Commun.
math.
Phys.
41,
273—279
(1975)
© by
Springer-Verlag
1975
On
Completeness
of
Eigenfunctions
of
the
One-Speed
Transport Equation
Tomaz
Kline
Department
of
Mechanical Engineering,
University
of
Ljubljana,
Ljubljana,
Yugoslavia
Received
August
3,
1974
Abstract.
It is
shown that
the set of
Case's
eigenfunctions
of the one
speed transport equation
is
complete
in the
rigged Hubert space
W±
([-
1,
1])C
L
2
(-
1,
1)C
W
2
~
l
([-
1,
1]).
1.
Introduction
Case's
method
of
singular eigenfunction expansions
for
solving
the
transport
equation
[1]
seeks,
by
separation
of
variables,
to
construct
a
sufficiently
rich
set of
solutions,
called elementary solutions, which would enable
one to
expand
an
"arbitrary"
solution
of the
equation into
a
Fourier series
in
terms
of
this set.
An
important point
in
this method
is the
completeness proof
for the set of
ele-
mentary
solutions. Originally
it was
shown [2],
by
means
of the
theory
of
singular
integral
equations, that
the
expansion
coefficients
are
uniquely determined
for the
class
of
Holder-continuous functions, which, within this class,
proves
the
com-
pleteness.
An
alternative
to
this constructive approach
is the
demonstration
of the
closure relation
for the set of
elementary solutions
[1].
Unfortunately, either
proof
has to be
carried
out
separately
for
each particular
from
of the
transport
equation
under consideration,
and
moreover, there remains some doubt
as to
whether
the
obtained result
is the
strongest possible.
According
to an
idea
by A.
Skumanich,
commented upon
in
Ref. [1],
the
completeness proof
for
Case's
elementary solutions should
be
based
on
more
general arguments, provided
by the
functional-analytic properties
of the
under-
laying transport operator. This would lead
to the
completeness proof
for a
whole
class
of
operators which have certain common properties.
The
functional
analytic approach
to the
problem
was
considered
by
Hangel-
broek
[3] and by
Larsen
and
Habetler [4], where
it was
essentially shown that
the
Case eigenfunction expansion formula represents
the
resolution
of the
identity
of
a
transport operator,
but
again only
after
resorting
to a
kind
of
Holder con-
tinuity
requirement. There remains some ambiguity about
the
notion
of the
eigen-
function,
which
is
also
referred
to by
Baird
and
Zweifel
[5],
and the
structure
of
the
space
of
eigenfunctions
remains unclear.
Here
we
propose
a
completeness proof which
is
based
on the
theory
of
eigen-
function
expansions
for
self adjoint operators
in
rigged Hubert spaces,
as ex-
pounded
in the
treatise
by
Berezanskiϊ
[6].
A
rigged
Hubert
space,
of the
type
to be
considered,
is a
triple
of
separable Hubert spaces
H
+
C
H
C
H_,
where
H
+
,
the
positive space,
is
dense
in
H,
and
H_,
the
negative space,
is
isometric
to the
274 T.
Kline
dual space
of
H
+
and
contains
H as a
dense subspace. Rigged Hubert spaces
are
the
most appropriate
spaces
for
spectral decomposition
of
self adjoint operators
[7],
this being
due to the
following properties.
Any
continuous linear
functional
on
H
+
may be
represented
by
some element
of
H_
by
means
of the
scalar product
in
//,
instead
of the
scalar product
in
H
+
itself.
If
H
+
consists
of
finitely
differentiable
functions,
then
the
elements
of
H_
are
generalized
functions
of
finite
order.
If the
embedding
of
H
+
into
H is
quasi-nuclear,
i.e.
the
embedding operator
is
Hubert-
Schmidt, then
any
self
adjoint operator
in
H,
which admits
the so
called rigged
extension,
possesses
a
complete
set of
eigenfunctions which
are
elements
of
//_.
The
proposed approach
is
undertaken here
for the
special case
of the
one-
speed
transport equation with isotropic scattering,
and
c,
the
number
of
seconda-
ries
per
collision, smaller than
1. We
were unable
to
extend
it to the
case with
c
=
1?
even though
no
difficulties
are
encountered here
in the
constructive proofs.
It
is
applicable
to the
self adjoint eigenvalue problem
of the
form
Tf =
v/,
or to the
eigenvalue problem
Af =
vB,
with self adjoint
A and
positive
definite
B.
2.
The
Eigenvalue Problem
Consider
the
one-speed steady-state transport equation with isotropic
scattering
in
plane-parallel geometry
[1,2]
μ-j-ψ(χ,μ)+ψ(
X9
μ)=j-
}
Ψ(x,μ)dμ
9
(1)
with
0<c<
1.
By
seeking
its
solutions
in the
form
where
v is a
separation parameter
to be
determined,
the
task
of
solving
Eq.
(1)
is
reduced
to the
formal eigenvalue problem
posed
by the
equation
c
(Pv(μ)-
γ
ί
Ψv(p)dμ\.
(2)
c
-i
It
may be
shown that
the
integral
of
φ
v
cannot vanish unless this
function
is
identically
equal
to
zero.
The
eigenvalue problem
(2)
will
be
placed
into
the
Hubert space
L
2
(—
1,
1),
with
the
usual scalar product
to be
denoted
by ( ,
)
0
.
By
introducing
two
operators
A
and
5,
Af=μf(μ)
Bf
=
f(μ)--^
}
f(μ)dμ,
z
-i
where
/
is in
L
2
(—
1,
1),
the
solutions
of Eq. (2) may be
sought
as
eigenfunctions
of
the
eigenvalue problem
Aφ =
v
Bφ
. (3)
Completeness
of
Transport
Eigenfunctions
275
The
operator
A,
multiplication
by the
independent variable
in
L
2
(—
1,1),
is
continuous
and
self adjoint,
and has a
purely continuous spectrum consisting
of
the
closed interval
[—1,1].
The
operator
B is
continuous, one-to-one, and,
for 0 < c <
1,
positive
definite,
since
(Bf,
/)
0
>(1
—
c)(f,
/)
0
.
By
writing
B as E —
cP,
where
E is the
identity
operator
and P the
projection
the
positive square roots
B*
and
B~
^
may be
evaluated
by
means
of the
Neumann
series,
and one
gets
This enables
us to
rewrite
Eq. (3) in the
form
of an
equivalent
self
adjoint
eigenvalue
problem
TΦ
=
vΦ,
(4)
where
and
Φ
=
B^φ.
Explicitly this equation
reads
as
l-c)-*-l]
}
(μ
+
μ')Φ(μ')dμ'
=
vΦ(μ).
(5)
The
continuous
and
self
adjoint operator
T is, as
evident
from
Eq.
(5), equal
to
the sum of the
operator
A and a
completely continuous integral operator.
The
operator
T has the
same continuous spectrum
as the
operator
A,
since this part
of
the
spectrum
is
conserved under completely continuous perturbations
ac-
cording
to the
Weyl-von
Neumann theorem
[8].
The
integral operator
in Rq. (5)
contributes
two
real eigenvalues
to the
spectrum
of T, and it may be
shown that
these eigenvalues
satisfy
the
equation
l-Cvlog--=0.
(6)
To
summarize,
the
spectrum
of T
consists
of a
continuous part, being
the
closed interval
[—1,
1],
and of two
eigenvalues,
to be
denoted
by
v
0
and —
v
0
,
which
are the
roots
of Eq.
(6).
The
eigenfunctions
Φ
vo
and
Φ-
Vo
of T,
which correspond
to
eigenvalues
v
0
and —
v
0
,
are
orthogonal,
and the
orthogonality relation
may be
written
as
(Bφ
VQ
,
φ_
vo
)o
= 0,
where
φ
±VQ
=
B~*Φ
±VQ
are the
eigenfunctions
of the
eigen-
value problem (3).
By
taking into account
Eq.
(3), this relation
may be
written
in
the
familiar
form
of
The
operator
Tdoes
not
have
a
complete
set of
eigenfunction
within
L
2
(—
1,
1).
However, such
a set may be
found
in the
more general setup
of a
rigged Hubert
space.
276 T.
Kline
3.
The
Rigged
Hubert
Space
The
self
adjoint eigenvalue problem
(4)
will
now be
considered
in the
rigged
Hubert space
-UX^MC-U]).
(7)
The
Sobolev space
W^
([—1,1]),
the
positive space
of the
rigged Hubert space (7),
is
a
Hubert space obtained
by
completion
of the
space
C
1
([—
1,
1])
of
functions
with
continuous
first
derivative
on the
closed interval
[— 1, 1], by
means
of the
scalar product
(u,
v)
+
=
(u,
V)Q
+
(u'
9
ι/)
0
,
where
u and v are in
C
1
([—
1,
1]).
The
space
W^
1
([—
1,
1]),
the
negative space
of
the
rigged Hubert space (7),
is
isometric
to the
dual space
of
W^
([
—
1,1]).
It is
obtained
by
completion
of
L
2
(—
1,
1) by
means
of the
sclar
product
[9]
i i
ί ί
-1 -1
(/,£)_=
ί f
G(μ,μ')f(μ)g(μ')dμdμ',
where
/ and g are in
L
2
(—
1,
1), and G is the
Green
function
of the
boundary
value problem
—
u"(μ)
+
u(μ)
=
0,
The
space
W
/
2
1
([—
1,
1])
is
dense
in
L
2
(—
1,
1),
and its
embedding
is
quasi-
nuclear
[10].
The
same holds
for
L
2
(—
1, 1)
with respect
to
PF
2
~
1
([—
1,
1]).
Ac-
cording
to the
Sobolev embedding theorem
[11]
^([—1,
1])
is a
subspace
of
C([—
1,1]),
the
space
of
continuous
functions
on the
closed interval
[—1,1].
The
elements
of
H
7
2
1
([—
1,1])
are
characterized
as
follows
[12]:
a
function
is in
W\
([—1,1])
if and
only
if it is in
L
2
(—
1, 1), and has a
derivative,
in the
generalized
sense
of
Sobolev, which
is in
L
2
(—
1,1).
The
elements
of
W
/
2
~
1
([—
1,
1])
are
generalized
functions
of the
first
order.
Any
continuous linear
functional
on
^(["l*
1])
can
t>
e
uniquely represented
by
some element
α
of
H^"
1
^—
1,
1])
by
means
of the
scalar product
(w,
α)
0
,
where
u is in
VF
2
([—
1,
1]).
The
space
H
/
2
~
1
([—
1,
1)]
contains
the
Dirac
delta-function
δ(μ
— v),
which
is
continuous
with
respect
to v on [— 1, 1], and
whose operation
on
elements
of
W^(\_—
1,
1])
is
given
by
(u,
δ (μ —
v))
0
= u
(v),
ve[—
1,1]
[13].
The
scalar product (δ(μ
— v),
δ(μ—
v'))_
is
equal
to
G(v,
v'),
where
G is the
above mentioned Green's
function.
The
space spanned
by
delta-functions
δ(μ — v),
with
v in a
dense subset
of the
inter-
val
(-1,
1),
is
dense
in
WΓ
*([-!,
1]).
The
space
H^
2
~
1
([-
1,
1])
contains
the set
of
all
real-valued measures
of
bounded variation defined
on the
Borel
subsets
of
the
closed interval
[—
1,
1]
[14].
The
choice
of the
rigged Hubert space
(7) was
motivated
by the
desire
to
keep
it
as
tight
as
possible,
so
that
the
negative space
is
only
as big as
necessary. Since
the
functions
in the
positive space
are
only continuous
on
[—1,
1], the
negative
space does
not
contain generalized
functions
of
higher order, such
as the
derivatives
of
the
delta-function.
Completeness
of
Transport
Eigenfunctions
277
4.
Eigenfunctions
and
Completeness
The
self adjoint operator
T
admits
a
rigged extension
[15]
in the rigged
Hubert space
(7):
it
continuously maps
the
space
W^d—
1,1])
into
itself.
There
exists
the rigged
extension
f of T,
which continuously maps
Wf
1
^—
1?
1])
into
itself,
so
that
for
all
u
in
W\
([-1,1])
and all Φ in
W^
1
([-
1,
1]).
This extension
is
defined
by
)o],
(8)
with
Φ in
W^d
-l,
1]).
Analogously,
the
operators
A,
B,
and
B
±
^
have rigged extensions which
are
given
by
AΦ
=
μΦ
,
Φ-i(l,Φ)o,
(9)
The
operator
f is
then equal
to B
*
AB
%
and the
inverse
of
β**
is
To the
self adjoint operator
T in the
rigged
Hubert space
(7)
then apply
the
completeness
and
eigenfunction expansion theorems
[16],
which
can be
sum-
marized
as
follows.
There exists
a
non-negative
finite
measure
ρ, the
spectral measure
of the
operator
T,
defined
on the
Borel
subsets
of the
real line, with support
on the
spectrum
of T, and
ρ-almost
everywhere there exists
the
operator-valued
function
P(v)
from
W^ft—i,
1])
into
W
2
~
1
(l—i
9
1]),
whose values
are
positive Hubert-
Schmidt operators.
The
operator
P(v)
is the
derivative,
in
terms
of the
Hubert
operator norm,
of the
resolution
of the
identity
E
v
of
T
with respect
to the
measure
ρ.
Operators
P(v)
project
W^d—i,
1])
into
W
2
~
1
([—i,
1]),
and
this projection
is
orthogonal:
if
(P(v)
v
9
u}
Q
= 0 for all v in
W%([-
1,1]),
then P(v)
u = 0.
The
range
of
P(v)
is the
generalized eigenspace
of the
operator
T
corre-
sponding
to the
eigenvalue
v. Its
elements
Φ
v
=
P(v)
v,ve
W\
([—
1,1]),
are
such
that
(Φ
v
,(Γ-vE)ιι)
0
= 0
(10)
for
all u in
W^ίE—1,1]).
Equivalently, they
are
eigenfunctions
of the
rigged
extension
T,
TΦ
v
=
vΦ
v
.
(11)
The set of
eigenvalues
of f is the
spectrum
of T. The
eigenvalues
are
non-
degenerate,
as can be
verified
from
Eq.
(8),
so
that
the
range
of
P(v)
is
one-di-
mensional.
Equation
(11)
has two
continuous solutions
Φ
vo
and
Φ_
vo
which
correspond
to
eigenvalues
v
0
and —
v
0
,
respectively.
For v on
[—1,1]
the
solutions
Φ
v
are
generalized
functions
from
W^
1
^—
1,1]).