On the pseudospectrum of elliptic quadratic differential operators
ON THE PSEUDOSPECTRUM OF ELLIPTIC QUADRATIC
DIFFERENTIAL OPERATORS
Karel Pravda-Starov
University of California, Berkeley
Abstract. We study the pseudospectrum of a class of non-selfadjoint differential
operators. Our work consists in a detailed study of the microlocal properties, which
rule the spectral stability or instability phenomena appearing under small pertur-
bations for elliptic quadratic differential operators. The class of elliptic quadratic
differential operators stands for the class of operators defined in the Weyl quantiza-
tion by complex-valued elliptic quadratic symbols. We establish in this paper a simple
necessary and sufficient condition on the Weyl symbol of these operators, which en-
sures the stability of their spectra. When this condition is violated, we prove that
it occurs some strong spectral instabilities for the high energies of these operators,
in some regions which can be far away from their spectra. We give a precise geo-
metrical description of them, which explains the results obtained for these operators
in some numerical simulations giving the computation of “false eigenvalues” far from
their spectra by algorithms for eigenvalues computing.
Key words. Spectral instability, pseudospectrum, semiclassical quasimodes, non-
selfadjoint operators, non-normal operators, condition (Ψ), subellipticity.
2000 AMS Subject Classification. 35P05, 35S05.
1. Introduction
1.1. Miscellaneous facts about pseudospectrum. In recent years, there has been
a lot of interest in studying the pseudospectrum of non-selfadjoint operators. The
study of this notion has been initiated by noticing that for certain problems of sci-
ence and engineering involving non-selfadjoint operators, the predictions suggested by
spectral analysis do not match with the numerical simulations. This fact lets thinking
that in some cases the only knowledge of the spectrum of an operator is not enough to
understand sufficiently its action. To supplement this lack of information contained
in the spectrum, some new subsets of the complex plane called pseudospectra have
been defined. The main idea about the definition of these new subsets is that it is
interesting to study not only the points where the resolvent of an operator is not de-
fined, i.e. its spectrum, but also where this resolvent is large in norm. This explains
the following definition of the ε-pseudospectrum σε(A) of a matrix or an operator A,
σε(A) =
z ∈ C, ‖(zI −A)−1‖ ≥ 1
for any ε > 0, if we write by convention that ‖(zI − A)−1‖ = +∞ for every point z
belonging to the spectrum σ(A) of the operator.
Let us mention that there exists an abundant literature about this notion of pseu-
dospectrum. We refer here for the definition and some general properties of pseu-
dospectra to the paper [15] of L.N. Trefethen. Let us also point out the more recently
published book [16], which draws up a wide all-round view of this topic and gives a
lot of illustrations.
According to the previous definition, studying the pseudospectra of an operator is
exactly studying the level lines of the norm of its resolvent. What is interesting in
studying such level lines is that it gives some information about the spectral stability
http://arxiv.org/abs/0704.0324v1
of the operator. Indeed, pseudospectra can be defined in an equivalent way in term of
spectra of perturbations of the operator. For instance, we have for any A ∈ Mn(C),
σε(A) = {z ∈ C, z ∈ σ(A +B) for some B ∈ Mn(C) with ‖B‖ ≤ ε}.
It follows that a complex number z belongs to the ε-pseudospectrum of a matrix A if
and only if it belongs to the spectrum of one of its perturbations A+B with ‖B‖ ≤ ε.
More generally, if A is a closed unbounded linear operator with a dense domain on a
complex Hilbert space H , the result of Roch and Silbermann in [13] gives that
σε(A) =
B∈L(H), ‖B‖L(H)≤ε
σ(A +B),
where L(H) stands for the set of bounded linear operators on H . From this second
description, we understand the interest in studying such subsets if we want for example
to compute numerically some eigenvalues of an operator. Indeed, we start to do
it by discretizing this operator. This discretization and inevitable round-off errors
will generate some perturbations of the initial operator. Eventually, algorithms for
eigenvalues computing will determine the eigenvalues of a perturbation of the initial
operator, i.e. a value in a ε-pseudospectrum of the initial operator but not necessarily
a spectral one. This explains why it is important in such numerical computations to
understand if the ε-pseudospectra of studied operators contain more or less deeply
their spectra.
Let us first notice that this study is a priori non-trivial only for non-selfadjoint
operators, or more precisely for non-normal operators. Indeed, we have for a normal
operator A an exact expression of the norm of its resolvent given by the following
classical formula (see for example (V.3.31) in [8]),
(1.1.1) ∀z 6∈ σ(A), ‖(zI −A)−1‖ = 1
z, σ(A)
where d
z, σ(A)
stands for the distance between z and the spectrum of the operator,
when A is a closed unbounded linear operator with a dense domain on a complex
Hilbert space. This formula proves that the resolvent of a normal operator cannot
blow up far from its spectrum. It ensures the stability of its spectrum under small
perturbations because the ε-pseudospectrum is exactly equal in this case to the ε-
neighbourhood of the spectrum
(1.1.2) σε(A) =
z ∈ C : d
z, σ(A)
Nevertheless it is well-known that this formula (1.1.1) is no more true for non-normal
operators. For such operators, it can occur that their resolvents are very large in
norm far from their spectra. This induces that the spectra of these operators can be
very unstable under small perturbations. To illustrate this fact, let us consider the
case of the rotated harmonic oscillator and the following numerical computation of its
spectrum. The rotated harmonic oscillator is a simple example of elliptic quadratic
differential operator
Hc = D
x + cx
2, Dx = i
−1∂x,
with c = eiπ/4. The numerical computation is performed on the matrix discretization
(HcΨi,Ψj)L2(R)
1≤i,j≤N
where N is an integer taken equal to 100 and (Ψj)j∈N∗ stands for the basis of L
composed by Hermite functions. The black dots appearing on this computation stand
for the numerically computed eigenvalues. We can notice on this numerical simulation
that the computed low energies are very close to theoretical ones since the spectrum
Figure 1. Computation of some level lines of the norm of the resol-
vent ‖(Hc− z)−1‖ = ε−1 for the rotated harmonic oscillator Hc with
c = eiπ/4. The right column gives the corresponding values of log10 ε.
0 20 40 60 80 100 120 140 160
dim = 100
of the rotated harmonic oscillator is only composed of eigenvalues regularly spaced
out on the half-line eiπ/8R∗+,
σ(Hc) = {eiπ/8(2n+ 1) : n ∈ N}.
However we notice that it is no more true for the high energies. It occurs for them
some strong spectral instabilities, which lead to the computation of “false eigenvalues”
far from the half-line eiπ/8R∗+. Let us mention that some comparable computations
can be found in [3]. In this paper, we are interested in studying when and how this
kind of phenomena occurs in the class of elliptic quadratic differential operators.
1.2. Elliptic quadratic differential operators. We study here the class of elliptic
quadratic differential operators. It is the class of pseudodifferential operators defined
in the Weyl quantization
(1.2.1) q(x, ξ)wu(x) =
(2π)n
ei(x−y).ξq
(x+ y
u(y)dydξ,
by some symbols q(x, ξ), where (x, ξ) ∈ Rn×Rn and n ∈ N∗, which are some complex-
valued elliptic quadratic forms i.e. complex-valued quadratic forms verifying
(1.2.2) (x, ξ) ∈ Rn × Rn, q(x, ξ) = 0 ⇒ (x, ξ) = (0, 0).
Let us first notice that since the symbols of these operators are some quadratic forms,
these are only some differential operators, which are a priori non-selfadjoint because
their Weyl symbols are complex-valued. As mentioned before, the rotated harmonic
oscillator is an example of such an operator since we have
D2x + e
iθx2 = (ξ2 + eiθx2)w, 0 < θ < π,
if Dx = i
−1∂x. This operator is a very simple example of non-selfadjoint operator for
which we have noticed on the previous numerical simulation that it occurs some strong
spectral instabilities under small perturbations for its high energies. These phenomena
have been studied in several recent works. We can mention in particular the works of
L.S. Boulton [1], E.B. Davies [3], K. Pravda-Starov [10] and M. Zworski [18], which
have given a good understanding of these phenomena.
A question, which has been at the origin of this work, has been to study if these
phenomena peculiar to the rotated harmonic oscillator are representative, or not, of
what occurs more generally in the class of elliptic quadratic differential operators in
every dimension. We have tried to answer to the following questions:
- Does it always occur some strong spectral instabilities under small perturba-
tions for the high energies of these operators ?
- If it is not the case, is it possible to give a necessary and sufficient condition on
the Weyl symbols of these operators, which ensures their spectral stability ?
- Can we precisely describe the geometry, which separates the regions of the
resolvent sets where the resolvents of these operators blow up in norm from
the ones where one keeps a control on their sizes ?
To understand these spectral stability or instability phenomena, we need to study
the microlocal properties, which rule these phenomena in the class of elliptic quadratic
differential operators. Let us mention that it is M. Zworski who first underlined in [18]
the close link between these questions of spectral instabilities and some results of
microlocal analysis about the solvability of pseudodifferential operators.
1.3. Semiclassical pseudospectrum. To answer to these previous questions, it is
interesting to use a semiclassical setting and to study a notion of pseudospectrum
in this new setting. We define for a semiclassical family (Ph)0<h≤1 of operators on
L2(Rn), with a domain D, the following notions of semiclassical pseudospectra.
Definition 1.3.1. For all µ ≥ 0, the set
Λscµ (Ph) =
z ∈ C : ∀C > 0, ∀h0 > 0, ∃ 0 < h < h0, ‖(Ph − z)−1‖ ≥ Ch−µ
is called semiclassical pseudospectrum of index µ of the semiclassical family (Ph)0<h≤1.
The semiclassical pseudospectrum of infinite index is defined by
Λsc∞(Ph) =
Λscµ (Ph).
With this definition, the points in the complement of the semiclassical pseudospectrum
of index µ are the points of the complex plane where we have the following control of
the resolvent’s norm for sufficiently small values of the semiclassical parameter h,
(1.3.1) ∃C > 0, ∃h0 > 0, ∀ 0 < h < h0, ‖(Ph − z)−1‖ < Ch−µ.
To prove the existence of semiclassical pseudospectrum of index µ, we will study the
question of existence of semiclassical quasimodes
(1.3.2) ∀C > 0, ∀h0 > 0, ∃ 0 < h < h0, ∃uh ∈ D,
‖uh‖L2(Rn) = 1 and ‖Phuh − zuh‖L2(Rn) ≤ Chµ,
in some points z of the resolvent set, which can be considered as some “almost eigen-
values” in O(hµ) in the semiclassical limit. Let us notice that the definition chosen
here for the notions of semiclassical pseudospectra differ from the one given in [5] for
a semiclassical pseudodifferential operator. In fact, we have chosen a definition for
semiclassical pseudospectra inspired by the remark made p.388 in [5], because this
definition only depends on the properties of the semiclassical operator rather than on
its symbol.
The interest of working in a semiclassical setting is a matter of geometry. We can
explain this choice by the fact that it is easier for an elliptic quadratic differential oper-
ator q(x, ξ)w to describe the geometry of semiclassical pseudospectra of its associated
semiclassical operator (q(x, hξ)w)0<h≤1, than to describe directly the geometry of its
ε-pseudospectra. The semiclassical setting is particularly well-adapted for the study
of elliptic quadratic differential operators because there exists a simple link between
this semiclassical setting and the quantum one. Indeed, using that the symbols of
these operators are some quadratic forms q, we obtain from the change of variables,
y = h1/2x with h > 0, the following identity between the quantum operator q(x, ξ)w
and its associated semiclassical operator (q(x, hξ)w)0<h≤1,
(1.3.3) q(x, ξ)w − z
q(y, hη)w − z
if z ∈ C. This identity allows to get some information about the resolvent’s norm
behaviour of the quantum operator
q(x, ξ)w − z
if we have some information about semiclassical pseudospectra for its associated semi-
classical operator. Let us mention for example that if a non-zero complex number z
belongs to the semiclassical pseudospectrum of infinite index of the operator
(q(x, hξ)w)0<h≤1,
the identity (1.3.3) induces that the resolvent’s norm of the quantum operator blows
up along the half-line zR+ with a rate faster than any polynomials
(1.3.4) ∀N ∈ N, ∀C > 0, ∀η0 ≥ 1, ∃η ≥ η0, ‖
q(x, ξ)w − zη
)−1‖ ≥ CηN ,
and this, even if this half-line zR+ does not intersect the spectrum of the opera-
tor q(x, ξ)w. Conversely, in the case where z 6∈ Λscµ
q(y, hη)w
, z 6= 0 and 0 ≤ µ ≤ 1,
the identity (1.3.3) shows that we can find some positive constants C1 and C2 such
that the resolvent of the operator q(x, ξ)w remains bounded in norm in some regions
of the resolvent set of the shape
(1.3.5)
u ∈ C : |u| ≥ C1, d(∆, u) ≤ C2|proj∆u|1−µ
∩ C \ σ
q(x, ξ)w
where ∆ = zR+ and proj∆u stands for the orthogonal projection of u on the closed
half-line ∆. Indeed, we obtain from (1.3.1) and (1.3.3) that
∃C > 0, ∃η0 ≥ 1, ∀η ≥ η0,
q(x, ξ)w − ηeiargz
∥ < Cηµ−1,
which induces that for all v ∈ D
q(x, ξ)w
and η ≥ η0,
q(x, ξ)w − ηeiargz
L2(Rn)
≥ C−1η1−µ‖v‖L2(Rn),
q(x, ξ)w
stands for the domain of the operator q(x, ξ)w . Then, we can find a
constant η̃0 ≥ 1 such that if z̃ belongs to
u ∈ C : |u| ≥ η̃0, d(eiargzR+, u) ≤ 2−1C−1|projeiargzR+u|
∩ C\σ
q(x, ξ)w
|projeiargzR+ z̃| ≥ η0.
This induces using the previous estimates and the triangular inequality that if z̃
belongs to
u ∈ C : |u| ≥ η̃0, d(eiargzR+, u) ≤ 2−1C−1|projeiargzR+u|
∩ C\σ
q(x, ξ)w
we have for all v ∈ D
q(x, ξ)w
q(x, ξ)w − z̃
q(x, ξ)w − projeiargzR+ z̃
eiargzR+, z̃
‖v‖L2
≥ 2−1C−1|projeiargzR+ z̃|
1−µ‖v‖L2
≥ 2−1C−1η1−µ0 ‖v‖L2,
because µ ≤ 1. This last estimate shows that the resolvent of the operator q(x, ξ)w is
bounded in norm by 2Cη
0 on the set
u ∈ C : |u| ≥ η̃0, d(eiargzR+, u) ≤ 2−1C−1|projeiargzR+u|
∩ C\σ
q(x, ξ)w
We notice that depending directly on the value of the index µ, 0 ≤ µ < 1, the previous
set contains more or less deeply in its interior the half-line
{u ∈ C : |u| ≥ η̃0, u ∈ zR+}.
This fact explains why in the following we will precise carefully the index of the
semiclassical pseudospectrum to which a point does not belong when there is no
semiclassical pseudospectrum of infinite index in that point.
2. Statement of the results
2.1. Some notations and some preliminary facts about elliptic quadratic
differential operators. Let us begin by giving some notations and recalling known
results about elliptic quadratic differential operators. Let q be a complex-valued
elliptic quadratic form
q : Rnx × Rnξ → C
(x, ξ) 7→ q(x, ξ),
with n ∈ N∗, i.e. a complex-valued quadratic form verifying (1.2.2). The numerical
range Σ(q) of q is defined by the subset in the complex plane of all values taken by
this symbol
(2.1.1) Σ(q) = q(Rnx × Rnξ ),
and the Hamilton map F ∈ M2n(C) associated to the quadratic form q is uniquely
defined by the identity
(2.1.2) q
(x, ξ); (y, η)
(x, ξ), F (y, η)
, (x, ξ) ∈ R2n, (y, η) ∈ R2n,
where q
stands for the polar form associated to the quadratic form q and σ is the
symplectic form on R2n,
(2.1.3) σ
(x, ξ), (y, η)
= ξ.y − x.η, (x, ξ) ∈ R2n, (y, η) ∈ R2n.
Let us first notice that this Hamilton map F is skew-symmetric with respect to σ.
This is just a consequence of the properties of skew-symmetry of the symplectic form
and symmetry of the polar form
(2.1.4) ∀X,Y ∈ R2n, σ(X,FY ) = q(X ;Y ) = q(Y ;X) = σ(Y, FX) = −σ(FX, Y ).
Under this assumption of ellipticity, the numerical range of a quadratic form can
only take some very particular shapes. It is a consequence of the following result
proved by J. Sjöstrand (Lemma 3.1 in [14]),
Proposition 2.1.1. Let q : Rnx × Rnξ → C a complex-valued elliptic quadratic form.
If n ≥ 2, then there exists z ∈ C∗ such that Re(zq) is a positive definite quadratic
form. If n = 1, the same result is fulfilled if we assume besides that Σ(q) 6= C.
This proposition shows that the numerical range of an elliptic quadratic form can only
take two shapes. The first possible shape is when Σ(q) is equal to the whole complex
plane. This case can only occur in dimension n = 1. The second possible shape is
when Σ(q) is equal to a closed angular sector with a top in 0 and an opening strictly
lower than π.
Figure 2. Shape of the numerical range Σ(q) when Σ(q) 6= C.
Σ(zq)
Indeed, if Σ(q) 6= C, using that the set Σ(q) is a semi-cone
tq(x, ξ) = q(
tξ), t ∈ R+, (x, ξ) ∈ R2n,
because q is a quadratic form, we have
Σ(q) = R+z
if z is the non-zero complex number given by the proposition 2.1.1 and I is the compact
interval
I = 1 + i Im(zq)(K),
where K is the following compact subset of R2n,
(x, ξ) ∈ R2n : Re(zq)(x, ξ) = 1
The compactness of K is a direct consequence of the fact that Re(zq) is a positive
definite quadratic form.
Elliptic quadratic differential operators define some Fredholm operators (see Lemma 3.1
in [6] or Theorem 3.5 in [14]),
(2.1.5) q(x, ξ)w + z : B → L2(Rn),
where B is the Hilbert space
(2.1.6)
u ∈ L2(Rn) : xαDβxu ∈ L2(Rn) if |α+ β| ≤ 2
with the norm
‖u‖2B =
|α+β|≤2
‖xαDβxu‖2L2(Rn).
The Fredholm index of the operator q(x, ξ)w + z is independent of z and is equal to 0
if n ≥ 2. In the case where n = 1, this index can take the values −2, 0 or 2. More
precisely, this index is always equal to 0 if Σ(q) 6= C.
In the following, we will always assume that Σ(q) 6= C. Under this assumption,
J. Sjöstrand has proved in the theorem 3.5 in [14] (see also Lemma 3.2 and Theorem 3.3
in [6]) that the spectrum of an elliptic quadratic differential operator
q(x, ξ)w : B → L2(Rn),
is only composed of eigenvalues with finite multiplicity
(2.1.7) σ
q(x, ξ)w
λ∈σ(F ),
−iλ∈Σ(q)\{0}
rλ + 2kλ
(−iλ) : kλ ∈ N
where F is the Hamilton map associated to the quadratic form q and rλ is the dimen-
sion of the space of generalized eigenvectors of F in C2n belonging to the eigenvalue
λ ∈ C. Let us notice that the spectra of these operators is always included in the
numerical range of their Weyl symbols.
To end this review of preliminary properties of elliptic quadratic differential oper-
ators, let us underline that the property of normality in this class of operators can be
easily checked by computing the Poisson bracket of the real part and the imaginary
part of their symbols
(2.1.8) {Re q, Im q} =
∂Re q
∂Im q
∂Re q
∂Im q
Proposition 2.1.2. An elliptic quadratic differential operator
q(x, ξ)w : B → L2(Rn), n ∈ N∗,
is normal if and only if the quadratic form defined by the Poisson bracket of the real
part and the imaginary part of its symbol is equal to zero
(2.1.9) ∀(x, ξ) ∈ R2n, {Re q, Im q}(x, ξ) = 0.
Proof of Proposition 2.1.2. This proposition is a direct consequence of the composition
formula in Weyl calculus (see Theorem 18.5.4 in [7]), which induces that the Weyl
symbol of the commutator
[qw, (qw)∗] = [qw, qw] = −2i[(Re q)w, (Im q)w],
is equal to
−2i(Re q ♯ Im q − Im q ♯ Re q) = −2{Re q, Im q},
because Re q and Im q are some quadratic forms. The notation Re q ♯ Im q stands
for the Weyl symbol of the operator obtained by composition (Req)w(Imq)w. �
Remark. Let us notice that the symplectic invariance of the Poisson bracket (see
(21.1.4) in [7]),
(2.1.10) {(Re q) ◦ χ, (Im q) ◦ χ} = {Re q, Im q} ◦ χ,
if χ stands for a linear symplectic transformation of R2n, implies that the condition
(2.1.9) is symplectically invariant.
2.2. Statement of the main results. Let us consider an elliptic quadratic differ-
ential operator
q(x, ξ)w : B → L2(Rn).
We know from (2.1.7) that the spectrum of this operator is contained in the numerical
range of its symbol Σ(q). The following proposition gives a first localization of the
regions where the resolvent can blow up in norm and where spectral instabilities can
occur.
Proposition 2.2.1. Let q : Rn × Rn → C, n ∈ N∗, be a complex-valued elliptic
quadratic form. We have
∀z 6∈ Σ(q),
q(x, ξ)w − z
∥ ≤ 1
z,Σ(q)
where d
z,Σ(q)
stands for the distance from z to the numerical range Σ(q).
This result shows that the resolvent of an elliptic quadratic differential operator
cannot blow up in norm far from the numerical range of its symbol. We are now
going to study what kind of phenomena can occur in this particular set. There are
two cases to separate according to the property of normality or non-normality of the
operator.
2.2.1. Case of a normal operator. Let us consider a normal elliptic quadratic differ-
ential operator
q(x, ξ)w : B → L2(Rn).
Let us recall that according to the proposition 2.1.2 this property of normality is
exactly equivalent to the fact that
∀(x, ξ) ∈ R2n, {Re q, Im q}(x, ξ) = 0.
In this case, we have the classical formula (1.1.1) for its resolvent’s norm
(2.2.1) ∀z 6∈ σ
q(x, ξ)w
q(x, ξ)w − z
z, σ(q(x, ξ)w)
which induces that the ε-pseudospectrum of this operator is exactly equal to the
ε-neighbourhood of its spectrum
q(x, ξ)w
z ∈ C : d
z, σ(q(x, ξ)w)
, ε > 0.
This classical formula (2.2.1) ensures that the resolvent cannot blow up in norm far
from the spectrum and induces that the spectrum of such an operator is stable under
small perturbations.
Example 1. The operator
(2.2.2) q1(x, ξ)
w = −(1 + i)∂2x1 − ∂
+ 4(−1 + i)x1∂x1 + 2(−1 + i)x2∂x1 + 6ix2∂x2
+ 2ix1∂x2 + (6 + 5i)x
1 + (11 + i)x
2 + (10 + 4i)x1x2 − 2 + 5i,
is an example of a normal elliptic quadratic differential operator. Its spectrum is given
q1(x, ξ)
(2k1 + 1) + (2k2 + 1)
4 : (k1, k2) ∈ N2
Figure 3. Spectrum and a ε-pseudospectrum of the operator q1(x, ξ)
∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗
∗ ∗ ∗
Σ(q1)
Example 2. Let us notice that when the numerical range Σ(q) is reduced to a closed
half-line, the elliptic quadratic differential operator q(x, ξ)w is always normal since
{Re q, Im q} = |z|2{Re(z−1q), Im(z−1q)} = 0,
if z ∈ C∗ is chosen such that Im(z−1q) = 0. In fact, the operator q(x, ξ)w can in this
particular case be reduced after a conjugation by a unitary operator on L2(Rn) to the
operator
+ x2j),
where λj > 0 for all j = 1, ..., n.
Figure 4. Example of a normal elliptic quadratic differential operator.
2.2.2. Case of a non-normal operator. Let us consider a non-normal elliptic quadratic
differential operator
q(x, ξ)w : B → L2(Rn), n ∈ N∗.
We assume in the following that the numerical range Σ(q) is distinct from the whole
complex plane
(2.2.3) Σ(q) 6= C.
As mentioned in the section 2.1, this additional assumption is always fulfilled in
dimension n ≥ 2. It only excludes some very particular one-dimensional elliptic
quadratic differential operators (see the remark following the proposition 2.2.2 for
more precision about these operators).
Under this additional assumption, the numerical range Σ(q) is always a closed
angular sector with a top in 0 and a positive opening strictly lower than π.
2.2.2.a. On the pseudospectrum at the interior of the numerical range. Let us consider
the associated semiclassical elliptic quadratic differential operator
(q(x, hξ)w)0<h≤1.
We can build in every point of the interior of the numerical range Σ̊(q) some semi-
classical quasimodes.
Theorem 2.2.1. If the elliptic quadratic differential operator
q(x, ξ)w : B → L2(Rn), n ∈ N∗,
is non-normal and verifies Σ(q) 6= C then for all z ∈ Σ̊(q) and N ∈ N, there exist
h0 > 0 and a semiclassical family (uh)0<h≤h0 ∈ S(Rn) such that
‖uh‖L2(Rn) = 1 and ‖q(x, hξ)wuh − zuh‖L2(Rn) = O(hN ) when h → 0+.
This result induces the existence of semiclassical pseudospectrum of infinite index in
every point of the interior of the numerical range Σ̊(q).
According to (1.3.4), this result in the semiclassical setting induces that the resol-
vent’s norm of the quantum operator q(x, ξ)w blows up fastly along all the half-lines
belonging to the interior of the numerical range Σ̊(q),
(2.2.4) ∀z ∈ Σ̊(q), ∀N ∈ N, ∀C > 0, ∀η0 ≥ 1, ∃η ≥ η0, ‖
q(x, ξ)w − zη
)−1‖ ≥ CηN .
We deduce from (2.1.7) that as soon as an elliptic quadratic differential operator is
non-normal its resolvent blows up in norm in some regions of the resolvent set far
from its spectrum. This fact induces that the high energies of such an operator are
very unstable under small perturbations as we have already noticed on the numerical
computation performed for the rotated harmonic oscillator. It follows that in the class
of elliptic quadratic differential operators1 the property of spectral stability is exactly
equivalent to the property of normality:
σ(q(x, ξ)w) is stable under ⇔ q(x, ξ)w is a normal ⇔ {Re q, Im q} = 0.
small perturbations operator
By spectral stability, we mean here that the resolvent of these operators cannot blow
up in norm far from their spectra. Let us add that it is not very surprising to have
this property of spectral stability under the assumption of normality, but it is worth
1If we exclude the one-dimensional particular cases previously mentioned.
noticing that as soon as this property is violated, it occurs in this class of operators
some strong spectral instabilities under small perturbations for their high energies.
Examples. The two following operators
(2.2.5) q2(x, ξ)
w = −∂2x1 − 2∂
+ 4ix2∂x2 + 2x
1 + (4 + i)x
2 + 4x1x2 + 2i
(2.2.6) q3(x, ξ)
w = −(1 + i)∂2x1 − 2∂
+ 4(−1 + i)x1∂x1 + 2(1− i)x2∂x1 − 4ix1∂x2
+ (9 + 4i)x21 + (2 + i)x
2 − 4(1 + i)x1x2 − 2 + 2i,
are some examples of non-normal elliptic quadratic differential operators.
2.2.2.b. On the pseudospectrum at the boundary of the numerical range. Let us now
study what occurs on the boundary of the numerical range ∂Σ(q) for a non-normal
elliptic quadratic differential operator
q(x, ξ)w : B → L2(Rn).
Let us mention that we always assume that Σ(q) 6= C. Under these assumptions, the
boundary of the numerical range is composed of the union of the origin 0 and two
half-lines ∆1 and ∆2,
(2.2.7) ∂Σ(q) = {0} ⊔∆1 ⊔∆2,
that we can write
(2.2.8) ∆1 = z1R
+ and ∆2 = z2R
+ with z1, z2 ∈ ∂Σ(q) \ {0}.
We need to define a notion of order for the symbol q(x, ξ) on these two half-lines ∆j ,
j = 1, 2. Let us begin by recalling the classical definition of the order k(x0, ξ0) of a
symbol p(x, ξ) at a point (x0, ξ0) ∈ R2n (see section 27.2, chapter 27 in [7]). This
order k(x0, ξ0) is an element of the set N ∪ {+∞} defined by
(2.2.9) k(x0, ξ0) = sup
j ∈ Z : pI(x0, ξ0) = 0, ∀ 1 ≤ |I| ≤ j
where I = (i1, i2, ..., ik) ∈ {1, 2}k, |I| = k and pI stands for the iterated Poisson
brackets
pI = Hpi1Hpi2 ...Hpik−1 pik ,
where p1 and p2 are respectively the real and the imaginary part of the symbol p,
p = p1 + ip2. The order of a symbol q at a point z is then defined as the maximal
order of the symbol p = q − z at every point (x0, ξ0) ∈ R2n verifying
p(x0, ξ0) = q(x0, ξ0)− z = 0.
Let us underline that the symplectic invariance of the Poisson bracket (2.1.10) induces
the same property for the order of a symbol at a point.
Since here the symbol q is a quadratic form, all the iterated Poisson brackets are
also some quadratic forms. This property of degree two homogeneity of these Poisson
brackets induces that the symbol q has the same order at every point of each half-line
∆j , j = 1, 2. This allows to define the order of the symbol q on the half-line ∆j by
defining this order by this common value. Let us mention that this order can be finite
or infinite.
Examples. One can easily check that the Weyl symbol
ξ2 + eiθx2, 0 < θ < π,
of the rotated harmonic oscillator has an order equal to 2 on the both half-lines R∗+
and eiθR∗+, which composes the boundary of its numerical range. The symbol q2 of
the operator defined in (2.2.5) has an order equal to 2 on iR∗+ and to 6 on R
Σ(q2) = {z ∈ C : Re z ≥ 0, Im z ≥ 0}.
On the other hand, we can verify that the symbol q3 of the operator defined in (2.2.6)
is of infinite order on the half-line R∗+ and has an order equal to 2 on e
iπ/4R∗+,
Σ(q3) = {0} ∪ {z ∈ C∗ : 0 ≤ arg z ≤ π/4}.
In the case where the symbol is of finite order on a half-line ∆j , j = 1, 2, we have
the following result.
Theorem 2.2.2. If the Weyl symbol q(x, ξ) of a non-normal elliptic quadratic differ-
ential operator is of finite order kj on the half-line
∆j , j ∈ {1, 2}, ∆j ⊂ ∂Σ(q) \ {0},
then this order is necessary even and there is no semiclassical pseudospectrum of
index kj/(kj + 1) on ∆j for the associated semiclassical operator
∆j ⊂ C \ Λsckj/(kj+1)
q(x, hξ)w
Remark. Let us mention that we can more precisely establish that in dimension n ≥ 1,
the order kj is an even integer verifying
2 ≤ kj ≤ 4n− 2.
This result is proved in [12].
By rephrasing this result in a quantum setting, it follows from (1.3.5) and (2.1.7)
that when the symbol q of a non-normal elliptic quadratic differential operator q(x, ξ)w
is of finite order kj on a half-line
∆j , j ∈ {1, 2}, ∆j ⊂ ∂Σ(q) \ {0},
then the resolvent of this operator remains bounded in norm in a set of the following
(2.2.10)
u ∈ C : |u| ≥ C1, d(∆j , u) ≤ C2|proj∆ju|
where C1 and C2 are some positive constants.
As we will see in its proof, this absence of semiclassical pseudospectrum is linked
to some properties of subellipticity. Let us just underline for the moment that the
index kj/(kj + 1), which appears in this result is exactly equal to the loss appearing
in the subelliptic estimate hidden behind this result.
About the case of infinite order, the situation is much more complicated. Never-
theless, we can first notice in this case that we cannot expect to prove a stronger result
than an absence of semiclassical pseudospectrum of index 1. Indeed, we can easily
check on the example of the operator q3(x, ξ)
w defined in (2.2.6) that its spectrum is
given by
q3(x, ξ)
(2k1 + 1)
2 + (2k2 + 1)3
8 : (k1, k2) ∈ N2
We recall that the spectrum of this operator is only composed of eigenvalues and that
its symbol is of infinite order on R∗+. It follows from the structure of the spectrum and
(1.3.5) that if there is no semiclassical pseudospectrum of infinite index in a point of
the half-line R∗+, there is necessary no semiclassical pseudospectrum of index µ with
an index µ ≥ 1. In fact, we can prove by using a result of exponential decay in time
for the norm of contraction semigroups generated by elliptic quadratic differential
operators (see [12]) that there is never some semiclassical pseudospectrum of index 1
on all these half-lines of infinite order. Let us mention that this result of exponential
decay will not be proved here but it will be explained in the following how it induces
the absence of semiclassical pseudospectrum of index 1.
2.2.3. About the geometry of ε-pseudospectra for elliptic quadratic differential opera-
tors. Let us now explain what are the consequences of these results on the geometry
of ε-pseudospectra for elliptic quadratic differential operators. Let us begin by con-
sidering the one-dimensional case which is a bit particular. In dimension n = 1, an
elliptic quadratic differential operator can be reduced after a similitude and a conju-
gation by a unitary operator to the harmonic oscillator or to the rotated harmonic
oscillator.
Proposition 2.2.2. Let us consider q : R×R → C a complex-valued elliptic quadratic
form such that Σ(q) 6= C. For all h > 0, there exist a unitary operator (more precisely
a metaplectic operator) Uh on L
2(R), which is an automorphism of the spaces S(R)
and B, z ∈ C∗ and θ ∈ [0, π[ such that
∀h > 0, q(x, hξ)w = zUh
(hDx)
2 + eiθx2
U−1h .
Remark. In the case where Σ(q) = C, an elliptic quadratic differential operator
q(x, ξ)w can be reduced after a similitude and a conjugation by a unitary operator on
L2(Rn) to the operator defined in the Weyl quantization by the symbol
(ξ + ix)(ξ + ηx) with η ∈ C, Im η > 0,
(ξ − ix)(ξ + ηx) with η ∈ C, Im η < 0,
depending on the value of its Fredholm index, which is equal to −2 in the first case
and to 2 in the second one.
As we will see in the following, this proposition allows us to reduce the study of a
one-dimensional non-normal elliptic quadratic differential operator verifying
Σ(q) 6= C,
to the one of the rotated harmonic oscillator
Hθ = D
x + e
iθx2, 0 < θ < π.
Let us mention that the previous results (Theorem 2.2.1 and Theorem 2.2.2) were
already known in the particular case of the rotated harmonic oscillator. Indeed,
the existence of semiclassical quasimodes inducing the presence of semiclassical pseu-
dospectrum of infinite index in every point of the interior of the numerical range for
the associated semiclassical operator, is a direct consequence of a result proved by
E.B. Davies in [4] (Theorem 1). About the absence of semiclassical pseudospectrum
of index 2/3 on the boundary of the numerical range, this result has been proved for
the rotated harmonic oscillator in [10]2.
As proved in [10], this absence of semiclassical pseudospectrum allows to give a
proof of a conjecture stated by L.S. Boulton in [1]. It deals with the geometry of ε-
pseudospectra for the rotated harmonic oscillator. Let us now recall some facts about
this conjecture and some results proved by L.S. Boulton in [1].
2Let us recall that the value of the order is equal to 2 in this case.
L.S. Boulton has first proved (Theorem 3.3 in [1]) that the resolvent of the rotated
harmonic oscillator blows up in norm along all a family of curves of the following form
η 7→ bη + eiθηp,
where b and p are some positive constants verifying 1/3 < p < 3,
(2.2.11)
Hθ − (bη + eiθηp)
∥ → +∞ when η → +∞.
On the other hand, he also proved that the resolvent of this operator remains bounded
in norm on two half-stripes parallel to the half-lines R+ or e
iθR+. More precisely, he
proved that there exist some positive constants d and Md such that
(2.2.12) sup
, 0≤b≤d
Hθ − (η + ib)
∥ ≤ Md,
(2.2.13) sup
, 0≤b≤d
Hθ − eiθ(η − ib)
∥ ≤ Md.
These bounds provide some information about the shape of ε-pseudospectra of the
operator Hθ. Indeed, L.S. Boulton has proved using these results that for all suffi-
ciently small value of the positive parameter ε, the ε-pseudospectra of the rotated
harmonic oscillator is contained in the shaded set appearing on the following figure.
The eigenvalues appear on this figure marked by some ⋄.
Figure 5. A first localization of the ε-pseudospectra of the rotated
harmonic oscillator.
More precisely, L.S. Boulton proved that for all 0 < δ < 1 and m ∈ N, there exists
a positive constant ε0 such that for all 0 < ε < ε0,
(2.2.14) σε(Hθ) ⊂
{z ∈ C : |z − λn| < δ} ∪
λm+1 − δeiθ/2 + Sθ
where
λn = e
iθ/2(2n+ 1), n ∈ N
Sθ = {z ∈ C∗ : 0 ≤ arg z ≤ θ} ∪ {0}.
In fact, in view of some numerical calculations performed by E.B. Davies in [3],
L.S. Boulton has conjectured that the index p = 1/3 appearing in (2.2.11) is the
critical one in the following sense:
Let us consider 0 < p < 1/3, 0 < δ < 1 and m ∈ N. If bm,p and E are some positive
constants verifying
bm,pE + e
iθEp = λm and ∀η > E, arg zη < θ/2,
where zη = bm,pη + e
iθηp, let us set
Ωm,p =
|zη|eiα ∈ C : η ≥ E, arg zη ≤ α ≤ arg(zηeiθ)
L.S. Boulton has conjectured the following result.
Boulton’s conjecture. There exists ε0 > 0 such that for all 0 < ε < ε0,
(2.2.15) σε(Hθ) ⊂
{z ∈ C : |z − λn| < δ} ∪ Ωm,p.
The absence of semiclassical pseudospectrum of index 2/3 on the boundary of the
numerical range ∂Σ(q)\{0} for the rotated harmonic oscillator3 given by the theorem
2.2.2 shows that this index 1/3 is actually the critical one. Indeed, we can deduce
(2.2.15) from (2.2.10) (see [10] for more details) since here kj = 2, j ∈ {1, 2}. As
we will see, this theorem 2.2.2 is a consequence of a subelliptic estimate for gen-
eral semiclassical pseudodifferential operators proved by N. Dencker, J. Sjöstrand and
M. Zworski in [5] (Theorem 1.4). In the particular case of the rotated harmonic oscil-
lator, a more elementary proof of this result using only some non-trivial localization
scheme in the frequency variable is given in [10].
Let us notice that this inclusion (2.2.15) allows to give a sharp description of the ε-
pseudospectra of the rotated harmonic oscillator, which is optimal in view of (2.2.11).
Figure 6. Shape of the ε-pseudospectra of the rotated harmonic oscillator.
By coming back to the case of an arbitrary dimension n ≥ 1, let us finally underline
that using the theorem 2.2.2, we can give similar descriptions of the ε-pseudospectra
for non-normal elliptic quadratic differential operators, to the one given by L.S. Boul-
ton for the rotated harmonic oscillator, when the symbols of these operators are of
finite order on the two open half-lines, which compose the boundary of their numerical
ranges. The only difference with the particular case of the rotated harmonic oscillator
is that the critical indices, which appear in this description can be different. Indeed,
3The order of the rotated harmonic oscillator’s symbol is equal to 2 on ∂Σ(q) \ {0}.
these critical indices depend directly according to (2.2.10) on the order of the symbols
on the two half-lines composing the boundary of their numerical ranges. We refer the
reader to [10] for more details about the way of getting from (2.2.10) such descriptions
of ε-pseudospectra.
3. The proofs of the results
Before giving the proofs of the results stated in the previous section, let us begin by
recalling the symplectic invariance property of the Weyl quantization (see Theorem
18.5.9 in [7]). This symplectic invariance is actually the most important property of
the Weyl quantization.
For every affine symplectic transformation χ of R2n, there exists a unitary trans-
formation U on L2(Rn), uniquely determined apart from a constant factor of modulus
1, such that U is an automorphism of the spaces S(Rn), B and S ′(Rn), where B is
the Hilbert space defined in (2.1.6), and
(3.0.1) (a ◦ χ)(x, ξ)w = U−1a(x, ξ)wU,
for all a ∈ S ′(R2n). The operator U is a metaplectic operator associated to the affine
symplectic transformation χ.
This symplectic invariance of the Weyl quantization induces the same property for
the semiclassical pseudospectra of elliptic quadratic differential operators in the sense
that if
q : Rnx × Rnξ → C,
is a complex-valued elliptic quadratic form and χ is a linear symplectic transformation
of R2n, we have for all µ ∈ [0,∞],
(3.0.2) Λscµ
(q ◦ χ)(x, hξ)w
= Λscµ
q(x, hξ)w
To prove this fact, let us begin by noticing that for all a ∈ S ′(R2n) and h > 0, we
U−1h a(x, ξ)
wUh = a(h
−1/2x, h1/2ξ)w,
where
Uhf(x) = h
n/4f(h1/2x),
since according to the proof of Theorem 18.5.9 in [7], Uh is a metaplectic operator
associated to the linear symplectic transformation
(x, ξ) 7→ (h−1/2x, h1/2ξ).
Let us now consider the case where the symbol a is a quadratic form. The homogeneity
property of such a symbol implies that
∀h > 0, a(h−1/2x, h1/2ξ) = 1
a(x, hξ),
∀h > 0, U−1h a(x, ξ)
wUh =
a(x, hξ)w.
If q : Rnx × Rnξ → C is a complex-valued elliptic quadratic form and χ is a linear
symplectic transformation of R2n, we can notice that
(q ◦ χ)(x, hξ)w , h > 0,
is actually an elliptic quadratic differential operator since the symbol q◦χ is an elliptic
quadratic form. Let z ∈ C and U be a metaplectic operator associated to the linear
symplectic transformation χ. Using that U and Uh are some automorphisms of the
Hilbert space B and
(3.0.3) U−1h U
−1Uhq(x, hξ)
wU−1h UUh = U
−1hq(x, ξ)wUUh
= hU−1h (q ◦ χ)(x, ξ)
wUh = (q ◦ χ)(x, hξ)w ,
we obtain that
U−1h U
q(x, hξ)w − z
U−1h UUh =
(q ◦ χ)(x, hξ)w − z
Using finally that U−1h U
−1Uh is a unitary transformation of L
2(Rn), this identity
implies that
(q ◦ χ)(x, hξ)w − z
q(x, hξ)w − z
which proves (3.0.2). In the following, this property of symplectic invariance will
allow us to reduce certain symbols to some normal forms by choosing new symplectic
coordinates. We can now begin to prove the results stated in the previous section.
Let us start by the proof of the proposition 2.2.1.
Proof of Proposition 2.2.1. If the numerical range is equal to the whole complex plane,
there is nothing to prove. If Σ(q) 6= C, we have seen in the previous section that the
numerical range is necessary a closed angular sector with a top in 0 and an opening
strictly lower than π.
Let us consider z 6∈ Σ(q) and denote by z0 its orthogonal projection on the non-
empty closed convex set Σ(q). According to the shape of the numerical range, it
follows that z0 belongs to its boundary and that we can find a complex number
z1 ∈ C∗, |z1| = 1 such that
Σ(z1q) ⊂
z ∈ C : Re z ≥ 0
(3.0.4) z1z ∈
z ∈ C : Re z < 0
z,Σ(q)
= d(z1z, iR).
Using now that the operator i[Im(z1q)]
w is formally skew-selfadjoint, we obtain that
for all u ∈ S(Rn),
z1q(x, ξ)
wu− z1zu, u
L2(Rn)
= d(z1z, iR)‖u‖2L2(Rn) +
z1q(x, ξ)
L2(Rn)
.(3.0.5)
Then, since the quadratic form Re(z1q) is non-negative, we deduce from the symplectic
invariance of the Weyl quantization and the theorem 21.5.3 in [7] that there exists a
metaplectic operator U such that
z1q(x, ξ)
= U−1
+ x2j) +
j=k+1
with k, l ∈ N and λj > 0 for all j = 1, ..., k. By using that U is a unitary operator on
L2(Rn), we obtain that the quantity
z1q(x, ξ)
L2(Rn)
‖DxjUu‖2L2(Rn) + ‖xjUu‖
L2(Rn)
j=k+1
‖xjUu‖2L2(Rn),
is non-negative. Then, we can deduce from the Cauchy-Schwarz inequality, (3.0.4)
and (3.0.5) that for all u ∈ S(Rn),
z,Σ(q)
‖u‖L2(Rn) ≤ |z1| ‖q(x, ξ)wu− zu‖L2(Rn).
Finally, using the density of the Schwartz space S(Rn) in B and the fact that |z1| = 1,
we obtain that
∀z 6∈ Σ(q),
q(x, ξ)w − z
∥ ≤ 1
z,Σ(q)
since according to (2.1.7), σ
q(x, ξ)w
⊂ Σ(q). �
We now consider the one-dimensional case, which is a bit particular.
3.1. The one-dimensional case. In dimension n = 1, we can reduce the study of
complex-valued elliptic quadratic forms to exactly three normal forms after a simili-
tude and a real linear symplectic transformation.
Lemma 3.1.1. Let q : Rx × Rξ → C be a complex-valued elliptic quadratic form in
dimension 1. Then, there exists a linear symplectic transformation χ of R2 such that
the symbol q ◦ χ is equal to one of the following normal forms:
(i) α(ξ2 + eiθx2) with α ∈ C∗, 0 ≤ θ < π.
(ii) α(ξ + ix)(ξ + ηx) with α ∈ C∗, η ∈ C, Im η > 0.
(iii) α(ξ − ix)(ξ + ηx) with α ∈ C∗, η ∈ C, Im η < 0.
In the two last cases (ii) and (iii), the numerical range Σ(q) is equal to the whole
complex plane, Σ(q) = C.
Proof of Lemma 3.1.1. Let q : R2 → C be a complex-valued elliptic quadratic form.
Let us first consider the case where Σ(q) 6= C. We deduce from the proposition 2.1.1
that we can reduce our study to the case where Re q is a positive definite quadratic
form. Then, using Lemma 18.6.4 in [7], we can find a real linear symplectic transfor-
mation to reduce the quadratic form Re q to the normal form
λ(x2 + ξ2), with λ > 0.
It follows that there exist some real constants a, b and c such that
q(x, ξ) = λ
x2 + ξ2 + i(ax2 + 2bxξ + cξ2)
Then, we can choose an orthogonal matrix P ∈ O(2,R) diagonalizing the real sym-
metric matrix associated to the quadratic form ax2 + 2bxξ + cξ2,
with λ1, λ2 ∈ R. If P ∈ O(2,R) \ SO(2,R), we have
if σ0 is the matrix with determinant equal to −1,
and P̃ = Pσ0. It follows that we can always diagonalize the real symmetric matrix
associated to the quadratic form λ−1Im q by conjugating it by an element of SO(2,R).
Since the symplectic group is equal in dimension 1 to the group SL(2,R), we can after
a linear symplectic transformation of R2 reduce the quadratic form q to
x2 + ξ2 + i(γ1x
2 + γ2ξ
= α(ξ2 + reiθx2),
where γ1, γ2 ∈ R, α ∈ C∗, r > 0 and θ ∈] − π, π[. Let us notice that the elliptic-
ity of q actually implies that θ 6≡ π[2π]. Finally, using the real linear symplectic
transformation (x, ξ) 7→ (r−1/4x, r1/4ξ), we get a symbol of type (i),
αr1/2(ξ2 + eiθx2),
if 0 ≤ θ < π. If −π < θ < 0, we need to use besides the real linear symplectic
transformation (x, ξ) 7→ (ξ,−x) to obtain a symbol of type (i),
2 eiθ(ξ2 + e−iθx2).
Let us now assume that Σ(q) = C. Since the dimension is equal to 1, we can factor the
symbol q on C as a polynomial function of degree 2 in the variable ξ. Thus, according
to the dependence in the variable x of the polynomial function’s coefficients, we can
find some complex numbers λ1, λ2 and α ∈ C∗ such that
q(x, ξ) = α(ξ − λ1x)(ξ − λ2x).
The ellipticity assumption for the quadratic form q induces that
Im λj 6= 0,
if j = 1, 2. Using now the linear symplectic transformation (x, ξ) 7→ (x, ξ + Re λ1x),
we can assume that
(3.1.1) q(x, ξ) = α(ξ − irx)(ξ + bx),
with r ∈ R∗ and Im b 6= 0. Let us now check that the assumption Σ(q) = C induces
that r Im b < 0. Since
(ξ − irx)(ξ + bx) = ξ2 + (b − ir)xξ − irbx2,
the condition Σ(q) = C implies that for all (v, w) ∈ R2, there exists a solution
(x0, ξ0) ∈ R2 of the system
(3.1.2)
ξ2 +Re b xξ + r Im b x2 = v
xξ(Im b− r) − r Re b x2 = w.
Let us first notice that the second equation of (3.1.2) is fulfilled for all w ∈ R only if
Im b 6= r.
If w 6= 0, it follows from the second equation of (3.1.2) that x0 6= 0 and
(3.1.3) ξ0 =
w + r Re b x20
(Im b− r)x0
Let us consider the case where v = 0. Using (3.1.3) and the first equation of (3.1.2),
we obtain that
(w + r Re b x20)
2 +Re b (Im b− r)x20(w + r Re b x20) + r Im b (Im b− r)2x40 = 0.
We can rewrite this equation as fw(X0) = 0 if we set X0 = x
0 and
(3.1.4) fw(X) = r Im b
(Re b)2 + (Im b− r)2
X2 + w Re b (Im b+ r)X + w2.
Thus, the condition Σ(q) = C implies that there exists for all w 6= 0, a non-negative
solution X0 of the equation fw(X0) = 0. Since the quantity r Im b is assumed to be
non-zero, we first study the case where r Im b > 0. In this case, since
(3.1.5) f ′w(X) = 2r Im b
(Re b)2 + (Im b− r)2
X + w Re b (Im b+ r)
2r Im b
(Re b)2 + (Im b− r)2
because Im b 6= r, we have
(3.1.6) ∀X ∈ R+, fw(X) ≥ fw(0) = w2 > 0,
if w 6= 0 and
− w Re b (Im b+ r)
2r Im b
(Re b)2 + (Im b− r)2
) ≤ 0.
The estimate (3.1.6) shows that if r Im b > 0, the equation fw(X) = 0 has no non-
negative solution for all value of the parameter w 6= 0. This proves that the condition
Σ(q) = C induces that r Im b < 0. Using the linear symplectic transformation
(x, ξ) 7→ (|r|−1/2x, |r|1/2ξ),
we obtain the normal forms (ii) and (iii),
α|r|(ξ + ix)(ξ + ηx) with Im η > 0 and α|r|(ξ − ix)(ξ + ηx) with Im η < 0,
where η = |r|−1b. Finally, we can easily check that the numerical ranges of the normal
forms (ii) and (iii) are actually equal to the whole complex plane C. �
Let us notice that the proposition 2.2.2 and the remark following its statement are
some direct consequences of the symplectic invariance property of the Weyl quanti-
zation (see (3.0.3)) and the previous lemma. We can add that as proved after the
lemma 3.1 in [6], the Fredholm indices of the one-dimensional elliptic quadratic dif-
ferential operators with symbols of type (i), (ii) and (iii) are respectively equal to 0,
−2 and 2.
As we have mentioned in the previous section, the results of Theorem 2.2.1 and
Theorem 2.2.2 are already known in the particular case of the rotated harmonic oscil-
lator. The existence of semiclassical quasimodes inducing the presence of semiclassical
pseudospectrum of infinite index in every point of the interior of the numerical range
for the associated semiclassical operator, is a direct consequence of a result proved
by E.B. Davies in [4] (Theorem 1) and; the absence of semiclassical pseudospectrum
of index 2/3 on the boundary of the numerical range has been proved for the ro-
tated harmonic oscillator in [10]4. As we have previously mentioned (see (2.1.10) and
(3.0.2)), the property of non-normality, the order of symbols and the semiclassical
pseudospectra of elliptic quadratic differential operators are symplectically invariant.
These properties allow us to reduce by any real linear symplectic transformations the
symbols of the elliptic quadratic differential operators that we consider in our proof of
the theorem 2.2.1 and the theorem 2.2.2. By using the lemma 3.1.1, we deduce from
the results of the theorem 2.2.1 and the theorem 2.2.2 proved for the rotated harmonic
oscillator that they are therefore also fulfilled by all non-normal one-dimensional el-
liptic quadratic differential operators with a numerical range different from the whole
complex plane.
We now consider the multidimensional case. As we will see in the following, there is
a real jump of complexity between the one-dimensional case and the multidimensional
one. This jump is among other things a consequence of the complexity increase of
symplectic geometry in dimension n ≥ 2 and the larger diversity appearing in the
class of elliptic quadratic differential operators.
4Let us recall that the value of the order is equal to 2 in this case.
3.2. Case of dimension n ≥ 2. We only need to study the case of a non-normal
elliptic quadratic differential operator
(3.2.1) q(x, ξ)w : B → L2(Rn),
in dimension n ≥ 2. Let us recall that in this case, the numerical range Σ(q) is a
closed angular sector with a top in 0 and a positive opening strictly lower than π, and
that the proposition 2.1.2 gives that
(3.2.2) ∃(x0, ξ0) ∈ R2n, {Re q, Im q}(x0, ξ0) 6= 0.
Let us begin by studying what occurs at the interior of the numerical range Σ̊(q).
3.2.1. On the pseudospectrum at the interior of the numerical range. To prove the
existence of semiclassical quasimodes for the associated semiclassical operator given
by the theorem 2.2.1, we need a first purely algebraic step to characterize the points
belonging to the interior of the numerical range.
Let us consider the following decomposition of the numerical range
(3.2.3) Σ(q) = Ã ⊔ B̃,
where
(3.2.4) Ã =
z ∈ Σ(q) : ∃(x0, ξ0) ∈ R2n, z = q(x0, ξ0), {Re q, Im q}(x0, ξ0) 6= 0
(3.2.5) B̃ =
z ∈ Σ(q) : z = q(x0, ξ0) ⇒ {Re q, Im q}(x0, ξ0) = 0
The next section is devoted to give a geometrical description of these two sets. We
establish using purely algebraic arguments that
(3.2.6) Ã = Σ̊(q) and B̃ = ∂Σ(q).
This result is a consequence of the geometry induced by the quadratic setting to which
the studied symbols belong.
Let us begin by noticing that the symplectic invariance of the Poisson bracket
(2.1.10) induces the same property for the sets Ã and B̃. We can therefore use some
real linear symplectic transformation to reduce the symbol q. Since
{Re(zq), Im(zq)} = |z|2{Re q, Im q},
we deduce from this symplectic invariance, from the proposition 2.1.1 and the lemma
18.6.4 in [7] that after a similitude, we can reduce our study to the case where
(3.2.7) Re q(x, ξ) =
j + x
with λj > 0 for all j = 1, ..., n.
3.2.1.a. Geometrical description of the sets Ã and B̃. We begin by proving the fol-
lowing inclusion
(3.2.8) ∂Σ(q) ⊂ B̃.
Let us consider z ∈ ∂Σ(q) and (x0, ξ0) ∈ R2n such that z = q(x0, ξ0). This is
possible because the numerical range is a closed angular sector. If z = 0, the ellipticity
property of q implies that
(x0, ξ0) = (0, 0) and {Re q, Im q}(x0, ξ0) = 0,
because this Poisson bracket is also a quadratic form. This proves that z ∈ B̃. If
z ∈ ∂Σ(q) \ {0},
let us consider the global solution Y of the linear Cauchy problem
(3.2.9)
Y ′(t) = HRe q
Y (t)
Y (0) = (x0, ξ0),
associated to the Hamilton vector field of the symbol Re q,
HRe q =
(∂Re q
− ∂Re q
It is actually a linear Cauchy problem since Re q is a quadratic form. Setting
f(t) = Im q
Y (t)
a direct computation gives that
f ′(0) = {Re q, Im q}(x0, ξ0).
If f ′(0) 6= 0, we could find t0 6= 0 such that
|f(t0)| > |f(0)| = |Im z|.
Since Y is the flow associated to the Hamilton vector field of Re q, the quadratic form
Re q is constant under it. It follows that for all t ∈ R,
Y (t)
= Re q
Y (0)
= Re z
and provides a contradiction because, since z ∈ ∂Σ(q) \ {0}, this would imply in view
of the shape of the numerical range Σ(q) (see Figure 7) that
Y (t0)
6∈ Σ(q).
It follows that the Poisson bracket {Re q, Im q}(x0, ξ0) is necessary equal to 0 and
Figure 7.
q(Y (t
that z ∈ B̃. This ends the proof of the inclusion (3.2.8).
Let us now assume that
(3.2.10) ∂Σ(q) ⊂ B̃, ∂Σ(q) 6= B̃.
In this case, we could find
(3.2.11) z ∈ B̃ \ ∂Σ(q).
Let us first notice that z is necessary non-zero since 0 ∈ ∂Σ(q), and that Re z > 0,
since from (3.2.7),
(3.2.12) Σ(q) \ {0} ⊂ {z ∈ C∗ : Re z > 0}.
The fact that z belongs to the set B̃ implies that
(3.2.13)
Re q(x, ξ) = Re z
Im q(x, ξ) = Im z
=⇒ {Re q, Im q}(x, ξ) = 0.
We also know that there exists at least one solution to the system appearing in the
left-hand-side of (3.2.13). Since from (3.2.7), the quadratic form Re q is positive
definite, we can simultaneously reduce the quadratic forms Re q and Im q by finding
an isomorphism P of R2n such that in the new coordinates y = P−1(x, ξ),
(3.2.14) Re q(Py) =
y2j and Im q(Py) =
j with α1 ≤ ... ≤ αn.
Let us now consider the following quadratic form
(3.2.15) p(y) = {Re q, Im q}(Py).
We get from (3.2.13) and (3.2.14) that
(3.2.16)
j=1 y
j = Re z
j=1 αjy
j = Im z
=⇒ p(y) = 0.
Let us underline that the isomorphism P is not a priori a symplectic transformation
and that it does not preserve the Poisson bracket {Re q, Im q}.
We consider the two following sets
(3.2.17) E1 =
y ∈ R2n : r(y) = 0
where
(3.2.18) r(y) =
(3.2.19) E2 =
y ∈ R2n : p(y) = 0
The next lemma gives a first inclusion between these two sets E1 and E2.
Lemma 3.2.1. We have
(3.2.20) E1 ⊂ E2.
Proof of Lemma 3.2.1. Let y ∈ E1. If y = 0 then y belongs to E2 since from (3.2.15),
p is a quadratic form in the variable y. If y 6= 0, we set
y2j > 0 and ∀j = 1, ..., 2n, ỹj =
We recall from (3.2.12) that z ∈ B̃ \ ∂Σ(q) implies that Re z > 0. Then, since, on
one hand
ỹ2j = Re z,
and that, on the other hand, we have from (3.2.17) and (3.2.18) that
αj ỹ
y2j = Im z,
because y ∈ E1, we deduce from (3.2.16) and the homogeneity of degree 2 of the
quadratic form p that
p(ỹ) =
p(y) = 0.
According to (3.2.19), this proves that y ∈ E2 and ends the proof of the lemma 3.2.1.�
Then, we can notice from (3.2.14) that the boundary of the numerical range ∂Σ(q)
is given by
(3.2.21) (1 + iα1)R+ ∪ (1 + iαn)R+.
Since the numerical range Σ(q) is a closed set, the assumption
z ∈ B̃ \ ∂Σ(q) ⊂ Σ(q) \ ∂Σ(q) = Σ̊(q),
induces from (3.2.21) that
∈]α1, αn[.
This implies that the signature (r1, s1) of the quadratic form r defined in (3.2.18)
fulfills
(3.2.22) (r1, s1) ∈ N∗ × N∗ and r1 + s1 ≤ 2n.
Thus, we can assume after a new labeling that
(3.2.23) r(y) = a1y
1 + ...+ ar1y
− ar1+1y2r1+1 − ...− ar1+s1y
r1+s1
with aj > 0 for all j = 1, ..., r1+ s1. It follows from (3.2.17) and (3.2.23) that in these
new coordinates, the set E1 is the direct product of a proper cone C of R
r1+s1 and
R2n−r1−s1 ,
(3.2.24) E1 = C × R2n−r1−s1 .
Figure 8.
We are now going to prove that the two sets E1 and E2 are equal
(3.2.25) E1 = E2.
Let us reason by the absurd by assuming that it is not the case. Then, we could find
from the lemma 3.2.1,
(3.2.26) y0 ∈ E2 \ E1, y0 = (y′0, y′′0 ) with y′0 ∈ Rr1+s1 , y′′0 ∈ R2n−r1−s1 .
We deduce from (3.2.24) that y′0 6∈ C. Let us now recall an elementary geometrical
fact that we will use several times. This fact is that the intersection of a real line and
a real quadric surface is reduced to either 0, 1 or 2 points, or the line is completely
contained in the quadric surface. We first begin by proving that
(3.2.27) Rr1+s1 × {y′′ = y′′0} ⊂ E2.
Indeed, let us consider the affine subspace
F = {y ∈ R2n : y = (y′, y′′) ∈ Rr1+s1 × R2n−r1−s1 , y′′ = y′′0}.
We identify for more simplicity the space F to the space Rr1+s1 . We agree to say that
a point x′0 of R
r1+s1 belongs to the set E2 to mean that the point (x
0 ) belongs to
the set E2. With this convention, it is sufficient for proving the inclusion (3.2.27) to
consider some particular lines of Rr1+s1 , containing the point y′0 defined in (3.2.26)
and, which have an intersection with the cone C in at least two other different points
u′0 and v
0 (see Figure 9). These lines are necessary contained in the quadric surface
E2 because from the lemma 3.2.1,
E1 ⊂ E2,
and that there are at least three different points of intersection between these lines
and the quadric surface E2,
(u′0, y
0 ) ∈ C × R2n−r1−s1 = E1 ⊂ E2, (v′0, y′′0 ) ∈ C × R2n−r1−s1 = E1 ⊂ E2,
and (y′0, y
0 ) ∈ E2. Thus, we prove that the shaded disc appearing on the figure 10
is completely contained in the set E2. By using the cone structure of the set E2,
we can deduce that all the interior of the cone C (see Figure 11) is contained in E2.
Then, using again other particular intersections with some lines as on the figure 12,
we deduce from our identification of the space F to Rr1+s1 that the inclusion (3.2.27)
is fulfilled.
Figure 9.
We now prove that under these conditions, we have the identity
(3.2.28) E2 = R
Indeed, let us consider (ỹ′0, ỹ
0 ) ∈ R2n = Rr1+s1 × R2n−r1−s1 . If ỹ′0 ∈ C, then
(ỹ′0, ỹ
0 ) ∈ E2,
Figure 10.
These three points belong to E2.
The line D is contained in E2.
Figure 11.
because from (3.2.20) and (3.2.24), (ỹ′0, ỹ
0 ) ∈ E1 and E1 ⊂ E2. If, on the other hand
ỹ′0 6∈ C, we can choose a point u ∈ Rr1+s1 different from ỹ′0 such that u 6∈ C, and such
that the line containing ỹ′0 and u in R
r1+s1 , has an intersection with C in at least two
other different points v and w (see Figure 13). Thus, we can find some distinct real
numbers t1, t2 ∈ R \ {0, 1} such that
v = (1− t1)ỹ′0 + t1u ∈ C and w = (1− t2)ỹ′0 + t2u ∈ C.
Considering now the line
(1− t)(ỹ′0, ỹ′′0 ) + t(u, y′′0 ) : t ∈ R
we can notice that this real line contains at least three different points of E2:
(v, (1 − t1)ỹ′′0 + t1y′′0 ), (w, (1 − t2)ỹ′′0 + t2y′′0 ) and (u, y′′0 ).
Indeed, this is a consequence of the fact that v and w belong to C, and from (3.2.20),
(3.2.24) and (3.2.27). Thus, the line D is contained in the quadric surface E2. This
implies that (ỹ′0, ỹ
0 ) ∈ D ⊂ E2.
To sum up, we have proved that if the two sets E1 and E2 are different then the
set E2 is equal to R
2n. This fact induces in view of (3.2.19) that the quadratic form p
is identically equal to zero. By coming back to the first coordinates (x, ξ) = Py, it
Figure 12.
Figure 13.
follows from (3.2.15) that the quadratic form {Re q, Im q} is also identically equal to
zero, which contradicts (3.2.2). This proves the identity (3.2.25),
E1 = E2.
With this fact, we can resume our first reasoning by the absurd, which assume in
(3.2.11) the existence of a point z ∈ B̃ \ ∂Σ(q). Let us now consider y0 6∈ E1 = E2.
This is possible according to (3.2.2), (3.2.15) and (3.2.19). We deduce from (3.2.17)
and (3.2.19) that r(y0) and p(y0) are non-zero. By considering λ ∈ R∗ such that
p(y0) = λr(y0)
(3.2.29) r̃(y) = p(y)− λr(y),
it follows from (3.2.17), (3.2.19), (3.2.25) and (3.2.29) that
(3.2.30) E1 ⊂ {y ∈ R2n : r̃(y) = 0}.
This inclusion (3.2.30) is strict since
r̃(y0) = 0 and y0 6∈ E1.
By using now exactly the same reasoning as the one previously described to prove
(3.2.25), about the intersections of real lines and quadric surfaces, we prove that the
quadratic form r̃ is necessary identically equal to zero. Then, it follows from (3.2.29)
(3.2.31) p = λr.
By coming back to the first coordinates (x, ξ) = Py, we get using (3.2.14), (3.2.15),
(3.2.18) and (3.2.31) that for all (x, ξ) ∈ R2n,
(3.2.32) {Re q, Im q}(x, ξ) = λ
Im q(x, ξ)− Im z
Re q(x, ξ)
Let us now consider (x0, ξ0) ∈ R2n such that q(x0, ξ0) ∈ ∂Σ(q) \ {0}. This is possible
since the numerical range Σ(q) is a closed angular sector with a top in 0 and a positive
opening. We deduce from (3.2.5) and (3.2.8) that we necessarily have
{Re q, Im q}(x0, ξ0) = 0.
This induces from (3.2.32) that
(3.2.33) Im q(x0, ξ0) =
Re q(x0, ξ0),
because λ ∈ R∗. Since according to the shape of the numerical range Σ(q) and
(3.2.12),
q(x0, ξ0) ∈ ∂Σ(q) \ {0} ⊂ {z ∈ C : Re z > 0},
the identity (3.2.33) proves that the point z also belongs to the set ∂Σ(q), but it
contradicts the initial assumption
z ∈ B̃ \ ∂Σ(q).
Finally, this ends our reasoning by the absurd and proves (3.2.6).
3.2.1.b. Existence of semiclassical quasimodes at the interior of the numerical range.
To prove the existence of semiclassical quasimodes for the associated semiclassical
operator
(q(x, hξ)w)0<h≤1,
in every point of the numerical range’s interior (Theorem 2.2.1), we use an existence
result of semiclassical quasimodes for general pseudodifferential operators violating
the condition (Ψ)5. Let us mention that this result generalizes the two existence
results of semiclassical quasimodes given by E.B. Davies, in the case of Schrödinger
operators (Theorem 1 in [4]), and by M. Zworski in [17] and [18], for pseudodifferential
operators.
This existence result of semiclassical quasimodes can be stated as follows. Let us
consider a semiclassical symbol P (x, ξ;h) in S(〈(x, ξ)〉m, dx2 + dξ2) with m ∈ R+,
〈(x, ξ)〉2 = 1 + x2 + ξ2,
5The definition of the condition (Ψ) is recalled below.
where S(〈(x, ξ)〉m, dx2 + dξ2) stands for the following symbol class
S(〈(x, ξ)〉m, dx2 + dξ2) =
a(x, ξ;h) ∈ C∞(Rnx × Rnξ ,C) :
∀α ∈ N2n, sup
0<h≤1
‖〈(x, ξ)〉−m∂αx,ξa(x, ξ;h)‖L∞(R2n) < +∞
with a semiclassical expansion
(3.2.34) P (x, ξ;h) ∼
hjpj(x, ξ),
where for all j ∈ N, pj is a symbol of the class S(〈(x, ξ)〉m, dx2 + dξ2) independent
from the semiclassical parameter h.
Let z ∈ C, we assume that there exists a function q0 ∈ C∞b (R2n,C), where
C∞b (R
2n,C) stands for the set of bounded complex-valued functions on R2n with
all derivatives bounded, and a bicharacteristic curve, t ∈ [a, b] 7→ γ(t), of the real part
Re(q0(p0 − z)) of the symbol q0(p0 − z), with a < b, such that
(3.2.35) ∀t ∈ [a, b], q0
6= 0 and
q0(γ(a))
p0(γ(a))− z
> 0 > Im
q0(γ(b))
p0(γ(b))− z
Theorem 3.2.1. Under these assumptions (3.2.34) and (3.2.35), for all open neigh-
bourhood V of the compact set γ([a, b]) in R2n and for all N ∈ N, there exist h0 > 0
and (uh)0<h≤h0 a semiclassical family in S(Rn) such that
‖uh‖L2(Rn) = 1, FS
(uh)0<h≤h0
⊂ V and ‖P (x, hξ;h)wuh − zuh‖L2(Rn) = O(hN ),
when h → 0+.
The notation FS
(uh)0<h≤h0
stands for the frequency set of the semiclassical fam-
ily (uh)0<h≤h0 defined as the complement in R
2n of the set composed by the points
(x0, ξ0) ∈ R2n, for which there exists a symbol χ0(x, ξ;h) ∈ S(1, dx2 + dξ2) such that
χ0(x0, ξ0;h) = 1 and ‖χ0(x, hξ;h)wuh‖L2(Rn) = O(h∞),
when h → 0+.
This existence result of semiclassical quasimodes is an adaptation in a semiclassical
setting of the proof given by L. Hörmander in [7] for proving that the condition (Ψ) is a
necessary condition for the solvability of a pseudodifferential operator (Theorem 26.4.7
in [7]). The existence of this result has been first mentioned in [5]. A complete proof of
this adaptation in a semiclassical setting is given in [11]. This result shows that when
the principal symbol p0−z of the symbol P−z violates the condition (Ψ), there exists
in this point z some semiclassical quasimodes inducing the presence of semiclassical
pseudospectrum of infinite index for the semiclassical operator P (x, hξ;h)w.
Condition (Ψ). A complex-valued function p ∈ C∞(R2n,C) fulfills the condition (Ψ)
if there is no complex-valued function q ∈ C∞(R2n,C) such that the imaginary part
Im(qp) of the function qp changes sign from positive values to negative ones along
an oriented bicharacteristic of the symbol Re(qp) on which the function q does not
vanish.
By using the characterization given in the previous section for the interior of the
numerical range Σ̊(q) (see (3.2.4) and (3.2.6)), we are now going to prove that the
principal symbol q(x, ξ) − z of the semiclassical operator
q(x, hξ)w − z,
violates the condition (Ψ) for all z in Σ̊(q). This violation of the condition (Ψ) will
induce in view of the theorem 3.2.1 that for all z ∈ Σ̊(q) and N ∈ N, we can find a
semiclassical quasimode (uh)0<h≤h0 ∈ S(Rn), with h0 > 0, verifying
‖uh‖L2(Rn) = 1 and ‖q(x, hξ)wuh − zuh‖L2(Rn) = O(hN ) when h → 0+,
which will end the proof of Theorem 2.2.1.
Let us consider z ∈ Σ̊(q). We are now going to prove that there is actually a
violation of the condition (Ψ) for the symbol q − z. According to (3.2.4) and (3.2.6),
there are two cases to separate.
Case 1. Let us assume that there exists (x0, ξ0) ∈ R2n such that
(3.2.36) z = q(x0, ξ0), {Re(q − z), Im(q − z)}(x0, ξ0) = {Re q, Im q}(x0, ξ0) < 0.
By considering the solution of the following Cauchy problem
(3.2.37)
Y ′(t) = HRe q
Y (t)
Y (0) = (x0, ξ0),
we define the following function
(3.2.38) f(t) = Im q
Y (t)
− Im q(x0, ξ0).
As mentioned before, (3.2.37) is a linear Cauchy problem. It follows that its solution
Y is global and that the function f is well-defined on R. A direct computation using
(3.2.37) and (3.2.38) gives that for all t ∈ R,
(3.2.39) f ′(t) = {Re q, Im q}
Y (t)
Since from (3.2.36), (3.2.37), (3.2.38) and (3.2.39),
f(0) = 0, f ′(0) = {Re q, Im q}(x0, ξ0) < 0
and HRe q−Re z = HRe q, we deduce in this first case that the imaginary part of the
function q − z changes sign, at the first order, from positive values to negative ones
along the oriented bicharacteristic Y of the symbol Re q−Re z. This proves that the
symbol q − z actually violates the condition (Ψ).
Case 2. Let us now assume that there exists (x0, ξ0) ∈ R2n such that
(3.2.40) z = q(x0, ξ0), {Re(q − z), Im(q − z)}(x0, ξ0) = {Re q, Im q}(x0, ξ0) > 0.
We consider as in the previous case, the global solution Y of the Cauchy problem
(3.2.37) and the function f defined in (3.2.38). Since from (3.2.37), (3.2.38), (3.2.39)
and (3.2.40),
(3.2.41) f(0) = 0, f ′(0) = {Re q, Im q}(x0, ξ0) > 0,
we deduce this time that the imaginary part of the function q − z also changes sign,
at the first order, along the oriented bicharacteristic Y of the symbol Re q − Re z.
Nevertheless, this change of sign is done in the “wrong” way. It is a change of sign
from negative values to positive ones, which does not induce directly a violation of
the condition (Ψ). To check that there is actually a violation of the condition (Ψ)
in this second case, we need to study more precisely the behaviour of the function
Im q − Im z along this bicharacteristic Y .
We deduce from (3.2.41) that there exists ε > 0 such that
∀t ∈ [−ε, ε], f ′(t) > 0,
which induces that
(3.2.42) f(ε) > 0 and f(−ε) < 0,
since from (3.2.41), f(0) = 0. By using the following lemma, we obtain that for all
δ > 0, there exists a time t0(δ) > ε such that
(3.2.43) |Y
t0(δ)
− Y (−ε)| < δ.
Figure 14.
q(Y (�"))
z = q(Y (0))
q(Y ("))
Lemma 3.2.2. If Y (t) = (x(t), ξ(t)) is the C∞(R,R2n) function solving the linear
system of ordinary differential equations
Y ′(t) = HRe q
Y (t)
where Re q is the symbol defined in (3.2.7), then we have
∀t0 ∈ R, ∀ε > 0, ∀M > 0, ∃T1 > M, ∃T2 > M,
|Y (t0)− Y (t0 + T1)| < ε and |Y (t0)− Y (t0 − T2)| < ε.
Proof of Lemma 3.2.2. If Y (t0) = (a1, ..., an, b1, ..., bn) ∈ R2n, we deduce from (3.2.7)
that the function Y (t) = (x(t), ξ(t)) solves the following Cauchy problem
∀j = 1, ..., n,
x′j(t) = 2λjξj(t)
ξ′j(t) = −2λjxj(t)
xj(t0) = aj
ξj(t0) = bj.
It follows that for all j = 1, ..., n and t ∈ R,
(3.2.44)
xj(t) = bj sin
2(t− t0)λj
+ aj cos
2(t− t0)λj
ξj(t) = bj cos
2(t− t0)λj
− aj sin
2(t− t0)λj
Setting βj = λj/π for all j = 1, ..., n, we need to study two different cases.
Case 1: ∀j ∈ {1, ..., n}, βj ∈ Q. In this case, the function Y is periodic and the
result of Lemma 3.2.2 is obvious.
Case 2: (β1, ..., βn) 6∈ Qn. In this second case, we use the following classical result of
rational approximation: ∀ε > 0, ∀(θ1, ..., θn) ∈ Rn \Qn, ∃p1, ..., pn ∈ Z, ∃q ∈ N∗ such
0 < sup
j=1,...,n
If 0 < ε1 < 1/2, we can therefore find some integers p1,1, ..., p1,n ∈ Z and qε1 ∈ N∗
such that
0 < sup
j=1,...,n
|qε1βj − p1,j | < ε1.
j=1,...,n
|qε1βj − p1,j | > 0,
using again this result of rational approximation, we can find some other integers
p2,1, ..., p2,n ∈ Z and qε2 ∈ N∗ such that
0 < sup
j=1,...,n
|qε2βj − p2,j | < ε2.
By using this process, we build some sequences (pm,j)m∈N∗ of Z for j = 1, ..., n,
(εm)m∈N∗ of R
+ and (qεm)m∈N∗ of N
∗ such that for all m ≥ 2,
(3.2.45) 0 < sup
j=1,...,n
|qεmβj − pm,j | < εm =
j=1,...,n
∣qεm−1βj − pm−1,j
(3.2.46) 0 < εm <
The elements of the sequence (qεm)m∈N∗ are necessary two by two different. Indeed,
if qεk = qεl for k < l, this would imply according to (3.2.45) and (3.2.46) that
∀j = 1, ..., n, |pk,j − pl,j| ≤ |qεkβj − pk,j |+ |qεlβj − pl,j | < εk + εl < 1,
because 0 < ε1 < 1/2, which would induce that ∀j = 1, ..., n, pk,j = pl,j because pk,j
and pl,j are some integers; and would contradict (3.2.45) because
0 < sup
j=1,...,n
|qεlβj − pl,j | < εl ≤
j=1,...,n
|qεkβj − pk,j |.
Since the sequence (qεm)m∈N∗ is composed of integers two by two different, we can
assume after a possible extraction that qεm → +∞ when m → +∞. We deduce from
(3.2.44), (3.2.45) and (3.2.46) that
Y (t0 + qεm) → Y (t0) when m → +∞.
Then, considering (β̃1, ..., β̃n) = (−β1, ...,−βn), we obtain by using the same method
a sequence (q̃εm)m∈N∗ of integers such that q̃εm → +∞ and
Y (t0 − q̃εm) → Y (t0) when m → +∞.
This ends the proof of Lemma 3.2.2. �
Since from (3.2.42), f(−ε) < 0, we deduce from (3.2.38) and (3.2.43) that there
exists t0 > ε such that f(t0) is arbitrarily close to f(−ε). It follows in particular that
we can find t0 > ε such that f(t0) < 0. Since from (3.2.42), f(ε) > 0 and f(t0) < 0,
we deduce from (3.2.38) and (3.2.40) that the function
t 7→ Im q
Y (t)
− Im z,
changes sign from positive values to negative ones on the interval [ε, t0]. This proves
that the imaginary part of the function q−z actually changes sign from positive values
to negative ones along the oriented bicharacteristic Y of the symbol Re q−Re z; and
that the symbol q − z also violates in this second case the condition (Ψ). This ends
the proof of Theorem 2.2.1.
3.2.1.c. Another proof for the existence of semiclassical quasimodes. In the following
lines, we give another proof for the existence of semiclassical quasimodes in some
points of the numerical range’s interior. The result proved in this section is weaker
than the one given by the theorem 2.2.1, since we prove the existence of semiclassical
quasimodes in every point of the numerical range’s interior without a finite number
of particular half-lines.
Let us consider a non-normal elliptic quadratic differential operator
(3.2.47) q(x, ξ)w : B → L2(Rn),
in dimension n ≥ 2. We assume, as before, that (3.2.7) is fulfilled. Using that
the quadratic form Re q is positive definite, we can simultaneously reduce the two
quadratic forms Re q and Im q by choosing an isomorphism P of R2n such that in
the new coordinates y = P−1(x, ξ),
(3.2.48) r1(y) = Re q(Py) =
y2j , r2(y) = Im q(Py) =
with α1 ≤ ... ≤ αn. Let us study when the differential forms dr1(y) and dr2(y) are
linearly dependent on R i.e. when there exist (λ, µ) ∈ R2 \ {(0, 0)} such that
(3.2.49) λdr1(y) + µdr2(y) = 0.
It follows from (3.2.48) and (3.2.49) that for all j = 1, ..., 2n,
(3.2.50) (λ+ µαj)yj = 0.
If y 6= 0, then there exists j0 ∈ {1, ..., 2n} such that yj0 6= 0. This implies that
(3.2.51) λ+ µαj0 = 0.
We deduce from (3.2.50) and (3.2.51) that yj = 0 if αj 6= αj0 . Thus, we obtain that if
z ∈ Σ̊(q) \
(1 + iα1)R
+ ∪ ... ∪ (1 + iαn)R∗+
then the differential forms dRe q and dImq are linearly independent on R in every
point of the set q−1(z).
Figure 15.
(1 + i�
(1 + i�
(1 + i�
(1 + i�
(1 + i�
Let us consider such a point
z ∈ Σ̊(q) \
(1 + iα1)R
+ ∪ ... ∪ (1 + iαn)R∗+
Since the dimension n ≥ 2, we can apply the lemma 3.1 in [5] (see also the lemma 8.1
in [9]). It follows that for any compact, connected component Γ of q−1(z), we have
(3.2.52)
{Re q, Im q}(ρ)λq,z(dρ) = 0,
where λq,z stands for the Liouville measure on q
−1(z),
λq,z ∧ dRe q ∧ dIm q =
The set q−1(z) is a non-empty submanifold of codimension 2 in R2n. We deduce from
(3.2.4) and (3.2.6) that there exist (x0, ξ0) ∈ q−1(z) such that
(3.2.53) {Re q, Im q}(x0, ξ0) 6= 0.
Then, it follows from (3.2.52) and (3.2.53) that there necessary exists (x̃0, ξ̃0) ∈ q−1(z)
such that
(3.2.54) {Re q, Im q}(x̃0, ξ̃0) < 0.
Under this condition (3.2.54), we can use the reasoning given in the first studied case
(see (3.2.36)) to prove that the imaginary part of the function q−z changes sign, at the
first order, from positive values to negative ones along an oriented bicharacteristic of
the symbol Re q−Re z. This induces that the symbol q−z violates the condition (Ψ);
and we can conclude by using the theorem 3.2.1. Let us mention that we can also
directly use the existence result of semiclassical quasimodes given by M. Zworski in
[17] and [18]. This second proof gives the existence of semiclassical quasimodes in
every point belonging to the set
Σ̊(q) \
(1 + iα1)R
+ ∪ ... ∪ (1 + iαn)R∗+
3.2.2. On the pseudospectrum at the boundary of the numerical range. In this section,
we give a proof of the theorem 2.2.2. Let us consider a non-normal elliptic quadratic
differential operator
q(x, ξ)w : B → L2(Rn),
in dimension n ≥ 1. We assume that Σ(q) 6= C, and that its Weyl symbol q(x, ξ) is of
finite order kj on a half-line ∆j , j ∈ {1, 2} (See the definition given in (2.2.9)), which
composes the boundary of its numerical range
(3.2.55) ∂Σ(q) = {0} ⊔∆1 ⊔∆2.
As we have already done several times, we can reduce our study to case where (3.2.7)
is fulfilled.
Proof of Theorem 2.2.2. Let us consider the following symbol belonging to the
C∞b (R
2n,C) space, composed of bounded complex-valued functions on R2n with all
derivatives bounded
(3.2.56) r(x, ξ) =
q(x, ξ) − z
1 + x2 + ξ2
with z ∈ ∆j . Setting Σ̃(r) = r(R2n), we can first notice that
z ∈ ∂Σ(q) \ {0} ⇒ 0 ∈ ∂Σ̃(r).
Let us also notice that the symbol r fulfills the principal-type condition in 0. Indeed,
if (x0, ξ0) ∈ R2n was such that r(x0, ξ0) = 0 and dr(x0, ξ0) = 0, we would get from
(3.2.56) that
(3.2.57) dq(x0, ξ0) = 0.
Since from (3.2.7) and (3.2.57), we have
dRe q(x0, ξ0) = 2
(x0)jdxj + (ξ0)jdξj
this would imply that
(x0, ξ0) = (0, 0), q(x0, ξ0) = 0,
because q is a quadratic form and that λj > 0 for all j = 1, ..., n. On the other hand,
since r(x0, ξ0) = 0, we get from (3.2.56) that q(x0, ξ0) = z 6= 0 because
z ∈ ∆j ⊂ ∂Σ(q) \ {0},
which induces a contradiction. It follows that the symbol r actually fulfills the
principal-type condition in 0. Let us notice that, since symbol q is of finite order kj
in z, this induces in view of (3.2.56) that the symbol r is also of finite order kj in 0.
On the other hand, we deduce from (3.2.7) and (3.2.56) that the set
{(x, ξ) ∈ R2n : r(x, ξ) = 0} = {(x, ξ) ∈ R2n : q(x, ξ) = z},
is compact. Under these conditions, we can apply the theorem 1.4 in [5], which proves
that the integer kj is even and gives the existence of positive constants h0 and C1
such that
(3.2.58) ∀ 0 < h < h0, ∀u ∈ S(Rn), ‖r(x, hξ)wu‖L2(Rn) ≥ C1h
kj+1 ‖u‖L2(Rn).
Remark. We did not check the dynamical condition (1.7) in [5], because this assump-
tion is not necessary for the proof of Theorem 1.4. Indeed, this proof only use a part
of the proof of lemma 4.1 in [5] (a part of the second paragraph), where this condition
(1.7) is not needed.
By using some results of symbolic calculus given by Theorem 18.5.4 in [7] and (3.2.56),
we can write
(3.2.59) r(x, hξ)w(1 + x2 + h2ξ2)w = q(x, hξ)w − z + hr1(x, hξ)w + h2r2(x, hξ)w ,
(3.2.60) r1(x, ξ) = −ix
(x, ξ) + iξ
(x, ξ)
(3.2.61) r2(x, ξ) = −
(x, ξ) − 1
(x, ξ).
We can easily check from (3.2.56) that these functions r1 and r2 belong to the space
C∞b (R
2n,C), and we deduce from the Calderón-Vaillancourt theorem that there exists
a positive constant C2 such that for all u ∈ S(Rn) and 0 < h ≤ 1,
(3.2.62) ‖r1(x, hξ)wu‖L2 ≤ C2‖u‖L2 and ‖r2(x, hξ)wu‖L2 ≤ C2‖u‖L2.
It follows from (3.2.58), (3.2.59), (3.2.62) and the triangular inequality that for all
u ∈ S(Rn) and 0 < h < h0,
kj+1 ‖(1 + x2 + h2ξ2)wu‖L2(Rn)
≤ ‖r(x, hξ)w(1 + x2 + h2ξ2)wu‖L2(Rn)
≤ ‖q(x, hξ)wu− zu‖L2(Rn) + C2h(1 + h)‖u‖L2(Rn).
Since from the Cauchy-Schwarz inequality, we have for all u ∈ S(Rn) and 0 < h ≤ 1,
‖u‖2L2(Rn) ≤ ‖u‖2L2(Rn) + ‖xu‖2L2(Rn) + ‖hDxu‖2L2(Rn)
(1 + x2 + h2ξ2)wu, u
L2(Rn)
≤ ‖(1 + x2 + h2ξ2)wu‖L2(Rn)‖u‖L2(Rn),
we obtain that for all u ∈ S(Rn) and 0 < h < h0,
(3.2.63) C1h
kj+1 ‖u‖L2(Rn) ≤ ‖q(x, hξ)wu− zu‖L2(Rn) + C2h(1 + h)‖u‖L2(Rn).
Since kj ≥ 1, we deduce from (3.2.63) that there exist some positive constants h′0 and
C3 such that for all 0 < h < h
0 and u ∈ S(Rn),
‖q(x, hξ)wu− zu‖L2(Rn) ≥ C3h
kj+1 ‖u‖L2(Rn).
Using that the Schwartz space S(Rn) is dense in B and that the operator
q(x, hξ)w + z,
is a Fredholm operator of index 0, we obtain that for all 0 < h < h′0,
q(x, hξ)w − z
∥ ≤ C−13 h
kj+1 ,
which ends the proof of Theorem 2.2.2. �
About the case of infinite order, the situation is much more complicated. As
mentioned before, we cannot expect to prove a stronger result than an absence of
semiclassical pseudospectrum of index 1, but we can actually prove that there is never
some semiclassical pseudospectrum of index 1 on every half-line of infinite order, by
using a result of exponential decay in time for the norm of contraction semigroups
generated by elliptic quadratic differential operators proved in [12].
The result proved in [12] shows that the norm of a contraction semigroup
‖etq(x,ξ)
‖L(L2), t ≥ 0,
generated by an elliptic quadratic differential operator q(x, ξ)w with a Weyl symbol
verifying
Re q ≤ 0, ∃(x0, ξ0) ∈ R2n, Re q(x0, ξ0) 6= 0,
decreases exponentially in time
(3.2.64) ∃M,a > 0, ∀t ≥ 0, ‖etq(x,ξ)
‖L(L2) ≤ Me−at.
Let us consider a non-normal elliptic quadratic differential operator
q(x, ξ)w : B → L2(Rn),
in dimension n ≥ 1 such that Σ(q) 6= C. We explain in the following lines how (3.2.64)
allows to prove that there is never some semiclassical pseudospectrum of index 1 on
any open half-lines composing the boundary of the numerical range ∂Σ(q) \ {0}.
Let z ∈ ∂Σ(q)\{0}. Since the numerical range Σ(q) is a closed angular sector with
a top in 0 and a positive opening strictly lower than π, we can find ε ∈ {±1} such
(3.2.65) Re(εiz−1q) ≤ 0, ∃(x0, ξ0) ∈ R2n, Re(εiz−1q)(x0, ξ0) 6= 0.
Using the theorem 2.8 in [2], we obtain that for all η ∈ R,
q(x, ξ)w − ηz
= − iz−1ε
εiη − εiz−1q(x, ξ)w
= − iz−1ε
e−iεηsesεiz
−1q(x,ξ)wds.(3.2.66)
It follows from (3.2.64) and (3.2.65) that for all η ∈ R,
q(x, ξ)w − ηz
∥ ≤ |z|−1
‖esεiz
−1q(x,ξ)w‖L(L2)ds
≤ |z|−1
Me−asds = |z|−1M
< +∞,
which proves the absence of semiclassical pseudospectrum of index 1 on the half-line
zR∗+. We can actually use the theorem 2.8 in [2] because
iR ⊂ C \ σ
εiz−1q(x, ξ)w
Indeed, if it was not the case, we would deduce from (2.1.7) that there exists u0 ∈
B \ {0} and λ0 ∈ R such that
εiz−1q(x, ξ)wu0 = iλ0u0.
Since from (3.2.65), the quadratic form −Re(εiz−1q) is non-negative, we deduce from
the symplectic invariance of the Weyl quantization and the theorem 21.5.3 in [7] that
there exists a metaplectic operator U such that
(3.2.67) −
εiz−1q(x, ξ)
= U−1
+ x2j ) +
j=k+1
with k, l ∈ N and λj > 0 for all j = 1, ..., k. By using that U is a unitary operator on
L2(Rn), we obtain that
0 = − Re(iλ0u0, u0)L2
= − Re
εiz−1q(x, ξ)wu0, u0
εiz−1q(x, ξ)
u0, u0
‖DxjUu0‖2L2 + ‖xjUu0‖2L2
j=k+1
‖xjUu0‖2L2 ,
which induces that u0 = 0, because from (3.2.65) and (3.2.67), k + l ≥ 1. It follows
from (2.1.7) that there exists ε0 > 0 such that
εiz−1q(x, ξ)w
⊂ {z ∈ C : Re z ≤ −ε0}.
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Department of Mathematics, University of California, Evans Hall, Berke-
ley, CA 94720, USA
E-mail address: karel@math.berkeley.edu
	1. Introduction
	1.1. Miscellaneous facts about pseudospectrum
	1.2. Elliptic quadratic differential operators
	1.3. Semiclassical pseudospectrum
	2. Statement of the results
	2.1. Some notations and some preliminary facts about elliptic quadratic differential operators
	2.2. Statement of the main results
	3. The proofs of the results
	3.1. The one-dimensional case
	3.2. Case of dimension n 2
	References
