A NNALES DE L ’I. H. P., SECTION A
A. V. S OBOLEV
D. R. Y AFAEV
On the quasi-classical limit of the total scattering cross-section in nonrelativistic quantum mechanics
Annales de l’I. H. P., section A, tome 44, n
o2 (1986), p. 195-210
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195
On the quasi-classical limit
of the total scattering cross-section in nonrelativistic quantum mechanics
A. V. SOBOLEV and D. R. YAFAEV (*) Leningrad Branch of Mathematical Institute (LOMI),
Fontanka 27, Leningrad, 191011 USSR
Vol. 44, n° 2, 1986, Physique theorique
ABSTRACT. - The total
scattering
cross-section for the operator- A +
gV
is studied forlarge coupling
constants g andhigh energies
K2.RESUME. - La section efficace totale de diffusion pour
l’opérateur
- 0 +
gV
est etudiee pour lesgrandes
constantes decouplage
g et les hautesenergies
K2.1. INTRODUCTION. A GENERAL SURVEY The total
scattering
cross-section for theSchrodinger equation
where XE
~83,
K >0,
g > 0 and(1)
(*) Currently visiting the Laboratoire de Physique Theorique et Hautes
Energies,
Batiment 211, Universite de Paris-Sud, F-91405 Orsay cedex (Laboratoire associe au CNRS).
(1) By C and c we denote different positive constants. If necessary, its dependence on
some parameters is specified in the notation, i. e. C = C(K).
l’Institut Henri Poincaré - Physique theorique - 0246-0211
Vol. 44/86/02/195/16/X 3,60/(~) Gauthier-Villars
196 A. V. SOBOLEV AND D. R. YAFAEV
can be defined in the
following
way. Denoteby S(K, g)
thescattering
matrix for the
equation (1.1) (for
theprecise
definition seebelow).
ThenS(K, g) -
I is anintegral
operator in the spaceL2(~2)
with a kernelThe function
f
is called theamplitude
ofscattering
in the direction ~p for the direction cv of theincoming plane
wave. In terms of
f
the totalscattering
cross-section for theincoming
direction 03C9 is defined
by
When
averaged
over 03C9where ~.~2
is the Hilbert-Schmidt norm.The aim of this paper is to discuss the
asymptotic
behavior ofK, g)
as the energy K2 of the
particle
and thecoupling
constant g both tend toinfinity.
Note that theequation, formally
moregeneral
than(1.1),
+
gV(x)u
= K2u isobviously
reducible to(1.1).
Inparticular,
thequasi-classical
limit as h --+0,
g and Kfixed,
is the same as that for(1.1)
as g = cK2 --+ oo, c = const.
However,
we do not restrict ourselves to thisparticular
case andpermit
different relations between g and K asg --+ oo, K --+ oo .
Though
the total cross-section is one of the basicobjects
in aquantum scattering problem,
there was littlestudy
of it untilrecently.
Even at a heuristic level somequite general properties
ofK, g)
seem not to becommonly
known. Weemphasize
that theasymptotics
ofK, g)
as K --+ oo, g -> oodepends crucially
on therelation between K
and g
and on the rate of the fall-off ofV(x)
atinfinity.
We
begin
our discussion with thesimple
case wheng/K
--+ 0. As iswell-known
(see
e. g.[7])
as K --+ oo, gfixed,
or g --i0,
Kfixed,
theasymptotics
of differentscattering
characteristics for theequation (1.1)
may be obtained
by perturbation theory (Born approximation
inphysical terms).
These formulae can berigorously justified [2]-[~] and,
moreover,if
g/K
--+ 0they
are valid also in case K -i> oo, g --+ oo . Inparticular,
for the total cross-section the Born
approximation
shows that undersuitable
assumptions
on V :where b E
f~2, ~ b, cc~ ~
==0, x
= b + Thus in caseg/K -~
0 the asymp- totics ofdepends
on the values ofV(x)
for allThe case
g/K -~
oo(or
the intermediate caseg/K
=const)
cannot bestudied
by perturbation theory
and theasymptotics
of as~/K -~
oo is very sensitive to the behavior ofV(x) as x ~ I
~ oo. Letde Henri Poincaré - Physique théorique
firstly
V have acompact
support.If g = cK2
-+ oo, c = const, it is shown in[5] ]
that under certainassumptions (including assumptions
on thecorresponding
classicalsystem)
where is a total classical cross-section for the direction co of the incident beam of
particles.
Moreprecisely,
in[5 it
is proven that(1. 6)
holds if its left-hand side is
averaged
over some small energy interval.Note that
generically
does notdepend
on the energy andequals
the square of theprojection
of supp V on theplane orthogonal
to M. The
simpler
case g --+ oo, K --+ oo, g y >0,
was discussedin
[6] ]
where theasymptotics ( 1. 6) (without averaging
overenergy)
wasjustified
with thehelp
of the so-called eikonalapproximation.
In thisregion
one can evaluate also the(logarithmically growing) asymptotics
of
K, g)
forexponentially decaying potentials.
Outside of theregion
g
cK2 only
upper bounds onK, g)
are known.Namely,
in[7 ], [~] ]
it is shown that :
where C
depends only
on the size of thesupport
of V but not on K and g.In
[8] the problem
wasposed
to obtain the bound( 1. 7)
withoutaveraging
over energy.
However,
as found in[9] ]
suchsharp-energy
bound fails ingeneral
to be true.Namely,
forspherically-symmetric (radial) V(x)
=V(r),
with a non-trivial
negative
part and fixedK,
a sequence gl = oo was constructed in[9] ]
such thatThe
growth
ofu(K, g)
for some sequence ~ has a resonant nature, which is discussed morethoroughly
in section 2. On the contrary, forrepulsive
av .
(not
necessaryradial) potentials V,
~|x|~ 0,
the total cross-section is bounded[9] uniformly
in thecoupling
constant:This result is
improved
in section 4 where(1.9)
is established for all nonnegative potentials.
Now we turn to the case of
potentials
with apower-like
behavior atinfinity.
Under theassumption (1.2)
the upper bound198 A. V. SOBOLEV AND D. R. YAFAEV
is proven in
[7 ~, [8 ].
In[1 D it
wasconjectured
thatasg/K
oo the asymp- totics ofK, g)
is determinedonly by
theasymptotics
ofV(x)
asI
-~ oo.Namely,
assume thatThe
hypothesis
of[7~] is
thatwhere
and
~w
is a unit circle in theplane Aw.
For radialpotentials
thevalidity
of
(1.12), (1.13)
was asserted in the book[1] on
the basis of the asympto- tics ofphase
shifts03B4l(K, g)
forlarge l,
K and g.By analogy
with the radialcase the formulae
( 1.12), ( 1.13)
wereformally
derived in[70] ]
from theasymptotics
of theeigenvalues = 1,
:t > 0 of thescattering
matrixS(K, g).
However in[70] ]
theasymptotics
of1;(K,g)
as n oo was established for fixed K
and g
so that the calculations of[70] may
beregarded only
as heuristic. At present theasymptotics (1.12)
for
V(x) obeying (1.11) is rigorously
proven in case~/K -~ cK2,
where c =
c(V)
issufficiently
small number(and
inparticular cK2-y,
y >
0).
This result wasreported
in[6] and
its detailedproof
based on theeikonal
approximation
will bepublished
elsewhere.In section 3 we
give
theprecise
formulation and a sketch of theproof (the
details may be found in[77])
of(1.12)
for radialpotentials.
It turnsout that for non
negative
V the conditions ofvalidity
of( 1.12)
are muchbroader than in
general
case but thequasi-classical
limit isalways permitted.
We
emphasize
that ourproof
of(1.12)
does notrequire
anyassumptions
on the classical system
corresponding
to(1.1).
Theproof
of(1.12)
is basedon the
analysis
ofphase
shifts~~(K, g)
in theregion
where 1°‘-1 is of the sameorder as The
technique developped
herepermits
also to find theasymptotics
ofg)
forpotentials
with astrong positive singularity V(r) ~
i~o >0, /~
>2,
as r ~ 0 in theregion ~/K -~ 0,
00(and
inparticular
in thehigh-energy
limit K -~ oo, gfixed).
Note that forsuch
potentials
theright-hand
sideof (1. 5)
is infinite so that the formula(1. 5)
fails
certainly
to be true. We show that in contrast toregular potentials
the
asymptotics
of6(K, g)
is determinedonly by
thesingularity
ofV(r)
at r = 0 and as
g/K ~
0 the total cross-section isvanishing
slower than in aregular
case.In a final section 4 we prove some
sharp-energy
upper bounds oncr(K, g).
Annales de ll’Institut Henri Poincaré - Physique theorique
This work is in progress now and we report two results here.
Firstly,
weshow that under
assumption (1.2)
Secondly,
forpositive potentials
with compact support we establish the above-mentioned bound( 1. 9).
In conclusion of this introduction note that the condition x E
[R3
is in essential and all results can beeasily
carried over to m >_ 2.2. ON THE RESONANT SCATTERING BY A NEGATIVE POTENTIAL
Let us
firstly give
aprecise
definition of the totalscattering
cross-section for a radialpotential V(x)
=V(r),
r=
Consider a set ofequations
describing particles
with orbital quantum numbers l =0,1, 2,
... Assumethat
V(r)
is bounded away from thepoint
r =0, V(r)
=O(r-P), ~8 2,
as r -~ 0 andThe
equation (2.1)
has a realregular
solution _K, g) satisfying
=
O(r
1 +1 ),
r -+ 0. As r -+ o0where
phase
shifts~l
=~1(K, g)
are determinedby (2 . 3)
up to a summand 7~n
integer.
In terms ofphase
shiftswhich is consistent with
(1.3), (1.4)
for ageneral
case. The total cross-section is finite if a > 2 but even for
potentials
with compact supports it is notuniformly
bounded as g oo .THEOREM 1. 2014 Let
Vex)
=V(r)
have acompact support
andV(r)
0on a set of a
positive Lebesgue
measure in Then for each K > 0 there exists a sequence gj=~/(K) -~
oo as -~ oo such that the lower bound(1. 8)
holds. The constant C =
C(K)
in(1.8)
can be chosen common for allKe(0,KoLKo
oo .Vol. 44, n° 2-1986.
200 A. V. SOBOLEV AND D. R. YAFAEV
The
proof
of this theorem relies on thefollowing :
LEMMA 1.
2014 Under theassumptions
of Th. 1 for allsufficiently large
lthere exists a
coupling
constant gl =gl(K)
such that03B4l(K, gl) _ Tc/2
andC 1 l _ g~ C212, Cj
> 0. ConstantsCj
=C/K)
may be chosen commonfor all
Ke(0,KoL Ko
oo .Once Lemma 1 is proven, the total cross-section
r(K, gl)
can be boundedbelow
by
which proves Th. 1. The detailed
proof
of Lemma 1 isgiven
in[9 ].
Herewe note
only
that the numbergl(K)
may be chosen in thefollowing
way.Let
V(r)
= 0 for r > p, and let N be the Neumann function. Consider theboundary-value problem
with g as a
spectral parameter.
Underassumptions
of Th. 1 thisproblem
has a discrete spectrum with + oo
(and possibly -
ooalso)
as an accumu-lation
point.
Let nowgj(K)
be the firstpositive eigenvalue
of(2.5), (2.6).
Then satisfies all the
requirements
of Lemma 1. Note that Th. 1 canbe
easily generalized
topotentials
withnon-compact supports.
Here we discuss the
phenomena
of thegrowth
of the total cross-section from thephysics point
of view. To makethings transparent
assume thatV(r)
0. For orbitalmomentum l, l(l
+1)
2K2 p2,
the effectivepotential
does not allow a classical
particle
with an energy K2 to penetrate into theregion r ~
ro where ro =[1(l
+1) ]1/2K -1
is aturning point.
Thus forlarge
a classicalparticle
does not « feel » thepotential
well
gV(r) supported
in r p. Thecentrifugal term 1(1
+1)r-2 separates
this well from theclassicaly
allowedregion by
thepotential
barrier with theheight
of anorder l2
and the width of an Order 1. In contrast to aclassical,
a
quantum particle
canpenetrate through
such a barrier on account oftunneling.
Thus thepotential gV(r) perturbates
allphase
shifts~l.
butwith the
growth
of the barrier the influenceof gV(r)
isgenerically quickly vanishing
so that5/
are small forlarge l. However,
ouranalysis
shows thatfor the constructed
g(K)
the behavior of aquantum particle
in a fieldVeff(r)
is very different from that in a « free » field +1)r-2
if the energy K’2 is close to K2. Thejump
of thephase
shift up toTc/2
can benaturally explained
Annales de l’Institut Henri Poincaré - Physique theorique
by
the existence of thequasistationary
state in the fieldVeff.
When such astate exists a quantum
particle, tunneling through
abarrier,
is detained ina
potential
well for alarge
time(in corresponding
quantumunits).
Sincean energy of a
particle
ispositive,
a proper bound state does not exist sothat a
particle
tunnelsagain
toinfinity contributing
to the total cross-section. This ensures the
sharp (resonant) peak
of6(K’,
as K’ -~ K.Thereby
thispeak
getssharper
as l ~ oo.In a different situation the described
phenomena
of the resonantgrowth
of the total cross-section is well-known
(Gamow’s theory)
and is discussede. g. in the book
[72].
In thistheory
one considers a quantumparticle
ina
deep potential
well(due
to nuclearforces)
which forlarge
r is screenedby
arepulsive
Coulombpotential.
In contrast to thisproblem
ourpotential gV(r)
isnegative
and apositive
barrier arisesonly
on account of a centri-fugal
termby
aseparation
of variables. Fornegative potentials
the existence of resonantpeaks
of the total cross-section was not, as far as weknow,
discussed earlier.
3. THE ASYMPTOTICS OF THE TOTAL CROSS-SECTION FOR RADIAL POTENTIALS
The aim of this section is to find the conditions of the
validity
of theasymptotics ( 1.12), ( 1.13)
and itsrigorous proof
for radialpotentials V(x)
=V(r).
The detailedexposition
of the results below can be found in[11 ].
Setagain x
=2(a - 1)-1
andTHEOREM 2. 2014 Assume that
V(r)
=~3 2,
as r ~ 0 and Thenas
~/K -~
oo,g3 -aK2(a- 2) ~
oo. Under the additionalassumption V(r)>0
the relation(3 . 3)
holds in a broaderregion ~/K -~
~ oo.Let us discuss the
assumptions
of Th. 2.Though
not assumedexplicitly,
the
condition ~ -~
oo is a consequence oo, oo . On the contrary, for oc 3 and for V ~ 0 wepermit
bounded K and even thecase K ~ 0. Thus for
nonnegative potentials
theasymptotics (3. 3)
holdstrue in a
large coupling limit ~ -~
oo, K fixed. In ageneral
case we canassert its
validity only
for a E(2, 3).
This is connected with the existence of the lower bound( 1. 8)
which shows that(3 . 3)
iscertainly destroyed
for a >
5, when xa 1/2.
For 03B1 ~[3, 5] the validity
of(3 . 3)
as g ~ oo, Vol. 44,202 A. V. SOBOLEV AND D. R. YAFAEV
K
fixed,
is an openquestion.
What aquasi-classical limit g
=cK2
~ oois
concerned,
it isalways permitted by
the conditions of Th. 2. Weemphasize
that no
assumptions
on thecorresponding
classical system arethereby
necessary. Note also that the value
(3.1)
of theasymptotic
coefficient x«is consistent with the
general
formula(1.13).
Our
proof
of Th. 2 is based on thestudy
in its conditions of the asymp- totics of forl°‘ -1
>_ > 0 fixed. We remind thatby (2 . 3)
the
phase
shiftb(K, g)
is determinedonly
up to a summandinteger.
This
ambiguity
isstandardly
eliminated.Namely, g)
forfixed g
and lmay be chosen continuous in K and
~l(K, g)
~ ~cn as K -~ oo.Taking
n = 0 we get the
unique phase
shift~1(K, g)
which is considered below. SetTHEOREM 3. 2014 Let V
satisfy
condition of Th. 2 and let d be any fixedpositive
number. Thenas
g/K -~
oo,g~ ~K.~~ 2) ~ ~
Under the additionalassumption
V > 0
(3 . 4)
holds in theregion ~/K -~
oo, ~ oo.Once Th. 3 is proven the
asymptotics (3.3)
can be derived from the definition(2.4). Namely, according
to(3.4)
one canreplace asymptoti-
in
(2 . 4) by -
+1/2) - °‘ + 1
and the sum over lby
the
corresponding integral. Evaluating
thisintegral
we arrive at(3 . 3), (3 .1).
We do not dwell here upon the
details,
which can be found in[77].
At the heuristic level the
asymptotics (3.4)
can be deduced from the well-knownquasi-classical
relationwhere
rl(K, g)
is thelargest
of roots of twoequations l(l
+1)r - 2 +gV(r)
= K2and
1(1 + 1)r- 2
=K2.
Inparticular, (3 . 5)
ensures that theasymptotics
of
~(K,g)
is determinedonly by
aclassically
allowedregion
The last assertion is
easily justified
fornonnegative potentials
but in thegeneral
case it isprecisely
at thisplace
where the additional restriction~3-~2(~-2) _~ ~ ~ required.
Formulae(3.5)
and hence(3.4)
can bemost
probably justified
with thehelp
of theWKB-approximation,
whathowever demands
assumptions
on the behavior ofV’(r)
andV"(r)
atinfinity.
To avoid these unnaturalassumptions
we use the so-called variablephase approach (see
e. g.[13 ] ;
note that the variablephase approach
was
already
used in[8] to
obtain the two-side bounds ong)
for V > 0Annales de l’lnstitut Henri Poincaré - Physique theorique
and g -~ oo, K
fixed)
whichreplaces
theSchrodinger equation by
thefirst-order nonlinear
equation. Namely,
let v = vl := l +1/2
ande,,( p)
=K, g)
be thephase
shift for the cut-offpotential gVp(r),
whereVp(r)
=V(r)
for r ~ p andVp(r)
= 0 for r > p. Then = as r ~ 0 andDefine now = +
where ’
(2 -1 ~cx) 1 ~2~,,(x),
and 0’ Nv
stand 0for Bessel and o Neumann functions. It can
be easily
derived 0 from the Schro-dinger equation (1.1)
that satisfies the variablephase
’equation
It follows from
(3.7)
thatand 03B8v ~
0 for V 2 o.Our
proof
of theasymptotics (3.4)
relies on thephase equation (3.7).
We describe here
only
its essential steps.According
to(3.6), (3.7)
with
and
By (3 . 9)
theproof
of(3.4)
issplit
up into two steps.Firstly
we must prove that what theasymptotics
ofðv-l/2(K, g)
is concerned the summandBv(v/K, K, g)
isinsignificant,
i. e.under the
assumptions
of Th. 3.Essentially, (3.11)
means that the classi-cally
forbiddenregion
does not contribute to theasymptotics
of thephase
shifts. The
proof
of(3.11)
is based on theappropriate
two-side boundson
0y(r, K, g).
The lower bound(3 . 8)
is validirrespective
of thesign
of V.Combined
with ~
0 itspermits easily
to prove(3.11)
for the case V ~ 0.Vol. 44, n° 2-1986.
204 A. V. SOBOLEV AND D. R. YAFAEV
In the
general
case the proper upper bound for9v
is obtained with thehelp
of thefollowing :
LEMMA 2. 2014 Set
Assume that for some ro > 0 and all r E
(0, ro)
Then for r E
(0, ~o)
To deduce
(3.11)
from(3 . 8), (3.13)
we use well-known[7~] ]
uniformasymptotic
formulae forfv(r), J~(r), N,(r), N(r)
as r --+ oo, v --+ oo,r v. We
emphasize
that it isjust by verifying (3.12)
that we need anextra
assumption g3 -aK2(a-2) ~
oo.The second part of the
proof
of(3 . 4)
is the evaluation of theasymptotics
of the
integral (3.10).
This can beperformed
under theassumptions g/K
--+ --+ ooirrespective
of thesign
of V. Due to the conditiong/K
--+ oo we canreplace
hereV(r) by
itsasymptotics
Forbv(vs)
weuse the relation
which is valid
uniformly
in s for every compact subinterval of( 1, 00).
The relation
(3.14)
is a consequence of theasymptotics
ofJ ir), N)
asr -+ oo, v -+ oo,. r > v
(see [7~]).
Thusas
~/K -~
oo, oo andFinally
wereplace
heresin2 ~pv
by
2 -1 - 2 -1 cosIntegration by
parts shows that on account of thephase equation (3 . 7)
the summand with cos does not contribute to theasymptotics (3.4)
and thereforeCombining
this with(3 . 9), (3 .11)
we conclude theproof
of Th. 3.The
analysis
ofphase
shifts turns out to be useful also for thestudy
ofthe
high-energy asymptotics
of6(K, g)
forpotentials V(r)
with astrong positive singularity
at r = 0. Assume thatV(r)
is bounded away from thepoint
r = 0 and as r -~ 0Henri Poincaré - Physique theorique
The
numbers 03BA03B2
=2( (3 - 1 ) -1
and ~03B2(see (3 .1 ))
are determined nowby
the
singularity
ofV(r).
THEOREM 4. 2014 Let V
satisfy (3.15)
as r -~ 0 andV(r)
= a >2,
as r ~ oo. Then
as
g/K ~ 0,
oo .Note that under the
assumptions
of Th. 4necessarily
K ~ oo . Onthe contrary, if K ~ oo the conditions of Th. 4 are fulfilled
if g
= constoo, g =
0(K).
Thefall-off g
0 is alsopermitted though
itcannot be too fast because of the condition 00. This is
qualita- tively
different from the conditions of thevalidity
of the Born formula(1. 5)
in a
non-singular
case where g -~ 0only helps.
For a
purely
powerpotential V(r)
vo >0,
a >2,
the asympto- tics(3 . 3) (or (3 .16),
what is thesame)
is true under theunique assumption
oo. The conditions
g/K -~
00 org/K --~
0 were usedonly
toreplace
apotential V(r) by
itsasymptotics
as r ~ 00 or r ~ 0. In conclu-sion of this section we note that the
analysis
ofphase
shiftspermits
alsoto find the
asymptotics
of the forwardscattering amplitude
under assump- tions similar to those of Theorems 2 and 4.4. UPPER BOUNDS
ON THE TOTAL SCATTERING CROSS-SECTION
Let us define
firstly
thescattering
matrixS(K, g)
for theSchrodinger equation (1.1)
in a non-radial case. We assume here(1. 2).
Set X= V 11/2,
U =
sgn V, Ho
= - 0 andRo(z) = (Ho -
in the space ~ =L2(1R3).
Then. as is well-known, the operator function
is
analytic
in a Hilbert-Schmidtnorm ~ ~ . ~ ~ 2
and is continuous in thisnorm as z
approaches
the cutover ~ + . Moreover,
thepoint -
1 doesnot
belong
to the spectrum of the operatorgA(K2
±iO)U,
K > 0. Definenow on
L2(M~)
n the operatorZK:
Then the operator
ZKX
is extended to a compact operator from ~f to=
L2(~2)
andVol. 44, n° 2-1986.
206 A. V. SOBOLEV AND D. R. YAFAEV
which is a usual relation between a
spectral
measure andboundary
valuesof a resolvent.
Now the
scattering
matrixS(K, g)
can becorrectly
definedby
theequality
Note that the
right-hand
side of(4.2)
is aproduct
of bounded operators and henceS(K, g)
is a bounded operator in the space A.Moreover,
as iswell-known
(and
can beeasily
seen from(4.1))
the operatorS(K,g)
isunitary
in In terms ofS(K, g)
theaveraged
total cross-section is definedby (1.4),
where theright
hand side is finite under theassumption (1.2).
We need also the concept
of s-numbers,
orsingular
numbers(see
e. g.[15 ])
of a bounded operator A in a Hilbert space Jf. Let
%n
be a set of all n-dimen- sional operators. Thenby
definitionNote the
inequality
For a compact A the
singular
numbers,l(A)
is theeigenvalue (multi- plicities
taken intoaccount)
of apositive
compact operator(A * A)1/2.
Sincewe consider here not
only
compact A but also operators withcompact
difference A -I,
it is convenient to use the moregeneral
definition(4.3).
Moreover,
we need thefollowing
lemma which we were unable to find in the literature.LEMMA 3. 2014 Let A be compact, - 1 does not
belong
to itsspectrum
and sn + 1 Y 1. Thensn + 1 ((I
+A) -1 ) ( 1 - y) -1.
Proof
2014 The operator A can bedecomposed
as A =Kn
+Bm
whereKn~ %n and ~Bn~
y. It follows that I + A =(I
+Bn)(I
+Ln)
withOperators
I + A and are invertiblesimultaneously
and(I
+ I +Mn with Mn
= -LI
+Thus
( 1
+A) -1 - (I
+Mn)(I
+B,)’’
andby (4 . 4)
since +
Mn)
= 1according
to(4.3). ~
Now we can prove the bound
( 1.14).
Note thatwith A
belonging
to the Hilbert-Schmidt class iff theright-hand
side isfinite.
Annales de l’Institut Henri Poincare - Physique theorique
THEOREM 5. - Let the condition
( 1. 2)
with a > 2 be fulfilled. Then the bound( 1.14)
holds.Proof
2014 Denoteby sn(K, g)
the .s-numbers of the operatorS(K, g) -
I.Then
by
definition( 1. 4)
.
Recollect that
( -
A - K2 T is anintegral operator
with a kernel(47c! x - x’ ~ ) -1
exp( ±
and thuswith C
independent
of K. Letan(K)
:=sn(A(K2
+io)U).
The bound(4. 6)
ensures that
Choose now the number n so that
1/2
and considersums (4.5)
over m nand m ~
separately.
Sincesm(K, g)
2by unitarity
ofS(K, g) Cg2 by (4.7),
we have thatTo estimate the sum
(4.5)
over m ~ n we use the definition(4.2)
and theinequality
(4.4):By
the choice of n the s-numbersn(gA(K2
+1/2
and thusby
Lemma 3
It follows now that
Relations
(4.1)
and(4.6)
ensure that theright-hand
side here is boundedby Cg2. Combining
it with(4.8),
we conclude theproof.
IIRemark. 2014 The
proof
of Th. 5 shows that the bound(1.14)
isessentially
Vol. 44, n° 2-1986.
208 A. V. SOBOLEV AND D. R. YAFAEV
of an abstract nature.
Namely,
letS(~,)
be ascattering
matrix for somepair
ofself-adjoint
operatorsHo,
H =Ho
+ V(if Ho ~ 0,
~, = K2 andS(A)
=S(K, 1)
in ourprevious notation).
Then up to some technical assump- tionsif the
right-hand
side istmitc ;
here C is some universal constant.Moreover,
the same method
applies
to prove thatThe bound
(4 .10)
can becompared
with then= 1
Birman-Krein
inequality [16 ]
where the
integral
is taken over theabsolutely
continuous spectrum ofHo.
In contrast to
(4.11)
the bound(4.10)
holds true for a fixed value of aspectral parameter ~, (without averaging
over~).
Theproof of (4 .10) requires
_that ~, is not an
eigenvalue of H,
whereS(À)
ispossibly
notproperly
defined.However, (4.10)
holdsuniformly
in ~, as ~,approaches
the isolatedeigen-
value of H.
The bound
( 1.14)
can beconsiderably improved
if we assume fasterfall-off of
V(x)
atinfinity.
Here we reportonly
one result of this type.THEOREM 6. - Let V >_ 0 and let V have a compact support. Then the bound
( 1. 9),
withC(K) depending only
on K and a size of a support ofV,
holds.Proof.
2014 We startagain
from the definition(4.2),
where U = I now, and use thesimple identity
with
Note that the
right-hand
side of(4.12)
has the same structure as its left-hand side with
Yg playing
the role ofgX.
Thereforecopying
theproof
of Th. 5 one can establish that
Annales de l’Institut Henri Poincaré - Physique theorique
So it remains to prove
that !! YgRo(K2
+i0)Yg~2
is boundeduniformly
in g or
according
to(4.6),
thatfor some y > 1.
By
the Heinzinequality
it is sufficient to prove thatLet
now (
ECO(~3)
and~(x)
= 1 on a support of V.By commuting operators
ofmultiplications
in coordinate and momentumrepresentations
of ~fit is easy to prove the boundedness of the operator
(for
anyy).
Nowby
definition(4.13)
and therefore
since Tg ~ 0.
IIREMARK 1. 2014 Instead of the
identity (4.12) by
theproof
of(4.14)
wecould have used an abstract bound
(4. 9)
for thepair (Ho+I)’B (H o
+gV
+I) -1
combined with the invarianceprinciple.
REMARK 2. 2014 Th. 1 shows that the condition V > 0 of Th. 6 cannot be omitted.
Further results on the upper bounds for
6(K, g)
will be described else- where.ACKNOWLEDGMENTS
The second of authors
(D.
R.Y.)
expresses hisdeep gratitude
to Pro-fessors K.
Chadan,
M. Combescure and J. Ginibre for theirhospitality during
his stay inOr say,
where the final draft of the paper was written.[1] L. D. LANDAU, E. M. LIFSHITZ, Quantum mechanics, Moscow, Fizmatgiz, 1963 (Russian).
[2] W. HUNZIKER, Potential scattering at high energies, Helv. Phys. Acta, t. 36, 1963,
p. 838-858.
[3] V. S. BUSLAEV, Trace formulae and some asymptotic estimates of the resolvent kernel for the three-dimensional Schrödinger operator, Topics in Math. Physics, t. 1, 1966, p. 82-101 (Russian).
Vol. 44, n° 2-1986.
210 A. V. SOBOLEV AND D. R. YAFAEV
[4] A. JENSEN, The scattering cross-section and its Born approximation at high energies,
Helv. Phys. Acta, t. 53, 1980, p. 398-403.
[5] K. YAJIMA, The quasi-classical limit of scattering amplitude I. Finite range potentials.
Preprint. Tokyo, 1984.
[6] D. R. YAFAEV, The eikonal approximation for the Schrödinger equation, Uspehi
Mat. Nauk, t. 40, n° 5, 1985, p. 240 (Russian).
[7] W. AMREIN, O. PEARSON, A time-dependent approach to the total scattering cross- section, J. Phys. A : Math. Gen., t. 12, 1979, p. 1469-1492.
[8] V. ENSS, B. SIMON, Finite total cross-sections in nonrelativistic quantum mechanics, Commun. Math. Phys., t. 76, 1980, p. 177-209.
[9] D. R. YAFAEV, On the resonant scattering by a negative potential, Notes of Lomi seminars, t. 138, 1984, p. 184-193 (Russian).
[10] M. Sh. BIRMAN, D. R. YAFAEV, The asymptotics of the spectrum of the s-matrix
by a potential scattering, Dokl. Acad. Nauk USSR, t. 255, 1980, p. 1085-1087
(Russian).
[11] A. V. SOBOLEV, D. R. YAFAEV, Phase analysis for scattering by a radial potential,
Notes of Lomi seminars, t. 147, 1985, p. 155-178 (Russian).
[12] A. I. BAZ, A. M. PERELOMOV, Ya. B. ZELDOVITCH, Scattering, reactions and decay in
nonrelativistic quantum mechanics, Moscow, Nauka, 1971 (Russian).
[13] F. CALOGERO, Variable phase approach to potential scattering, New York, London, Academic Press, 1967.
[14] M. ABROMOWITZ, I. A. STEGUN, Handbook of mathematical functions, Dover Inc., 1972.
[15] I. C. GOHBERG, M. G. KREIN, Introduction à la théorie des opérateurs linéaires non auto-adjoints dans un espace hilbertien, Paris, Dunod, 1971.
[16] M. Sh. BIRMAN, M. G. KREIN, On the theory of wave operators and scattering opera- tors, Dokl. Acad. Nauk. USSR, t. 144, 1962, p. 475-478 (Russian).
(Manuscrit reçu le 7 août 1985.)
Annales de Henri Poincaré - Physique theorique