Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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u
0
(t) = exp (iH
0
t )f
0
. Equation [2] leads to a
connection between the corresponding initial data
f
0
and f given by
f ¼ lim
t!1
expðiHtÞexpðiH
0
tÞf
0
½3
If f is an eigenvector of H, that is, Hf = f , then
obviously u(t) = e
it
f . On the contrary, if f belongs
to the (absolutely) continuous subspace of H, then
necessarily u(t) has the free asymptotics as t !1.
This result is known as asymptotic completeness.
The Schro¨ dinger operator H = þ V(x)inthe
space H= L
2
(R
d
) with a real potential V decaying
at infinity is a typical Hamiltonian of scattering
theory. The operator H describes a particle in an
external potential V or two interacting particles.
Asymptotically (as t !þ1 or t !1), particles
may either form a bound state or be free (a
scattering state). Of course, a bound (scattering)
state at 1 remains the same at þ1.Tobemore
precise, suppose that
jVðxÞj Cð1 þjxjÞ
½4
where >1. Then relation [2] can be justified with
the kinetic energy operator H
0
= playing the
role of the unperturbed operator.
As discussed in Landau and Lifshitz (1965) (see
also Amrein et al. (1977), Pearson (1988),andYafaev
(2000)), in scattering experiments one sends a beam
of particles of energy >0 in a direction !.Sucha
beam is described by the plane wave
0
ðx; !; Þ¼expðikh!; xiÞ;¼ k
2
> 0
(which satisfies of course the free equation
0
=
0
). The scattered particles are described
for large distances by the outgoing sph erical wave
að
^
x;!; Þjxj
ðd1Þ=2
expðikjxjÞ
Here
^
x = xjxj
1
is the direction of observation and
the coefficient a(
^
x, !; ) is known as the scattering
amplitude. This means that quantum particles
subject to a potential V(x) are described by the
solution of eqn [5] with asymptotics [6] at infinity:
þ VðxÞ ¼ ½5
ðx; !; Þ¼expðikh!; xiÞ
þ að
^
x;!; Þjxj
ðd1Þ=2
expðikjxjÞ
þ o jxj
ðd1Þ=2
½6
The existence of such solutions requires of course a
proof. The differential scattering crosssection
defined by eqn [7] gives us the part of particles
scattered in a solid angle d
^
x:
dð
^
x;!; Þ¼jað
^
x;!; Þj
2
d
^
x ½7
As discussed below, the temporal asymptotics
of solutions of the time-dependent Schro¨ dinger
equation [1] are closely related to the asymptotics
at large distances of solutions of the stationary
Schro¨ dinger equation [5].
Time-Dependent Scattering Theory
and Møller Operators
If V(x) !0asjxj!1, then the essential spectrum
of the Schro¨ dinger operator H = þ V(x) covers
the whole positive half-line, whereas the negative
spectrum of H consists of eigenvalues accumulating,
perhaps, at the point zero only.
Scattering theory requires a more advanced
classification of the spectrum based on measure
theory. Consider a self-adjoint operator H defined
on domain D(H) in a Hilbert space H. Let E be its
spectral family. Then the space H can be decom-
posed into the orthogonal sum of invariant sub-
spaces H
(p)
, H
(sc)
and H
(ac)
. The subspace H
(p)
is
spanned by eigenvectors of H and the subspaces
H
(sc)
, H
(ac)
are distinguished by the condition that
the measure (E(X)f , f ) (here X R is a Borel set) is
singularly or absolutely continuous with respect to
the Lebesgue measure for all f 2H
(sc)
or f 2H
(ac)
.
Typically (in applications to quantum-mechanical
problems) the singularly continuous part is absent,
that is, H
(sc)
= {0}. We denote by H
(ac)
the restriction
of H on its absolutely continuous subspace H
(ac)
and
by P
(ac)
the orthogonal projection on this subspace.
The same objects for the operator H
0
will be
endowed with the index ‘‘0.’’
Equation [3] motivates the following fundamental
definition. The wave, or Møller, operator
W
= W
(H, H
0
) for a pair of self-adjoint operators
H
0
and H is defined by eqn [8] provided that the
corresponding strong limit ex ists:
W
¼ s-lim
t!1
expðiHtÞexpðiH
0
tÞP
ðacÞ
0
½8
The wave operator is isometric on H
(ac)
0
and enjoys
the intertwining property
HW
¼ W
H
0
½9
Therefore, its range Ran W
iscontainedinthe
absolutely continuous subspace H
(ac)
of the operator H.
The operator W
(H, H
0
) is said to be complete if
eqn [10] holds:
Ran W
ðH; H
0
Þ¼H
ðacÞ
½10
252 Quantum Mechanical Scattering Theory
It is easy to see that the completeness of W
(H, H
0
)
is equivalent to the existence of the ‘‘inverse’’ wave
operator W
(H
0
, H). Thus, if the wave operator
W
(H, H
0
) exists and is complete, then the opera-
tors H
(ac)
0
and H
(ac)
are unitarily equivalent. We
emphasize that scattering theory studies not arbi-
trary unitary equivalence but only the ‘‘canonical’’
one realized by the wave operators.
Along with the wave operators an important role
in scattering theory is played by the scattering
operator defined by eqn [11] where W
þ
is the
operator adjoint to W
þ
:
S ¼ SðH; H
0
Þ¼W
þ
ðH; H
0
ÞW
ðH; H
0
Þ½11
The operator S commutes with H
0
and hence
reduces to multiplication by the operator function
S() = S(; H, H
0
) in a representation of H
(ac)
0
which
is diagonal for H
(ac)
0
. The operator S() is known as
the scattering matrix. The scattering operator [11] is
unitary on the subspace H
(ac)
0
provided the wave
operators W
(H, H
0
) exist and are complete. The
scattering operator S(H, H
0
) connects the asympto-
tics of the solutions of eqn [1] as t !1 and as
t !þ1 in terms of the free problem, that is
S(H, H
0
):f
0
7!f
þ
0
, where f
0
are the same as in eqn
[2]. The scattering operator and the scattering
matrix are usually of great interest in mathematical
physics problems, because they connect the ‘‘initial’’
and the ‘‘final’’ characteristics of the process
directly, bypassing its consideration for finite times.
The definition of the wave operators can be
extended to self-adjoint operators acting in different
spaces. Let H
0
and H be self-adjoint operators in
Hilbert spaces H
0
and H, respectively, and let
‘‘identification’’ J : H
0
!H be a bounded operator.
Then the wave operator W
= W
(H, H
0
; J) for the
triple H
0
, H, and J is defined by eqn [12] provided
again that the strong limit there exists:
W
¼ s-lim
t!1
expðiHt ÞJ expðiH
0
tÞP
ðacÞ
0
½12
Intertwining property [9] is preserved for wave
operator [12]. This operator is isometric on H
(ac)
0
if
and only if
lim
t!1
kJ expðiH
0
tÞf
0
k¼kf
0
k
for all f
0
2H
(ac)
0
. Since
s-lim
jtj!1
K expði H
0
t ÞP
ðacÞ
0
¼ 0
for a compact operator K, wave operators [12]
corresponding to identifications J
1
and J
2
coincide if
J
2
J
1
is compact or, at least, the operators (J
2
J
1
)
E
0
(X) are compac t for all bounded intervals X.
Consideration of wave operators [12] with J 6¼ I
may of course be of interest also in the case H
0
= H.
It suffices to verify the existence of limits [8] or
[12] on some set dense in the absolutely continuous
subspace H
(ac)
0
of the operator H
0
. The following
simple but convenient condition for the existence of
wave operators is usually called Cook’s criterion.
Suppose that H
0
= H
(ac)
0
and that the operator J
maps domain D(H
0
) of the operator H
0
into D(H).
Let
Z
1
0
kðHJ JH
0
ÞexpðiH
0
tÞf kdt < 1
for all f from some set D
0
D(H
0
) dense in H
0
.
Then the wave operator W
(H, H
0
; J) exists.
This result is often useful in applications since the
operator exp (iH
0
t) is known explicitly. For
example, it works with J = I for the pair
H
0
¼; H ¼ H
0
þ VðxÞ½13
if V(x) satisfies estimate [4] with >1. On the
other hand, different proofs of the existence of the
wave operators W
(H
0
, H; J
) require new mathe-
matical tools. There are two essentially different
approaches in scattering theory: the trace-class and
smooth methods.
Time-Independent Scattering Theory
The approach in scattering theory relying on
definition [8] is called time dependent. An alter-
native possibility is to change the definition of wave
operators replacing the unitary groups by the
corresponding resolvents R
0
(z) = (H
0
z)
1
and
R(z) = (H z)
1
. They are related by a simple
identity
RðzÞ¼R
0
ðzÞR
0
ðzÞVRðzÞ
¼ R
0
ðzÞRðzÞVR
0
ðzÞ½14
where V = H H
0
and Im z 6¼ 0. In the stationary
approach in place of limits [8] one has to study
the boundary values (in a suitable topology) of the
resolvents as the spectral parameter z tends to the
real axis. An important advantage of the stationary
approach is that it gives convenient formulas for the
wave operators and the scattering matrix.
Let us discuss here the stationary formulation of
the scattering problem for operators [13] in the
Hilbert space H= L
2
(R
d
)intermsofsolutionsofthe
Schro¨ dinger equation [5].IfV(x) satisfies estimate [4]
with >(d þ 1)=2, then for all >0 and all unit
vectors ! 2 S
d1
, eqn [5] has the solution (x; !, )
with asymptotics [6] as jxj!1. Moreover, the
scattering amplitude a(
^
x, !; ) belongs to the space
Quantum Mechanical Scattering Theory 253
L
2
(S
d1
) in the variable
^
x uniformly in ! 2 S
d1
,and
it can be expressed via (x; !, ) by the formula
að; !; Þ¼
d
ðÞ
Z
R
d
e
ikh;xi
VðxÞ ðx; !; Þdx
where
d
ðÞ¼e
iðd3Þ=4
2
1
ð2Þ
ðd1Þ=2
ðd3Þ=4
Let us define two sets of scattering solutions, or
eigenfunctions of the continuous spectrum, by the
formulas
ðx; !; Þ¼ ðx; !; Þ and
þ
ðx; !; Þ
¼
ðx; !; Þ
In terms of boundary values of the resolvent, the
functions
(!, ) can be constructed by the formula
ð!; Þ¼
0
ð!; ÞRð i0ÞV
0
ð!; Þ½15
Obviously, functions [15 ] satisfy eqn [5]. Using
resolvent identity [14], it is easy to derive the
Lippmann–Schwinger equation
ð!; Þ¼
0
ð!; ÞR
0
ð i0ÞV
ð!; Þ
for
(!, ). Asymptotics [6] can be deduced from
the formula
ðR
0
ð i0Þf ÞðxÞ¼c
ðÞð
0
ðÞf Þð
^
xÞjxj
ðd1Þ=2
expðikjxjÞ þ Oðjxj
ðdþ1Þ=2
Þ
where f 2 C
1
0
(R
d
), c
() =
1=2
1=4
e
i(d3)=4
and
the operator
0
() defined by eqn [16] is (up to the
numerical factor) the restriction of the Fourier
transform
^
f = F
0
f onto the sphere of radius
1=2
:
ð
0
ðÞf Þð!Þ¼2
1=2
ðd2Þ=4
^
f ð
1=2
!Þ;!2S
d1
½16
The wave operators W
(H,H
0
) can be con-
structed in terms of the solutions
. Set =
1=2
!
( is the momentum variable), write
(x,) instead
of
(x;!,), and consider two transformations
ðF
f ÞðÞ¼ð2Þ
d=2
Z
R
d
ðx;Þf ðxÞdx ½17
(defined initially, e.g., on the Schwartz class S(R
d
))
of the space L
2
(R
d
) into itself. The operators F
can
be regarded as generalized Fourier transforms, and
both of them coincide with the usual Fourier
transform F
0
if V = 0. It follows from eqns [5],
[17] that under the action of F
the operator H goes
over into multiplication by jj
2
, that is,
ðF
Hf ÞðÞ¼jj
2
ðF
f ÞðÞ
Moreover, with the help of eqn [15], it can be
shown that F
is an isomet ry on H
(ac)
, it is zero on
HH
(ac)
, and its range Ran F
= L
2
(R
d
). This is
equivalent to eqns [18]:
F
F
¼ P
ðacÞ
; F
F
¼ I ½18
Hence any function f 2H
(ac)
admits the expansion
in the generalized Fourier integral
f ðxÞ¼ð2Þ
d=2
Z
R
d
ðx;ÞðF
f ÞðÞd
It can also be deduced from eqn [6] that the vector
F
F
0
expðijj
2
tÞ
^
f
tends to zero as t !1for an arbitrary
^
f 2 L
2
(R
d
).
This implies the existence of the wave operators
W
= W
(H, H
0
) for pair [13] and gives the
representation
W
¼F
F
0
½19
Completeness of W
follows from eqn [19] and
the f irst equation in [18]. The second equality in
[18] is equivalent to the isometricity of W
.
Formula [19] is an example of a stationary
representation for the wave operator. It f ormally
implies that
W
0
ð!; Þ¼
ð!; Þ
which means that each wave operator establishes a
one-to-one correspondence between eigenfunctions of
the continuous spectrum of the operators H
0
and H.
The main ideas of the stationary approach go
back to Friedrichs (1965), and Povzner. The inverse
problem of reconstruct ion of a potential V given the
scattering amplitude a (see eqn [6])istreatedin
Faddeev (1976).
The Trace-Class Method
Recall that the class S
p
, p 1, consists of compact
operators T such that the norm
kTk
p
¼
X
n
p
n
ðjTjÞ
!
1=p
; jTj¼ðT
TÞ
1=2
is finite. Eigenvalues
n
(jTj) =: s
n
(T) of a non-
negative operator jTj are called singular numbers
of T . In particular, S
1
is the trace class and S
2
is the
Hilbert–Schmidt class.
The trace-class method (see Reed and Simon (1976)
or Yafaev (1992) for a detailed presentation) makes no
assumptions about the ‘‘unperturbed’’ operator H
0
.Its
basic result is the following theorem of Kato and
Rosenblum. If V = H H
0
belongs to the trace class
254 Quantum Mechanical Scattering Theory
S
1
, then the wave operators W
(H, H
0
)existandare
complete. In particular, the operators H
(ac)
0
and H
(ac)
are unitarily equivalent. This can be considered as a far
advanced extension of the H Weyl theorem, which
states the stability of the essential spectrum under
compact perturbations.
The condition V 2 S
1
in the Kato–Rosenblum
theorem cannot be relaxed in the framework of
operator ideals S
p
. This follows from the Weyl–von
Neumann–Kuroda theorem. Let H
0
be an arbitrary
self-adjoint operator. For any p > 1 and any ">0
there exists a self-adjoint operator V such that V 2 S
p
,
kVk
p
<" and the operator H = H
0
þ V has purely
point spectrum. Of course, such an operator H has no
absolutely continuous part. At the same time, the
operator H
0
may be absolutely continuous. In this
case, the wave operators W
(H, H
0
) do not exist.
Although sharp in the abstract framework, the
Kato–Rosenblum theorem cannot directly be applied
to the theory of differential operators where a
perturbation is usually an operator of multiplication
and hence is not even compact. We mention its two
generalizations applicable to this theory. The first,
the Birman–Kato–Kre
ın theorem, claims that the
wave operators W
(H, H
0
) exist and are complete
provided
R
n
ðzÞR
n
0
ðzÞ2S
1
for some n = 1, 2, ... and all z with Im z 6¼ 0. The
second, the Birman theorem, asserts that the same is
true if D(H) = D(H
0
)orD(jHj
1=2
) = D(jH
0
j
1=2
) and
EðXÞðH H
0
ÞE
0
ðXÞ2S
1
for all bounded intervals X.
The wave operators enjoy the following property
known as the Birman invariance principle. Suppose
that ’(H) ’(H
0
) 2 S
1
for a real function ’ such
that its derivative ’
0
is absolutely continuous and
’
0
() > 0. Then the wave operators W
(H, H
0
) exist
and eqn [20] holds:
W
ðH; H
0
Þ¼W
ð’ðHÞ;’ðH
0
ÞÞ ½20
A direct generalization of the Kato–Rosenblum
theorem to the operators acting in different spaces is
due to Pearson. Suppose that H
0
and H are self-
adjoint operators in spaces H
0
and H, respectively,
J : H
0
!H is a bounded operator and V = HJ
JH
0
2 S
1
. Then the wave operators W
(H, H
0
; J)
and W
(H
0
, H; J
) exist.
Although rather sophisticated, the proof relies
only on the following elementary lemma of Rosen-
blum. For a self-adjoint ope rator H, consider the set
R H
(ac)
of elements f such that
r
2
H
ðf Þ :¼ ess sup dðEð Þf ; f Þ=d<1
If K : H!G (G is some Hilbert space) is a Hilbert–
Schmidt operator, then for all f 2 R
Z
1
1
kK expðiHtÞf k
2
dt 2r
2
H
ðf ÞkKk
2
2
½21
Moreover, the set R is dense in H
(ac)
.
The Pearson theorem allows to simplify consider-
ably the original proofs of different generalizations
of the Kato–Rosenblum theorem.
A typical application of the trace-class theory is
the following result. Suppose that
H¼L
2
ðR
d
Þ; H
0
¼ þV
0
ðxÞ; H ¼H
0
þVðxÞ½22
where the functions V
0
and V are real, V
0
2L
1
(R
d
)
and V satisfies estimate [4] for so me >d. Then the
wave operators W
(H,H
0
) exist and are complete.
The Smooth Method
The smooth method (see Kuroda (1978), Reed and
Simon (1979),orYafaev (1992), for a detailed
presentation) relies on a certain regularity of the
perturbation in the spectral representation of the
operator H
0
. There are different ways to understand
regularity. For example, in the Friedrichs–Faddeev
model H
0
acts as multiplication by independent
variable in the space H= L
2
(; N ), where is an
interval and N is an auxiliar y Hilbert space. The
perturbation V is an integral operator with suffi-
ciently smooth kernel.
Another possibility is to use the concept of H-
smoothness introduced by Kato. An H-bounded
operator K is called H-smooth if, for all f 2D(H),
Z
1
1
kK expðiHtÞf k
2
dt Ckf k
2
½23
(cf. eqns [21] and [23]). Here and below, C are different
positive numbers whose precise values are inessential.
It is important that this definition admits equivalent
reformulations in terms of the resolvent or of the
spectral family. Thus, K is H-smooth if and only if
sup
2R;">0
kKðRð þ i"ÞRð i "ÞÞK
k < 1
or if and only if
sup jXj
1
kKEðXÞk
2
< 1
for all intervals X R.
In applications the assumption of H-smoothness
of an operator K imposes too stringent conditions
on the operator H. In particular, the operator H is
necessarily absolutely continuous if kernel of K is
trivial. This assumption excludes eigenvalues and
other singular points in the spectrum of H, for
Quantum Mechanical Scattering Theory 255
example, the bottom of the continuous spectrum for
the Schro¨ dinger operator with decaying potential or
edges of bands if the spectrum has the band
structure. The notion of local H-smoothness sug-
gested by Lavine is considerably more flexible. By
definition, K is called H-smooth on a Borel set X R
if the operator KE(X)isH-smooth. Note that, under
the assumption
sup
2X;">0
kKðRð þ i"ÞRð i"ÞÞK
k < 1½24
the operator K is H-smooth on the closure of X.
The following Kato–Lavine theorem is simple but
very useful. Suppose that
HJ JH
0
¼ K
K
0
where the operators K
0
and K are H
0
-smooth and
H-smooth, respectively, on an arbitrary compact
subinterval of some interval . Then the wave
operators
W
ðH; H
0
; JE
0
ðÞÞ and W
ðH
0
; H; J
EðÞÞ
exist (and are adjoint to each other).
This result cannot usually be applied directly since
the verification of H
0
- and especially of H-smooth-
ness may be a difficult problem. Let us briefly
explain how it can be done on the example of pair
[10], where the potential V(x) satisfies estimate [4]
for some >1. Let us start with the operator
H
0
= . Denote by L
(l)
2
= L
(l)
2
(R
d
) the Hilbert
space with the norm kf k
l
= khxi
l
f k, wher e hxi= (1 þ
jxj
2
)
1=2
. Let the operator
0
() be defined by
eqn [16], and let X (0, 1) be some compact
interval. Set N = L
2
(S
d1
). If f 2 L
(l)
2
with l > 1=2,
then, by the Sobolev trace theorem,
k
0
ðÞf k
N
Ckf k
l
k
0
ðÞf
0
ð
0
Þf k
N
Cj
0
j
kf k
l
½25
for an arbitrary l 1=2, <1 and all ,
0
2 X.
These estimates imply that the function
ðE
0
ðÞf ; f Þ¼
Z
jj
2
<
j
^
f ðÞj
2
d ½26
is differentiable and the derivative
dðE
0
ðÞf ; f Þ=d ¼k
0
ðÞf k
2
N
; f 2 L
ðlÞ
2
; l > 1=2
is Ho¨ lder-continuous in >0 (uniformly in f,
kf k
l
1). Theref ore, applying the Privalov theorem
to the Cauchy integral
ðR
0
ðzÞf ; f Þ¼
Z
1
0
ð zÞ
1
dðE
0
ðÞf ; f Þ
we obtain that the analytic operator function
R
0
ðzÞ¼hxi
l
R
0
ðzÞhxi
l
; l > 1=2
considered in the space H, is continuous in norm in
the closed complex plane C cut along (0, 1) with
possible exception of the point z = 0. This implies
H
0
-smoothness of the operator h xi
l
, l > 1=2, on all
compact intervals X (0, 1).
To obtain a similar result for the operator H,
we proceed from the resolvent identity [14].
Let R(z) = hxi
l
R(z)hxi
l
, and let B be the operator
of multiplication by the bounded function
(1 þjx j)
V(x). If
f þR
0
ðzÞBf ¼ 0
then = R
0
(z)hxi
l
Bf satisfies the Schro¨ dinger equa-
tion H = z .SinceH is self-adjoint, this implies that
= 0andhencef = 0. Using eqn [14], we obtain that
RðzÞ¼ðI þR
0
ðzÞBÞ
1
R
0
ðzÞ; Im z 6¼ 0 ½27
because the inverse operator here exists by the
Fredholm alternative.
The operator function (I þR
0
(z)B)
1
is analytic
in the complex plane cut along (0, 1) with possible
exception of poles (coinciding with eigenvalues of H)
on the negative half-axis. Moreover, (I þR
0
(z)B)
1
is continuous up to the cut except the set N(0, 1 )
of where at least one of the homogeneous equations
f þR
0
ð i0ÞBf ¼ 0 ½28
has a nontrivial solution. It follows from eqn [27]
that the same is true for the operator function R( z).
It can be shown that the set N is closed and has the
Lebesgue measure zero. Let =(0, 1)nN; then
= [
n
n
where
n
are disjoint open intervals. By
condition [24], the operator hxi
l
, l > 1=2, is
H-smooth on any strictly interior subinterval of
every
n
. Applying the Kato–Lavine theorem, we see
that the wave operat ors W
(H, H
0
; E
0
(
n
)) and
W
(H
0
, H; E(
n
)) exist for all n. Since E
0
() = I
and E() = P
(ac)
, this implies the existence of
W
(H, H
0
)andW
(H
0
, H). Thus, the wave opera-
tors W
(H, H
0
) for pair [13] exist and are complete
if estimate [4] holds for some >1.
Compared to the trace-class method, conditions on
the perturbation V(x) are less restrictive, while the
class of admissible ‘‘free’’ problems is essentially more
narrow (in eqn [22] V
0
(x) is an arbitrary bounded
function). It is not known whether the wave
operators W
(H, H
0
) exist for all pairs [22] such
that V
0
2 L
1
and V satisfies [4] for some >1.
It is important that the smooth method allows one
to prove the absence of the singular continuous
spectrum. Note first that the continuity of R(z)
implies that the operator H is absolutely contin uous
on the subspace E()H. Therefore, the singular
256 Quantum Mechanical Scattering Theory
positive spectrum of H is necessarily contained in N.
To prove that its continuous part is empty, it suffices
to check that the set N consists of eigenvalues of the
operator H. In terms of u = hxi
l
Bf , l = =2, eqn
[28] can be rewritten as
u þ VR
0
ð i0Þu ¼ 0 ½29
Multiplying this equation by R
0
( i0)u and taking
the imaginary part of the scalar product, we see that
dðE
0
ðÞu; uÞ=d ¼ ImðR
0
ð i0Þu; uÞ¼0
According to eqn [26], this implies that
^
uðÞ¼0 for jj¼
1=2
½30
It follows from eqn [29] that
¼ R
0
ð i0Þu ½31
that is,
ˆ
() = (jj
2
i0)
1
^
u(), is a formal
(because of the singularity of the denominator)
solution of Schro¨ dinger equation [5].Therefore,one
needs only to verify that 2 L
2
(R
d
). Since u 2 L
(l)
2
,
where l = =2, this is a direct consequence of [25] and
[30] if >2. In the general case, one uses that under
assumption [30] the function (jj
2
)
1
^
u()belongs
to the space L
(p)
2
for any p < l 1. By virtue of
condition [4] where >1, eqn [29] now shows that
actually u 2 L
(p)
2
for any p < l þ 1. Repeating
these arguments, we obtain, after n steps, that u 2
L
(p)
2
for any p < l þ n( 1). For n large enough, this
implies that u 2 L
(p)
2
for p > 1, and consequently
function [31] belongs to L
2
(R
d
).
Similar arguments show that eigenvalues of H
have finite multiplicity and do not have positive
accumulation points. For the proof of boundedness
of the set of eigenvalues, one uses additionally the
estimate
kR
0
ð i0Þk ¼ Oð
1=2
Þ;!1 ½32
Actually, according to Kato theorem the Schro¨ din-
ger operator H does not have positive eigenvalues.
There exists also a pur ely time- dependent
approach, the Enss method (see Perry (1983)),
which relies on an advanced study of the free
evolution operator exp (iH
0
t).
The Scattering Matrix
The operator H
0
= can of course be diagona-
lized by the classical Fourier transform . To put it
slightly differently, set
ðF
0
f ÞðÞ¼
0
ðÞf
where the operator
0
() is defined by eqn [16].
Then
F
0
: L
2
ðR
d
Þ!L
2
ðR
þ
; N Þ; N ¼ L
2
ðS
d1
Þ
is a unitary operator and (F
0
H
0
f )() = (F
0
f )().
Under assumption [4] where >1, the scattering
operator S for pair [13] is defined by eqn [11].Itis
unitary on the space H= L
2
(R
d
) and commutes
with the operator H
0
. It follows that (F
0
Sf )() =
S()(F
0
f )(), >0, where the unitary operator
S():N !N is known as the scattering matrix. The
scattering matrix S() for the pair H
0
, H can be
computed in terms of the scattering amplitude.
Namely, S() acts in the space L
2
(S
d1
), and S()
I is the inte gral operator whose kernel is the
scattering amplitude. More precisely,
ðSðÞf ÞðÞ
¼ f ðÞþ2i
1=2
d
ðÞ
Z
S
d1
að; !; Þf ð!Þd!
In operator notation, this representation can be
rewritten as
SðÞ¼I 2 i
0
ðÞðV VRð þ i0ÞVÞ
0
ðÞ½33
The right-hand side here is correctly defined as a
bounded operator in the space N and is continuous
in >0. Moreover, the operator S() I is compact
since
0
()hxi
l
: H!N is compact for l > 1=2by
virtue of the Sobolev trace theorem.
It follows that the spectrum of the operator S()
consists of eigenvalues of finite multiplicity, except
possibly the point 1, lying on the unit circle and
accumulating at the point 1 only. In the general
case, eigenvalues of S() play the role of scattering
phases or shifts considered often for radial potent ials
V(x) = V(jxj).
The scattering amplitude is singular on the
diagonal = ! only. Moreover, this singularity is
weaker for potentials with faster decay at infinity
(for bigger). If >(d þ 1)=2, then the operator
S() I belongs to the Hilbert–Schmidt class. In this
case the total scattering cross section
ð!; Þ¼
Z
S
d1
jað; !; Þj
2
d
is finite for all energies >0 and all incident
directions ! 2 S
d1
.If>d, then the operator
S() I belongs to the trace class. In this case, the
scattering amplitude a( , !; ) is a continuous func-
tion of , ! 2 S
d1
(and >0). The unitarity of the
operator S() implies the optical theorem
ð!; Þ¼
1=2
Im
1
d
ðÞað!; !; Þ
Quantum Mechanical Scattering Theory 257
Using resolvent identity [14], one deduces from
eqn [33] the Born expansion
SðÞ¼I 2 i
X
1
n¼0
ð1Þ
n
0
ðÞVðR
0
ð þ i0ÞVÞ
n
0
ðÞ
This series is norm-convergent for small potentials V
and according to estimate [32] for high energies .
Long-Range Interactions
Potentials decaying at infinity as the Coulomb
potential
VðxÞ¼jxj
1
; d 3
or slower are called long-range. More precisely, it is
required that
j@
VðxÞj Cð1 þjxjÞ
jj
;2ð0; 1½34
for all derivatives of V up to some order. In the
long-range case, the wave operators W
(H, H
0
)do
not exist, and the asymptotic dynamics should be
properly modified. It can be done in a time-
dependent way either in the coordinate or momen-
tum representations. For example, in the coo rdinate
representation, the free evolution exp (iH
0
t)
should be replaced in definition [8] of wave
operators by unitary operators U
0
(t) defined by
ðU
0
ðtÞf ÞðxÞ¼expði ðx; tÞÞð2itÞ
d=2
^
f ðx=ð2tÞÞ
where
^
f is the Fourier transform of f. For short-
range potentials we can set (x, t) = (4t)
1
jxj
2
. In the
long-range case the phase function (x, t) should be
chosen as a (perhaps, approximate) solution of the
eikonal equation
@=@t þjrj
2
þ V ¼ 0
In particular, we can set
ðx; tÞ¼ð4tÞ
1
jxj
2
t
Z
1
0
VðsxÞds
if >1=2in[34]. For the Coulomb potential,
ðx; tÞ¼ð4tÞ
1
jxj
2
tjxj
1
ln jtj
(the singularity at x = 0 is inessential here). Thus,
both in short- and long-range cases solutions of the
time-dependent Schro¨ dinger equation ‘‘live’’ in a
region of the configuration space where jxj is of
order jtj. Long-range potentials change only asymp-
totic phases of these solutions.
Another possibility is a time-independent modifi-
cation in the phase space. Let us consider wave
operators W
(H, H
0
; J), where J is a pseudodiffer-
ential operator,
ðJf ÞðxÞ¼ð2Þ
d=2
Z
R
d
e
ihx;i
e
iðx;Þ
ðx ;Þ
^
f ðÞd
with oscillating symbol exp (i(x, ))(x, ). Due to the
conservation of energy, we may suppose that (x, )
contains a factor (jj
2
)with 2 C
1
0
(0, 1). Set
’ðx;Þ¼hx;iþðx;Þ
The perturbation HJ JH
0
is also a pseudodiffer-
ential operator, and its symbol is short-range (it is
O(jxj
1"
), ">0, as jxj!1) if exp (i’(x, )) is an
approximate eigenfunction of the operator H corre-
sponding to the ‘‘eigenvalue’’ jj
2
. This leads to the
eikonal equation
jr
x
’ðx;Þj
2
þ VðxÞ¼jj
2
The notorious difficulty (for d 2) of this method is
that the eikonal equation does not have (even
approximate) solutions such that jr
x
(x, )j!0as
jxj!1 and the arising error term is short-range.
However, it is easy to construct functions ’ = ’
satisfying these conditions if a conical neighbor hood
of the direction is removed from R
d
. For
example,
ðx;Þ¼2
1
Z
1
0
Vðx ÞVðÞðÞd
if >1=2ineqn [34]. Then the cutof f function
(x, ) =
(x, ) should be homogeneous of order
zero in the variable x and it should be equal to zero
in a neighborhood of the direction . We empha-
size that now we have a couple of different
identifications J = J
.
The long-range problem is essentially more diffi-
cult than the short-range one. The limiting absorp-
tion principle remains true in this case, but its proof
cannot be performed within perturbation theory.
The simplest proof relies on the Mourre estimate
(see Cycon et al. (1987)) for the commutat or i[H, A]
of H with the generator of dilations
A ¼i
X
d
j¼1
ðx
j
@
j
þ @
j
x
j
Þ
The Mourre estimate affirms that, for all >0,
iEð
Þ½H; AEð
ÞcðÞEð
Þ; cðÞ > 0 ½35
if
= ( ", þ ")and" is small enough. For the
free operator H
0
, this estimate takes the form
i[H
0
, A] = 4H
0
and can be regarded as a commutation
relation. Estimate [35] means that the observable
ðAe
iHt
f ; e
iHt
f Þ
258 Quantum Mechanical Scattering Theory
is a strictly increasing function of t for all f 2H
(ac)
.
The H-smoothness of the operator hxi
l
, l > 1=2, is
deduced from this fact by some arguments of
abstract nature (they do not really use concrete
forms of the operators H and A).
However, the limiting absorption princ iple is not
sufficient for construction of scatter ing theory in the
long-range case, and it should be supplemented by
an additional estimate. To formulate it, denote by
ðr
?
uÞðxÞ¼ðruÞðxÞjxj
2
hðruÞðxÞ; xix
the orthonal projection of a vector (ru)(x) on the
plane orthogonal to x. Then the operator
K = hxi
1=2
r
?
is H-smooth on any compact X
(0, 1). This result is formulated as an estimate
(either on the resolvent or on the unitary group of
H), which we refer to as the radiation estimate. This
estimate is not very astonishing from the viewpoint
of analogy with the classical mechanics . Indeed, in
the case of free motion, the vector x(t)ofthe
position of a particle is directed asymptotically as its
momentum . Regarded as a pseudodifferential
operator, r
?
has symbol jxj
2
h, xix, which
equals zero if x = for some 2 R. Thus, r
?
removes the part of the phase space where a classical
particle propagates. The proof of the radiation
estimate is based on the inequality
K
K C
0
½H;@
r
þC
1
hxi
1
;@
r
¼ @=@jxj
which can be obtained by a direct calculation. Since
the integral
i
Z
t
0
ð½H;@
r
e
iHs
f ; e
iHs
Þf Þds
¼ð@
r
e
iHt
f ; e
iHt
f Þð@
r
f ; f Þ
is bounded by C (X )kf k
2
for f 2 E(X)f and
the operator hxi
(1þ)=2
is H-smooth on X, this
implies H-smoothness of the operator KE(X).
Calculating the perturbation HJ
J
H
0
, we see
that it is a sum of two pseudodifferential operators.
The first of them is short-range and thus can be
taken into account by the limiting absorption
principle. The symbol of the second one contains
first derivatives (in the variable x) of the cutoff
function
(x, ) and hence decreases at infinity as
jxj
1
only. This operator factorizes into a product of
H
0
- and H-smooth operators according to the
radiation estimate. Thus, all wave operators
W
(H, H
0
; J
) and W
(H
0
, H; J
) exist. These
operators are isometric since the operators J
are in some sense close to unitary operators.
The isometricity of W
(H
0
, H; J
) is equivalent to
the completeness of W
(H, H
0
; J
).
Although the modified wave operators enjoy
basically the same properties as in the sh ort-range
case, properties of the scatter ing matrices in the
short- and long-range cases are drastically different.
Here we note only that for long-range potentials,
due to a wild diagonal singularity of kernel of the
scattering matrix, its spectrum covers the whole unit
circle.
Different aspects of long-range scattering are
discussed in Derezin
´
ski and Ge´rard (1997), Pearson
(1988), Saito
¯
(1979),andYaf aev (2000).
See also: N-Particle Quantum Scattering; Quantum
Dynamical Semigroups; Random Matrix Theory in
Physics; Scattering in Relativistic Quantum Field Theory:
Fundamental Concepts and Tools; Schro
¨
dinger
Operators; Spectral Theory for Linear Operators.
Further Reading
Amrein WO, Jauch JM, and Sinha KB (1977) Scattering Theory in
Quantum Mechanics. New York: Benjamin.
Cycon H, Froese R, Kirsh W, and Simon B (1987) Schro¨ dinger
Operators, Texts and Monographs in Physics. Berlin:
Springer.
Derezin
´
ski J and Ge´rard C (1997) Scattering Theory of Classical
and Quantum N Particle Systems. Berlin: Springer.
Faddeev LD (1976) Inverse problem of quantum scattering theory
II. J. Soviet. Math. 5: 334–396.
Friedrichs K (1965) Perturbation of Spectra in Hilbert Space.
Providence, RI: American Mathematical Society.
Kuroda ST (1978) An Introduction to Scattering Theory, Lecture
Notes Series No. 51. Aarhus University.
Landau LD and Lifshitz EM (1965) Quantum Mechanics.
Pergamon.
Pearson D (1988) Quantum Scattering and Spectral Theory.
London: Academic Press.
Perry PA (1983) Scattering Theory by the Enss Method. vol. 1.
Harwood: Math. Reports.
Reed M and Simon B (1979) Methods of Modern Mathematical
Physics III. New York: Academic Press.
Saito
¯
Y (1979) Spectral Representation for Schro¨ dinger Operators
with Long-Range Potentials. Springer Lecture Notes in Math,
vol. 727.
Yafaev DR (1992) Mathematical Scattering Theory. Providence,
RI: American Mathematical Society.
Yafaev DR (2000) Scattering Theory: Some Old and New
Problems, Springer Lecture Notes in Math., vol. 1735.
Quantum Mechanical Scattering Theory 259
Quantum Mechanics: Foundations
R Penrose, University of Oxford, Oxford, UK
ª 2006 Elsevier Ltd. All rights reserved.
The Framework of Quantum Mechanics
In 1900, Max Planck initiated the quantum revolu-
tion by presenting the hypothesis that radiation is
emitted or absorbed only in ‘‘quanta,’’ each of
energy h, for frequency (where h was a new
fundamental constant of Nature). By this device, he
explained the precise shape of the puzzling black-
body spectrum. Then, in 1905, Albert Einstein
introduced the concept of the photon, according to
which light, of frequency would, in appropriate
circumstances, behave as though it were constituted
as individual particles, each of energy h, rather
than as continuous waves, and he was able to
explain the conundrum posed by the photoelectric
effect by this means. Later, in 1923, Prince Louis de
Broglie proposed that, conversely, all particles
behave like waves, the energy being Planck’s h
and the momentum being h
1
, where is the
wavelength, which was later strikingly confirmed in
a famous experiment of Davisson and Germer in
1927. Some years earlier, in 1913, Niels Bohr had
used another aspect of this curious quantum
‘‘discreteness,’’ explaining the stable electron orbits
in hydrogen by the assumption that (orbital) angular
momentum must be quantized in units of h (= h=2).
All this provided a very remarkable collection of
facts and co ncepts, albeit somewhat disjointed,
explaining a variety of previously baffling physical
phenomena, where a certain discreteness seemed to
be entering Natu re at a fundamental level, where
previously there had been continuity, and where
there was an overriding theme of a confusion as to
whether – or in what circumstances – waves or
particles provide better pictures of reality. More-
over, no clear and consistent picture of an actual
‘‘quantum-level reality’’ as yet seemed to arise out of
all this. Then, in 1925, Heisenberg introduced his
‘‘matrix mechanics,’’ subsequently developed into a
more complete theory by Born, Heisenberg and
Jordan, and then more fully by Dirac. Some six
months after Heisenberg, in 1926, Schro¨ dinger
introduced his very different-looking ‘‘wave
mechanics,’’ which he subsequently showed was
equivalent to Heisenberg’s scheme. These be came
encompassed into a compr ehensive framework
through the transformation theory of Dirac, which
he put together in his famous book The Principles of
Quantum Mechanics, first published in 1930. Later,
von Neumann set the framework on a more rigorous
basis in his 1932 book, Mathematische Grundlageen
der Quantenmechanik (later translated as Mathe-
matical Foundations of Quantum Mechanics, 1955).
This formalism, now well known to physicists, is
based on the presence of a quantum state j i
(Dirac’s ‘‘ket’’ notation being adopted here). In
Schro¨ dinger’s description, j i is to evolve by unitary
evolution, according to the Schro¨ dinger equation
ih
@j i
@t
¼ Hj i
where H is the quantum Hamiltonian. The totality of
allowable stat es j i constitutes a Hilbert space H
and the Schro¨ dinger equation provides a continuous
one-parameter family of unitary transformations of H.
The letter U is used here for the ‘‘quantum-level’’
evolution whereby the state j i evolves in time
according to this unitary Schro¨ dinger evolution.
However, we must be careful not to demand an
interpretation of this evolution similar to that
which we adopt for a classical theory, such as is
provided by Maxwell’s equations for the electro-
magnetic field. In Maxwell’s theory, the evolution
that his equations provide is accepted as very
closely mirroring the actual way in which a
physically real electromagnetic field evolves with
time. In quantum mechanics, however, it is a highly
contentious matter how we should regard the
‘‘reality’’ of the unitarily evolving state j i.
One of the key difficulties resides in the fact that
the world that we actually observe about us rather
blatantly does not accord with such a unitarily
evolving j i. Indeed, the standard way that the
quantum formalism is to be interpreted is very far
from the mere following of such a picture. So long
as no ‘‘measurement’’ is deemed to have been taking
place, this U-evolu tion procedure would be adopted,
but upon measurem ent, the state is taken to behave
in a very different way, namely to ‘‘jump’’ instanta-
neously to some eigensta te ji of the quantum
operator Q which is taken to represent the measure-
ment, with probability given by the Born rule
jhj ij
2
if we assume that both j i and ji are normalized
(h j i= 1 = hji); otherwise we can express this
probability simply as
hj ih ji
h j ihji
(The operator Q is normally taken to be self-adjoint,
so that Q = Q
and its eigenvalues are real, but more
260 Quantum Mechanics: Foundations
generally complex eigenvalues are accommodated if
we allow Q to be normal, that is, QQ
= Q
Q.In
each case we require the eigenvectors of Q to span
the Hilbert space H.) This ‘‘evolution procedure’’ of
the quantum state is very different from U,owing
both to its discontinuity and its indeterminacy. The
letter R will be used for this, standing for the
‘‘reduction’’ of the quantum state (sometimes referred
to as the ‘‘collapse of the wave function’’). This
strange hybrid, whereby U and R are alternated, with
U holding between measurements and R holding at
measurements, is the standard procedure that is
pragmatically adopted in conventional quantum
mechanics, and which works so marvelously well,
with no known discrepancy between the theory and
observation. (In his classic account, von Neumann
(1932, 1955),‘‘R’’ is referred to as his ‘‘process I’’
and ‘‘U’’ as his ‘‘process II.’’) However, there appears
to be no consensus whatever about the relation
between this mathematical procedure and what is
‘‘really’’ going on in the physical world. This is the
kind of issue that will be of concern to us here.
Quantum Reality
The discussion here will be given only in the
Schro¨ dinger picture, for the reason that the issues
appear to be clearer with this description. In the
Heisenberg picture, the state j i does not evolve in
time, and all dynamics is taken up in the time
evolution of the dynamical variables. But this
evolution does not refer to the evolution of specific
systems, the ‘‘state’’ of any particular system being
defined to remain constant in time. Since the
Schro¨ dinger and Heisenberg pictures are deemed to
be equivalent (at least for the ‘‘normal’’ systems that
are under consideration here), we do not lose
anything substantial by sticking to Schro¨ dinger’s
description, whereas there does seem to be a
significant gain in understanding of what the
formalism is actually telling us.
There are, however, many different attitudes that
are expressed as to the ‘‘reality’’ of j i. (There is an
unfortunate possibility of confusion here in the two
uses of the word ‘‘real’’ that come into the discussion
here. In the quantum formalism, the state is mathe-
matically a ‘‘complex’’ rather than a ‘‘real’’ entity,
whereas our present concern is not directly to do
with this, but with the ‘‘ontology’’ of the quantum
description.) According to what is commonly regarded
as the standard – ‘‘Copenhagen’’ – interpretation of
quantum mechanics (due primarily to Bohr,
Heisenberg, and Pauli), the quantum state j i is not
taken as a description of a quantum-level reality at all,
but merely as a description of the observer’s
knowledge of the of the quantum system under
consideration. According to this view, the ‘‘jumping’’
that the quantum state undergoes is regarded as
unsurprising, since it does not represent a sudden
change in the reality of the situation, but merely in the
observer’s knowledge, as new information becomes
available, when the result of some measurement
becomes known to the observer. According to this
view, there is no objective quantum reality described
by j i. Whether or not there might be some objective
quantum-level reality with some other mathematical
description seems to be left open by this viewpoint, but
the impression given is that there might well not be any
such quantum-level reality at all, in the sense that it
becomes meaningless to ask for a description of
‘‘actual reality’’ at quantum-relevant scales.
Of course some connection with the real world is
necessary, in order that the quantum formalism can
relate to the results of experiment. In the Copenha-
gen viewpoint, the experimenter’s measuring appa-
ratus is taken to be a classical-level entity, which can
be ascribed a real ontological statu s. When the
Geiger counter ‘‘clicks’’ or when the pointer
‘‘points’’ to some position on a dial, or when the
track in the cloud chamber ‘‘becomes visible’’ –
these are taken to be real event s. The intervening
description in terms of a quantum state vector j i is
not ascribed a reality. The role of j i is merely to
provide a calculational procedure whereby the
different outcomes of an experiment can be assigned
probabilities. Reality comes about only when the
result of the measurem ent is manifested, not before.
A difficulty with this viewpoint is that it is hard to
draw a clear line between those entities which are
considered to have an actual reality, such as the
experimental apparatus or a human observer, and
the elemental constituents of those entities, which
are such thing s as electrons or protons or neutrons
or quarks, which are to be treated quantum
mechanically and therefore, on the ‘‘Copenhagen’’
view, their mathematical descriptions are denied
such an honored ontological status. Moreover, there
is no limit to the number of particles that can
partake in a quantum state. According to current
quantum mechanics, the most accurate mathemati-
cal procedure for describing a system with a large
number of particles would indeed be to use a
unitarily evolving quantum state. What reasons can
be presented for or against the viewpoint that this
gives us a reasonable description of an actual
reality? Can our perceived reality arise as some
kind of statis tical limit when very large numbers of
constituents are involved?
Before entering into the more subtle and con-
tentious issues of the nature of ‘‘quantum reality,’’ it
Quantum Mechanics: Foundations 261