A NNALES DE L ’I. H. P., SECTION A
M ARY B ETH R USKAI
Improved estimate on the number of bound states of negatively charged bosonic atoms
Annales de l’I. H. P., section A, tome 61, n
o2 (1994), p. 153-162
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153
Improved estimate
onthe number of bound states of negatively charged bosonic atoms
Mary
Beth RUSKAI*Department of Mathematics,
University of Massachusetts-Lowell, Lowell, MA 01854, USA
Dedicated to , G. M. Zhislin in honor of his 60th Birthday
Ann. Inst. Henri Poincare,
Vol. 61, n° 2, 1994, Physique theorique
ABSTRACT. - It is shown that the number of bound states of an atom whose "electrons"
satisfy
bosonicsymmetry
conditions is bounded aboveby C,~ (log where, /~
andC,~
are constants, the nuclearcharge
Z >
ZR
for somme constantZa
and the number of "electrons" N satisfiesN > (1+03B2)Z+1
with/?
> 0. The constant 03BA isuniversal,
butZ03B2 depends
upon
{3
andC,~
isinversely proportional
to a power of(3.
On montre que Ie nombre d’etats lies d’un atome dont les
«
electrons
»seraient
desbosoas, est
bornesupérieurement par C03B2 (log Z)03BA
ou tb et
C f3
sont des constantes, pourvu que lacharge
nucleaire Z satisfasse Z >ZR
pour une certaine constanteZ03B2
et que Ie nombre N « d’électrons » satisfasse constante 03BA est universelle maisZ03B2 depend
de{3
etC f3
est inversementproportionnelle
a unepuissance positive
de~3.
* Supported by National Science Foundation Grants DMS-89-08125 and HRD-91-03315.
Annales de l’!nstitut Henri Poincaré - Physique theorique - 0246-0211
Vol. 61/94/OZJ$ 4.OU/0 Gauthier-Villars
154 M. B. RUSKAI
1. INTRODUCTION
In a recent paper,
Bach, Lewis,
Lieb andSiedentop [2]
studied thenumber of bound states, i.e. discrete
eigenvalues,
of the Hamiltonian foran atom
consisting
of a nucleus with infinite mass andcharge
Z and Nnegatively charged spinless
bosonsinteracting
with a Coulombpotential.
This
system
is describedby
the Hamiltonianrestricted to the
symmetric sub space
of(g)[L2(R~)]~ i.e.,
Theeigenfunctions
of H arerequired
to besymmetric
underinterchange
of coordinates of any two
particles.
If the number of bound statesof such a
system
is denoted as vb(N, Z), they
showed that when( 1
+/?)
Z + 1 N(1
+/?’) Z, then vb ( N, Z) CZk
where theconstants C and k may
depend
upon/3
and{3’. Although
their estimatesare neither
optimal
noreasily
extended tofermions,
this result issignificant
because it is the first
explicit
bound on the actual number of bound states ofa
multi-particle
atomicsystem.
Theonly previous
results gave conditionson Nand Z for which the number of bound states was zero,
finite,
or infinite.(See [9]
forreferences.)
In
[2], configuration
spaceR3N
is divided into tworegions,
an "innerball"
{x : II x 1100 R~
in which coordinates of all electrons are withina distance R from the
nucleus,
and an "outer ball" >R~
in .which at least one electron is further than a distance R from the
nucleus,
where x denotes x2, ... , sothat ~x~~
=max, They
thenshow that under suitable conditions there are no bound states
supported
onthe outer ball in the sense.
where
Eo ( N -1, Z )
in theground
state energy of H( N -1, Z ) .
With thisterminology,
the main result of Section 3 of[2]
can be stated as follows.THEOREM 1. -
If
the Hamiltonian H(N, Z)
has nobound
statessupported
on an outer ballof
radiusR,
then vb(N, Z)
Cn~
wheren =
r (cl Z2
+Z/ ~
x1)3/2 d3
x and Cl and C are constants.Annales de l’Institut Henri Poincare - Physique theorique
155
NUMBER OF BOUND STATES OF NEGATIVELY CHARGED BOSONIC ATOMS
If we now write R =
f (Z)/Z and f (Z)
>1,
thenso that Theorem 1 can be restated as
THEOREM 2. -
If the
Hamiltonian H(N, Z)
has no bound statessupported
on an outer ball
of radius R
=f (Z)/Z, then vb (N, Z) C f (Z)]3k for
some constant C.
In
[2]
it was shown that thehypothesis
of the theorem could be satisfiedfor N in the range
(1 + /3)
Z + 1 N( 1
+/3’)
Z andsufficiently large
Z with R = = which
yields
vb(N, Z)
In thisnote we show that it is
possible
to find constantsC {3
andZ~
such thatthe
hypothesis
is satisfied if Z >Z~
and R =C {3 log Z/Z,
whichimplies
vb
( N, Z) C{3 ( log Z ) 3k . Moreover,
we show that this holdsprovided
N >
(1 + (3)
Z+1,
i.e. we show that the upper limit on N can beremoved,
and discuss the
dependence of vb (N, Z)
on/3.
Our final result can be stated as follow.THEOREM 3. - For all
/3
> 0 there existsZ03B2
such that whenever Z >Z03B2,
and N >
,Q Z,
thenZ) C (log Z)~//3~
where C and areconstants.
2. GENERAL REMARKS ON LOCALIZATION ERROR We now discuss some heuristic about localization error.
Except
forSimon’s enhancement of the
"Sigal-Simon
localization trick" the contents of this section are well-known.However,
becausecomputing
the localizationerror
precisely
is rathertedious,
it is useful to review someaspects
of itsgeneral
behavior. Since details are available in the literature(and
one canalways
check aspecific
caseexplicitly by differentiation),
we concentrateon heuristics here and refer the reader to
[3, 7, 12]
for further details.In many
problems
one wants to break upconfiguration
space intopieces,
i. e. one wants to consider -
However,
this leads to technical difficulties unless 03A8 = 0 on theboundary
of each In order to achieve this one needs a set of
localizing
functionsVol. 61, n° 2-1994.
156 M. B. RUSKAI
~
such that0 ~ F~ :::; 1,
supp(Fjb)
CH~
and2: F~==
1. then instead~
of
(3),
one hasThe
quantity
LE= i / 1 12 12
is called the localizationerror and there is term
containing i
and0394i
for eachLaplacian -~i
withcoefficient in the Hamiltonian H. Now suppose there are M
localizing
functions
Fk (i.e.
Jk = 1...M)
and Ngradient
terms(i.e.
i = 1...N).
Ifone is
dealing
with aproblem
withpermutational symmetry,
oneexpects
each of these terms to be
roughly
the same size(except possibly
for a fewexceptional
terms which will not affect thelarge M,
Nbehavior)
so thatone would
expect
the total localization error(LE)
to beproportional
toM N.
Therefore,
it is somewhatsurprising that,
at least for atomicproblems,
it is often
possible
to chooseFk
so that LE growsonly
like(log M)2 !
Before
sketching
the construction whichyields
thisbehavior,
we reviewa few other facts.
By
asimple scaling argument,
it is clear that LE isproportional
to1/(distance)2.
SinceF~
isusually
chosen so thatF~
= 1 onmost of is 0 outside and decreases
smoothly
from 1 to 0 near theboundary
of it will have a non-zero derivativeonly
near theboundary.
Now one
typically
chooses the localization so that the boundaries ofSZ~
are
precisely
theplaces
where certain critical distances areequal,
or atleast
comparable
in awell-specified
sense.Therefore,
one has a naturalparameter
r sothat) ~i Fk|2
isproportional
to1 /r2 .
It is often easier to first find functions
Gk
with 0G~ 1
supp(Gk)
Cí!k
andG2k ~
1. One then definesFk
=and
easily
verifies thatNow one
typically
choosesG K
to have the formGjk ( ac )
== X(! x ~ ~ ~ ~ ~
where,
x is oftenreplaced by
aspecified
subset of .,~~some chosen value of
to .
Annales de l’Institut Henri Poincaré - Physique theorique
157
NUMBER OF BOUND STATES OF NEGATIVELY CHARGED BOSONIC ATO
We now show how to remove ’ the
dependence
" of LE on ~V. Since "Vi II
x!!p
=’
x1I~-1,
it follows that ifGk
has the aboveform,
thenfor i =I k
ifp>2.
SinceVi 0 =} II
xlip
>211
x one can choose eitherr or r
== II
xlip
asappropriate.
In either case,rather than
NC/r2.
The reason for theapparent disappearance
of theexpected growth
of LE with N isthat, especially
when p islarge, only
one derivative of the form
lip (i
= 1...N)
dominates in anyregion
of
R 3N.
Localizations with p-norms(and
p 2014~oo)
seems to have first been used in[7] (Zhislin [ 13]
and othershaving previously
used localizationswith p
==2);
Simon[3] subsequently
advocatedsacrificing
some smoothnessand
using
sup norms, for which it is somewhat clearerthat,
for a fixed&,
theregions
on whichVi G~ (i
= 1N)
is non-zero do notoverlap.
We now describe the
"Sigal-Simon
Localization Trick" forminimizing
the
dependence
of LE on the number oflocalizing
functions M. Letwhere a > 1
and 03C8 (t)
=1 -
4 t - 3j. [The precise
formof 03C8
and thecut-off
at -
are chosen forsimplicity
and are not essential.Moving
thecutoff
from -
toto
closer to 1 may be desirable for some purposes butwill increase the localization error
by
a factor of1/(1 - to)2.] With X given by (8)
Vol. 61, n° 2-1994.
158 M. B. RUSKAI
Now consider
(as
a functionof ~
>0)
theexpression
which has its maximum when
r~a
=~/4 (a - 1)
so thatIf one then uses
this x
to construct a set oflocalizing
functionsF~
viaG~
asabove,
one findsIf one now
chooses 11
=1 and combines this estimate with thoseabove,
one finds a net localization errorsatisfying
LEC a2 M1/a
1.
Writing
M =ba,
orequivalently, choosing a
=(log M)b
onegets
the final estimate LE C(log M)2
for some constant CThis trick was first used
by Sigal [ 11, 12]
in thespecial
case a =2,
whichyields
a localization errorproportional
toB/M (and
for which the bound in( 10)
can be obtainedby
asimple squaring argument).
The ideaof
reducing
this to M03C3 with 03C3 > 0arbitrarily
small is due toSimon;
itis sketched in
[12] (see
Sect.5)
and alluded to in[3] (see
the remark onp.
47)
and in[6, 9]; however,
the details have notpreviously appeared
inthe literature and the reduction to
log
M seems new.Finally
it is worthremarking
that thisanalysis
isessentially unchanged
if
Gk
is aproduct
ofa fixed
number of ~. Ifhowever,
one needs aproduct
of N
functions,
as in[4, 8],
thisapproach
does not work.3. SHRINKING THE SIZE OF THE INNER BALL
We now show how to
apply
these ideas toimprove
the estimates in[ 12] . Although
this involvesonly
the localization for the secondpartition
in Section 2 of
[2],
which we willsubsequently
show(see
Section 4Annales de l’Institut Henri Poincaré - Physique théorique
NUMBER OF BOUND STATES OF NEGATIVELY CHARGED BOSONIC ATOMS 159
below)
canactually
beeliminated,
theargument
issimple
and illustrates the power of theSigal-Simon
localization trick.Moreover,
the final estimates obtained here will beslightly
better than those in Section 4.Let
Jk~
= X(I
~~1/11
xk1100)
X(I
whereX~
denotes xwith ~~
removed, etc., x
is as in(8)
and thelocalizing
functions A andJn+i
are defined in[2].
We need hereonly
the facts that A and makea contribution of
C/&2 R2
toLE,
and that issupported
on aregion
where at least two "electrons" are outside a ball of radius R 8 so that
supp Jkl
C{x :
>k,
land (12)
where 8
,~
and R will be chosen later. Now the number ofJkl
isM=~V(~V-1)/2~~/2.
As discussed in Section 2above,
oneeasily
checks that the localization error from thispartition
satisfiesLE
Thus,
eq.(10)
of[2]
now can bereplaced by
thefollowing
estimate which is valid onwhere
10 (N - 1, Z) = Eo (N - 2, Z) - Eo (N - 1, Z)
denotes the io- nizationpotential
of the(N - 1) - st
"electron". If one now usesBach’s estimate
[1]
thatZ) >
for some constant M, it isthen evident that one can find constants
C {3
andZo
such that theright
side of
(13)
ispositive
if R =C{3 log Z/Z
and Z >Zo Using
this inTheorem 2
gives
the final boundof vb (N, Z) C{3 (log Z)3k
asexplained
in the introduction. Note that the choice R =
C {3 log Z/Z implies
thatI
>-2 Z/b R.
>-(constant) Z2/ log
Z. Since the LE termsare also of the
form -Z2/log Z,
it will be necessary to choose eitherC {3
or
Zo extremely large
to ensure thatthey
are dominatedby
Thus far we, like BLLS
[2],
haveignored
thequestion
of how Randv
(N, Z) depend
upon the lower limit/3. Using
the fact that one must have6 ~3
andchoosing & _ /3/2,
one finds that the estimates aboverequire
R >
Clog Z//?
Z.However,
thepositivity
of eq.(8)
of[2] requires
thestronger
condition R >Clog Z//~3
Z sothat / (Z)
=clog Z/~j3
whichyields
a net bound ofVol. 61, n° 2-1994.
160 M. B. RUSKAI
Since one must have
{3 {3c
~ 0.21 this estimate islarge
even near,Q~
(since {3;9
>l06!)
and becomes enormous as/?
2014~ 0.4. RANGE OF N-REMOVING THE UPPER BOUND
The upper limit of N
(1 + {3’)
Z in[2]
arises because of the need for the bound on the ionizationpotential Eo (N - 1, Z) - Eo (N - 2, Z) > ~ Z2,
which
requires
that N be in the range whereH (N, Z)
still has bound states.However,
the number of bound states isexpected
to decrease as Napproaches
its upper limit of~3~
~ ~ 1.21 Z so that this upper limit on N should not be necessary. We now show how to remove it.In
[2],
theiranalysis
of the outerregion begins
with a localization onwhich the effective
potential
of theelectron, ~~ = 2014Z/r~+Y~
2~~
is bounded below
(see
eq.(8)
of[2])
aswhich will be
positive only
if/3
> 8. Since this is valid on aregion
onwhich >
J~,
and since,C~ 0.21,
suchregions
will not coverthe outer
region (which
wouldrequire 8
>1)
and theresulting
localization will notgive
apartition
ofunity.
In order to treat theremaining region they
make an additional
partition
intoregions
on which r~. > ril,
as discussed in Section 3 above. In this
region they
need Bach’s estimateof Z2
on the ionizationpotential
whichrequires
N - Z,Q’
Z/3c
Z.In order to eliminate the second
partition,
wereplace
the estimate( 15) by
a more refined electrostatic estimate usedby Lieb, Sigal,
Simon andThirring [6]
togive
the firstproof
ofasymptotic neutrality. They
show thatfor any f > 0 there exists
NE
andregions SZ~
which coverR3N
when TV >NE
and on whichThey
also showed that one can find a localizationcorresponding
to theseregions
with L E In theterminology
ofSection 2
above,
the contribution of1/6~ corresponds
to1 / ( 1 - to ) 2;
the factor
Vii corresponds
toM1/a
when a = 2 and can bereplaced by (log N)2;
and the(log N)2 already present
is an additional factorAnnales de l’Institut Henri Poincaré - Physique théorique
NUMBER OF BOUND STATES OF NEGATIVELY CHARGED BOSONIC ATOMS 161
that arises because of a cut-off
parameter needed,
asexplained
in[6]).
Thus one can
improve
this bound(as already
remarked in[6])
toLE
C (log N)4/EZ ~~ x ~~~.
Then on the effectivepotential
and LE can be estimated as
where we have used the fact
[5]
that N 2 Z + 1(but
wereally only
need that N Z’n for some m toreplace log
Nby log Z).
The first term will bepositive
> E.Choosing
E =/?/2,
onecan conclude that LE > 0 if R > C
(log Z)4/~~3
Z.Using
this inTheorem 2 with
f (Z)
= C(log Z)4//~3 gives
a final bound ofvalid whenever
Z + 1 > N - Z > ,Q Z
and Z >Z~ .
Theprice
one pays forremoving
the restricctionN - Z ~3’ Z ,Q~ Z ~
0.21 Z is ahigher
power oflog
Z. A more seriousprice
is that we do not have any information about howdepend
upon/?.
Inaddition,
we have not been able toimprove
thedependence of vb ( N, Z)
on1//3,
even for,Q
nearIn
[2]
the restriction that theparticles
are bosons was used in the outerregion only
to estimate the ionizationpotential
Since we haveeliminated the need for
this,
theargument
above should also work forfermions,
and it does.Unfortunately,
itonly
works for N >/3
Z and wealready
know[4, 6, 10] that,
because fermionic atoms areasymptotically neutral,
vb( N, Z)
= 0 in thisregion,
i.e. for fermions theonly region
ofinterest is the very delicate
region
with 03C3 1.(By [ 10]
o-5/7.) Indeed,
the estimates sketched in this section combined with theeasily proved
fact that no bound states havesupport
such that all electrons lie within a ball of radius 0(7V’~)
suffice to proveasymptotic
neutrality
and this isessentially
theargument
in[6].
ACKNOWLEDGEMENTS
This work was done while the author was
visiting
theMittag-Leffler
Institute, University
ofTrondheim,
Ceremade(Universite Paris-IX),
andVol. 61, n° 2-1994.
162 M. B. RUSKAI
the
University
of Leiden. The author isgrateful
to these institutions for thehospitality
andsupport.
It is apleasure
to also thank Professor H.Siedentop
for
suggesting
that the authortry
toimprove
the localization error estimates in[2],
R. Brummelhuis forstimulating
discussions about Section4,
andB. Simon for
explaining
his localization trick to the author some years ago.[1] V. BACH, Ionization Energies of Bosonic Coulomb System, Lett. Math. Phys., Vol. 21, 1991, pp. 139-149.
[2] V. BACH, R. LEWIS, H. SIEDENTOP, On the Number of Bound States of a Bosonic N-Particle Coulomb System, Zeits f. Math., Vol. 214, 1993, pp. 441-460.
[3] H. L. CYCON, R. FROESE, W. KIRSCH, B. SIMON, Schrödinger Operators, Springer-Verlag,
1987.
[4] C. L. FEFFERMAN, L. A. SECO, Asymptotic Neutrality of Large Ions, Commun. Math. Phys.,
Vol. 128, 1990, pp. 109-130.
[5] E. H. LIEB, Bound on the Maximum Negative Ionization of Atoms and Molecules, Phys.
Rev. A, Vol. 29, 1984, pp. 3018-3028.
[6] E. H. LIEB, I. M. SIGAL, B. SIMON, W. THIRRING, Asymptotic Neutrality of Large-Z Ions, Commun. Math. Phys., Vol. 116, 1988, pp. 635-644.
[7] M. B. RUSKAI, Absence of Discrete Spectrum of Highly Negative Ions, Commun. Math.
Phys., Vol. 82, 1982, pp. 457-469.
[8] M. B. RUSKAI, Limits on Stability of Positive Molecular Ions, Lett. Math. Phys., Vol. 18, 1989, pp. 121-132.
[9] M. B. RUSKAI, Limit on the Excess Negative Charge of Dynamic Diatomic Molecule, Ann.
Inst. H. Poincaré A: Physique Théorique, Vol. 52, 1990, pp. 397-414.
[10] L. A. SECO, I. M. SIGAL, P. SOLOVEJ, Bound on the Ionization Energy of Large Atoms, Commun. Math. Phys., Vol. 131, 1990, pp. 307-315.
[11] I. M. SIGAL, Geometric Methods in the Quantum Many-Body Problem. Nonexistence of
Very Negative Ions, Commun. Math. Phys., Vol. 85, 1982, pp. 309-324.
[12] I. M. SIGAL, How Many Electrons Can A Nucleus Bind?, Ann. Phys., Vol. 157, 1984, pp. 307-320.
[13] G. M. ZHISLIN, On the Finiteness of the Discrete Spectrum of the Energy Operator of Negative Atomic and Molecular Ions, Teor. Mat Fiz, Vol. 7, 1971, pp. 332-341; Eng.
trans. Theor. Math. Phys., Vol. 7, 1971, pp. 571-578.
(Manuscript received May 14, 1993;
revised version received September 2, 1993. )
Annales de l’Institut Henri Poincare - Physique " theorique "