
(z − zj)|
≤ C logM − C log(ηe
)N≤ C(1 + log
(ηe
)) logM
and
Reg(0) ≥ log |f(0)| −N∑j=1
log |zj|
≥ −C(1 + log〈R〉) logM.
Proof of Theorem 13. Let iϕj(z) = ϕ(zj) + 2∂zϕ(zj)(z − zj), then
ϕ(z) = Re (iϕj(z)) +O((z − zj)2)
and
∂zϕj(z) =2
i∂zϕ(z) +O((z − zj)).
Let
vj(z) = u(z)e−iϕj(z)/h,
then
e−Cεh ≤ |vj(z)| ≤ e
Cεh
NOTES ON RANDOM PERTURBATION OF NON-SELF-ADJOINT OPERATORS 35
in the disc D(zj, C√ε). Let f(z) = vj(zj +
√ε(z− zj)), by our previous lemma we get
f(z) = eiθeg(z)N∏j=1
(z − zj)
for N . εh
and |g(z)| . εh. Now the number of zeros of u(z) in Ω is
1
2πi
∫∂Ω
u′(z)
u(z)dz =
1
2πi
∑j
∫γj
(i
hϕ′j(z) +
v′j(z)
vj(z)
)dz
=1
2πh
∫∂Ω
2
i∂zϕ(z)dz +O(
√ε
h)
=1
2πh
∫Ω
∆ϕ(z)dm(z) +O(
√ε
h).
Lemma 49. Let u(z) = eFδ(z)Eδ
−+(z), then the zeros of u(z) coincides with eigenvalues
of P δ with multiplicity.
Proof. By Lemma 13, we have
limγ→z0
tr
∫γ
P δ(z)−1dP δ(z) = limγ→z0
∫γ
Eδ−+(z)−1dEδ
−+(z)
= limγ→z0
∫γ
(eFδ(z)Eδ
−+(z))−1d(eFδ(z)Eδ
−+(z)).
Proof of Hager’s theorem. Use Theorem 13 for ϕ(z) = hF (z) and ε = h log(1δ), then
]u−1(0) ∩ Ω =1
2πh
∫Ω
∆ϕdm(z) +O(
√ε
h)
=1
2πh
∫Ω
(dξ+ ∧ dx+ − dξ− ∧ dx−) +O(
√ε
h)
=1
2πh
∫p−1(Ω)
dξ ∧ dx+O(
√ε
h).
4. Higher oder generalizations
4.0.1. Examples. Consider the operator
P = ∂x(sinx)∂x + ∂x.
36 ZHONGKAI TAO
The spectrum is discrete on the imaginary axis.
4.1. Basic constructions. Let
P (x, hDx, h) =∑h≤m
bk(x, h)(hDx)k,
we want to use WKB method to find an approximate eigenvalue.