Appendix B
Spectral theory
This chapter is devoted to the explanation of mathematical background of the spectral theory. Spectral theory of linear operators is quite widely used in linear stability or linear wave analyses in plasma physics. The mathematical basis is, however,rarely quoted in literatures, which might lead to the improper understanding of thecomplicated phenomena of plasmas.
The examples described here might be somewhat trivial, at least in mathematics.
However, detailed analyses of basic problems may sometimes help the physicists forbetter understandings. First, we will review the spectral theory of ﬁnite dimensional
linear matrix operator, which has been completed due to the great works by Jordan.
Next, we will treat inﬁnite dimensional diﬀerential operator, which is, in any sense,
far from complete theory unlike the ﬁnite dimensional one. Since whole part of thisthesis is devoted to the analyses of such inﬁnite dimensional operators, we will focus
on the simple description here and explain the basis of well-known methods widelyused in literatures.

B.1

Finite dimensional operators

Let us ﬁrst describe the complete classiﬁcation of the ﬁnite dimensional operators.
The ﬁnite dimensional operators can be classiﬁed in a suitable way for the spectral
theory as follows;
1. Hermitian (selfadjoint) matrix
All eigenvectors can be taken orthogonal and all eigenvalues are real.
2. Normal matrix (commutable with its adjoint)
All eigenvectors can be taken orthogonal, but some eigenvalues are complex.
123

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Appendix B: Spectral theory

Jordansemi-simple
normalHermiteFigure B.1: Classiﬁcation of ﬁnite dimensional linear matrix operators.
3. Semi-simple matrix
Eigenvectors cannot be taken orthogonal, but linear space is spanned only byeigenvectors.
4. Jordan matrix
Eigenvectors are not enough to span whole linear space.
The schematic view is shown in Fig. B.1. In general, any ﬁnite dimensional matrix
operator belongs to one of the above sets. We can recognize how small is the region
occupied by the Hermitian operators in the whole linear operators even in the ﬁnite
dimensional case. In some sense, there may expand rich varieties of unknown sea
out of Hermitian operators.

B.1.1

Spectral resolution of Hermitian matrices

Here we will discuss the solution of the abstract Schrödinger type evolution equationi∂t ψ = Aψ,(B.1)

where the generator A is assumed a Hermitian matrix deﬁned in N ( ∈ N) dimensional vector space V = CN . The deﬁnition of Hermitian matrix will be given later.
The scalar product for two elements φ, ψ ∈ V is deﬁned as(φ | ψ) =N%φ̄j ψj ,(B.2)

j=1

where φj (ψj ) denotes the j-th component of the vector φ (ψ). Then the vector
space V is found to be an N dimensional Hilbert space. Here, the bar denotes thecomplex conjugate.

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B.1: Finite dimensional operators

An eigenvector ϕ of the matrix A is deﬁned by a nonzero element of V which
satisﬁesAϕ = λϕ,(B.3)
where λ ∈ C is called the eigenvalue. Then, the zero vector and the totality of ϕ,which belong to the eigenvector corresponding to the eigenvalue λ, are denoted asW . This constitutes the A-invariant subspace of V . Here W is called an eigenspacecorresponding to the eigenvalue λ.
Adjoint matrix A∗ is a matrix which satisﬁes(A∗ φ | ψ) = (φ | Aψ),(B.4)

for any φ, ψ ∈ V . The A∗ is represented by the complex conjugate of the transposed
matrix. Hermitian (selfadjoint) matrix is the one which satisﬁes A∗ = A. It is readily
shown that all eigenvalues of the Hermitian matrix are real.1 Moreover, it is known
that all eigenvectors can be taken orthogonal and the totality of eigenspaces span
the original vector space:
V = W 1 ⊕ W 2 ⊕ · · · ⊕ W N,

(B.5)

where ⊕ denotes the direct orthogonal sum of the eigenspaces. It is noted that
normal matrices also allow the orthogonal decomposition of the vector space V bytheir eigenvectors, however, the semi-simple matrices allow the decomposition
V = W 1 +̇ W 2 +̇ · · · +̇ W N ,(B.6)

where +̇ denotes the simple direct sum in which eigenspaces may not be orthogonal.
The spectral resolution of the Hermitian matrix A will be described with use of
the commutability with its adjoint.2 Let Pn be the projection operator from V onto
W n , then it will be expressed as
Pn = ϕn (ϕn | · ),(B.7)

where ϕn ∈ V denotes the normalized eigenvector of the operator A which corresponds to the eigenvalue λn . Here the dimension of the eigenspace W n is assumed as unity for simplicity. It is noted that Eq. (B.7) is sometimes expressed as
Pn = |ϕn )(ϕn | in the quantum mechanics context by following Dirac [8]. It is shownthat any Hermitian operator A will be expressed in terms of the projector asA=N%

λn Pn ,(B.8)

n=1
1

For any eigenvalue λ and the corresponding eigenvector ϕ, (ϕ | Aϕ) = λ(ϕ | ϕ) holds. On theother hand, (ϕ | Aϕ) = (Aϕ | ϕ) = λ̄(ϕ | ϕ) also holds, which supports λ = λ̄.
2Therefore, the following orthogonal spectral resolution is applicable to more general normalmatrix which satisfies A∗ A = AA∗ .

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Appendix B: Spectral theory

which is called a spectral resolution of the Hermitian matrix operator A. Here theresolution of identity can be produced asN%Pn = I,

(B.9)

n=1

where I denotes the identity matrix, and the orthogonality of the projector
Pi Pj = 0 (i = j)(B.10)

holds due to the orthogonality of the eigenvectors.
Based on the above knowledges, we can solve Eq. (B.1) by means of spectral resolution method. Substituting the spectral resolution (B.8) into original Schrödinger
type equation (B.1) leads toi∂t ψ ==N%n=1
N%

λn Pn ψλn ϕn (ϕn | ψ).

(B.11)

n=1

Taking the scalar product of both sides with ϕi (i ∈ N) givesi∂t (ϕi | ψ) = λi (ϕi | ψ),(B.12)

due to the orthogonality of eigenvectors. Equation (B.12) is readily solved and weobtain the time evolution of each ‘mode’ as(ϕi | ψ)(t) = e−iλi t (ϕi | ψ)(0)(B.13)

Substituting it into resolution of ψ(t)
ψ(t) =N%

Pn ψ(t)n=1=N%ϕn (ϕn | ψ)(t),(B.14)

ϕn e−iλi t (ϕi | ψ)(0).

(B.15)

n=1

we obtain the general solution as
ψ(t) =N%
n=1

Since the whole linear vector space V is spanned by only eigenfunctions with real
eigenvalues for the Hermitian matrix A, the time evolution is written in the form of
the superposition of simple harmonic oscillators (eigenmodes).
It is noted that, up to semi-simple matrix, this method works with slight modiﬁcations. The time evolution of the whole system is determined by the exponentialfunction with eigenvalues of the operator as its exponent, even though the orthogonality of eigenmodes may not follow in the case of semi-simple matrix.

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B.1: Finite dimensional operators

B.1.2Algebraic instability of Jordan matrices

In this section, the pathology of applying the spectral resolution for Jordan matrices
is shown. We will invoke here a diﬀerent method from the previous section. Sinceﬁnite dimensional matrix operator is a bounded one, we can deﬁne the exponentialfunction of the operator aseA = I + A +1 2
A + ···,
2!

(B.16)

where I denotes the identity matrix. It is shown that this expression gives a convergent series. Using the exponential function of the matrix, we can write the solutionof the original Schrödinger equation (B.1) as
ψ(t) = e−itA ψ(0).

(B.17)

If A were a Hermitian (semi-simple) matrix, the linear space V would be spannedby eigenvectors. Therefore, the expansion
e−itA = e−iλt e−it(A−λI)
t2= e−iλt I − it(A − λI) − (A − λI)2 + · · · ,2(B.18)

applied for the component of the eigenspace corresponding to the eigenvalue λ, would
give no contribution except for the ﬁrst term in the square bracket. It made theproblem possible to represent the whole dynamics of the system by the superpositionof exponential time evolution.
However, the eigenvectors are not enough, in general, to span the whole linear
space V for the Jordan matrix. Let ϕ be one of eigenvectors in a wider sense
belonging to the eigenvalue λ, and (A − λI)n ϕ vanishes at ﬁrst for n ∈ N (n > 1).
That is shown as
= 0 for j < n.
(B.19)(A − λI)j ϕ= 0 for j ≥ nSuppose that the initial condition is taken as ϕ. From the expressions shown inEqs. (B.17) and (B.18), we haveψ(t) = e−iλt e−it(A−λI) ϕn−1%(−it)j(A − λI)j ϕ.
= e−iλtj!
j=0Thus, the fastest growing mode of the eigenvector in a wider sense belonging to theeigenvalue λ shows divergence of the dependence t(n−1) e−iλt , which corresponds toan instability even if the eigenvalue λ is real. The algebraic growth of amplitudes iscalled ‘secularity.’

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B.2

Appendix B: Spectral theory

Diﬀerential operator

In this section, we will show an example of spectral resolution for the diﬀerential
operator corresponding to Schrödinger equation, and compare two widely used methods for the spectral analysis, i.e. Fourier transformation and Laplace transformation.
Diﬃculties of constructing the complete spectral theory for diﬀerential operators arecoming from the inﬁnity of their dimensions, which leads to the appearance of continuous spectra, and unboundedness of their spectra. However, we will not discusssuch diﬃculties in this section. For readers who are interested in such profoundproblems, several mathematical books are useful [28, 11, 6]. It is also noted that thefollowing discussions may not follow with the terminology of modern mathematics.

B.2.1

Spectral resolution

First, we will consider the one dimensional Laplacian operator
A = ∂x2 ,(B.20)

deﬁned in the Sobolev space H01 [0, 1]. Here, H01 [0, 1] denotes the set of the once differentiable functions f (x) deﬁned in the region x ∈ [0, 1] which satisﬁes the boundary
condition f (0) = f (1) = 0. The scalar product in this functional space is deﬁned by 1ūv dx,(B.21)(u | v) =0for the elements u, v ∈ H01 [0, 1], where the bar denotes the complex conjugate.

Itis readily shown that the operator A is Hermitian (selfadjoint) in this functionalspace.
Sinced2sin(nπx) = −(nπ)2 sin(nπx)
dx2holds, the eigenvalues of the operator A are
λn = −(nπ)2(B.22)(n ∈ N),(B.23)and the corresponding normalized eigenfunctions are√
un = 2 sin(nπx).

(B.24)

It is known that this set of eigenfunctions un constitutes a complete orthogonalbasis in H01 [0, 1] with the scalar product (B.21). Therefore, an arbitrary function
φ ∈ H01 [0, 1] will be expanded as
φ(x) =∞%
n=1

an un (x),(B.25)

129

B.2: Diﬀerential operator

where an is given by
an = (un | φ).

(B.26)

It is noted here that the operator
Pn = |un )(un | = un (x) 10

un (x) · dx

(B.27)

describes the projection from H01 [0, 1] onto the subspace spanned by |un ) and thespectral resolution of A is denoted asA=∞%

λn Pn ,(B.28)

n=1which shows the formal equivalence between the matrix representation and the differential one in quantum mechanics [8, 22]. Furthermore,∞%

Pn = 1(B.29)

n=1

is called the resolution of identity for the operator A, which is obtained from theParseval’s equality∞%2|(un | φ)|2,(B.30)
.φ. =
n=1

where .φ. = (φ | φ) denotes the norm of the element φ. The possibility of constructing the resolution of identity by means of eigenfunctions is guaranteed onlyfor Hermitian operators (von Neumann theorem). Although this resolution is notalways expressed by the summation — in general, the Hermitian operator of theHilbert space contains the continuous spectra, we have taken such a space for simplicity. In the extension of the previous sections, it is quite natural to consider that
non-Hermitian operators may include eigenfunctions in a wider sense.
Based on the above knowledges, we can solve the following time evolution equation (Schrödinger equation for a free particle);i∂t ψ = ∂x2 ψ,(B.31)by means of the spectral resolution of the operator A. Expanding ψ by the eigenfunctions un of the operator A as
ψ(x, t) =∞%an (t)un (x),(B.32)

n=1and substituting it into Eq. (B.31), we have∞%n=1(i∂t an − λn an )un = 0.

(B.33)

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Appendix B: Spectral theory

Since un is orthogonal for diﬀerent n, the equation can be decomposed into the onefor each ‘mode’ to givei∂t an = λn an ,(B.34)
which leads to the solution of each mode asan (t) = an (0)e−iλn t .

(B.35)

Thus, the general solution for an arbitrary initial perturbation ψ(x, 0) can be obtained as∞%√2 2
ψ(x, t) =2an (0)ein π t sin(nπx),(B.36)
n=1

where an (0) is obtained byan (0) = (un | ψ(x, 0)).

B.2.2(B.37)

Fourier transformation

Let us solve the Schrödinger equation (B.31) by means of the Fourier transformation.
This method is intrinsically parallel to the spectral resolution.
Fourier transformation is formally deﬁned as 1 ∞ψ(x, t) e−i(kx−ωt) dt dx,ψ̂(k, ω) =0(B.38)−∞

where the inversion will be given by
1 %ψ(x, t) =2π k ∞ψ̂(k, ω) ei(kx−ωt) dω,(B.39)−∞

where the inversion with respect to k is expressed by the discrete summation since
we have taken the ﬁnite domain [0,1]. The wave number k is chosen as
k = nπ(n ∈ N).

(B.40)

We will apply the Fourier transformation to the Schrödinger equation and obtain(ω + k 2 )ψ̂(k, ω) = 0,(B.41)

which is readily solved for ψ̂ to giveψ̂(k, ω) = δ(ω + k 2 ).

(B.42)

ω = −k2 ,(B.43)

The relation

131

B.2: Diﬀerential operator

denoting the singularity of ψ̂, is called the ‘dispersion relation’ since only thesevalues which satisfy Eq. (B.43) will give the contribution when inverted.
The way of solving the initial value problem is described as follows. Since aninitial perturbation can be spatially Fourier transformed as 1
ψ(x, 0) e−ikx dx,(B.44)â(k) =0

the solution will be given by the superposition of singular eigenfunction ψ̂(k, ω)
multiplied by â(k), which leads to%  −∞ψ(x, t) =â(k)δ(ω + k 2 ) ei(kx−ωt) dω
=

k∞
%∞

ân exp[i(nπx + n2 π 2 t)],(B.45)

n=1

where we have deﬁned ân = â(k) with the relation (B.40). This expression exactly
coincides with the solution obtained by the spectral resolution (B.36) by taking thereal part of Eq. (B.45).
It should be noted that the Fourier transformation in time corresponds to theexpression with whole superposition on the spectra, which now has a discrete sumof projections onto point eigenvalues.

B.2.3

Laplace transformation

Here we will solve the same Schrödinger equation (B.31) with an another method,i.e. Fourier transformation in space and Laplace transformation in time.
Fourier-Laplace transformation of the perturbed ﬁeld is deﬁned as 1 ∞ψ̃(k, s) =ψ(x, t) e−ikx−st dt dx,0(B.46)

0where the real part of s is chosen to be larger than any temporal singularity of thefunction ψ(t) for the convergence of the integration. The inversion will be given by
1 %ψ(x, t) =2πik

 s0 +i∞ψ̃(k, s) eikx+st ds,(B.47)

s0 −i∞

where s0 = Re(s) > 0 and k satisﬁes the condition (B.40).
The Fourier-Laplace transformation of the Schrödinger equation (B.31) givesisψ̃ = −k2 ψ̃ + iψ̄(k, 0),(B.48)

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Appendix B: Spectral theory

which leads toψ̃(k, s) =ψ̄(k, 0).
s − ik 2

Here ψ̄(k, 0) denotes the spatially Fourier transformed initial value 1
ψ(x, 0) e−ikx dx.
ψ̄(k, 0) =(B.49)(B.50)

0

By inverting Eq. (B.49), we obtain
1 %ψ(x, t) =2πi=∞%k s0 +i∞s0 −i∞ψ̄(k, 0) ikx+steds,s − ik 2ψ̄(k, 0) exp[i(nπx + n2 π 2 t)],(B.51)

n=1

which again coincides with the previous two methods.
It is noted that the Laplace transformation method in time corresponds to theexpression with whole integration around the spectra, which is now rewritten by thediscrete sum of independent eigenmodes due to Cauchy’s integral theorem.