数理解析研究所講究録 1226 巻 2001 年 239-250

239
Non-Hermitian dynamics of vortices in a shear flow

東大新領域

吉田善章, 龍野智哉 (Zensho Yoshida, Tomoya Tatsuno)
Graduate School of Frontier Sciences, University of TokyoI ntroduction1

Non-0rthogonality of eigenfunctions (modes) is the determining characteristic of non-Hermitian systems, which brings about interactions among different modes. This resembles
the mode couplings in nonlinear systems, and hence, the diversity of transient behavior in
non-Hermitian system is rather rich.
consider an abstract autono nous evolution equation of the Schr\"odinger tvpe
Let
$1\mathrm{i}\mathrm{S}$$\dot{\uparrow,}\partial_{t}u=llu$

(1)

$\{$

$\iota x$(0)$=\iota\iota_{()}$is acertain linear operator. When we can generate an exponential function
where, we can write thc solution of (1) as(propagator)$\mathcal{H}$$e^{-it\mathcal{H}}$$u(t)=e^{-it\mathcal{H}}\iota\iota_{0}$, thenandWhenmathematics. For vectors
the standard power series$u\in \mathrm{C}$$\mathcal{H}$$\in \mathrm{C}$is nothing but the exponential function elementary, we call define
and alinear map $lt$ :$e^{-it?t}$$\in \mathrm{C}^{N}$$\mathrm{s}\iota$.
$()\mathrm{f}$$\mathrm{C}^{N}arrow \mathrm{C}^{N}$$c^{-i\mathrm{f}H}=$$n-,1 \sum_{--}^{\infty}\frac{(-it\mathcal{H})^{n}}{n!}$$e^{-it\mathcal{H}}\mathrm{b}.\mathrm{v}$(2),or the Cauchy integral (inverse Laplace transform)$e^{-it\mathcal{H}}= \frac{1}{2\pi i}\oint e^{-- it\lambda}(\lambda I-\mathcal{H})^{-1}d\lambda$(3).

of operators,
is an operator in I’. For some different
in aHilbert space 1 ,. For bounded operators, we can invoke the Dunford
we have theories to generate
integral that is similar to (3). Amost general theory of generating an exponential function
for positive (so-called semigroup theory) is due to Hille and Yosidafl].
of the typetheorythiswarrants the solvability of initial value problems for wide classof generators, understanding of the behavior of the solution is not simple. Indeed, theexponential functions of matrices or operators are not necessarily “exponential” in theconventional sense.
The von Neumallll theory for Hermit ian (self-acjjoint) operators provides adeep
which invokes the spectral resolution of the generator in term
into the structure of
For

$r$

$\mathrm{c}1\mathrm{a}\mathrm{s}‘ \mathrm{s}.\epsilon_{\iota}\backslash \mathrm{s}$$\mathcal{H}$$u$$\epsilon^{t}-it\mathcal{H}$$e^{tA}$$t$$\dot{\mathrm{e}}1$$\mathrm{A}1\mathrm{t}_{)}\mathrm{h}\mathrm{o}1\iota \mathrm{g}\mathrm{h}$$\mathrm{i}_{11\mathrm{S}},$$\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}$$e_{\backslash }^{-- it\mathcal{H}}$$\mathcal{H}$240
of acomplete set of orthogonal modes. The basic idea is that the
may be represented
as asum of independent harmonic oscillators, each of which is an eigenfunction of $H$ andthe corresponding eigenvalue (real number) gives the frequency of the oscillation. Unlike
the case of finite dimension vector space, however, the conventional eigenfunctions may not
be complete to span the Hilbert space. The most essential generalization needed to study
an infinite dimension space was the introduction of continuous spectra that correspond to
singular eigenfunctions. The spectral resolution ofis, in general, given by an integral
over the spectra (an example will be given in Sec. 3.1). The contribution to thefrom
the continuous spectra brings about the “phase mixing” of oscillations with continuous
frequencies, resulting in various types of damping. Hence, the reality of the spectra of an
Hermitian operator does not necessarily imply stationary (non-dumping) oscillations.
For alinear map in afinite dimension vector space, the spectral resolution yields the
Jordan canonical form, and the explicit representation of
can be constructed using
the canonical form. It is well known that anilpotent yields secular behavior of the corresponding generalized eigenvector. Therefore, even if every eigenvalue
is real, thecan describe “instabilities” (growth of oscillations).
In aHilbert space, however, such ageneral theory of spectral resolution is limited toeither compact operators or Hermitian operators. This paper is an attempt to obtain aspectral resolution of anon-Hermitian operator that is not included in the above mentioned
categories. This operator is related to an important physics problem (Sec. 2).
$e^{-it\mathcal{H}}$$\mathcal{H}$$e^{-it?t}$$e^{-it\mathcal{H}}$$e^{-it\mathcal{H}}$$\lambda_{j}$2Non-Hermitian dynamics of vortices

The vortex dynamics equation in
Liouville equation$\mathrm{R}^{2}$[the coordinates are denoted by ( , )] reads as a$x$$\partial_{t}\Psi+\{H, \Psi\}=0$

,

$y$

(4)

where 1is the vorticity, $H$ is the Hamiltonian (stream function) of an incompressible flowthat transports the vortices, and$v=(\partial_{y}H, -\partial_{x}H)^{t}$$\{a, b\}=(\partial_{y}a)(\partial_{x}b)-(\partial_{x}a)(\partial_{y}b)=-\nabla a\mathrm{x}$ $\nabla b\cdot\nabla z$

is the Poisson bracket.
When the Hamiltonian $H$ depends o, the evolution equation (4) is nonlinear. The
dynamics of
can couple with other fields when they are included in $H$ . The simplest
example of nonlinear vortex dynamics is that of the Euler fluid (incompressible ideal flow),where$\mathrm{n}$$\Psi$$\Psi$$-\triangle H=\Psi$or, denoting the Green operator of the,$\mathrm{L}\mathrm{a}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{a}\mathrm{n}-\Delta$$H=\mathcal{G}\Psi$

(5)

by

$\mathcal{G}$.

(6)

Let us linearize (4) with decomposing 1and $H$ into their ambient (denoted by subscript0) and fluctuation parts:$\Psi$

$H$

,

$=$

$\Psi_{0}+\psi$$=$$H_{0}+h=\mathcal{G}\Psi_{0}+\mathcal{G}\psi$.

241
Neglecting the second-0rder terms, (4) reads$\partial_{t}\psi+\{H_{0}, \psi\}-\vdash\{\triangle H_{0}, \mathcal{G}\psi\}=0$.

(7)

In this PaPer, we consider one-dimensional problem with
$H_{0}=H_{0}(x)$.

Since the ambient Hamiltonianis independent of , the wavenumber in is agood
quantum number, and we can replaceby $ik$ . In what follows, we assume $k\neq 0$ , andnormalize $k=1[2]$ . We write$H_{0}$$y$$y$$\partial_{y}$$v(x)=-\partial_{x}H_{0}(x)$,to obtain the standard Rayleigh equation$i\partial_{t}\psi=v(x)\psi+v’(x)\mathcal{G}\psi$

The Green operator$\mathcal{G}$(8).

is represented by aconvolution integral.

$( \mathcal{G}f)(x)=\int_{-\infty}^{+\infty}\frac{e^{-|x-\xi|}}{2}f(\xi)d\xi$(9)

In what follows, we denote by $G(x, \xi)$ the Green function;$G(x, \xi)=\frac{e^{-|x-\xi|}}{2}$

(10)

.

3Convection and oscillations
The generator of the vortex dynamics equation (8) consists of two terms, each of which
describes different mechanism of vortex motion. The first term on the right-hand side of (8)[originating from,in (7)] represents the transport of the vorticity by the ambient
flow $v(x)$ . An inhomogeneous (sheared) flow distorts vortices, and hence, no stationary
structure can persist in ashear flow
. Such adynamics is described byacontinuous spectrum (Sec. 3.1). On the other hand, the second term [originating fromin (7) describes the interaction between the perturbation and the ambient
field. When the ambient vorticity
has aspatial gradient, aflow induced by aperturbation yields alocal change of the vorticity. This term, hence, can create perturbed
vortices from the ambient field.
In this section, we study the role of both terms by formal calculations. In what follows,
it is convenient to generalize (8) with replacing $v’(x)$ by acertain “real” function $w(x)$that is independent of $v(x)$ . With assuming $k\neq 0$ , we consider$\{H_{0}$$\psi\}$$(v(x)\neq \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t})$$\{\triangle H_{0}, \mathcal{G}\psi\}$$]$$\Psi_{0}=-\triangle H_{0}$$i\partial_{t}\psi=v(x)\psi+\tau v(x)\mathcal{G}\psi$.

The case when $w(x)–v’(x)$ recovers the physically relevant equation (8)

(11)

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Convection –shear flow transport

3.1

Here, we assume $w(x)=0$ in (11) and consider(12)$i\partial_{t}\psi=v(x)\psi$with a“continuous” real function $v(x)$ , which reads as aSchr\"odinger equation with a
Hamiltonian $v(x)$ .
The formal eigenvalue and the corresponding eigenfunction of the generator of (12), withsetting$v(x)\psi=\omega\psi$(i.e.,$\psi(t)=e^{-i\omega t}\psi$), is given by$\omega$$=v(\mu)$

where is an arbitrary real number and
we write$\mu$,$\delta$)$\iota[$$=\delta(x-\mu)$

(13)

,

denotes the delta-measure. For the convenience,$(f(x), \delta(x-l^{l))=}\int_{-\infty}^{+\infty}f(x)\delta(x-\mu)dx=f(l^{l})$.

Aformal spectral resolution of the generator is written as$v(x)f(x)$$=$$\int_{-\infty}^{+\infty}v(\mu)(f, \delta(x-\mu))\delta(x-l^{l})d/\iota$$=$$\int_{-\infty}^{+\infty}v(_{l}\iota)f(\mu)\delta(x-\mu)d\mu$

(14)

.

Rigorous mathematical representation of this “continuous spectrum” is given by thespectral resolution of the coordinate operator:$xf(x)= \int_{-\infty}^{+\infty}\mu dE(\mu)f(x)$where$\{E(\mu);\mu\in \mathrm{R}\}$

(13)

,

is afamily of projectors (resolution of the identity) defined by$E(\mu)f(x)=\{$

for $x\leq\mu$0for $x>\mu$$f(x)$(16)

The projector $E(\mu)$ gives aresolution of the identity:$I= \int_{-\infty}^{+\infty}$(17)dE(\mu ).

Using this representation of the coordinate operator, we can write$v(x)f(x)= \int_{-\infty}^{+\infty}v(l^{l})dE(\mu)f(x)$

.

(18)

The solution of (12) with initial condition $\psi(x, 0)=\psi_{0}(x)$ is given by$\psi(x, t)=\int_{-\infty}^{+\infty}e^{-:tv(\mu)}dE(\mu)\psi_{0}(x)=e^{-:tv(x)}\psi_{0}(x)$

.

(19)

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3,2

Chandrasekhar model of surface-waves

Non-Hermitian property stems from the second term on the right-hand side of (11), because
the multiplication of $w(x)$ and the integral operator (; does not commute. As we have noted,
this term represents the interaction between the perturbed flow and the ambient vorticity.
Physically, the non-Hermitian property implies the non-conservation of the ”energy” of the. We also remark that the original nonlinear systemvorticity, i.e., the enstrophy(4) conserves the enstrophy, as well as all “Casimirs” $\int f(\Psi)dx(f$ is an arbitrary smoothfunction). The non-conservation of the enstrophy in the linearized system is due to theseparation of the vorticity into the perturbed component and the ambient field. Because, theof the interaction between these two parts, which is enabled by the term
perturbed component does not describe aclosed dynamical system.
in (11)] is most simply highlighted by
The role of the non-Hermitian term [Chandrasekhar’s model of ashear flow, which assumes apiece-wise linear flow $v(x)$ andthe corresponding delta measure $v’(x)[3]$ . Before giving amathematical justification, let
us examine formal solutions of this model.
In this subsection, we assume $v(x)=0$ and consider$\mathrm{J}^{\cdot}|\psi|^{2}dx$$\{h, \Psi_{0}\}$$\psi$$w(x)\mathcal{G}\psi$(20)$i\partial_{t}\psi=w(x)\mathcal{G}\psi$with$w(x)=A\delta(x-a)$$(A, a\in \mathrm{R})$

(21)

.

The formal eigenfunction of the generator, under the setting ofmined by$A \delta(x-a)\int_{-\infty}^{+\infty}G(x, \xi)\psi(\xi)d\xi=\omega\psi(x)$where $G(x, \xi)$ is the Green function of$\omega$$\mathcal{G}$$= \frac{A}{2}$,$i\partial_{t}=\omega$in (20), is deter(22),[see (10)]. Solving (22), we obtain$\psi=\delta(x-a)$

(23)

.

We thus have an oscillation of a“surface wave” that is localized at $x=a$ and has thewavenumber in the direction [4].
If we have multiple “sources” of the surface waves, these waves interact through spatial
couplings induced by perturbed flows. Let us consider $N$ (finite number) sources
$k$

$y$

$w(x)= \sum_{j=1}^{N}A_{j}\delta(x-a_{j})$$(A_{j}, a_{j}\in \mathrm{R}, j=1, \ldots, N)$

.

(24)

The frequencies of the coupled surface waves are given by solving$\sum_{j=1}^{N}A_{j}\delta(x-a_{j})\int_{-\infty}^{+\infty}G(x, \xi)\mathrm{s}\mathit{1})(\xi)d\xi=\omega\psi(x)$Substitutin$\psi=\sum_{j=1}^{N}\alpha_{j}\delta(x-a_{j})$

,

.

(25)

244
into (25), we obtain the “dispersion relation”$\alpha_{1}$$\Lambda I$$\{$$\alpha_{1}$.
.

–$=\omega$$\cdot$$\{$$\alpha_{N}/$

.
.

(26)

$\cdot$$\alpha_{N}/$with$M_{i,j}=A_{i}G(a_{i}, a_{j})=A_{i} \frac{e^{-|a-a_{j}|}}{2}$

.

(27)

The eigenvalue problem (26) determines the frequencies of the coupled oscillations. Obviously, the matrix $M$ is non-symmetric (except for the case of $A_{j}=C$ for all ), representing the non-Hermitian property of the generator. For some sets of coefficients$(j=1, \cdots, N)$ , the frequency
can be complex. The imaginary part of gives the growthrate of the unstable mode of oscillation which corresponds to the “Kelvin-Helmholtz in tability”.
$j$$A_{j}$$\omega$$\omega$

3.3

Coupling of the two generators

We have seen the dynamics of vortices induced by each of the two different generators in(11), separately. Now, we study the coupling of these two generators.
Let us first consider the case of single source; see. The eigenvalue problem associatedwith the generalized Rayleigh equation (11) reads$(2\mathrm{i})$$v(x) \psi+A\delta(x-a)\int_{-\infty}^{+\infty}G(x, \xi)\psi(\xi)d\xi=\omega\psi$

,

(28)

where $G(x, \xi)=e^{-|x-\xi|}/2$ is the Green function [see (10)]. Let us try to find aformal
solution with assuming$\psi=\alpha\delta(x-a)+\beta\delta(x-\mu)$

,

(29)

where is an arbitrary “fixed” real number [see (13) and (23)]. Substituting (29) into (28.),
we obtain an eigenvalue problem
$\mu$

$L$

$(\begin{array}{l}\alpha\beta\end{array})=\omega$(30)$(\begin{array}{l}\alpha\beta\end{array})$where$L=\{$ $v(a)+AG(a, a)0$

$AG(a,\mu)v(\mu))$

.

(31)

We can solve (30) to find aset of eigenvalues and eigenfunctions:$\omega$$= \Omega_{1}(a):=v(a)+\frac{A}{2}$,$(\begin{array}{l}\alpha\beta\end{array})=U_{1}:=(\begin{array}{l}10\end{array})$

,

(32)

and
$\omega$$=\Omega_{c}(\mu):=v(\mu)$,$(\begin{array}{l}\alpha\beta\end{array})=U_{c}:=m(\mu)(\frac{AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a),1},$

$)$

,

(33)

245
where the normalization factor is(34)$m( \mu)=[1+(\frac{AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a)})^{2}]-1/2$

The first eigenvalue $\Omega_{1}=v(a)+(A/2)$ gives the “Doppler-shifted” frequency of the surface
wave [see (23)]. The corresponding formal eigenfunction is exactly $\psi=\delta(x-a)$ . The
second eigenvalue $\Omega_{c}=v(\mu)$ represents the local flow velocity [see (13)], while the corresponding formal eigenfunction describes acombination of the surface wave and alocal
vortex.
By the transforms$T=(U_{1}U_{c})=(01$the matrix$L$$\frac{m(\mu)AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a),7n(\mu)},$$)$,$T^{-1}=(01$$- \frac{AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a),m(\mu)^{-1}},$

$)$

(35)

,

is diagonalized;.

$T^{-1}LT=(\begin{array}{ll}\Omega_{1} 00 \Omega_{c}\end{array})$

We note that is not aunitary transform, reflecting the fact that the generator is not a
Hermitian operator.
If the “resonance” $\Omega_{1}=\Omega_{c}[v(a)+A/2=v(\mu)]$ occurs, the second solution (33) degenerates into the first one (32). This is the case when the matrix of (30) transforms
into aJordan block. We introduce ageneralized eigenfunction belonging to the degenerate
eigenvalue
;
$T$

$L$

$\Omega_{1}$,$U_{c}’=(\begin{array}{l}1(AG(a,\mu))^{-1}\end{array})$(36)

which satisfies $(L-\Omega_{1}I)^{2}U_{c}’=0$ . By transforms$T’=(U_{1}U_{c}’)=(\begin{array}{ll}1 10 (\Lambda G(a,l^{l}))^{-1}\end{array})$

we can transform$L$,$T^{\prime-1}=(\begin{array}{lll}1 -AG(a \mu)0 AG(a,\mu) \end{array})$,

into aJordan canonical form
$T^{\prime-1}LT’=(\begin{array}{ll}\Omega_{1} 10 \Omega_{1}\end{array})$.

To unify both the non-resonant and resonant (nilpotent) cases, we define$m(\mu)$$\tilde{m}(\mu)=\{$$(AG(a, \mu))^{-1}$and combine$U_{c}$

and$U_{c}’$

if
if$\Omega_{c}(\mu)\neq\Omega_{1}(a)$(i.e. $m(\mu)=0$ ),$\Omega_{c}(\mu)=\Omega_{1}(a)$

(37)

as
$\tilde{U}_{c}(\mu)=(\frac{m(\mu)AG(a,\mu)}{\Omega_{\mathrm{c}}(l^{l})-\Omega_{1}(a),\tilde{m}(\mu)},$

$)$

.

(38)

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The transform$\tilde{T}=(U_{1}\tilde{U}_{c}(\mu))=(01$$\frac{n\iota(\mu)AG\prime(a,\mu)}{\Omega_{r}(\mu)-\Omega_{1}(a),\tilde{m}(\mu)},$

(39)

$)$

is regular for all .
Next, we study the case of multiple sources; see (24). We solve
$\mu$

(40)

$v(x) \psi+\sum_{j=1}^{N}A_{j}\delta(x-a_{j})\int_{-\infty}^{+\infty}G(x, \xi)\psi(\xi)d\xi=\omega\psi$with assuming$\psi=\sum_{j=1}^{N}\alpha_{j}\delta(x-a_{j})+\beta\delta(x-\mu)$.

To generalize the above calculations, we prepare notation [see (32)]$\Omega_{j}(a_{j})=v(a_{j})+\frac{A_{j}}{2}$$(j=1, \ldots, N)$ .

(41)

.

(42)

The dispersion relation is$L$

where the matrix$L$$(\begin{array}{l}\alpha_{1}\vdots\alpha_{N}\beta\end{array})=\omega$$(\begin{array}{l}\alpha_{1}\vdots\alpha_{N}\beta\end{array})$

generalizes (31) as$L=\{$$\mathrm{f}l_{1}(..\cdot a_{1})0^{\cdot}.$.

$A_{1}G(a_{1}.\cdot.’

a_{N})\Omega_{N}(a_{N})0$$A_{N}G(..\cdot a_{N},\mu)A_{1}G(a_{1},\mu)\Omega_{c}(\mu))$

.

(43)

$A_{N}G(a_{r\mathrm{V}}, a_{1})$We have two different classes of solutions. The first group, corresponding to (32), is obtained with setting $=0$ . Then, the eigenvalue problem (42) reduces into$\beta$$(\begin{array}{lll}\Omega_{1}(a_{1})\ddots A_{1}G(a_{1},a_{N})\vdots \ddots \vdots A_{N}G(a_{N},a_{1}) \Omega_{N}(a_{N})\end{array})(\begin{array}{l}\alpha_{1}\vdots\alpha_{N}\end{array})=\omega$ $(\begin{array}{l}\alpha_{1}\vdots\alpha_{N}\end{array})$

(44)

,

which reads as the dispersion relation that is Doppler shifted from (26). The second class. The eigenvalue isof eigenvectors, corresponding to (33), is given by setting$\beta\neq 0$$\Omega_{c}(\mu)=v(\mu)$,and the corresponding eigenfunction is determined by$(\begin{array}{lll}\Omega_{1}(a_{1})-\Omega_{c}(\mu) A_{1}G(a_{1},a_{N})\vdots \ddots \vdots A_{N}G(a_{N},a_{\mathrm{l}}) \Omega_{N}(a_{N})-\Omega_{c}(\mu)\end{array})(\begin{array}{l}\alpha_{1}\vdots\alpha_{N}\end{array})=-\beta$$(\begin{array}{l}A_{1}G(a_{1},\mu)\vdots A_{N}G(a_{N},\mu)\end{array})$

.

(45)

As discussed above, the resonances $\Omega_{j}(a_{j})=\Omega_{c}(\mu)(j=1, \cdots, N)$ yield singularities in
the matrix of (45), and then, we must consider nilpotents

247
Spectral resolution of coupled non-Hermitian generator

4

In this section, we formulate the vortex dynamics equation (11) with the delta-measure
field (24) as an evolution equation in an appropriate Hilbert space, and give aspectral
resolution of the generator. The generator reads$\mathcal{L}\psi=v(x)\psi+\sum_{j=1}^{N}A_{j}\delta(x-a_{j})\int_{-\infty}^{+\infty}G(x, \xi)\psi(\xi)d\xi$

(46)

,

, $a_{j}\in \mathrm{R}(j=1, \ldots, N)$ , and $G(x, \xi)=e^{-|x-\xi|}/2$ is the Green
where $v(x)\in C(\mathrm{R})$ ,function [see (10)]. In what follows, we assume $|v(x)|<c(\forall x)$ with some finite number .
is not amember of the Lebesgue space, we encounter
Since the delta measure
Hilbert space.
adifficulty in formulating the problem in the conventional$A_{j}\in \mathrm{R}$$c$$\delta(x-a_{j})$$L^{2}$

4,1

Mathematical formulation of the generator

Let us consider aHilbert space$V=\mathrm{C}^{N}\oplus L^{2}(\mathrm{R})$

where

on

(47)

,

is the complex Lebesgue spaceis the unitary space of dimension $N$ , andendowed with the standard inner product. The member of is written as$L^{2}(\mathrm{R})$$\mathrm{C}^{N}$$V$$\mathrm{R}$$\psi=(\varphi(x)\alpha)$

The inner product of$V$$[\alpha\in \mathrm{C}^{N}, \varphi(x)\in L^{2}(\mathrm{R})]$

(48)

.

is, thus, defined as(49)$\langle\psi, \psi’\rangle=(\alpha, \alpha’)+(\varphi, \varphi’)=\sum_{j=1}^{N}\alpha_{j}\overline{\alpha}_{j}’+\int_{-\infty}^{+\infty}\varphi(x)\overline{\varphi}’(x)dx$

We identify$\uparrow\int J=(\varphi(x)\alpha)\Leftrightarrow\psi(x)=\sum_{j=1}^{N}\alpha_{j}\delta(x-a_{j})+\varphi(x)$

(50)

.

It is essential to decompose the delta-measure part (representing the surface waves) from thetotal vorticity . Although the supports (in the sense of distributions) of both components$\delta(x-a_{j})$ andmay overlap, we separate them into different degrees of freedom.
for all $\psi\in V$ , the generator is abounded operator on .
Because
Following (50), the generator of (46) is now written in amatrix form [see (43)]$\psi$$\varphi(x)$$V$$\mathcal{L}$$\mathcal{G}\psi\in C(\mathrm{R})$$\mathcal{L}$$\mathcal{L}\psi=\{$$\Omega_{1}(...a_{1})0^{\cdot}.$$A_{N}G(a_{N}, a_{1})$.

$A_{1}G(a_{1}.\cdot.’

a_{N})0$$\int A_{N}G(a_{N}.\cdot.,x)\cdot dx\int A_{1}G(a_{1},x)\cdot dx\Omega_{c}(x))$$(\begin{array}{l}\alpha_{1}\vdots\alpha_{N}\varphi(x)\end{array})$(51)$\Omega_{N}(a_{N})$In the previous section, we dealt delta functions in aformal way and did calculations[see (13) and (29)]. We note that such formal
using $\delta(x-\mu)$ with an arbitrary
functions are not the member of the Hilbert space . In this section, they are justified as
generalized eigenfunctions corresponding to “continuous spectr\"a”.
$\mu\in \mathrm{R}$

$V$

248
4.2

Spectral resolution of the generator

First, we consider the simple case of single “source”, i.e., $w(x)=A\delta(x-a)$ [see (21)]. The
surface wave mode has only one degree of freedom $(N=1)$ . Here, the generator of (51)simplifies as$\mathcal{L}$$\mathcal{L}=($.

$\Omega_{1}(a)0$$dx$$\int AG(a,x)\Omega_{c}(x)$

).

(52)

As we have shown in Sec. 3.3, there are two different classes of formal eigenfunctions [see(32) and (33) ; In the form consistent to the notation of (48), they read$]$$\Omega_{1}(a)=v(a)+\frac{A}{2}$$\Omega_{c}(\mu)=v(\mu)$,,$U_{1}=(\begin{array}{l}\mathrm{l}0\end{array})$$(53\grave{J}$$\tilde{U}_{c}(\mu)=(\tilde{m}’\mu)\frac{m(\mu)AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a),(\mu)\delta(x-})$

.

(54)

The first eigenfunction represents the surface wave. The second one includes an arbitrary
real number , corresponding to the continuous spectrum, and asingular function.
We must integrate (54) overto span the complete basis of . Formally, we cangeneralize the transform of (39) as$\delta(x-\mu)$

$\mu$

$V$

$\mu\in \mathrm{R}$$\tilde{T}$$\mathcal{T}=(U_{1}\int(\cdot, \delta(x-\mu))\tilde{U}_{c}(\mu)d\mu)=(01$$\int(\cdot,\delta(x-\mu))\tilde{m}’\mu)d\mu\int(\cdot,\delta(x-\mu))\frac{m(\mu)AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a),(\mu)\delta(x-}d\mu)$

.

(55)

To cast this formal expression in an appropriate mathematical representation, we invoke
the resolution of the identity (17). The formal correspondence is$\int_{-\infty}^{+\infty}(u(x), \delta(x-\mu))\delta(x-\mu)d\mu=\int_{-\infty}^{+\infty}$dE(\mu )u $=u$ .

We also define$F( \mu)u=\int_{-\infty}^{\mu}u(x)dx$

,

(56)

which gives$dF(\mu)u=u(\mu)d\mu$ .

Using this notation, we can write$\int f(\mu)dF(\mu)u(x)=\int f(\mu)u(\mu)d\mu=\int f(x)u(x)dx$ .

The operator$\mathcal{T}$is now written in arigorous form of$\mathcal{T}=($$01$$\int\frac{m(\mu)AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a),\int\tilde{m}(\mu)dE},dF(\mu)(\mu))=(01$$\int\frac{m(x)AG(a,x)}{\Omega_{c}(x)-\Omega_{1}(a),\tilde{m}(x)},$$\cdot dx)$

(57)

Reflecting the non-Hermitian property of the generator , the operator is not aunitarytransform. By combing both non-resonant and resonant (nilpotent) cases [cf. (39)], thisis aregular transform. The inverse operator is$\mathcal{L}$$\mathcal{T}$$\mathcal{T}$. $dx$$\mathcal{T}^{-1}=($$01$$- \int(\frac{m(x)}{\overline{m}(x)})(\frac{AG(a,x)}{\Omega_{c}(x)-\Omega_{1}(a),(x)^{-1}},)\tilde{m}$

).

(581

249
Using the transforms$\mathcal{T}$and$\mathcal{T}^{-1}$, we obtain the Jordan canonical form of$\mathcal{T}^{-1}\mathcal{L}\mathcal{T}=$$(\Omega_{1}0$$=$$\rho(x)=\{$;$\int\Omega_{c}(\mu)dE(\mu)\int\rho(\mu)dF(\mu))$$(\begin{array}{llll}\Omega_{1} /\backslash \rho(x)\cdot dx0 \Omega_{c}(x) \end{array})$where$\mathcal{L}$1if$\Omega_{c}(\mu)=\Omega_{1}(a)$0if$\Omega_{c}(\mu)\neq\Omega_{1}(a)$

(59)

,

can have afinite measure when the resonance condition
The support ofholds on a finite interval of .
$\rho(x)$$\Omega_{c}(\mu)=\Omega_{1}(a)$

$x$

4.3

Spectral representation of the propagator

The propagator$e^{-it\mathcal{L}}$is defined by solving the initial value problem for (11)$i\partial_{t}\psi=\mathcal{L}\psi$

,

(60)

$\{$

$\psi(0)=\psi_{0}$and writing the solution as$(/)(t)=e^{-it\mathcal{L}}\psi_{J_{0}}$Defining$\psi=\mathcal{T}\chi$.

, we transform (60) into
$i\partial_{t}\chi=\mathcal{T}^{-1}\mathcal{L}\mathcal{T}\chi$

,

(61)

$\{$

$\chi(0)=\mathcal{T}^{-1}\psi_{0}$.

Using the spectral resolution (59), the solution of (61) is given by$e^{-it\mathcal{T}^{-1}\mathcal{L}\mathcal{T}}$$=$$- \int ite^{-it\Omega_{1}}\rho(\mu)dF(\mu)\int e^{-il\Omega_{c}(\mu)}dE(\mu))$$(c_{0}^{\mathrm{J}}-it\Omega_{1}$($=$$e_{0}^{-it\Omega_{1}}$-/$\cdot$$ite^{-it\Omega_{1}},\rho(x)e^{-it\Omega_{c}(x)}\cdot d.c$

).

(62)

The solution of (60) is given by$\psi(t)=\mathcal{T}[e^{-it\mathcal{T}^{-1}\mathcal{L}\mathcal{T}}]\mathcal{T}^{-1},\psi_{0}$.

Using (57) and (58), we obtain$e^{-it\mathcal{L}}$

$=$

$=$

$\mathcal{T}$($e_{0}^{-it\Omega_{1}}$$- \int ite^{-it\Omega_{1}},\rho(x)e^{-it\mathrm{f}\mathit{1}_{c}(x)}\cdot dx$$(\begin{array}{ll}e^{-il\Omega_{1}} \lrcorner \mathrm{Y}0 e^{-it\Omega_{c}(x)}\end{array})$,)$\mathcal{T}^{-1}$

(63)

250

$\yen$[ee$\ovalbox{\tt\small REJECT} t\mathrm{O}_{c}(x)$r)$\ovalbox{\tt\small REJECT} t0_{1}(a)]_{\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}}1\mathrm{C}\mathrm{I}\ovalbox{\tt\small REJECT}(a,$ile$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}\ldots$$AG(a_{\ovalbox{\tt\small REJECT}}x)p(x)$. dx,$\mathrm{O}_{c}(\ovalbox{\tt\small REJECT} \mathrm{r})-\mathrm{O}.(a)$and we have used the relations$\frac{m(x)}{\tilde{m}(x)}=1-\rho(x)$$\{$$\frac{\rho(x)}{\tilde{m}(x)}=AG(a, x)\rho(x)$The off-diagonal part $X$ of the matrix operator (63) represents the mode interactions
originating from the non-Hermitian property of the generator. The $X$ consists of two parts;
one is the contribution from the non-resonant flow in the region of the support of $1-\rho(x)$ ,and the other is from the resonant flow in that of. The latter produces secular behavior(represented by the factor).
$l^{y}(x)$$ite^{-it\Omega_{1}}$References[1] K. Yosida, Functional Analysis (Springer-Verlag, Berlin, 1995).

[2] By changing the scale of y, we can normalize k to 1. On the contrary, to take k,
we translate y $arrow ky$ , v $arrow kv$ , w $arrow kw$ and $e^{-|x-\xi|}/2arrow e^{-k|x-\xi|}/(2k)$ in the later
calculations; see (8), (11) and (9).
$\neq 1$

[3] S. Chandrasekhar, Hydrodynamic and hydromagnetic stability (Clarendon, Oxford,
1961).
[4] There are rich examples of relevant phenomena; The Rossby waves of perturbations
in geological jet streams, the diocotron waves in non-neutral plasmas, and the drift
waves in neutral plasmas