BASIC MATRIX PERTURBATION THEORY
BENJAMIN TEXIER
Abstract. In this expository note, we give the proofs of several results in finitedimensional matrix perturbation theory: continuity of the spectrum, regularity
of the total eigenprojectors, existence and computation of one-sided directional
derivatives of semi-simple eigenvalues, and Puiseux expansions of coalescing eigenvalues. These results are all classical, at least in the case of one-dimensional,
analytical perturbations; a standard reference is the treatise of T. Kato, Perturbation theory for linear operators. In contrast with Kato, we consider perturbations which are not necessarily smooth, in arbitrary finite dimension, and in
the coalescing case do not use the notion of multi-valued function. The proofs
use Rouché’s theorem, representations of projectors as contour integrals, and the
description of conjugacy classes of connected covering maps of the punctured disk.
Consider a family of matrices
(1)
M:
x ∈ Ω → M (x) ∈ CN ×N ,
where Ω is open in Rd .
1. Continuity of the eigenvalues
Proposition 1. If x → M (x) is continuous, then the spectrum of M is continuous,
in the following sense: given x0 ∈ Ω, given λ0 ∈ sp M (x0 ) with multiplicity m as
a root of the characteristic polynomial of M (x0 ), for any small enough r > 0 there
exists a neighborhood U of x0 in Ω, such that for all x ∈ U, the matrix M (x) has m
eigenvalues (counting multiplicites) in B(λ0 , r).
Proof. This is a consequence of Rouché’s theorem (see for instance the Corollary
to Theorem 20 in chapter 4 of [1]), which states that if f, g are holomorphic in
B̄(λ0 , r) ⊂ C, and if |f − g| < |g| in ∂B(λ0 , r), then f and g have the same number
of zeros (counting multiplicities) in the open ball B(λ0 , r).
Let Π(λ, x) = det(λ − M (x)), holomorphic in λ and continuous in x. By finiteness
of the spectrum, if r > 0 is small enough, then Π(·, x0 ) has only one zero in the
closed ball B̄(λ0 , r). In particular, |Π(λ, x0 )| > 0 on the boundary ∂B(λ0 , r), and
the inequality
(2)
h(x) = max |Π(·, x) − Π(·, x0 )| − |Π(·, x0 )| < 0
∂B(λ0 ,r)
holds at x = x0 . The map h is continuous. Hence (2) holds in some neighborhood
U of x0 . By Rouché’s theorem, applied with f = Π(·, x) and g = Π(·, x0 ), with x
fixed in U, the function Π(·, x) has the same number of zeros as Π(·, x0 ) in B(λ0 , r),
meaning that M (x) has exactly m eigenvalues in B(λ0 , r), for all x ∈ U.
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BENJAMIN TEXIER
We assume continuity of M in the following. In particular, Proposition 1 applies.
Let
[
S :=
sp M (x) × {x} = (λ, x) ∈ C × Ω, det(λ − M (x)) = 0
x∈Ω
be the spectrum of the family of matrices M, and let the projection
π : (λ, x) ∈ S −→ x ∈ Ω,
so that the spectrum of matrix M (x) is the fiber π −1 ({x}).
The multiplicity of a point (λ, x) ∈ S is the algebraic multiplicity of λ in sp M (x),
that is the order of λ as a root of the characteristic polynomial of M (x).
A point (λ0 , x0 ) ∈ S is said to have constant multiplicity if locally around (λ0 , x0 ),
there exists only one eigenvalue of M (x), not counting multiplicity.
Corollary 2. Around a point of constant multiplicity, the projection π is a local
homeomorphism. If the whole spectrum of M (x0 ) has constant multiplicity, then π
is a covering map at x0 , and the number of sheets is equal to the number of distinct
eigenvalues around x0 .
Proof. If (λ0 , x0 ) has constant multiplicity, the continuous branch of eigenvalues
λ given by Proposition 1 is a continuous section of the projection p, such that
λ(x0 ) = λ0 . Thus in restriction to a neighborhood of (λ0 , x0 ), the projection p
is a homeomorphism. If the whole spectrum {λ1 , . . . , λp } of M (x0 ) has constant
multiplicity, then in addition the fibers have constant cardinal, equal to p, around
x0 . Thus π is a covering map.
If a point in S does not have constant multiplicity, it is said to be a coalescing
point in the spectrum. The associated multiplicity is strictly greater than one.
Coalescing points in the spectrum are not necessarily isolated, even if M is
smooth. Consider for instance the case Ω = R, and let F be a closed set
in R.
0
1
There exists a smooth a ≥ 0 such that F = a−1 ({0}). Then for
every
a(x) 0
point in {0} × F is a coalescing point in the spectrum.
Proposition 3. If Ω ⊂ R, or if Ω is an open subset of C, and if M (x) is a polynomial
in x ∈ Ω, then the spectrum has a finite number of coalescing points.
Proof. We may work with irreducible components Πj of the characteristic polynomial Π (a polynomial in two variables, λ and x). For every such component, Πj and
∂λ Πj are relatively prime. In particular (see for instance Theorem 3 in chapter 8 of
[1]), there is a finite number of x such that Πj (·, x) and ∂λ Πj (·, x) have a common
root λ(x). These common roots (x, λ(x)) are precisely the coalescing points in the
spectrum.
We say that (λ, x) is a regular point in the spectrum (of M, (1)) if either it has
constant multiplicity or it is an isolated coalescing point, in the sense
that there
exists a neighborhood U of (λ, x) in C × Ω such that U \ {(λ, x)} ∩ S comprises
only points of constant multiplicity.
BASIC MATRIX PERTURBATION THEORY
3
Corollary 4. If (λ0 , x0 ) is an isolated coalescing point in the spectrum, then if ε > 0
is small enough, the restriction of the projection π : π −1 (B(x0 , ε)∗ ) → B(x0 , ε)∗ is
a covering map, where π −1 (B(x0 , ε)∗ ) is the inverse image of the ball B(x0 , ε)∗ .
Proof. Identical to the proof of Corollary 2, since the fibers above the (connected)
punctured ball have constant cardinal.
At a coalescing point in the spectrum, eigenvalues may fail to be differentiable,
even if M is smooth. The canonical example is
0 1
(3)
,
x ≥ 0.
x 0
2. Cauchy formulas
We use notation S for the spectrum of the family of matrices M, and the notion
of regular point in the spectrum, as defined in Section 1.
Proposition 5 (Cauchy formula for total eigenprojectors). Let (λ0 , x0 ) be a regular
point in S, and γ a closed, positively oriented curve in C, which does not intersect
sp M (x0 ), and the interior of which intersects sp M (x0 ) at λ0 only. Then for x close
to x0 ,
Z
1
(4)
P (x) =
(λ − M (x))−1 dλ
2iπ γ
is the sum of the projectors onto the generalized eigenspaces associated with eigenvalues of M (x) which lie in the interior of γ. In particular, P is as regular as M.
Above and below, projectors onto generalized eigenspaces (equivalently, generalized eigenprojectors) are implicitly parallel to the direct sum of the other generalized
eigenspaces.
Proof. For fixed x near x0 , consider a spectral decomposition of M (x) :
X
(5)
M (x) =
(λj (x) + Nj (x))Pj (x),
1≤j≤J(x)
where the λj are distinct eigenvalues, the Pj are projectors onto generalized eigenspaces,
such that
X
(6)
Id =
Pj (x),
1≤j≤J(x)
and the Nj are the associated nilpotent components, such that Nj Pj = Pj Nj .
The integer J(x) is the total number of distinct eigenvalues of M (x) near x0 . Since
(λ0 , x0 ) is regular, for some neighborhood U of x0 the number of distinct eigenvalues
near (λ0 , x0 ) is constant on U \ {x0 }. Let m ≤ J(x) be this number, and, for all
x ∈ U \ {x0 }, let λ1 (x), . . . , λm (x) be the distinct eigenvalues of M (x) near λ0 .
By (5) and (6), there holds for x ∈ U and λ ∈
/ sp M (x) :
X
−1
(7)
(λ − M (x))−1 =
λ − λj (x) − Nj (x) Pj (x),
1≤j≤J(x)
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BENJAMIN TEXIER
which we may rewrite, the matrix Id −µNj being invertible for all µ :
(λ − M (x))−1
−1
X
−1 Pj (x),
=
λ − λj (x)
Id −(λ − λj (x))−1 Nj (x)
1≤j≤J(x)
and, expanding in inverse powers of λ − λj (x),
(λ − M (x))−1
X −1
=
(λ − λj (x)
+ O(λ − λj (x))−2 Pj (x).
(8)
1≤j≤J(x)
Given now a closed curve γ as in the statement of the Proposition, for x close
enough to x0 , and different from x0 , the spectrum of M (x) does not intersect γ,
and intersects the interior of γ on exactly the set {λ1 (x), . . . , λm (x)}. We compute
residues:
Z
1
(λ − λj (x))−1 Pj (x) dλ = Pj (x),
1 ≤ j ≤ m,
2iπ γ
Z
1
(λ − λj (x))−1 Pj (x) dλ = 0,
m + 1 ≤ j ≤ J(x),
2iπ γ
Z
1
O(λ − λj (x))−2 Pj (x) dλ = 0,
for all j.
2iπ γ
X
Thus P =
Pj satisfies representation (4).
j
Corollary 6. Around a point (λ0 , x0 ) of constant multiplicity in the spectrum, the
associated eigenvalue and generalized eigenprojector are as regular as M, and there
holds
Z
1
(9)
(λ(x) + N (x))P (x) =
λ(λ − M (x))−1 dλ,
2iπ γ
where x → λ(x) is the local branch of eigenvalues such that λ(x0 ) = λ0 , P is the
associated projector, and N the associated nilpotent.
Note that in the case m = 1, regularity of the eigenvalue follows directly from
the implicit function theorem, since λ0 is then a simple root of the characteristic
polynomial of M.
Proof. The constant multiplicity hypothesis implies that the total eigenprojector
P (x) from Proposition 5 is the generalized eigenprojector onto the unique eigenvalue
λ(x) of M (x) near λ0 . Thus P is as regular as M.
Next we use a spectral decomposition of M (x) in order to express λ(λ − M (x))−1 ,
for λ ∈ C, as a sum of projectors, as we did in (7) in the proof of Proposition 5:
−1
λ(λ − M (x))−1 = λ λ − λ(x) − N (x) P (x)
X
−1
+
λ − λj (x) − Nj (x) Pj (x),
2≤j≤J(x)
BASIC MATRIX PERTURBATION THEORY
5
where λ(x) is the eigenvalue of M (x) which is equal to λ0 at x0 , and the λj (x) are
the other eigenvalues of M (x). In particular, near x0 , the eigenvalues λj (x) are far
from λ0 . Then, if the interior of γ contains λ0 and is small enough, there holds
Z
Z
1
1
−1
λ(λ − M (x)) dλ =
λ(λ − λ(x) − N (x))−1 P (x) dλ.
(10)
2iπ γ
2iπ γ
We now expand the integrand in the above right-hand side in powers of (λ−λ(x))−1 ,
going here one step further than in (8) because of the extra λ prefactor:
λ(λ − λ(x) − N (x))−1
= λ(λ − λ(x))−1 + λ(λ − λ(x))−2 N (x) + λO(λ − λ(x))−3 .
We compute residues:
1
2iπ
Z
1
2iπ
Z
λ(λ − λ(x))−1 P (x) dλ = λ(x)P (x),
γ
λ(λ − λ(x))−2 N (x)P (x) dλ = N (x)P (x),
γ
and
Z
1
λO(λ − λ(x))−3 P (x) dλ = 0.
2iπ γ
With (10), this implies representation (9), from which we deduce that the map
x → (λ(x)+N (x))P (x) is as regular as M. Taking the trace, we find that x → mλ(x)
is as regular as M, where m ≥ 1 is the multiplicity of λ.
Corollary 7. If (λ0 , x0 ) is an isolated coalescing point in the spectrum, with multiplicity m > 1, then there holds
Z
X
1
(11)
(λj (x) + Nj (x))Pj (x) =
λ(λ − M (x))−1 dλ,
2iπ γ
0
1≤j≤m
where x → λj (x), for 1 ≤ m0 ≤ m, are the distinct branches of eigenvalues such that
λj (x0 ) = λ0 , the matrices Pj are the associated projectors, and Nj the associated
nilpotents.
Proof. For all x ∈ U \ {x0 }, where U is some neighborhood of x0 , the matrix M (x)
has the same number of distinct eigenvalues in a neighborhood of λ0 . This number is
less than or equal to m, the multiplicity of λ0 . Let λ1 , . . . , λm0 be these eigenvalues.
It suffices to reproduce the computations of the proof of Corollary 6, where each λj
plays the same role as λ in the proof of Corollary 6, to arrive at (11).
3. Regularity of the eigenvalues
Proposition 8. If M is differentiable at x0 , then for any local branch λ of eigenvalues of M around x0 , there holds the bound
(12)
|λ(x) − λ(x0 )| ≤ C(M )|x − x0 |1/m ,
locally around x0 , with C(M ) > 0, where m is the multiplicity of (λ(x0 ), x0 ).
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BENJAMIN TEXIER
If (λ(x0 ), x0 ) has constant multiplicity and M is locally Lipschitz, then by Corollary 6 the eigenvalues are actually Lipschitz, locally around x0 , which of course is
much better than (12) in the case m > 1. Estimate (12) however accurately describes
the eigenvalue behavior in the canonical coalescing case (3).
Proof of Proposition 8. We denote λ0 = λ(x0 ). The characteristic polynomial
Π(λ, x) = det λ − M (x)
factorizes into Π = Π0 Π1 , where Π1 (λ0 , x0 ) 6= 0, and Π0 (λ, x0 ) = (λ − λ0 )m . The
degree of Π0 is equal to m, the multiplicity of (λ0 , x0 ), and Π0 is unitary. We
may focus on Π0 in the following. Let λ be a branch of eigenvalues such that
λ(x0 ) = λ0 . Expanding Π0 in powers of λ(x) − λ0 , we find, since ∂λj Π0 (λ0 , x0 ) = 0
for 0 ≤ j ≤ m − 1 :
Π0 (λ(x), x0 ) = m!−1 (λ(x) − λ0 )m + O(|λ(x) − λ0 |)m+1 .
Besides, the matrices M being differentiable at x0 , the characteristic polynomial Π
is differentiable in x at x0 , and so is Π0 :
Π0 (λ(x), x) = Π0 (λ(x), x0 ) + O(|x − x0 )| ≡ 0.
Thus
m!−1 (λ(x) − λ0 )m + O(|λ(x) − λ0 |)m+1 = O(|x − x0 |),
which implies (12).
The estimate of Proposition 8 is much improved in the semi-simple case:
Proposition 9. If M is differentiable at x0 , and if (λ0 , x0 ) is an isolated coalescing
point such that λ0 is a semi-simple eigenvalue of M (x0 ), any local branch λ of
eigenvalues of M around x0 has a one-sided directional derivative in every direction,
and there holds, for all ~e ∈ Rd ,
lim
t→0
t>0
λ(x0 + t~e ) − λ(x0 )
∈ sp P (λ0 , x0 )M 0 (x0 ) · ~e P (λ0 , x0 ).
t
In particular, the eigenvalue are Lipschitz:
|λ(x) − λ(x0 )| ≤ C(M )|x − x0 |,
locally around x0 , with C(M ) > 0.
Proof. We denote λ0 = λ(x0 ). Let m be the multiplicity of λ0 , and λ1 , . . . , λm0 ,
2 ≤ m0 ≤ m, the distinct eigenvalues that coalesce at x0 with value λ0 . By Corollary
7, there holds
Z
X
−1
λ λ − M (x0 + h)
dλ =
λ̃j (h) + Ñj (h) P̃j (h),
γ
1≤j≤m0
where h ∈ Rd is small and γ is a suitable curve in C. Above, λ̃j (h) := λj (x0 + h),
and Ñj and P̃j are the corresponding nilpotent and projector. By Proposition 5,
BASIC MATRIX PERTURBATION THEORY
7
there holds
Z
λ0 (λ − M (x0 + h)−1 dλ = P̃ (h) =
γ
X
P̃j (h).
1≤j≤m0
Thus
Z
(λ − λ0 ) λ − M (x0 + h)
−1
dλ
γ
(13)
X
=
λ̃j (h) − λ0 + Ñj (h) P̃j (t),
1≤j≤m0
By differentiability of M at x0 :
−1
λ − M (x0 + h)
= (λ − M (x0 ))−1 + O(h),
the O(h) term being precisely
(λ − M (x0 ))−1 M 0 (x0 ) · h(λ − M (x0 ))−1 + o(h).
Since λ0 is semi-simple, the spectral decomposition at x0 is
M (x0 ) = λ0 P (λ0 , x0 ) + M (x0 )(Id −P (λ0 , x0 )),
where P (λ0 , x0 ) is the generalized eigenprojector. Thus
(λ − M (x0 ))−1 = (λ − λ0 )−1 P (λ0 , x0 ) + (λ − M (x0 ))−1 (Id −P (λ0 , x0 )),
and, computing residues, we find for a suitable curve γ :
Z
1
(λ − λ0 )(λ − M (x0 ))−1 dλ = 0,
2iπ γ
and
1
2iπ
Z
(λ − λ0 )(λ − M (x0 ))−1 (Id −P (λ0 , x0 )) dλ = 0.
γ
From (13) and the above, we deduce
X
1≤j≤m0
λj (x0 + h) − λ0 + Ñj (h)
|h|
= P (λ0 , x0 )M 0 (x0 ) ·
!
P̃j (h)
h
P (λ0 , x0 ) + o(1),
|h|
Equating spectra, evaluating at h = t~e, for t > 0, and taking the limit t → 0 implies
the result.
We could trade in Proposition 9 the assumption that (λ0 , x0 ) is an isolated coalescing point for an assumption of regularity of M, by reasoning as in the proof of
the upcoming Proposition 11.
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BENJAMIN TEXIER
4. Puiseux expansions
We describe eigenvalues around a coalescing point, following the approach of [9].
Consider a point (λ0 , x0 ) with multiplicity m in the spectrum, and suppose that
M is q ≥ m times differentiable at x0 , so that
M (x) = M0 (x − x0 ) + o(x − x0 )q ,
where M0 is a polynomial of degree q. In particular, M0 has an extension to Cd . Let
~e ∈ Rd be a fixed spatial direction, and consider
S0 := (λ, z) ∈ C × B(0, ε), det(M0 (x0 + z~e) − λId) = 0 ,
where B(0, ε) ⊂ C is the open disk centered at 0 and with radius ε > 0 in the
complex plane. We denote π0 the projection
π0 : (λ, z) ∈ S0 −→ z ∈ B(0, ε).
By Proposition 3, if ε is small enough then S0 has only (λ0 , 0) has a coalescing point.
Thus by Corollary 4, the restriction of π0 to π0−1 (B(0, ε)∗ ) is a covering map if ε
is small enough. Let V be a connected component of S0 which intersects R∗+ . We
denote π̃0 the restriction
π̃0 :
(λ, z) ∈ V ∩ π0−1 (B(0, ε)∗ ) −→ z ∈ B(0, ε)∗ .
Then π̃0 is a connected covering of the punctured disk B(0, ε)∗ , in the sense that
the total space V ∩ π0−1 (B(0, ε)∗ ) is connected.
0
Lemma 10. The covering map π̃0 is conjugated to the covering p : z → z m of
B(0, ε)∗ for some m0 ≤ m. That is, there exists a homeomorphism ψ such that the
following diagram is commutative:
ψ
V ∩ π0−1 (B(0, ε)∗ ) −→ p−1 (B(0, ε)∗ )
π̃0 @
R
@
p
B(0, ε)∗
Proof. Let λ1 (x0 + z~e), . . . , λm0 (x0 + z~e) be the eigenvalues of M0 which takes values in V0 for z ∈ B(0, ε)∗ . The numbers of these eigenvalues is constant over the
connected punctured disk B(0, ε)∗ . In particular, π̃0 is an m0 -sheeted covering of
B(0, ε)∗ . Connected coverings of a punctured ball in C are determined, up to isomorphism, by their numbers of sheets (see for instance [6], chapter V, Theorem 6.6).
Thus π̃0 is conjugated to p, by a homeomorphism ψ.
Based on Lemma 10, we may give Puiseux expansions of eigenvalues around a
coalescing point:
Proposition 11. If (λ0 , x0 ) is a coalescing point in the spectrum of M, with multiplicity m, and if M is q ≥ m times differentiable at x0 , then for any local branch
λ of eigenvalues of M which coalesce at x0 with value λ0 , any ~e ∈ Rd , there exists
BASIC MATRIX PERTURBATION THEORY
9
a smooth map φ defined in [0, t0 ], for some t0 > 0, and a positive integer m0 ≤ m,
where m is the multiplicity of λ0 , such that
(14)
0
λ(x0 + t~e ) = φ(t1/m ) + o(tq/m ),
for 0 ≤ t ≤ t0 .
Proof. Let
M (x) = M0 (x − x0 ) + |x − x0 |q R(x0 , x),
be the Taylor expansion of M around x0 at order q. The family of matrices R(x0 , x)
converges to 0 as x → x0 . Associated with the family of matrix polynomials M0 ,
consider the spectrum S0 and the connected covering π̃0 , as described above Lemma
10.
Let µ be a section of π̃0 , that is a branch of eigenvalues of M0 (x0 + z~e). There
holds π̃0 (µ) ≡ Id, hence, by Lemma 10, p ◦ ψ −1 ◦ µ ≡ Id. Thus ψ −1 ◦ µ is a section
of p, meaning an m0 -th root of unity:
0
µ(z) = φ(ωz 1/m ),
(15)
where φ is the first component of ψ, and ω is a given m0 -th root of unity. Equality
(15) holds for all z ∈ π̃0 (V ). By definition, V is a connected component of the total
−1
∗
∗
space
p (B(0, ε) ) which
intersects R+ . If t0 > 0 belongs to π0 (V ), then all of the
arc (µ(t), t), 0 < t < t0 belongs to V, by connectedness of V and continuity of µ.
Thus t belongs to π̃0 (V ), if t > 0 is small enough, and equality (15) holds for small
enough t > 0. In particular,
0
µ(tm ) = φ(ωt),
for 0 < t ≤ t0 .
implying that φ is as regular as µ, hence analytical. Thus φ is analytical in 0 < t < t0
and bounded around t = 0, implying that φ is analytical in [0, t0 ], so that (15) holds
for all 0 ≤ t ≤ t0 , with µ(0) = φ(0). Let now
M(x, y) = M0 (x − x0 ) + y,
2
y ∈ CN ,
and let λ be a local branch of eigenvalues of M, such that, for small t,
λ(x0 + t~e , 0) = µ(t),
and for small |x − x0 | :
λ x0 + t~e , |x − x0 |q R(x0 , x) = λ(x0 + t~e ),
where λ is a local branch of coalescing eigenvalues of M. By Proposition 8 (where
2
2
the variable y ∈ CN is seen as a real variable y ∈ R2N ), there holds
λ(x0 + t~e , y) − λ(x0 + t~e , 0) = O(|y|)1/m .
Thus
λ(x0 + t~e ) = µ(t) + O |x − x0 |q |R(x0 , x|)
which implies (14), via (15), since R = o(1).
1/m
,
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BENJAMIN TEXIER
Bibliographical note. The Cauchy formula of Proposition 5 is found in equation
(1.16), paragraph 1.4, chapter 2, in [3]. The existence of directional derivatives
(Proposition 9) is found in Theorem 2.3, paragraph 2.3, chapter 2, in [3]. Kato
refers to Knopp [4], without proof, for details on Puiseux expansions (see [3], chapter 2, paragraph 1.2). So do Reed and Simon ([7], XII.1). Knopp’s discussion is
limited to polynomials in two variables, the roots of which are described as multivalued analytical functions; here eigenvalues around a coalescing point are seen as
(perturbations of) sections of a ramified covering of a disk in the complex plane.
Remark 12 (On hyperbolic polynomials). If the spectrum of M (x) is real for all
x ∈ Ω, then the family M is said to be hyperbolic. The eigenvalues are then locally
Lipschitz; see Brohnstein [2], or Kurdyka and Paunescu [5]. In one space dimension,
Rellich’s theorem [8] states that analytic family of hermitian matrices have analytic
eigenvalues and eigenvectors.
Remark 13 (On geometric optics). An important consequence of Proposition 9 is
that the amplitude of a wave-packet is transported by a hyperbolic system at group
velocity; this is a crucial step in the derivation of amplitude equations in geometric
optics, see [9] and references therein.
Similar formulas exist for higher derivatives (see [9], Proposition 2.6 and Remark
2.7, and Kato [3], paragraphs 2.1 and 2.2, chapter 2). The corresponding identity
for second-order derivatives describes the Schrödinger correction to the transport
along rays for distances of propagation equal to the inverse of the wavelength.
References
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[2] M. D. Brohnstein. Smoothness of roots depending on parameter, Siberian Math. J. 20
(1979), 347–352.
[3] T. Kato. Perturbation theory for linear operators. Reprint of the 1980 edition. Classics in
Mathematics. Springer-Verlag, Berlin, 1995. xxii+619 pp.
[4] K. Knopp. Theory of Functions. II. Applications and Continuation of the General Theory.
Dover Publications, New York, 1947. x+150 pp.
[5] K. Kurdyka, L. Paunescu. Hyperbolic polynomials and multiparameter real analytic perturbation theory, Duke Math. J. 141 (2008), 123–149.
[6] W. S. Massey. A basic course in algebraic topology. Graduate Texts in Mathematics, 127.
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[7] M. Reed, B. Simon. Methods of modern mathematical physics. IV. Analysis of operators.
Academic Press, New York-London, 1978. xv+396 pp.
[8] F. Rellich. Perturbation theory of eigenvalue problems. Assisted by J. Berkowitz. With a
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9 (2004), no. 1, 1-52.
Institut de Mathématiques de Jussieu-Paris Rive Gauche UMR CNRS 7586, Université Paris-Diderot
E-mail address: [email protected]