SCATTERING THEORY (LAX-PHILLIPS APPROACH)
VADYM ADAMYAN
Abstract. In this lectures we consider the Lax-Phillips approach to the scattering theory originated by problems for the classical wave equation. Despite
of some narrownes of this model based on the classical wave eqution it brought
about a wealth of striking results not only in the quantum theory of scattering
but also in analysis, the theory of stochastic processes, control theory etc.
1. Abstract model
1.1. Scattering operator. Scattering theory compares the asymptotic behavior
of an evolving system in the remote past ( t → −∞) with that in the distant future
( t → +∞). It is well facilitated for studying systems obtained from a simpler one
by imposition of some perturbations provided that the influence of perturbations
on waves or particles at large |t| is negliable in the sense that any motion of the
perturbed system as |t| → ∞ is indistinguishable from a motion of unperturbed
system.
Let U (t) and U0 (t) denote the operators relating the state vectors of perturbed
and unperturbed systems at zero time with their respective states at time t. It
assumed that for each state vector ψ of the perturbed system there are two state
vectors ψ± of unperturbed system such that
U (t)ψ → U0 (±t)ψ± .
t→∞
The scattering operator is defined as the mapping:
S : ψ− → ψ+ .
The aim of scattering theory is to prove the existence of scattering operator and
link its properties to the nature of the perturbation. If the scattering operator
contains the only physically observable data on the evolution of system, then the
main task of scattering theory is reduced to the inverse problem of reconstruction
of the scatterer by scattering data. We introduce the principal notions of scattering
theory by the example of scattering for one-dimensional wave equation.
1.2. Evolution operators U0 (t) for one-dimensional wave equation. The
propagation of elastic, sound and electromagnetic waves in one dimension is one
way or another governed by the wave equation
(1.1)
ψtt = ψxx .
We will consider this equation on the positive half axis x > 0 with the boundary
condition
(1.2)
ψ(0, t) = 0
This text is a synopsis of short lecture course for postgraduates of the Department of Applied
Mathematics of the Polytechnic University of Valencia, Spain given in May of 2006.
1
2
VADYM ADAMYAN
at x = 0 and initial data
(1.3)
ψ(x, 0)l = f1 (x),
ψt (x, 0) = f2 (x).
Multiplying the both sides of ( 1.1) by ψt (x, 0) and integrating by parts one can
verify that the total energy
Z
1 ∞ 2
(1.4)
E [ψ] :=
ψx + ψt2 dx
2 0
is conserved; that is E is independent of time and, as follows
Z
1 ∞ 02
f1 + f22 dx.
(1.5)
E [ψ] =
2 0
Writing ψ in the general solution of the wave equation
(1.6)
ψ(x, t) = u(t + x) + v(t − x)
and choosing u and v to fit the boundary and initial data we get the general solution
of the problem (1.1)-(1.3). The initial data require that
(1.7)
u(x) + v(−x) = f1 (x) (x > 0),
u0 (x) + v 0 (−x) = f2 (x) (x > 0).
(1.7) determines u(x) for x > 0 and v(x) for x < 0 up to a constant, which we are
free to adjust the conditions u(0) = v(0) = 0. The condition (1.2):
(1.8)
u(t) + v(t) = 0
determines u(x) for x < 0 and v(x) for x > 0.
From now on we admit that the functions f1 , f2 may be complex. Writing initial
data {f1 , f2 } as column vector
f1
f=
f2
let us consider the mapping F:
f1
0
→ ϕ(s) = u (−s) =
(1.9)
F
f2
1
2
1
2
[f10 (s) − f2 (s)] , s > 0,
.
[f10 (−s) + f2 (−s)] , s < 0
For initial data {f1 , f2 } with finite energy we have
Z
Z ∞
i
1 ∞h 0 2
2
2
(1.10)
E = E [ψ] =
|f1 | + |f2 | dx =
|ϕ(s)| ds.
2 0
−∞
Hence if we measure the linear set of initial data by the energy norm:
Z
i
1 ∞h 0 2
2
2
kf kE =
|f1 | + |f2 | dx
2 0
the mapping F is isometric operator from the Hilbert space H0E of initial data with
finite energy to the space L2 (−∞, ∞). Actually F is bijective since its range is
L2 (−∞, ∞) and it is invertible.
We denote further by U0 (t) the operator relating the initial data {f1 , f2 } to the
ψ(x, t), −∞ < t < ∞, as follows:
f1
ψ(x, t)
(1.11)
U0 (t)
=
.
f2
ψt (x, t)
SCATTERING THEORY
By (1.6)-(1.9) and (1.11)
f1
(1.12) FU0 (t)
= ϕt (s) =
f2
1
2
3
1
2
[ψs (s, t) − ψt (s, t)] , s > 0,
= ϕ(s − t).
[ψs (−s, t) + ψt (−s, t)] , s < 0
We see that the operator F transforms U0 (t) into the right shift operator
(1.13)
(Rt ϕ) (s) := ϕ(s − t), ϕ ∈ L2 (−∞, ∞).
The operators Rt , −∞ < t < ∞, are unitary and have properties:
Z ∞
2
2
1)Rt1 Rt2 = Rt1 +t2 ; 2) kRt ϕ − ϕk =
|ϕ(s − t) − ϕ(s)| ds → 0.
−∞
t→0
By definition such operators form a group of unitary operators.
Hence the resolving operators U0 (t) in H form a group of unitary operators,
which is isomorphic to the group of right shifts operators Rt .
1.3. Incoming and outgoing representations. The starting point of the LaxPhillips approach to scattering theory is a pair of subspaces D− and D+ associated
with the group of operators U0 (t) . D− consists of the initial data of those solutions
of the wave equation, which are zero in the interval x < −t, t < 0 and D+ consists
of the initial data of those solutions, which are zero in the interval [x < t, t > 0].
These are called incoming and outgoing subspaces. By (1.9)
(1.14)
Observe that
(1.15)
D± = {f : f10 (x) = ∓f2 (x)} .
FD− = ϕ ∈ L2 (−∞, ∞) : ϕ(s) = 0, s > 0 ,
FD+ = ϕ ∈ L2 (−∞, ∞) : ϕ(s) = 0, s < 0 .
We denote the image of D± under U0 (t) by D± (t). By (1.13) and (1.15)
(1) D+ (t) is a decreasing and D− (t) an increasing family of subspaces that is,
for t > s, D+ (t) ⊂ D+ (s) and D− (s) ⊂ D− (t);
(2) As t →= +∞ [−∞], the subspaces D+ (t)[D− (t)] shrink to the null vector;
(3) As t →= −∞ [+∞], the subspaces D+ (t)[D− (t)] fill out the Hilbert
0;
spaceHE
;
(4) D− and D+ are orthogonal to each other.
The subspaces D+ and D− satisfying the above conditions 1) - 4) with respect
to some group of unitary operators {U (t)} are called outgoing and incoming, respectively.
P. Lax and R. Phillips proved that for any pair of such subspaces in some Hilbert
space H it is possible to develop an abstract scattering theory. This may be accomplished in the following way.
Theorem 1.1. (Ja. Sinai) If D+ is an outgoing space for the group {U (t)} in some
Hilbert space H, then H can be represented isometrically as L2 (−∞, ∞; N+ ), where
N+ is some auxiliary Hilbert space, so that U (t) acts as translation to the right by
t and so that D+ can be represented by L2 (0, ∞; N+ ).
Proof. Instead of a one-parameter group of unitary operators we consider first a
single unitary operator U and all of its powers. The subspace D+ is called outgoing
with respect of U if
(1.16)
1) U D+ ⊂ D+ , 2) ∩ U k D+ = {0}, 3) ∪U k D+ = H.
4
VADYM ADAMYAN
For this case L2 (−∞, ∞; N+ ) in the statement of the Sinai theorem should be
replaced by l2 (−∞, ∞; N+ ), the space of sequences {gk ; −∞ < k < ∞} whose values
P
2
lie in an auxiliary Hilbert space N+ and for which
kgk k < ∞, U corresponds to
the right shift and D+ is l2 (−∞, ∞; N+ ).
Let us take as N+ the orthogonal complement of U D+ in D+ ,
N+ = D+ U D+ .
Note that N+ contains non-zero vectors. Otherwise the condition 2) in (1.16) would
be violated. It follows from the condition 1) in (1.16) that U k+1 D+ ⊂ U k D+ and
further from the unitarity of U that
U k N+ = U k D+ U k+1 D+
and the subspaces U k N+ are mutually orthogonal. Let us consider the subspace
X
G=
⊕U k N+ ⊂ D+ .
k≥0
If G is a proper subspace of D+ , there exist a non-zero h ∈ D+  G. But since
h is orthogonal to the all subspaces U k N+ , k ≥ 0, then h ∈ ∩U k D+ = {0}, a
contradiction. Hence G = D+ and the condition 3) in (1.16) implies that
H=
∞
X
⊕U k N+ .
−∞
Therefore each x ∈ H has a unique decomposition of the form
x=
∞
X
⊕U k gk , gk ∈ N+ ,
−∞
and
2
kxkH =
∞
X
2
kgk kN+
−∞
P∞
2
and each sequence of gk ’s from N+ such that −∞ kgk kN+ < ∞ defines in this way
some vector x ∈ H. It is clear that that the mapping x → {gk } is an isometry of
H onto l2 (−∞, ∞; N+ ) under which D+ transforms to l2 (0, ∞; N+ ). Further, it is
clear that
∞
∞
X
X
Ux =
⊕U k+1 gk =
⊕U k gk−1 → {gk−1 },
−∞
−∞
so that U goes under this mapping into the right shift operator. Observe that the
obtained representation is unique up to an isomorphism of N+ .
The analogous representation can be obtained for an incoming subspace D−
defined similarly with property 1) in (1.16) replaced by U −1 D− ⊂ D− and 2),
3) remain unchanged. Note that that if D− [D+ ] is incoming [outgoing], then the
orthogonal complement of D− [D+ ] is outgoing[incoming]. We should only arrange
the mapping so that D− would transform to l2 (−∞, 0; N− ) and U −1 turn into left
shift in l2 (−∞, 0; N− ), U −1 D− ⊂ D− .
We turn now to the representation problem for a strongly continuous oneparameter group {U (t)} which we reduce to the representation of a single unitary
SCATTERING THEORY
5
operator by means of Cayley transform (see below) of the infinitesimal generator
A of the group {U (t)}:
U (t) − I
x.
Ax = lim
t↓0
it
Remind that for any strongly continuous group of unitary operators the infinitesimal
generator A is a self-adjoint operator the domain of which, namely D(A),consists
of all vectors x, for which the above limit exists. As A is a self-adjoint operator the
points ±i belong to its resolvent set. Consequently the Cayley transform
−1
(1.17)
U = (I + iA) (I − iA)
is a mapping with domain and range equal to all of H. Therefore for each x ∈ H
there is y ∈ D(A) such that
(1.18)
x = y − iAy, U x = y + iAy.
Making use of selfadjointness of A we see that
2
2
2
2
kxk = ky − iAyk = kyk − i [(Ay, y) − (y, Ay)] + kAyk =
2
2
2
2
kyk + kAyk = ky + iAyk = kU xk
and since the domain and range of U are equal to all of H it follows that U is
a unitary operator. We will call the Cayley transform of A the co-generator of
the group {U (t)}. The generator A can be recovered from the co-generator U by
equations (1.18):
(1.19)
y=
1
1
(x + U x), Ay = (U x − x).
2
2i
Note that by (1.18) (−1) is not an eigenvalue of U . Therefore we can write A as
the following mapping
−1
A = i (I − U ) (I + U )
with the domain D(A) = (I + U ) H. Note also that the Cayley transform of U −1
is simply −A.
Lemma 1.2. For the group {U (t)} and its co-generator U the following conditions
are equivalent:
a) U (t)D+ ⊂ D+ for all t > 0; b) U D+ ⊂ D+ .
Proof. a) → b) : Making use of the representation
Z ∞
−1
(I − iA) =
e−t U (t)dt
0
we get
Z
−1
(1.20) U x = (I + iA) (I − iA)
x = 2 (I − iA)
−1
x−x =
∞
e−t [2U (t)x − x] dt.
0
If for x ∈ D+ we assume that U (t)x ∈ D+ for all t > 0, then it follows from (1.20)
that U x ∈ D+ .
b) → a) : Let us consider the function
1−z
, t > 0,
et (z) = exp −t
1+z
6
VADYM ADAMYAN
which is analytic and contractive in the disk |z| < 1. As follows it can be represented
as the power series
X
ak (t)z k , |z| < 1,
et (z) =
k=0
which converges uniformly on each circle |z| = r < 1 to the function
X
1 − reiθ
et (reiθ ) =
ak (t)rk eikθ = exp −t
, 0 ≤ θ < 2π.
1 + reiθ
k=0
Let us consider now the operator
(1.21)
et (rU ) :=
∞
X
ak (t)rk U k .
k=0
Applying the spectral theorem for unitary operators yields
Z2π
(1.22)
2
k(U (t) − et (rU )) xk =
wt (reiθ ) − wt (eiθ )2 d (Eθ x, x) , x ∈ H,
0
where Eθ is the spectral function of the unitary operator U . Since
2t
et (reiθ ) − et (eiθ ) =
(1 − r) + O((1 − r)2 ), θ 6= π,
(1.23)
r↑1 |1 + eiθ |2
then et (reiθ ) converges uniformly as r ↑ 1 to et (eiθ ) for each arc
1 + eiθ > ε > 0.
Making use of this fact and taking into account that (−1) is not an eigenvalue of
U we deduce that
(1.24)
limet (rU )x = U (t)x, x ∈ H.
r↑1
If for x ∈ D+ we assume that U k x ∈ D+ for all integer k > 0, then it follows
from (1.21) and (1.24) that U (t)x ∈ D+ for any t > 0.This complete the proof of
the lemma.
Lemma 1.3. A closed subspace D is outgoing (incoming) for a group of unitary
operators {U (t)} if and only if it is outgoing (incoming) for its co-generator U .
Proof. We give the proof only for the outgoing D; the case of incoming D reduces
to the former by time reversal. The equivalence of the properties U (t)D ⊂ D for all
t > 0 and U D ⊂ D is simply the statement of 1.2. In order to prove the equivalence
(1.25)
∩ U (t)D = {0} ∩ U k D = {0}
t≥0
k≥0
we consider the subspaces
M = ∩ U (t)D and M 0 = ∩ U k D.
t≥0
k≥0
k
0
0
Evidently, U (t)M = M for all t and U M = M for all k. But by 1.2 this means
that U (t)M 0 = M 0 for all t and U k M = M for all k. Since M and M 0 are subspaces
of D, we have
M = ∩ U k M ⊂ ∩ U k D = M 0 = ∩ U (t)M 0 ⊂ ∩ U (t)D = M.
k≥0
k≥0
t≥0
t≥0
SCATTERING THEORY
7
Consequently, M = M 0 , which implies (1.25). The property
k
∪U (t)D = H ∪U D = H
t
k
can be verified in a similar way.
The established facts allow us to use the already developed outgoing translation
representation for U for getting of the desired translation representation for {U (t)}.
Note that if D+ is an outgoing subspace with respect to the unitary operator U ,
then H can be represented isometrically as L2 (0, 2π; N+ ) so that U goes into multiplication operator by eiθ and D+ is mapped onto the Hardy subspace H2 (0, 2π; N+ )
of vector functions from L2 (0, 2π; N+ ) whose kth Fourier coefficients vanishes for
all negative k’s.
Indeed, the mapping
{gk } ∈ l2 (−∞, ∞; N+ ) → f (θ) ≡
X
gk eikθ ∈ L2 (0, 2π; N+ )
with
Z 2π
X
1
2
2
2
kf (θ)kN+ dθ =
kgk k = k{gk }kl2
2π 0
defines an isomorphism of l2 (−∞, ∞; N+ ) onto L2 (0, 2π; N+ ) transforming the right
shift operator into multiplication by exp(iθ) and l2 (0, ∞; N+ ) onto H2 (0, 2π; N+ ).
We can get now from the obtained L2 (0, 2π; N+ ) spectral representation of U
the appropriate L2 (−∞, ∞; N+ ) spectral representation of {U (t)} transforming this
group into the group of the multiplication operators by {exp(iλt)}. To this end we
make use of a fractional linear transformation taking the unit disk into the upper
half-plane:
1−w
1 + iz
(1.26)
z=i
w=
1+w
1 − iz
2
kf kL2 =
In particular,
eiθ → λ = i
1 − eiθ
1 + eiθ
and
dθ = 2(1 + λ2 )dλ.
The following mapping
(1.27)
b
f (eiθ ) ∈ L2 (0,
2π; N+ ) → f (λ) =
π −1/2 (1 − iλ)−1 f
1+iλ
1−izλ
∈ L2 (−∞, ∞; N+ )
is an isometry:
1
2π
Z2π
0
iθ 2
f (e ) dθ =
N+
Z∞ b 2
f (λ) dλ.
−∞
N+
The subspaces H2 (0, 2π; N+ ) and H2− (0, 2π; N+ ) := L2 (0, 2π; N+ ) H2 (0, 2π; N+ )
are boundary values in the L2 sense of analytic N+ - valued functions inside (outside) of the unit disk whose square integrals on concentric circles are uniformly
bounded. It follows from this and the explicit form of the mapping (1.27) that the
images H2+ (N+ ) of the subspace H2 (0, 2π; N+ ) ( H2− (0, 2π; N+ ) ) under the mapping
(1.27) is the Hardy subspaces H2+ (N+ ) (H2− (N− ) := L2 (−∞, ∞; N− )  H2+ (N− ))
8
VADYM ADAMYAN
of boundary values of functions fb(λ) analytic in the half-plane Imz > 0
( Imz > 0 ) and such that
Z ∞
Z ∞
2
b
2
sup
kϕ(λ
b − iτ )kN+ dλ < ∞ .
f (λ + iτ ) dλ < ∞ sup
τ >0
−∞
We note that
N+
τ >0
−∞
λt
1 − eiθ
(U (t)f ) (eiθ ) = exp −
t
f (eiθ ) → ei fb(λ)
iθ
1+e
and that for the obtained representation of {U (t)} the image of the subspace D+
is just the Hardy subspace H2+ (N+ ). From such a spectral representation one can
obtain by Fourier transformation
Z ∞
1
b
eiλs fb(λ)dλ, fb ∈ L2 (−∞, ∞; N+ )
F f (s) = √
2π −∞
a translation
so that U (t) acts as the group of the right shifts
representation
iλt b
b
F e f (s) = F f (s − t) in L2 (−∞, ∞; N− ) and D+ is represented according to the Paley-Wiener theorem by the subspace L2 (0, ∞; N− ). This complete the
proof of the Sinai theorem.
An analogous representations holds with respect to D− with L2 (0, ∞; N+ ) replaced by L2 (−∞, 0; N− ) and the Hardy space H2+ (N+ ) replaced by the Hardy
space H2− (N− ) := L2 (−∞, ∞; N− )  H2+ (N− ) in the corresponding spectral representation.
Note that in our example with the one-dimensional wave equation L2 is the usual
space of scalar functions that is dim N+ = dim N+− = 1.
1.4. Scattering operator and matrix. We call the representations described
above for the group {U (t)} and the pair of spaces D− and D+ , the incoming
and outgoing translation representations and the representations obtained from
the by Fourier transformation the incoming and outgoing spectral representations.
It follows from the above property 3) that the dimensions of auxiliary subspaces
N− and N+ entering in the incoming and outgoing representations are equal and
therefore they can be regarded as identical.
Let F+ and F− the operators which assign to a given f ∈ H its outgoing and
incoming representer, respectively. The operator
(1.28)
S = F+ F−1
−
is called abstract scattering operator associated with the group {U (t)} and the
pair of subspaces D+ and D− .
Theorem 1.4. The operator S defined as in (1.28)
a) is unitary,
b) commutes with translations,
c) maps L2 (−∞, 0; N) into itself.
Proof. Since F± are isometric mappings of H onto L2 (−∞, ∞; N), then S is a
unitary operator in L2 (−∞, ∞; N). b) follows because F+ and F− both have the
property
F± U (t) = Rt F±
SCATTERING THEORY
9
−1
−1
and hence SRt = F+ F−1
− Rt = F+ U (t)F− = Rt F+ F− . c) follows from the orthog−1 2
onality of subspaces D± because F− L (−∞, 0; N) = D− and
2
2
SL2 (−∞, 0; N) = F+ F−1
− L (−∞, 0; N) = F+ D− ⊂ F+ (HD+ ) = L (−∞, 0; N).
Go over now to the spectral representation by applying the Fourier transformation F . The scattering operator goes into S = F SF −1 and Theorem 1.4 implies
the following for S:
a0 ) S is unitary.
b0 ) S commutes with multiplication by scalar functions.
c0 ) S maps the Hardy space H2− (N) into itself.
An operator with the properties a0 ) - c0 ) can be represented as a multiplication
by operator valued function S(λ) mapping N into N and satisfying the conditions:
a00 ) S is the boundary value of an operator valued function S(z) analytic for
Imz < 0.
b00 ) kS(z)k ≤ 1 for all z with Imz < 0.
c00 ) S(k) is unitary for almost all real λ.
S(z) is celebrated Heisenberg scattering matrix.
1.5. A semigroup of contraction operators related to a scattering system. There is a relation between the asymptotic of solutions of a perturbed system
and the analytic continuation of the corresponding scattering matrix to the complex plain. Let us consider the following set of operators pertaining to the local
properties of solutions:
(1.29)
T (t) := P+⊥ U (t)P−⊥ (t ≥ 0),
⊥
where P±⊥ = I − P± are orthogonal projectors in H onto subspaces D±
= H  D± ,
respectively, P+ and P− are orthogonal projections onto subspaces D+ and D− and
I is the unity operator. Evidently,
(1.30)
kT (t)k ≤ P+⊥ kU (t)k P−⊥ = 1.
It follows from properties of D+ and D− that T (t), t ≥ 0, annihilates D+ and D− .
Indeed, if f ∈ D+ , then f ⊥ D− and therefore P−⊥ f = f . Besides, U (t)f ∈ D+ for
t ≥ 0. Hence
T (t)f = P+⊥ U (t)P−⊥ f = P+⊥ U (t)f = P+⊥ P+ U (t)f = 0.
For D− the corresponding assertion is obvious. Note also that
(1.31)
P+⊥ U (t)P+ = P− U (t)P−⊥ = 0.
Let K denote the subspace H  [D+ ⊕ D− ] and let PK be the orthogonal projector
onto K. By definition,
(1.32)
P− + PK + P+ = I, P+⊥ = P− + PK , P−⊥ = PK + P+ .
Applying (1.31) and (1.32) yields
(1.33)
T (t) = (P− + PK ) U (t)P−⊥ = PK U (t)P−⊥ ,
T (t) = P+⊥ U (t) (P+ + PK ) = P+⊥ U (t)PK .
10
VADYM ADAMYAN
Besides, for s, t ≥ 0 by (1.33) we have
(1.34)
T (t)T (s) = P+⊥ U (t)PK U (s)P−⊥ = P+⊥ U (t)U (s)P−⊥ −
+ P+ ) U (s)P−⊥ = P+⊥ U (t + s)P−⊥ = T (t + s).
P+⊥ U (t) (P−
We conclude from (1.30), (1.33) and (1.34) for t ≥ 0 that
1) {T (t)} forms a one-parameter semigroup of operators mapping K into K;
2) T (t) is a contraction operator, that is kT (t)k ≤ 1.
By our assumption subspaces D+ (−t) for t > 0 dilate and as t → ∞ they fill
out the whole space H. Taking into account that U (−t)P+ U (t) : P+ (−t) is the
orthogonal projector onto subspace D+ (−t) we see that for any f ∈ K
2
2
kT (t)f k = P+⊥ U (t)f = ((I − P+ ) U (t)f, U (t)f ) =
2
2
2
kf k − (U (−t)P+ U (t)f, f ) = kf k − kP+ (−t)f k .
2
2
But by our assumption lim kP+ (−t)f k = kf k for any f ∈ H. Hence
t→∞
(1.35)
2
2
2
lim kT (t)f k = kf k − lim kP+ (−t)f k = 0, f ∈ K.
t→∞
t→∞
We proved that T (t) tends strongly to zero as t → ∞.
Let us consider the action of T (t) in the outgoing spectral representation where
U (t) corresponds to right translation by t and P+⊥ corresponds restriction to the
negative half-axis. The action of T (t) is translation followed by restriction to the
⊥
image of D+
. In this representation D+ corresponds to L2 (0, ∞; N) so that its
orthogonal complement is represented by L2 (−∞, 0; N). As we showed D− is represented by SL2 (−∞, 0; N) and therefore
(1.36)
K = L2 (−∞, 0; N)  SL2 (−∞, 0; N).
Since T (t) maps K onto itself, then the subspace representing K in the translation
representation is invariant under the right shift followed by translation.
In the outgoing spectral representation K corresponds to the subspace
(1.37)
K =H2− (N) SH2− (N).
Note that S is a unitary in L2 (−∞, ∞; N) because it is the multiplication operator
by the unitary valued operator function. Therefore subspaces SH2− (N) and SH2+ (N)
are orthogonal. As S(λ) is the boundary value of analytic i the half-plane Imz < 0
operator function S(z) then the operator function S(λ)∗ = S(λ)−1 is the boundary
value of analytic in the half-plane Imz > 0 operator function S(z)∗ . According
to (1.37) S ∗ K =S ∗ H2− (N) H2− (N). As follows K consists of those and only those
vector functions fb ∈ H2− (N) for which S(λ)∗ fb(λ) are functions from H2+ (N).
The spectral structure of semigroup of contraction operators{T (t)} are intimately connected with analytic properties of the scattering matrix S(λ). We touch
on this problem using the representation of K in the outgoing spectral representation
of {U (t)}.
Let B be the infinitesimal generator of {T (t)},
1
Bf = lim [T (t)f − f ] .
it
t↓0
Remind that the resolvent set of a closed linear operator A in a Hilbert space H
consists of those complex numbers z for which A−zI, where I is the unity operator,
has a bounded inverse. Other points of the complex plain form the spectrum of
SCATTERING THEORY
11
A. In particular, z0 belongs to the point spectrum of A if the null-space of A − zI
contains non-zero vectors. By the formula
Z∞
−1
(1.38)
(B − zI) f = −i e−izt T (t)f dt, Imz < 0, f ∈ K,
0
the resolvent set of B contains the lower half-plane. (This follows also from the
fact that the infinitesimal generator B of any contractive semigroup {T (t)} has the
property: Im(Bf, f ) ≥ 0 for any f ∈ H.) We can get the spectral representation
of (B − zI)−1 f for Imz < 0 taking into account that in the outgoing spectral
representation fb ∈ K ⊂H2− (N), U (t) is the multiplication operator by eiλt and
T (t)f = P+⊥ U (t)f, t > 0. We see from (1.38) that the image of (B − zI)−1 f for
Imz < 0 in the outgoing spectral representation is the projection onto H2− (N ) of the
vector function
1 b
f (λ),
λ−z
that is
i
1 hb
\
(1.39)
(B −
zI)−1 f (λ) =
f (λ) − fb(z) , f ∈ K, Imz < 0.
λ−z
From (1.39) and evident relations
h
i
1
gb(λ) = λ−z
fb(λ) − fb(z) , f ∈ K, Imz < 0
(1.40)
c
Bg(λ)
= λb
g (λ) + fb(z),
we get the following
Proposition 1.5. The domain of B in K consists of those and only those vector
functions gb(λ) for which there is a (unique) vector h ∈ N such that λb
g (λ) + h
2
c
belongs to L (−∞, ∞; N ) and Bg(λ) is just λb
g (λ) + h.
Contrary to the lower half-plane the upper half-plane contain the point spectrum
of B.
Theorem 1.6. If Imz0 > 0, then z0 belongs to the point spectrum of B if and only
if S(z0 )∗ has a non-trivial null-space.
Proof. Let gb ∈ K be an eigenfunction of B:
c
Bg(λ)
= z0 gb(λ), Imz0 > 0.
By 1.5 there is a non-zero h ∈ N such that
1
gb(λ) = −
h ∈ H2− (N ) ∩ K.
λ − z0
It follows from the outgoing spectral representation K of K that at the same time
1
(1.41)
S(λ)∗ gb(λ) = −
S(λ)∗ h ∈ H2+ (N ).
λ − z0
Since S(λ)∗ is the boundary value of the analytic in the upper half-plane operator
function S(z)∗ , (1.41) holds if and only if S(z0 )∗ h = 0. This proves the theorem
and a little more:
Corollary 1.7. dim [null − space of (B − z0 )] = dim [null − space of S(z0 )∗ )] .
12
VADYM ADAMYAN
2. One-dimensional perturbed wave equation
We now perturb the concerned wave equation replacing (1.1) by
(2.1)
ψtt = ψxx − q(x)ψ, x ≥ 0,
where q(x) ≥ 0 is a non-negative continuous function such that q(x) = 0 for x >
a > 0.
We will consider this equation as before with the boundary condition
(2.2)
ψ(0, t) = 0
and initial data
(2.3)
ψ(x, 0)l = f1 (x),
ψt (x, 0) = f2 (x).
In this case we have the following expression for the total energy
Z
Z
1 ∞ 2
1 a
(2.4)
E [ψ] :=
ψx + ψt2 dx +
q(x)ψ 2 dx.
2 0
2 0
E [ψ] is conserved in time and since it is a positive functional on solutions of (2.1)
satisfying (2.2). Those solutions are uniquely determined by the initial data. We
denote by {U (t)} the perturbed group of operators relating initial data to data
at any time t. The group {U (t) acts in the Hilbert space HE of initial data f =
{f1 , f2 } measured by the norm
Z
Z
i
1 ∞h 0 2
1 a
2
2
2
(2.5)
kf kE =
|f1 | + |f2 | dx +
q(x) |f1 | dx.
2 0
2 0
As incoming and outgoing subspaces for the perturbed group{U (t)} we take the
initial data of those solutions of the wave equation (2.1) which are zero in the
interval x < −t+a, t < 0 [x < t+a, t > 0]. Since equations (1.1) and (2.1) coincide
a
= U0 (±a)D±
for x > a, actually these subspace coincide with the subspaces D±
introduced above for the unperturbed wave equation (1.1). Observe that
a
U (±t)f = U0 (±t)f, f ∈ D±
, t > 0,
0
and that the metrics of spaces HE and HE
coincide on the linear sum of subspaces
a
a
a
D± . As follows, the subspaces D± (t) = U (t)D±
satisfy conditions
a
a
(1) D+ (t) is a decreasing and D− (t) an increasing family of subspaces that is,
a
a
a
a
(t);
(s) ⊂ D−
for t > s, D+
(t) ⊂ D+
(s) and D−
a
a
(t)] shrink to the null vector;
(2) As t →= +∞ [−∞], the subspaces D+ (t)[D−
a
a
(3) D− and D+ are orthogonal to each other.
It can be proved also that for non-negative potentials q(x) that the subspaces
a
a
D+
(t)[D−
(t)] fill out the Hilbert space HE as t →= −∞ [+∞]. So the perturbed
wave equation with these incoming and outgoing subspaces falls into a pattern
of the Lax-Phillips model. Our next goal is to construct the outgoing spectral
representation for {U (t)} and calculate the scattering matrix for the equation (2.1).
In accordance with general spectral theory the spectral representation of a given
f1
f=
∈ HE
f2
ought to be given as a scalar product in HE with eigenfunction or generalized
eigenfuction eλ of {U (t)},
(2.6)
fb(λ) = (f, eλ )HE ,
SCATTERING THEORY
where eλ satisfies the eigenvalue equation
(2.7)
(U (t)eλ ) (x) = e
In our case
iλt
eλ1 (x)
eλ2 (x)
13
.
ψλ (x, t)
,
(U (t)eλ ) (x) =
∂t ψλ (x, t)
where ψλ (x, t) satisfies the equation (2.1), boundary condition (2.2) and the initial
data
ψλ (x, 0) = eλ1 (x), ∂t ψλ (x, 0) = eλ2 (x).
Hence, eλ (x) is a (generalized) eigennfunction of the following eigenvalue boundary
problem:

eλ2 (x) = iλeλ1 (x),

e00λ1 (x) − q(x)eλ1 (x) = iλeλ2 (x),
(2.8)

eλ1 (0) = 0,
or
(2.9)
−e00λ1 (x) + q(x)eλ1 (x) = λ2 eλ2 (x), eλ1 (0) = 0.
Thus
(2.10)
eλ (x) =
1
iλ
D(λ)χ(x, λ),
where χ(x, λ) satisfies the system (2.9) and the additional condition χ0 (0, λ) = 1
and D(λ) is so far an indefinite coefficient. Note that for x ∈ (a, ∞), where q(x) = 0
the solution (2.10) can be represented in the form
1
(2.11)
eλ (x) =
D(λ) C+ (λ)eiλx + C− (λ)e−iλx
iλ
with coefficients
(2.12)
1 −iλa
C+ (λ) = 2iλ
e
[χ0 (a, λ) + iλχ(a, λ)] ,
1 iλa
C− (λ) = 2iλ e [−χ0 (a, λ) + iλχ(a, λ)]
found from the continuity conditions at x = a. We see that the the spectral
representation fb(λ) of any f ∈ HE and such that f (x) = 0 for x < a has the form
R∞
fb(λ) = −iλD(λ)C+ (λ) [−f10 (x) + f2 (x)] e−iλx dx−
a
(2.13)
R∞ 0
iλD(λ)C− (λ) [f1 (x) + f2 (x)] eiλx dx.
a
The first item in the right hand side of (2.13) depends only on the incoming component of f , while the second item depends only on the outgoing component of f .
a
a
The the restriction of the mapping (2.13) on D+
becomes an isometry of D+
onto
2
the Hardy space H+ if we put everywhere
(2.14)
D(λ) = √
1
2π2iλC− (λ)
a
Under the same mapping D−
is presented as the subspace SH2− , where S is the
multiplication operator by the function
χ0 (a, λ) − iλχ(a, λ)
(2.15)
S(λ) = 0
.
χ (a, λ) + iλχ(a, λ)
14
VADYM ADAMYAN
Remind that χ(a, λ) and χ0 (a, λ) are real valued functions of λ on the real axis and
they can be extended to the complex plane as entire functions of exponential type.
One can verify that the function S(λ) can be extended to the lower half-plane as
an analytic function satisfying the condition: |S(z)| < 1, Imz < 0. By definition
S(λ) is the scattering matrix for the perturbed wave equation.
For details and further reading I recommend the monograph:
P. Lax and R. Phillips. Scattering Theory (Pure and applied mathematics; v.
26), Academic Press, Inc. (1989)
Department of Theoretical Physics, Odessa National University, 65026 Odessa,
Ukraine
E-mail address: [email protected]