The generalized spectral theory
andits application to the Kuramoto conjecture
Advanced Institute for Materials Research, Tohoku University,
Sendai, 980-8577, Japan

Hayato CHIBA 1
Jan 18, 2024
Abstract
A spectral theory of linear operators based on a Gelfand triplet (rigged
Hilbert space) is developed under the assumptions that a linear operator T on
a Hilbert space H is a perturbation of a self-adjoint operator, and the spectral
measure of the self-adjoint operator has an analytic continuation near thereal axis. It is shown that for a suitable dense subspace X of H and its dual
space X 0 , for any ϕ ∈ X, the resolvent (λ − T )−1 ϕ of the operator T has ananalytic continuation from the lower half plane to the upper half plane as an
X 0 -valued holomorphic function even when T has a continuous spectrum on
R. The Gelfand triplet consists of three topological vector spaces X ⊂ H ⊂X 0 . Basic tools of the usual spectral theory, such as spectra, resolvents andRiesz projections are extended to those deﬁned on a Gelfand triplet. They
prove to have the same properties as those of the usual spectral theory. The
results are applied to estimate exponential decays of the semigroups of linear
operators and bifurcations of nonlinear dynamical systems. In particular, aconjecture on a bifurcation of the Kuramoto model (Kuramoto conjecture)
will be solved.

1

Kuramoto model

In the last few decades, the study of large / inﬁnite-dimensional dynamical systems
becomes more and more important. Let us consider the system of diﬀerential
equationsdxi= fi (x1 , · · · , xn ),dt

i = 1, · · · , N,

where xi = xi (t) moves on some phase space. This dynamical system assigns a
directed graph in the following way:1

E mail address : hchiba@tohoku.ac.jp

1

N
be the set of vortices. If the function fi depends on x j , we
Let V = {vi }i=1deﬁne the edge e ji from the vortex v j to vi , that means that x j aﬀects the dynamics
of xi . It is a challenging problem to understand how the graph structure is related
to the dynamics. For simplicity, we consider the system of the formdθi
= fi (θ1 , · · · , θn ),dt

i = 1, · · · , N,

(1.1)

where θi ∈ S 1 rotates on a circle. Thus, the phase space is an N-torus. A dynamicalsystem of the form is often called a system of coupled oscillators [21, 23]. For
example, if a given dynamical system on a phase space X has an N-torus as aninvariant manifold, we obtain a coupled oscillators by restricting the dynamics onit.
The Kuramoto model is one of the most famous coupled oscillators given byNKXdθi= ωi +sin(θ j − θi ),
dt
N j=1

i = 1, · · · , N,

(1.2)

which is well-known as a typical mathematical model for synchronization phenomena [20, 21, 26]. Here, ωi and K are constants called natural frequencies
and the coupling strength, respectively. When the coupling strength is zero, there
are no interactions between oscillators and they rotate with their own velocity ωi .
Hence, if ω j > ωi then θ j overtakes θi many times. However, if K is positive,
there are interactions between oscillators through the term sin(θ j − θi ) and we expect that if K is large enough, such an overtaking does not occur. Indeed, it is easy
to observe by numerics that there exists a threshold Kc such that when K > Kc , a
synchronized state appears; a subset of oscillators forms a cluster on a circle and it
behaves like a big oscillator without overtaking. The cluster consists of oscillators
whose natural frequency ωi is close to the average Ω of all natural frequencies. As
K increases, the number of oscillators that are entrained into the cluster gets larger(Fig.1).

r

Figure 1: (left) synchronization. (right) de-synchronization.

2

In order to observe that whether a synchronization occurs or not, it is convenient to introduce the order parameter deﬁned by
1 X iθ j (t)
e .
N j=1
N

η :=

(1.3)

This gives the center of mass of oscillators on a unit circle. Hence, when its
absolute value r := |η| is positive (resp. zero), a synchronization occurs (resp.
does not occur). Kuramoto performed a certain formal and technical calculation
using the order parameter, and reached the following result, though there are nomathematical proofs.
The Kuramoto conjecture [21, 26].
Suppose N → ∞ and the natural frequencies are independent and identically
distributed according to a probability density function g(ω). If g(ω) is an evenand unimodal function, a bifurcation diagram of the order parameter r = |η| isgiven as Fig. 2. This means that when K is smaller than Kc := 2/(πg(0)), the desynchronized state r = 0 is asymptotically stable. At K = Kc , a bifurcation (phasetransition) occurs and a stable synchronized state (r > 0) √exists for K > Kc . Near
the bifurcation point, r is approximately given by r ∼ O( K − Kc ).
r

Kc

K

Figure 2: A bifurcation diagram of the order parameter.
Note that by a translation of the coordinate θi 7→ θi + Ωt, we can assume
without loss of generality that the average value of g(ω) is zero. Then, “ g(ω)
is unimodal” means that when ω > 0 (resp. ω < 0), it is strictly monotonicallydecreasing (resp. increasing). The bifurcation point Kc := 2/(πg(0)) is often
called Kuramoto’s transition point. See [26] for Kuramoto’s formal calculation.
As explained in later sections, the diﬃculty of a mathematical approach to theKuramoto conjecture is that a certain linear operator obtained by the linearizationof the model has a continuous spectrum. Recently, the author developed the generalized spectrum theory of linear operators based on Gelfand triplets and proposed
an eﬀective method to investigate linear operators having continuous spectra [5].
3By applying this theory to the inﬁnite dimensional Kuramoto model, he proved
the Kuramoto conjecture under a suitable condition [3, 4]. In what follows, h(θ)
denotes a distribution of the initial values {θ j (0)}∞j=1 of oscillators.
Theorem 1.1 Suppose that g(ω) is the Gaussian distribution. When 0 < K < Kc ,there exists δ > 0 such that if h(θ) satisﬁes
Z 2π
ei jθ h(θ)dθ < δ, j = 1, 2, · · · ,0then the order parameter η(t) tends to zero as t → ∞ with an exponential rate.
Theorem 1.2 Suppose that g(ω) is the Gaussian distribution. There exist numbers ε0 , δ > 0 such that if h(θ) satisﬁes
Z 2π
ei jθ h(θ)dθ < δ, j = 1, 2, · · · ,0then for Kc < K < Kc + ε0 , the absolute value of the order parameter converges
to the following value as t → ∞s−16 p|η(t)| =K − Kc + O(K − Kc ).
πKc4 g00 (0)In particular, a bifurcation diagram of the order parameter is given as Fig. 2.
This result holds even if g(ω) is not Gaussian. The most essential assumption
is that g(ω) is analytic on R and it has an analytic continuation around the realaxis. On the other hand, the Kuramoto conjecture was proved when g(ω) is a C nfunction in [13] by a diﬀerent way. In this case, the decay rate (Thm.1.1) is notexponential but algebraic O(t−n ).
More general form of the Kuramoto model is given as
Ndθi
KX= ωi +
ai j sin(θ j − θi + α), i = 1, · · · , N,
(1.4)
dt
N j=1
and g(ω) is not unimodal, where α is a phase lag and ai j denotes the adjacency
matrix that determines the graph structure (we have an edge connecting θ j and θionly when ai j , 0). We can also consider other periodic function as the interaction
term such as sin 2(θ j − θi ). Even in these cases, we can obtain similar results asabove, see [4, 7, 8, 9, 10] for the details.
The generalized spectral theory used to prove the Kuramoto conjecture is also
applicable to any problems related to continuous spectra, such as Schrödinger
equations [6], chaos in symbolic dynamical systems [11], the onset of the humanbrain wave [12], and so on.
The purpose of this article is to illustrate the generalized spectral theory. For
it, we begin to investigate the Kuramoto model within the usual spectral theory inSec. 2 and 3.
4

2

Continuous limit

Since we assume N → ∞ for the Kuramoto conjecture, we ﬁrst start to deﬁne theinﬁnite-dimensional version of the Kuramoto model.
By substituting the deﬁnition of the order parameter η into the Kuramotomodel (1.2), we obtaindθi= ωi + Kr sin(ψ − θi ),dt
where we put η = reiψ . With this in mind, the continuous limit of the Kuramotomodel is deﬁned by the following system

∂
∂ρt



+ (vρt ) = 0, ρt = ρt (θ, ω),





 v∂t:= ω ∂θ
+ Kr sin(ψ − θ),
(2.1)


Z
Z 2π




iψ


eiθ ρt (θ, ω)dθ.
 η := re = g(ω)dω0R

Now inﬁnitely many oscillators rotate on a circle like a ﬂuid, and ρt denotes its
distribution. More precisely, ρt (θ, ω) implies a probability density function of θfor each time t and natural frequency ω. The ﬁrst line of the system is the equationof continuity (conservation law) of ρt . The velocity ﬁeld v is given by the second
line, that comes from the right hand side of the ﬁnite-dimensional one by removing
the subscript i. The third line is the deﬁnition of the inﬁnite-dimensional versionof the order parameter, that is obtained by replacing the summation in the ﬁnitedim model by the integral with respect to the measure g(ω)ρt (θ, ω)dωdθ, whereg(ω) is a given density function. In this article, we assume that it is the Gaussian
distribution for simplicity. It is easy to show that this system has a unique weaksolution for a given initial condition ρ0 (θ, ω) = h(θ, ω) for any t > 0.
Since the unknown function ρt is periodic in θ, we consider the Fourier seriesof it. The Fourier coeﬃcients are given by
Z 2π
Z j (t, ω) :=ei jθ ρt (θ, ω)dθ.
0

Rewriting the equation of ρt by Z j yields the system of equations of Z j as
dZ1
KK
= iωZ1 + η(t) − η(t)Z2 ,
dt
2
2
and
dZ j
jK
= i jωZ j +(η(t)Z j−1 − η(t)Z j+1 ),dt
25for j = 2, 3, · · · . Note that Z0 = 1 because of the normalization of a probability
density. This system has the trivial solution Z j = 0, j = 1, 2, · · · . In this caseρt = 1/(2π) is the uniform distribution on a circle which corresponds to the desynchronization state.
Let
R us investigate the stability of the trivial solution. Since η is written as
η = R Z1 g(ω)dω, the terms such as η(t)Z j in the system are nonlinear terms.
Therefore, the linearization of the system around the trivial solution is given by

dZ1
K 
= T 1 Z1 := iM + P Z1 ,
dt
2
dZ j
= i jMZ j , j = 2, 3, · · ·dtwhere the linear operators M and P are deﬁned by M : f (ω) 7→ ω f (ω) andZ
P f (ω) =f (ω)g(ω)dω.
R

R

The order parameter η = R Z1 g(ω)dω = PZ1 depends only on Z1 at least for
the linearized system. Hence, let us investigate the spectrum of the operator T 1 =iM + KP/2, that deﬁnes the linearized system for Z1 , as an operator on the Hilbert
space L2 (R, g(ω)dω) (weighted Lebesgue space).

3

Spectrum of linear operators

For the comparison with the generalized spectrum, we give a brief review of the(usual) spectral theory on a Banach space.
The spectrum set σ(T ) of a linear operator T on a Banach space X is the set ofthe singularities of the resolvent operator (λ − T )−1 . More precisely, it consists of;point spectrum σ p (T ). The set of point λ such that λ − T is not injective on X.
residual spectrum σr (T ). The set of point λ such that λ − T is injective on X but
its range is not a dense subspace of X.
continuous spectrum σc (T ). The set of point λ such that λ − T is injective andthe range is dense but the inverse (λ − T )−1 is not a continuous operator on
X.
The resolvent set is deﬁned by ρ(T ) = C\σ(T ).
The point spectrum is just the set of eigenvalues ; T v = λv has a solutionv , 0 in X. If X is a ﬁnite dimensional space, λ − T is surjective if and only if
6

it is injective, however, it is not true for inﬁnite one. Unfortunately, λ − T is notsurjective for most problems. Hence, we consider a more mild condition that therange of λ − T is dense or not dense in X. When it is not dense, the set of such
λ is the residual spectrum. If λ is neither in the point spectrum nor the residual
spectrum, the inverse (λ − T )−1 exists and the domain is dense. If (λ − T )−1 is acontinuous operator, we can continuously extend its domain to the whole space
X, and λ is a regular point. The continuous spectrum is the set of λ such that wecannot extend the domain of (λ − T )−1 to the whole space (recall the closed graphtheorem) 2 .
Let us consider the linear diﬀerential equation du/dt = T u, u ∈ X deﬁned ona Banach space X. The asymptotic behavior of a solution as t → ∞ is almost
characterized by the spectrum of T . Indeed, under a suitable assumption for T , itis known that a solution is expressed by the Laplace inversion formula
Z a+i∞1Tt
eλt (λ − T )−1 u(0)dλ,(3.1)
u(t) = e u(0) =2πi a−i∞for t > 0, where the integral path is a vertical straight line such that the spectrum
set of T is included in the left half plane Re(λ) < a. The operator eT t is called thesemigroup generated by T . For example when X is a ﬁnite dimensional space, theset of singularities of the integrant (λ − T )−1 consists only of the eigenvalues ofT . In this case, we can calculate the Laplace inversion formula by deforming the
integral path and using the residue theorem as is shown in Fig. 3. Hence, the realparts of eigenvalues completely determine the asymptotic behavior of solutions
because of the factor eλt . This is true even if X is an inﬁnite-dim space as long as
T is a bounded operator or a sectorial operator, that admit the deformation of theintegral path as in Fig. 3 3 .
2The concept of the spectrum makes sense only when T is a closed operator. Let us consider
the point λ ∈ ρ(T ). By the deﬁnition, the resolvent (λ − T )−1 is a continuous operator on X, inparticular it is a closed operator. It is known that the inverse of a closed operator is also closed.
Thus, ((λ − T )−1 )−1 = λ − T is also closed. Then, T = −(λ − T ) + λ is also closed. Consider thecontraposition. If T is not a closed operator, there are no λ ∈ ρ(T ); the whole complex plane is the
spectrum set.
3
Roughly speaking, a sectorial operator is an operator such that its spectrum set is included in asmall sector that is open toward the left direction (i.e. included in the > shape region). A bounded
operator is always sectorial because its spectrum set is compact.
If T is not sectorial, solutions u(t) may diverge as t → ∞ even when the spectrum set is includedin the left half plane. A typical situation is that the spectrum set is not bounded for imaginary
direction (so we can not take > shape region). This means that the spectrum set does not determine
the behavior of solutions [18].
This diﬃculty essentially comes from the fact that the spectral mapping theorem does not hold.
Let S (t) = eT t be a semi-group generated by T . If T is bounded or a self-adjoint operator on
a Hilbert space, we have the spectral mapping theorem, that states eσ(T )t = σ(S (t)). However,in general we only have eσ(T )t ⊂ σ(S (t)). This implies that S (t) has an information that is not

7

With this in mind, let us calculate the spectrum of the operator T 1 = iM +KP/2. We only give the sketch of proofs, see [3] for the detail.

Figure 3: A deformation of the integral path. × denotes an eigenvalue.

Proposition 3.1 T 1 is a densely deﬁned closed operator on L2 (R, g(ω)dω) satisfying(i) the continuous spectrum is σc (T 1 ) = i · supp(g),(ii) the residual spectrum is empty,(iii) an eigenvalue is given as a root of the equationZ1
2
g(ω)dω = , λ ∈ C\σc (T 1 ),(3.2)
K
R λ − iω
if it exists.
Since P is a compact operator, (i) and (ii) immediately follow from the perturbation theory of linear operators. If g is the Gaussian distribution, σc (T 1 ) = iR is
the whole imaginary axis. Let us derive the eigen-equation (3.2). Let P0 (ω) ≡ 1
be a constant function. By using the inner product on L2 (R, g(ω)dω), P is written
as P f = ( f, P0 )P0 . This gives
λv = T 1 v = iωv +=⇒ v =

K(v, P0 )P02

K(v, P0 )(λ − iω)−1 P0 .
2obtained from an information of σ(T ).
The domain D(T ) of an unbounded operator T is not the whole space X but its dense subspace.
However, the domain of its semigroup S (t) can be the whole space. Hence, if we take an initialcondition u(0) from X\D(T ), then the behavior of a solution u(t) is not captured by σ(T ). If wechoose u(0) from D(T ), we may obtain nice information about a solution from σ(T ), see [18] forthe detail.

8

By taking the inner product with P0 and dividing by (v, P0 ), we obtain (3.2). From
this calculation, it turns out that an eigenvector is given byv(ω) =

1
λ − iω

(3.3)

if an eigenvalue λ exists.
Proposition 3.2 Suppose that g is an even, unimodal and continuous function.
(i) When K > Kc := 2/(πg(0)), there exists a unique eigenvalue on the positive
real axis,(ii) it converges to the origin as K → Kc + 0 and(iii) when K ≤ Kc , there are no eigenvalues.
Proof. By setting λ = x + iy and decomposing (3.2) into the real and complex
parts, we have
 Z
x
2



g(ω)dω = ,22 ZR x + (ω − y)
K


ω
−
y



g(ω)dω = 0.
2
2
R x + (ω − y)Further the second line is written asZ ∞ω(g(y + ω) − g(y − ω)) dω = 0.
220 x +ω
If g is even, y = 0 satisﬁes it. By using that g is unimodal, we can verify thaty , 0 does not satisfy. The ﬁrst line shows that when K > 0, x is positive, whichmeans that if an eigenvalue exists, it should be on the positive real line. It also
follows from the ﬁrst line that an eigenvalue is unique if it exists. When |λ| = x
is suﬃciently large, the equation (3.2) is estimated as 1/λ + O(1/λ2 ) = 2/K,
which proves that the eigenvalue exists and given by λ ∼ K/2. On the other hand,since the left hand side of (3.2) is bounded on the right half plane, there are no
eigenvalues when K > 0 is suﬃciently small. This shows that there is a number
Kc > 0 such that x → +0 as K → Kc + 0. The value Kc is obtained from thewell-known formulaZxlimg(ω)dω = πg(0).
2x→+0 R x + ω2

This result proves that the trivial solution (de-synchro state) is unstable when
K > Kc because of the eigenvalue on the right half plane. The eigenvalue goes tothe left side as K decreases, and ﬁnally it is absorbed into the continuous spectrum
9

on the imaginary axis at K = Kc . When K ≤ Kc , there are no eigenvalues andthe spectrum set consists only of the continuous spectrum on the imaginary axis.
Since a point of the spectrum on the imaginary axis implies the neutral stability,
we cannot prove the asymptotic stability from the spectrum σ(T 1 ). This problem
will be resolved by introducing the generalized spectrum.

4

Gelfand triplet

To handle the diﬃculty caused by the continuous spectrum on the imaginary axis,
we develop the generalized spectral theory based on a Gelfand triplet. In thissection, we will illustrate how the triplet naturally arises by a simple example.
Let us consider the multiplication operator M : f (x) 7→ x f (x) on L2 (R). The
continuous spectrum is the whole real axis. Indeed, the resolvent is given by(λ − M)−1 f (x) =

1
f (x)
λ−x

and it is not included in L2 (R) when λ ∈ R. Nevertheless, we will show that there
exists a topological vector space larger than L2 (R) on which the resolvent operator
makes sense even if λ ∈ R.
To this end, we consider the L2 (R)-inner product with some functions ϕ, ψZ
1
−1∗((λ − M) ϕ, ψ ) =ϕ(x)ψ(x)dx,Rλ − xwhere ψ∗ (x) := ψ(x) is introduced to avoid the complex conjugate in the righthand side. The right hand side above is holomorphic in λ on the lower half plane{Im(λ) < 0}.
Next, suppose λ approaches the real axis from below
Z1
lim
ϕ(x)ψ(x)dx.
Im(λ)→0 R λ − x
The factor 1/(λ − x) diverges at x = λ ∈ R, however, it is known that as long as
ϕ and ψ are continuous functions on R, the above integral exists as an improper
integral and is continuous in λ ∈ R.
Further suppose that λ moves to the upper half plane. It is known that as longas ϕ and ψ are holomorphic on the region {Im(λ) ≥ 0}, the above function of λ hasan analytic continuation to the upper half plane given byZ
1
ϕ(x)ψ(x)dx + 2πiϕ(λ)ψ(λ), Im(λ) > 0.
Rλ − x
Now we have shown that if ϕ and ψ are holomorphic on the real axis and theupper half plane, the function ((λ − M)−1 ϕ, ψ∗ ) of λ has an analytic continuation10

from the lower to the upper half plane across the continuous spectrum on the realaxis. We denote it as
 Z 1



ϕ(x)ψ(x)dx,
Im(λ) < 0
 ZR λ − xR(λ; ϕ, ψ) := 

1



ϕ(x)ψ(x)dx + 2πiϕ(λ)ψ(λ),Im(λ) > 0.
Rλ − x
Motivated by this observation, let X be a dense subspace of L2 (R) consistingof some class of holomorphic functions and X 0 be its dual space, the vector spaceof continuous linear functionals on X. The mapping ϕ 7→ R(λ; ϕ, ψ) deﬁnes alinear functional on X, which is denoted by R(λ; •, ψ) ∈ X 0 . The topology on X isdeﬁned so that this functional is continuous. Then, the mapping ψ 7→ R(λ; •, ψ)
gives a linear mapping from X to X 0 , denoted by Rλ , that is holomorphic in λ ∈ C.
By the deﬁnition, Rλ = (λ − M)−1 when Im(λ) < 0. We call Rλ the generalized
resolvent of M.
This discussion is summarized as follows: As an operator from L2 (R) to L2 (R),the resolvent operator (λ − M)−1 is singular on the real axis because of the continuous spectrum. Nevertheless, if we regard it as an operator from X into X 0 , it has
an analytic continuation Rλ from the lower to the upper half plane. For any ψ ∈ X,
Rλ ψ is an X 0 -valued holomorphic function.
If X is a dense subspace of L2 (R) and the embedding is continuous, L2 (R) is
continuously embedded to the dual space X 0 . In this manner, we obtain the triplet
X ⊂ L2 (R) ⊂ X 0(4.1)

called the Gelfand triplet or rigged Hilbert space.
The spectrum set is also generalized as follows. Let H be a Hilbert space
and T a linear operator on H. Recall that the spectrum set of T is the set of
singularities of the resolvent (λ − T )−1 . Suppose that T has a continuous spectrum.
In a similar manner to the above, suppose that there exists a suitable subspace
X ⊂ H such that if we regard the resolvent as an operator from X to X 0 , then it
has an analytic continuation Rλ across the continuous spectrum. In general, theRiemann surface of Rλ is nontrivial. If the analytic continuation Rλ has a new
singularity on the Riemann surface diﬀerent from the original complex plane, wecall it a generalized spectrum. By the deﬁnition, it is not a true eigenvalue in Hsense, however, it is expected that it plays a similar role to a usual eigenvalue andprovides a new information that is not obtained from the framework of a Hilbert
space.
It is applied to the dynamics of the Kuramoto model as follows. Recall thatthe semigroup eT t generated by T is given by the Laplace inversion formula (3.1).
As explained, if T is a bounded operator, we can estimate the formula by deforming the integral path as shown in Fig. 3. However, for the operator T 1 obtained
11

from the Kuramoto model, any point on the imaginary axis is a singularity of theintegrant eλt (λ − T 1 )−1 and we cannot deform the integral path toward the left halfplane. Now we assume that there exists a subspace X ⊂ L2 (R, g(ω)dω) such thatthe resolvent (λ − T 1 )−1 has an analytic continuation Rλ from the right to the left
half plane as an operator from X into X 0 . Hence, we interpret (3.1) asZ a+iy1Tte ϕ = limeλt Rλ ϕ dλ, ϕ ∈ X.
(4.2)
y→∞ 2πi a−iy
Then, we can deform the integral path toward the left half plane (more precisely,
the second sheet of the Riemann surface), on which Rλ ϕ ∈ X 0 4 . A singularity ofRλ on the second Riemann sheet is called the generalized eigenvalue. By picking
up the residue of the generalized eigenvalue, we can estimate the asymptotic behavior of the semigroup. Recall that there is the eigenvalue on the positive real
axis when K > Kc , it moves to the left side as K decreases and is absorbed into
the continuous spectrum as K → Kc + 0, and disappears. Actually, the eigenvalue
does not disappear. It moves to the second Riemann sheet when K < Kc (now theimaginary axis is a branch cut of the Riemann surface). After getting across thebranch cut, it is not an eigenvalue in the usual sense but becomes a generalized
eigenvalue, that is not found in Hilbert space theory, see Fig. 4. We can deformthe integral path to the second Riemann sheet and calculate the residue aroundthe generalized eigenvalue, which prove the exponential decay of a solution of
dZ1 /dt = T 1 Z1 with respect to the topology of X 0 (asymptotic stability of the desynchronization when K < Kc ). Further, we can show that when the generalized
eigenvalue crosses the imaginary axis at K = Kc as K increases, a bifurcationfrom the de-synchro state to the synchronized state occurs. In this manner, theKuramoto conjecture was proved [3].

Figure 4: The motion of the (generalized) eigenvalue as K decreases. When
K > Kc , it is a usual eigenvalue in L2 -sense. When 0 < K < Kc , it is a generalized eigenvalue that lies on the second Riemann sheet diﬀerent from the originalcomplex plane.
4Thus, the limit lim in (4.2) is considered in weak sense (weak dual topology on X 0 ).
y→∞12

From the next section, the generalized spectral theory will be formulated in a
general setting. The reader can ﬁnd all omitted proofs in [5, 6]. Throughout thisarticle, D(·) and R(·) denote the domain and range of an operator, respectively. For
terminologies of topological vector spaces, refer to [27].

5

The generalized spectral theory

Let X be a locally convex Hausdorﬀ topological vector space over C, X 0 be its dual
space that is a vector space consisting of all continuous anti-linear functionals on
X.
For µ ∈ X 0 and ϕ ∈ X, µ(ϕ) is denoted by hµ | ϕi. For any a, b ∈ C, ϕ, ψ ∈ X
and µ, ξ ∈ X 0 , the equalities
hµ | aϕ + bψi = ahµ | ϕi + bhµ | ψi,
haµ + bξ | ϕi = ahµ | ϕi + bhξ | ϕi,hold. On the dual space X 0 , there are several ways to introduce a topology. The
most commonly used are the weak dual topology and the strong dual topology.
A sequence {µ j } ⊂ X 0 is said to be weakly convergent to µ ∈ X 0 if we havehµ j | ϕi → hµ | ϕi for each ϕ ∈ X. If we have hµ j | ϕi → hµ | ϕi uniformly on
arbitrary bounded subset in X, then {µ j } ⊂ X 0 is said to be strongly convergent toµ ∈ X0.
Let H be a Hilbert space with a Hermitian inner product (· , ·). Suppose X isa dense subspace of H and the embedding into H is continuous (i.e. the topology
of X is stronger than that of H). By considering their duals, it turns out that H 0 is
continuously embedded into X 0 . Since a Hilbert space is isomorphic to itself, wehave H 0 ' H ⊂ X 0 .
Deﬁnition 5.1 Assume that a locally convex Hausdorﬀ topological vector space
X is a dense subspace of a Hilbert space H and the topology of X is stronger thanthat of H. The triplet
X ⊂ H ⊂ X0
is called the Gelfand triplet or the rigged Hilbert space.
The embedding i : H → X 0 is deﬁned as follows: For ψ ∈ H, i(ψ) is denoted byhψ| and deﬁned asi(ψ)(ϕ) = hψ | ϕi = (ψ, ϕ),ϕ ∈ X.

In other words, the isomorphism H ' H 0 is deﬁned so that hψ | ϕi is compatible
with the inner product when ψ ∈ H. The embedding is injective and continuousby the assumption in Def. 5.1. A Gelfand triplet was proposed to generalize thetheory of Schwartz distribution [15], for which X = C0∞ (Rm ) and , H = L2 (Rm ).
13

I
Figure 5: The region Ω and the interval I.

5.1 Generalized eigenvalue
Let H be a Hilbert space over C, H be a self-adjoint operator densely deﬁned
on HR and {E(B)}B∈B its spectral measure; H admits the spectral representation
H = R ωdE(ω). Let K be another densely deﬁned operator on H. The purpose
here is to investigate spectral properties of T := H + K. For the Kuramoto model,
H = M, K = P, and for a Schrödinger operator, H is the Laplacian, K is apotential function [6].
Let Ω ⊂ C be a simply connected region included in the upper half plane, and¯
let I be the intersection of the closure of Ω and the real axis (we assume that it
is not empty and connected). Let I be an open interval obtained by removing theend points from I¯ (Fig. 5). Later we will see that our setting makes sense when I¯
is a (subset of) continuous spectrum of H. For a given operator T = H + K on H,
we assume that there exists a locally convex Hausdorﬀ topological vector spaceX(Ω) satisfying the following conditions.
(X1) X(Ω) is a dense subspace of H.
(X2) The topology of X(Ω) is stronger than that of H.
(X3) X(Ω) is a quasi-complete barreled space.
By (X1), (X2), the Gelfand triple
X(Ω) ⊂ H ⊂ X(Ω)0
is well-deﬁned. The deﬁnition of a barreled space is rather complicated [27].
It includes any Fréchet space, Banach space, Hilbert space, nuclear space and

14Montel space 5 . If X is a barreled space, the Banach-Steinhaus theorem 6 holds
and usual complex function theory (such as Cauchy theorem) is applicable for
X 0 -valued functions [5].
Next, we need the assumptions for the spectral measure E(B) of H.
(X4) For any ϕ ∈ X(Ω), the spectral measure (E(B)ϕ, ϕ) is absolutely continuous
on the interval I 7 . Its density function denoted by E[ϕ, ϕ](ω) has an analytic
continuation to the region Ω ∪ I.
(X5) For each point λ ∈ I ∪ Ω, the bilinear form E[ · , · ](λ) : X(Ω) × X(Ω) → C
is separately continuous.
By (X4) with the aid of the polarization identity, we can verify that (E(B)ϕ, ψ) is
absolutely continuous on I for any ϕ, ψ ∈ X(Ω). We denote its density function asE[ϕ, ψ](ω)：
d(E(ω)ϕ, ψ) = E[ϕ, ψ](ω)dω,ω ∈ I.

Then, the function E[ϕ, ψ](ω) is holomorphic in ω ∈ I ∪ Ω. For simplicity, we use
the notation E[ϕ, ψ](ω) for any ω ∈ R.
Deﬁne a linear operator A(λ) : X(Ω) → X(Ω)0 to be
 Z
1



E[ψ, ϕ](ω)dω + 2πiE[ψ, ϕ](λ) (λ ∈ Ω),



λ
−
ω

R


Z



1

lim
E[ψ, ϕ](ω)dω(λ = x ∈ I),(5.1)
hA(λ)ψ | ϕi = 
y→−0 R x + iy − ω



Z



1



E[ψ, ϕ](ω)dω(Im(λ) < 0).
λ−ωRWe can verify that the function hA(λ)ψ | ϕi is holomorphic in {Im(λ) < 0} ∪ Ω ∪ I.
In particular, if Im(λ) < 0 then hA(λ)ψ | ϕi = ((λ − H)−1 ψ, ϕ) and A(λ) coincides
5

If a locally convex topological vector space is barreled and has the property that “any closed
and bounded set is compact”, then it is called a Montel space (this property is often called theHeine-Borel property). A Montel space has a nice property that any weakly convergent series isalso strongly convergent. For a suﬃcient condition for a given space to be a Montel, refer to [16],[19]. For example, they are Montel spaces: the space of C ∞ functions, the space of C ∞ functions
with compact support, the space of rapidly decreasing C ∞ functions, the space of holomorphic
functions on an open region, and their dual spaces.
6
Banach-Steinhaus theorem.
Let X be a barreled space and X 0 its dual space. For a subset A ⊂ X 0 , the following conditions
are equivalent.
(i) A is bounded with respect to the weak dual topology.
(ii) A is bounded with respect to the strong dual topology.
(iii) A is equicontinuous as a family of mappings.
(iv) A is relatively compact with respect to the weak dual topology.
(i) ⇒ (ii) is well-known as the uniform boundedness principle when X is a Banach space.
7
Imagine the situation that I¯ is a spectrum of H. Otherwise (E(B)ϕ, ϕ) = 0 on I.

15

with the resolvent of H. This means that A(λ)ψ is an analytic continuation of theresolvent from the lower half plane to Ω through the interval I as an X(Ω)0 -valued
function. We can show that A(λ) : X(Ω) → X(Ω)0 is a continuous operator 8 , if
X(Ω)0 is equipped with the weak dual topology.
We need some notation for the next assumptions. Let Q be a densely deﬁned
linear operator on X(Ω). Its dual operator Q0 : D(Q0 ) → X(Ω)0 is deﬁned asfollows : The domain D(Q0 ) of Q0 is all elements µ ∈ X(Ω)0 so that the mapping
ϕ 7→ hµ | Qϕi from X(Ω) to C is continuous, and Q0 is deﬁned through the equality
hQ0 µ | ϕi = hµ | Qϕi. Next, for a densely deﬁned operator Q on H, its Hilbertadjoint Q∗ is deﬁned through (Qϕ, ψ) = (ϕ, Q∗ ψ). Moreover, if Q∗ is densely
deﬁned on X(Ω), its dual operator (Q∗ )0 can be considered and we denote is as
Q× . Then, Q× = (Q∗ )0 satisﬁes Q = Q× |D(Q) , which means that Q× is a naturalextension (lift) of Q from H to X(Ω)0 . For simplicity, we call Q× the dual operatorof Q.
For the operators H and K on H, we assume the following.
(X6) H (= H ∗ ) is densely deﬁned operator on X(Ω) (there is a dense subspace Yof X(Ω) such that HY ⊂ X(Ω)).
(X7) K is H-bounded and K ∗ is densely deﬁned on X(Ω).
(X8) For any λ ∈ {Im(λ) < 0} ∪ I ∪ Ω, we have K × A(λ)X(Ω) ⊂ X(Ω).
By (X6) and (X7), H × , K × and T × are densely deﬁned on X(Ω)0 (recall T :=H + K). If H and K are continuous on X(Ω), so are H × , K × and T × on X(Ω)0 ,but we do not assume it in general. An operator K is said to be H-bounded whenK(λ − H)−1 is a bounded operator on H for λ < σ(H). Recalling that A(λ) is ananalytic continuation of (λ − H)−1 , (X8) is in some sense the analytic continuation
version of (X7).
With these assumptions, we deﬁne a generalized eigenvalue. An eigenvalue
and eigenvector in the usual sense are deﬁned by (λ − T )v = 0. Since T = H +K now, it is rewritten as (id − (λ − H)−1 K)v = 0. Recalling that the analytic
continuation of (λ − H)−1 in X(Ω)0 is A(λ), we make the following deﬁnition.
Deﬁnition 5.2 If the equation(id − A(λ)K × )µ = 0(5.2)

have a solution 0 , µ ∈ X(Ω)0 for some λ ∈ Ω ∪ I ∪ {λ | Im(λ) < 0}, λ and µ are
called the generalized eigenvalue and generalized eigenvector of T , respectively.
Applying K × to (5.2), we obtain(id − K × A(λ))K × µ = 0.
8

(5.3)

Note that since X(Ω) may not be a Banach space, there is a gap between a continuous operator
and a bounded operator. The condition for two concepts to coincide other than a Banach space iscomplicated [1].

16

If K × µ = 0, then (5.2) gives µ = 0. Hence, λ is a generalized eigenvalue if andonly if id − K × A(λ) is not injective on X(Ω). Note that the operator K × A(λ) on
X(Ω) is well-deﬁned because of (X8).
Theorem 5.3 For a generalized eigenvalue λ of T and its generalized eigenvectorµ, the equality
T × µ = λµholds.
Sketch of a Proof. By the operational calculus, we can show D(λ − H × ) ⊃R(A(λ)) and (λ − H × )A(λ) = id|X(Ω) . This yields(λ − H × )(id − A(λ)K × )µ = (λ − H × − K × )µ = (λ − T × )µ = 0.
Thus, a generalized eigenvalue is a true eigenvalue of the dual operator T × ,although the converse statement is not true. An eigenvalue of T × is not always
a generalized eigenvalue. Since the dual space X(Ω)0 is too large, typically any
points in C become eigenvalues of T × .

5.2

Properties of A(λ)

For further discussion, let us investigate the properties of A(λ) in detail. For n =1, 2, · · · , we deﬁne an linear operator A(n) (λ) : X(Ω) → X(Ω)0 to be Z
1(−1)n−1 dn−1



E[ψ,
ϕ](ω)dω+2πiE[ψ, ϕ](z), (λ ∈ Ω),nn−1 z=λ


(λ
−
ω)
(n
−
1)!
dz
R


Z



1

(n)
hA (λ)ψ | ϕi = lim
E[ψ, ϕ](ω)dω, (λ = x ∈ I),
y→−0 R (x + iy − ω)n



Z



1



E[ψ, ϕ](ω)dω. (Im(λ) < 0)n
R (λ − ω)
By integration by parts, it is easy to show that hA(n) (λ)ψ | ϕi is an analytic continuation of ((λ − H)−n ψ, ϕ) from the lower half plane to the upper half plane. A(1) (λ)
is also denoted by A(λ) as before.
Proposition 5.4 For any integer j ≥ n ≥ 0, the operator A( j) (λ) satisﬁes(i) (λ − H × )n A( j) (λ) = A( j−n) (λ), where A(0) (λ) := id.
(ii) A( j) (λ)(λ − H × )n = A( j−n) (λ).
In particular, if (λ − H × )µ ∈ X(Ω) then A(λ)(λ − H × )µ = µ.
djhA(λ)ψ | ϕi = (−1) j j!hA( j+1) (λ)ψ | ϕi, j = 0, 1, · · · .
(iii)
j
dλ
17

(iv) For any ψ ∈ X(Ω), A(λ)ψ is expanded as
A(λ)ψ =∞
X

(λ0 − λ) j A( j+1) (λ0 )ψ,

(5.4)

j=0

and the right hand side converges with respect to the strong dual topology on
X(Ω)0 .
Sketch of a Proof. (i) and (ii) are easily proved by the operational calculus.
(iii) follows from the deﬁnition of A(λ). Since hA(λ)ψ | ϕi is holomorphic, (iii)yieldshA(λ)ψ | ϕi =∞X

(λ0 − λ) j hA( j+1) (λ0 )ψ | ϕi,

(5.5)

j=0

which means that A(λ)ψ is weakly holomorphic in X(Ω)0 . Since X(Ω) is barreled,a weakly holomorphic function is strongly holomorphic by the Banach-Steinhaus
theorem.
Next, we deﬁne an eigenspace and the multiplicity of a generalized eigenvalue.
In the usual spectral theory, the eigenspace of λ is deﬁned as the space spannedby solutions of (λ − T )n v = 0. For example n = 2, it is rearranged as(λ − H − K)(λ − H − K)v= (λ − H)2 (id − (λ − H)−2 K(λ − H)) ◦ (id − (λ − H)−1 K)v = 0.
Divided by (λ − H)2 , it gives(id − (λ − H)−2 K(λ − H)) ◦ (id − (λ − H)−1 K)v = 0.
Since the analytic continuation of (λ − H)−n is A(n) (λ), we may consider the equation(id − A(2) (λ)K × (λ − H × )) ◦ (id − A(λ)K × ) µ = 0.
Thus, let us deﬁne an operator B(n) (λ) : D(B(n) (λ)) ⊂ X(Ω)0 → X(Ω)0 by
B(n) (λ) = id − A(n) (λ)K × (λ − H × )n−1 .

(5.6)

Then, the above equation is simply written as B(2) (λ)B(1) (λ)µ = 0. The domain ofB(n) (λ) is the domain of A(n) (λ)K × (λ − H × )n−1 . The following equality(λ − H × )k B( j) (λ) = B( j−k) (λ)(λ − H × )k ,is easily proved.
18

j>k

(5.7)

Deﬁnition 5.5 The generalized eigenspace associated with a generalized eigenvalue λ is deﬁned by[Vλ =Ker B(m) (λ) ◦ B(m−1) (λ) ◦ · · · ◦ B(1) (λ),
m≥1and dimVλ is called the multiplicity of λ.
In particular, an element of Ker B(1) (λ) is a generalized eigenvector deﬁned inDef. 5.2. In the same way as Thm. 5.3, we can prove the next theorem.
Theorem 5.6 For any µ ∈ Vλ , there exists an integer M such that (λ − T × ) M µ = 0.
S
The theorem means that Vλ is a subspace of an eigenspaceS m≥1 Ker (λ − T × )mof T × . Since the dual space X(Ω)0 is too large, typically m≥1 Ker (λ − T × )mbecomes an inﬁnite dimensional, however, Vλ is ﬁnite dimensional for most applications (Thm. 5.16).

5.3

Generalized resolvent

Let Rλ = (λ − T )−1 be the resolvent operator of T . Since−1Rλ ψ = (λ − H)−1 id − K(λ − H)−1 ψ(5.8)

and the analytic continuation of (λ − H)−1 in X(Ω)0 is given by A(λ), we make thefollowing deﬁnition. In what follows, we put Ω̂ = Ω ∪ I ∪ {λ | Im(λ) < 0}.
Deﬁnition 5.7 When the inverse (id − K × A(λ))−1 exists on X(Ω), the generalized
resolvent Rλ : X(Ω) → X(Ω)0 of T is deﬁned by
Rλ = A(λ) ◦ (id − K × A(λ))−1 = (id − A(λ)K × )−1 ◦ A(λ),λ ∈ Ω̂.

(5.9)

The second equality follows from (id − A(λ)K × )A(λ) = A(λ)(id − K × A(λ)).
Note that id − K × A(λ) is an operator on X(Ω) because of (X8), and id − A(λ)K × isan operator on R(A(λ)). The former is injective if and only if so is the latter. Since
A(λ) is continuous as mentioned in Sec. 5.1, we require that Rλ : X(Ω) → X(Ω)0
is also continuous.
Deﬁnition 5.8 The set of λ ∈ Ω̂ satisfying the following two conditions is called
the generalized resolvent set ϱ̂(T )；There exists a neighborhood Vλ ⊂ Ω̂ of λ suchthat(i) For any λ0 ∈ Vλ , Rλ0 is a densely deﬁned continuous operator from X(Ω) into
X(Ω)0 , where X(Ω)0 is equipped with the weak dual topology.
(ii) For any ψ ∈ X(Ω), the set {Rλ0 (ψ)}λ0 ∈Vλ is a bounded set in X(Ω)0 9 .
9Because of the Banach-Steinhaus theorem, a weakly bounded set is strongly bounded. Thus,
we need not specify a topology here.

19

The complement σ̂(T ) = Ω̂\ϱ̂(T ) is called the generalized spectrum set of
T . The generalized point spectrum σ̂ p (T ) is the set of points λ ∈ σ̂(T ) at which
id − K × A(λ) is not injective (this is the set of generalized eigenvalues). The generalized residual spectrum σ̂r (T ) is the set of points λ ∈ σ̂(T ) such that the domainof Rλ is not dense in X(Ω). The generalized continuous spectrum is deﬁned tobe σ̂c (T ) = σ̂(T )\(σ̂ p (T ) ∪ σ̂r (T )).
By the deﬁnition, ϱ̂(T ) is an open set. This deﬁnition looks rather complicatedbecause X(Ω) is not a Banach space. To require the existence of the neighborhood Vλ ⊂ Ω̂ in the above deﬁnition was introduced by Waelbroeck [31] (see also
Maeda [18]) for the spectral theory on locally convex spaces. If ϱ̂(T ) were simply
deﬁned to be the set of points such that Rλ0 is a densely deﬁned continuous operator as in the Banach space theory, ϱ̂(T ) is not an open set in general. If X(Ω) isa Banach space, the deﬁnition coincides with the usual deﬁnition of the resolvent
set in a Banach space.
Theorem 5.9(i) For each ψ ∈ X(Ω), Rλ (ψ) is an X(Ω)0 -valued holomorphic function on ϱ̂(T ).
(ii) When Im(λ) < 0, Rλ = i ◦ (λ − T )−1 ( i is the embedding into X(Ω)0 ).
The second part (ii) implies that when Im(λ) < 0, the equality hRλ ψ | ϕi =((λ − T )−1 ψ, ϕ) holds for ψ, ϕ ∈ X(Ω). Thus, hRλ ψ | ϕi is an analytic continuationof ((λ − T )−1 ψ, ϕ).
Sketch of a Proof of (i).
Put ψλ = (id − K × A(λ))−1 (ψ). It is easy to conﬁrm thatRλ+h (ψ) − Rλ (ψ)= (A(λ + h) − A(λ))(ψλ ) + Rλ+h K × (A(λ + h) − A(λ))(ψλ ).
We show that it tends to zero as h → 0 with respect to the weak dual topology on
X(Ω)0 . Since A(λ) is holomorphic in λ, the ﬁrst term is easy to treat. To estimate
the second term, we need to estimate Rλ+h and K × A(λ). For the latter one, we canverify that K × A(λ) is also holomorphic in λ as an X(Ω)-valued function, so that
ϕh := K × (A(λ + h) − A(λ))(ψλ ) ∈ X(Ω)
tends to zero as h → 0. For the former one, the set {Rλ+h (ϕ); |h| : small } isbounded for any ϕ ∈ X(Ω) due to the condition (ii) in Def. 5.8. This shows thatRλ+h (ϕh ) → 0 weakly as h → 0 by the condition (i). Hence, Rλ+h (ψ) → Rλ (ψ) as
h → 0.
Repeating the same procedure after dividing by h, it turns out that Rλ (ψ) is
weakly holomorphic. Since X(Ω) is barreled, it is automatically strongly holomorphic.

20

Proposition 5.10 Rλ satisﬁes that(i) (λ − T × ) ◦ Rλ = id|X(Ω) ,(ii) when µ ∈ X(Ω)0 satisﬁes (λ − T × )µ ∈ X(Ω), then Rλ ◦ (λ − T × )µ = µ,(iii) T × ◦ Rλ = Rλ ◦ T × .
This proposition immediately follows from Prop. 5.4.

5.4

Generalized projection

Let Σ ⊂ σ̂(T ) be a bounded subset of the generalized spectrum set which is separated from the rest of the spectrum by a simple closed curve γ ⊂ Ω∪I∪{λ | Im(λ) <0}. Deﬁne an operator ΠΣ : X(Ω) → X(Ω)0 by
Z
1
Rλ ϕ dλ, ϕ ∈ X(Ω),(5.10)
ΠΣ ϕ =
2πi γ
where the integral is deﬁned as the Pettis integral 10 . Since the composition ΠΣ ◦ΠΣcannot be deﬁned, it is not a projection operator in the usual sense. Nevertheless,
it is reasonable to call ΠΣ the generalized projection because of the following
results.
Proposition 5.11 The following holdΠΣ (X(Ω)) ∩ (id − ΠΣ )(X(Ω)) = {0}X(Ω) ⊂ ΠΣ (X(Ω)) ⊕ (id − ΠΣ )(X(Ω)) ⊂ X(Ω)0
In particular, for any ϕ ∈ X(Ω), there are µ1 , µ2 ∈ X(Ω)0 such that ϕ is uniquely
decomposed asi(ϕ) = hϕ| = µ1 + µ2 ,µ1 ∈ ΠΣ (X(Ω)), µ2 ∈ (id − ΠΣ )(X(Ω)).

(5.11)

Proposition 5.12 ΠΣ is T × -invariant : ΠΣ ◦ T × = T × ◦ ΠΣ .
Theorem 5.13 Let λ0 be an isolated generalized eigenvalue, Π0 be the generalized projection for λ0 and V0 be the generalized eigenspace of λ0 (Def. 5.5). IfΠ0 X(Ω) is ﬁnite dimensional, Π0 X(Ω) = V0 .
In general, let X be a topological vector space, X 0 its dual space with the strong dual topology,
S compact Hausdorﬀ space and µ be a ﬁnite Borel measure on S . For a mapping f : S → X 0 , ifthere exists I( f ) ∈ X 0 satisfyingZhI( f ) | ϕi = h f | ϕidµ10

SR
for any ϕ ∈ X, then f is said to be Pettis integrable and I( f ) = S f dµ is called the Pettis integralof f . If X is barreled and f is holomorphic, it is Pettis integrable [5].

21In the usual spectral theory, these properties are proved by using Π ◦ Π = Π
and the resolvent identity. In our case, since these formulae do not hold becauseΠ and Rλ are mappings from X(Ω) into X(Ω)0 (i.e. the composition of them is not
deﬁned), the proof is rather technical [5].

5.5 Properties of of the generalized spectrum
Obviously the deﬁnition of the generalized spectrum depends on the choice ofthe space X(Ω). When we want to emphasize the choice, we denote σ̂(T ) asσ̂(T ; X(Ω)). If we have two spaces X1 (Ω) and X2 (Ω) satisfying (X1) to (X8),
there are two generalized spectra σ̂(T ; X1 (Ω)) and σ̂(T ; X2 (Ω)).
Proposition 5.14 ([6]) Suppose that X2 (Ω) is a dense subspace of X1 (Ω) and the
topology of X2 (Ω) is stronger than that of X1 (Ω). Then, the following statements
hold.
(i) σ̂(T ; X2 (Ω)) ⊂ σ̂(T ; X1 (Ω)),(ii) Let Σ , ∅ be a bounded subset of σ̂(T ; X1 (Ω)) which is separated from the restof the spectrum σ̂(T ; X1 (Ω)) by a simple closed curve γ. Then, there exists a pointof σ̂(T ; X2 (Ω)) inside γ. In particular, if λ is an isolated point of σ̂(T ; X1 (Ω)),
then λ ∈ σ̂(T ; X2 (Ω)).
Sketch of a Proof. Because of the assumption of the topology, the generalized
resolvent Rλ : X2 (Ω) → X2 (Ω)0 behaves “better” than Rλ : X1 (Ω) → X1 (Ω)0 ,which proves (i). For (ii), let ΠΣ be the generalized projection. By the assumption,ΠΣ X1 (Ω) , {0}. Since X2 (Ω) is dense in X1 (Ω), we have ΠΣ X2 (Ω) , {0}. □Due to this theorem, the existence of isolated generalized eigenvalues is independent of the choice of X(Ω) 11 .
For the next theorem, we deﬁne a uniformly compact operator. A linear operator L from a topological vector space X1 to another topological vector space X2
is said to be bounded if there exists a neighborhood U ⊂ X1 of the origin such that
LU ⊂ X2 is a bounded set. When L = L(λ) is parameterized by λ, it is said to bebounded uniformly in λ if such a neighborhood U is independent of λ. When thedomain X1 is a Banach space, L(λ) is bounded uniformly in λ if and only if L(λ)
is continuous for each λ (U is taken to be the unit sphere).
Similarly, L is called compact if there exists a neighborhood U ⊂ X1 of theorigin such that LU ⊂ X2 is relatively compact. When L = L(λ) is parameterizedby λ, it is said to be compact uniformly in λ if such a neighborhood U is independent of λ. When the domain X1 is a Banach space, L(λ) is compact uniformly in λ11

Historically, several deﬁnitions of the generalized eigenvalues had been proposed. It seems
that their results are the same because of this theorem. For example, for the study of Schrödinger
operators, the generalized eigenvalue is called the resonance pole. This is deﬁned by the analytic
continuation of a scattering matrix [24] or the method of complex deformation [17] and so on.

22

if and only if L(λ) is compact for each λ. When the range X2 is a Montel space, a(uniformly) bounded operator is (uniformly) compact because every bounded set
in a Montel space is relatively compact.
Put Ω̂ := {Im(λ) < 0} ∪ I ∪ Ω as before. In many applications, K × A(λ) is abounded operator on X(Ω). In such a case, the following proposition is useful toestimate the generalized spectrum.
Proposition 5.15 Suppose that for ﬁxed λ ∈ Ω̂, there exists a neighborhood Uλ ⊂Ω̂ of λ such that K × A(λ0 ) : X(Ω) → X(Ω) is a bounded operator uniformly in
λ0 ∈ Uλ . If id − K × A(λ) has a continuous inverse on X(Ω), then λ < σ̂(T ).
Sketch of a Proof. Check the condition of Def. 5.8. On the generalized
resolvent Rλ = A(λ) ◦ (id − K × A(λ))−1 , since A(λ) : X(Ω) → X(Ω)0 is continuous,
it is suﬃcient to show that there exists a neighborhood Vλ of λ such that the set{(id − K × A(λ0 ))−1 ψ}λ0 ∈Vλ exists and is bounded in X(Ω) for any ψ. For this purpose,
it is suﬃcient to show that the mapping λ0 7→ (id − K × A(λ0 ))−1 ψ is continuous in
λ0 ∈ Vλ . Since A(λ) is holomorphic, there is an operator D(λ, h) such thatid − K × A(λ + h) = id − K × A(λ) − hD(λ, h)= id − hD(λ, h)(id − K × A(λ))−1 ◦ (id − K × A(λ))
for small h ∈ C. Since K × A(λ) is uniformly bounded by the assumption, D(λ, h) isa uniformly bounded operator in h. Further, (id − K × A(λ))−1 is continuous by theassumption. Thus, D(λ, h)(id − K × A(λ))−1 is a bounded operator. Then, Bruyn’s
theorem [2] is applicable to show that id−hD(λ, h)(id−K × A(λ))−1 has a continuous
inverse that is continuous in h (when X(Ω) is a Banach space, Bruyn’s theorem isreduced to the existence of the Neumann series). This proves that id − K × A(λ0 )has a continuous inverse which is continuous in λ0 .
Theorem 5.16 Suppose that K × A(λ) : X(Ω) → X(Ω) is a compact operator uniformly in λ ∈ Ω̂. Then, the following statements hold.
(i) For any compact set D ⊂ Ω̂, the number of generalized eigenvalues in D is
ﬁnite (thus σ̂ p (T ) consists of a countable number of generalized eigenvalues andthey may accumulate only on the boundary of Ω̂ or inﬁnity).
(ii) For each λ0 ∈ σ̂ p (T ), the generalized eigenspace V0 is of ﬁnite dimensional(in particular Thm. 5.13 holds).
(iii) σ̂c (T ) = σ̂r (T ) = ∅.
This kind of result is well-known as the Riesz-Schauder theory for a Banach
space. Even if X(Ω) is not Banach but a general locally convex vector space, it
is known that the Riesz-Schauder theory is valid [25], which is used to prove theabove theorem.
23

5.6 Semigroup
Suppose that the operator iT√ = i(H +K) generates a C 0 -semigroup eiT t on H (here,
i is not the embedding but −1). It is expressed by the Laplace inversion formulaas
Z x−iy1iT t(e ψ, ϕ) =lim
eiλt ((λ − T )−1 ψ, ϕ)dλ, x, y ∈ R,
(5.12)
2πi x→∞ −x−iyfor ϕ, ψ ∈ H, where the integral path is the horizontal straight line below the
spectrum of T . If T has a continuous spectrum on the real axis, we cannot deformthe integral path from the lower to the upper half plane and it is diﬃcult to estimate
the asymptotic behavior of the semigroup as t → ∞. However, for ϕ, ψ ∈ X(Ω),
we can rewrite (5.12) asZ x−iy1
iT t
lim
eiλt hRλ ψ | ϕidλ.
(e ψ, ϕ) =
x→∞
2πi
−x−iy
As a result, the path can be deformed toward the Riemann surface of Rλ . In many
applications, the set of singularities of hRλ ψ | ϕi consists of isolated generalized
eigenvalues, and we can estimate the Laplace inversion formula by the residue
theorem. A residue is calculated by using the generalized projection. Let Π0be the projection associated with an isolated generalized eigenvalue λ0 with themultiplicity M. The residue of it is given by12πiZe hRλ ψ | ϕidλ =
iλt

γ0

M−1
X

eiλ0 tk=0(−it)kh(λ0 − T × )k Π0 ψ | ϕi,k!

where γ0 is a small simple closed curve enclosing λ0 , see Fig. 6. In particular,
if λ0 lies on the upper half plane, it induces an exponentially decaying term withrespect to the weak topology on X(Ω)0 , not the topology on H. This kind of decay
in the weak dual topology induced by the generalized eigenvalues is known inplasma physics as the Landau damping [14] and in Schrödinger equations as atunnel eﬀect [17, 24, 6].
In general, the decay of a semigroup occurs only during a transient state. To
see it, let λ0 be a generalized eigenvalue on the upper half plane and µ0 ∈ X(Ω)0
its generalized eigenvector. Let (eiT t )× be the dual operator. Since the equality(eiT t )× µ0 = ei λ0 t µ0 holds, if we consider µ0 as the initial condition, actually the
semigroup decays to zero exponentially. However, µ0 is an element of the dual
space, which may be not a suitable choice for an application. Since X(Ω) is adense subspace of X(Ω)0 , for any ε > 0, there are τ > 0 and a function ϕ0 ∈ X(Ω)
such that for 0 ≤ t ≤ τ, the inequality|h(eiT t )× ϕ0 | ψi − h(eiT t )× µ0 | ψi| < ε24

Figure 6: Deformation of the integral path γ to γ0 . The solid curve denotes thepath on the original complex plane, and the dotted one denotes the path on thesecond Riemann sheet.
holds. This means that for a ﬁnite time interval 0 ≤ t ≤ τ, we have(eiT t ϕ0 , ψ) ∼ eiλ0 t hµ0 | ψi,and the generalized eigenvalue gives a transient behavior.

6

Application to the Kuramoto model

6.1 The stability of the de-synchronization state
Let us apply the previous results to the linear operator T 1 = iM + KP/2 obtainedby the linearization of the Kuramotomodel (Sec. 2). Since the self-adjoint oper√
ator M is multiplied by i = −1, the right (resp. left) half plane play the samerole as the lower (resp. upper) half plane in the previous sections.
Since the continuous spectrum of T 1 is the whole imaginary axis, we cannot
determine the stability of the de-synchronization state within Hilbert space theory.
Thus, we suitably introduce the Gelfand triplet X ⊂ L2 (R, g(ω)dω) ⊂ X 0 so that if
the resolvent (λ−T 1 )−1 is regarded as an operator from X into X 0 , it has an analytic
continuation from the right half plane to the left half plane across the continuous
spectrum. We can show that the resolvent is calculated asK((λ − T 1 )−1 ϕ, ψ∗ ) = D[ϕ, ψ](λ) +D[ϕ, P0 ](λ) · D[P0 , ψ](λ),
2 − KD[P0 , P0 ](λ)
Z
1D[ϕ, ψ](λ) :=ϕ(ω)ψ(ω)g(ω)dω,R λ − iωwhere P0 (ω) ≡ 1 is a constant function and ψ∗ (x) := ψ(x). If D[ϕ, ψ](λ) has an25

analytic continuation from the right to the left half plane, it is given byZ
1
ϕ(ω)ψ(ω)g(ω)dω + 2πϕ(−iλ)ψ(−iλ)g(−iλ).
R λ − iωFor the existence of the second term, ϕ and ψ should be holomorphic on the upper
half plane. Thus, we deﬁne X to be some class of functions that are holomorphic
on the upper half plane and are in L2 (R, g(ω)dω) on the real axis (see [3] for the
precise deﬁnition). With a suitable topology on X, T 1 and X ⊂ L2 (R, g(ω)dω) ⊂X 0 satisfy the assumptions (X1) to (X8). Further, they satisfy the assumption forThm. 5.16 ; the generalized spectrum consists of discrete generalized eigenvalues
with ﬁnite multiplicities (actually the multiplicities are 1).
The generalized eigenvalues are deﬁned by (5.2). More convenient way toobtain them is as follows: An eigenvalue in the usual sense is given by the root of
the equation (3.2). The analytic continuation of this equation from the right to theleft half plane is given byZ
2
1
g(ω)dω + 2πg(−iλ) = , (Re(λ) < 0),(6.1)
K
R λ − iωwhose root gives a generalized eigenvalue. When g is the Gaussian distribution,
there are inﬁnitely many generalized eigenvalues on the left half plane. Recall thatwhen K > Kc , there exists a unique eigenvalue λ0 = λ0 (K), which is a root of (3.2),
on the positive real axis. For K ≤ Kc , it becomes a root of (6.1). This implies thatat K = Kc , λ0 (K) crosses the imaginary axis (now it is a branch cut of the Riemannsurface), goes to the second Riemann sheet and becomes a generalized eigenvalue(Fig. 4).
A generalized eigenvector is calculated in a similar manner. When K > Kc ,the eigenvector v ∈ L2 (R, g(ω)dω) of the usual eigenvalue λ is given by (3.3). By
the embedding i : L2 (R, g(ω)dω) → X 0 we regard v as an element in X 0 , which isdenoted by µλ . Its action on X is deﬁned byZ
1∗∗hµλ | ϕ i = (v, ϕ ) =ϕ(ω)g(ω)dω, (Re(λ) > 0).
R λ − iωThe generalized eigenvector µλ associated with a generalized eigenvalue on theleft half plane λ is given by the analytic continuation, that isZ1∗hµλ | ϕ i =ϕ(ω)g(ω)dω + 2πϕ(−iλ)g(−iλ), (Re(λ) < 0).
R λ − iωIt is not an element of L2 (R, g(ω)dω) but of X 0 .
∞Let {λn }∞n=0 , {µn }n=0 be the set of generalized eigenvalues and their generalized
eigenvectors, respectively. By deforming the integral path of the Laplace inversion
formula as in Fig. 3 and using the residue theorem, we can prove the next theorem.
26

Theorem 6.1 (Spectral decomposition) For any ϕ ∈ X, the dual operator of thesemigroup eT1 t generated by T 1 is expanded as(eT1 t )× ϕ =∞Xeλn t hµn | ϕiµn ,

(6.2)

n=0

where the right hand side converses in X 0 with respect to the strong dual topology.
When, 0 < K < Kc , Re(λn ) < 0 for all n = 0, 1, · · · , which proves that (eT1 t )× ϕ
converges to 0 as t → ∞ in X 0 (asymptotic stability of the de-synchronization).
Note that T 1 is not a self-adjoint nor compact operator. Thus, a spectral decomposition does not hold within Hilbert space theory. Nevertheless, it is possible
by using elements in X 0 .

6.2

Bifurcation to the synchronized state

The remaining task is to show a bifurcation from the de-synchro state to the synchro state. To investigate a bifurcation in dynamical systems, one of the mosteﬀective ways is to apply the center manifold reduction. In our problem, there isa continuous spectrum on the imaginary axis, so that the center manifold is not
well-deﬁned within Hilbert space. To treat this diﬃculty, by using generalized
eigenvalues λn and their generalized eigenvectors µn , we deﬁne the generalized
center subspace Ec by
Ec := span{µn | λn ∈ iR} ⊂ X 0 .
This is the range of the generalized projection associated with generalized eigenvalues on the imaginary axis. For the Kuramoto model, when K = Kc there is a
generalized eigenvalue λ0 = 0 with the multiplicity 1. Thus, there exists a corresponding 1-dimensional center subspace Ec . Furthermore, we can prove theexistence of a center manifold in X 0 which is tangent to Ec .
In Sec.2, we derive the system of equations for the Fourier coeﬃcients Z j . For
example, the equation of Z1 is
dZ1
K
K
K
= iωZ1 + η(t) − η(t)Z2 = T 1 Z1 − (Z1 , P0 )Z2 .
dt
222This deﬁnes an equation on L2 (R, g(ω)dω), however, it is diﬃcult to investigate
it in Hilbert space because of the continuous spectrum of T 1 . Thus, by using theembedding i : L2 (R, g(ω)dω) → X 0 , we regard this equation as an equation given
on X 0 . To this end, the operator T 1 is replaced by its dual, the inner product (Z1 , P0 )is replaced by the paring hZ1 | P0 i：
dZ1
K= T 1× Z1 − hZ1 | P0 iZ2 ,dt
227

Z j ∈ X0.

On the dual space, the center subspace Ec is well-deﬁned. Let ﬁx an elementµ0 ∈ Ec (generalized eigenvector of λ0 = 0). Let α be a coordinate of Ec andwe decompose Z1 into the direction of the center subspace and its complement as
Z1 = α(t)µ0 + Y1 , where Y1 is given by Y1 = (id − Π0 )Z1 by using the projection
Π0 . Since Y1 and Z2 , Z3 , · · · are vector that are outside the center subspace, we
can assume that they are of order O(α2 ). By substituting Z1 = α(t)µ0 + Y1 into theequation and after a long calculation, the dynamical system on the center manifold
is obtained asd
α = (K − Kc )p1 α + p3 α|α|2 + O(α5 ),(6.3)
dt
where p1 and p3 are constants given by
p1 =

D0
,
Kc

p3 =

πD0 Kc3 g00 (0),16

and D0 is a constant related to the residue around λ0 = 0. This equation is a normal
form of the pitchfork bifurcation. Hence, it is easy to see that when −p1 /p3 > 0
and K > Kc , there exists a steady state (ﬁxed point) approximately given by
r
−p1 p
|α| =
K − Kc + O(K − Kc ).
p3
Since r = |α| + O(α2 ), this result gives Kuramoto’s bifurcation diagram (Fig. 2).

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