Estimates on the number of eigenvalues of two-particle discrete Schrödinger operators

0.14 MB·English

by  Sergio Albeverio

ESTIMATES ON THE NUMBER OF EIGENVALUES OF TWO-PARTICLE
DISCRETE SCHRO¨DINGER OPERATORS
SERGIOALBEVERIO1,2,3,SAIDAKHMATN.LAKAEV4,5,ANDJANIKULI.ABDULLAEV5
5
0
0 ABSTRACT. Two-particle discrete Schro¨dinger operators H(k) = H0(k)−V onthe
2 three-dimensional latticeZ3,kbeingthetwo-particle quasi-momentum,areconsidered.
n AnestimateforthenumberoftheeigenvalueslyingoutsideofthebandofH0(k)viathe
numberofeigenvalues ofthepotential operatorV biggerthanthewidthofthebandof
a
J H0(k)isobtained.TheexistenceofnonnegativeeigenvaluesbelowthebandofH0(k)is
provenfornontrivialvaluesofthequasi-momentumk ∈ T3 ≡ (−π,π]3,providedthat
2
theoperatorH(0)haseitherazeroenergyresonanceorazeroeigenvalue.Itisshownthat
1 theoperatorH(k),k ∈ T3,hasinfinitelymanyeigenvaluesaccumulatingatthebottom
ofthebandfrombelowifoneofthecoordinatesk(j),j=1,2,3,ofk∈T3isπ.
1
v
6
3 SubjectClassification: Primary:81Q10,Secondary:35P20,47N50
0 Keywords and phrases: Spectral properties, two-particle, discrete Schro¨dinger oper-
1
ators, number of eigenvalues, band spectrum, Birman-Schwinger principle, zero energy
0
resonance,zeroeigenvalue,low-lyingexcitationspectrum.
5
0
/ 1. INTRODUCTION
h
p Thestudyofthelow-lyingexcitationspectrumforlatticeHamiltoniansofsystemswith
-
h aninfinitenumberofdegreesoffreedomhasrecentlyattractedconsiderableattention(see,
t e.g.,[7,10,20]).
a
m Seealso[3,4,5,6,9,11,13,19]forgeneralexpositionsandthediscussionofparticular
problemsofthetheoryofdiscreteSchro¨dingeroperatorsonlattices,includingapplications
:
v tosolidstatephysics.
i Themainaimofthepresentpaperistoprovideathoroughanalysisofthedependence
X
ofthenumberofeigenvaluesofatwo-particlelatticeSchro¨dingeroperatorwithemphasis
r
a onitsnon-trivialdependenceonthetotalquasi-momentumandthresholdphenomena.
Inthelatticecase,similarlyasinthecontinuouscase,thetwo-bodyproblemreducesto
aone-bodyproblemthroughtheusualsplit-offofthecenterofmassandtheintroduction
ofthe relativecoordinates. As a result, the two-bodyenergyoperatoris representedas a
directvonNeumannintegral,wherethefiberHamiltonianH(k)dependsparametricallyon
theinternalbindingthequasi-momentumk ∈T3 ≡(−π,π]3 (see[3,4,5,6,8,13,19]).
Onecanobservethatthenumberofeigenvaluesoftheself-adjointperturbationA−V
ofanabstractboundedself-adjointoperatorAintheHilbertspaceHdependsonthewidth
ofthespectrumofthenon-perturbedoperatorA(seeDefinition3.1belowforthisconcept).
Inthepresentpaperweobtainanestimateforthenumberofeigenvaluesoftheperturbed
operator A− V lying outside of the essential spectrum σ (A − V) depending on the
ess
numbern (w (A),|V|)ofeigenvaluesoftheperturbationoperatorV,whicharelargerin
+ s
absolutevaluethanthewidthoftheessentialspectrumw (A)oftheoperatorA(Theorem
s
3.2).
Date:February7,2008.
1
2 SERGIOALBEVERIO1,2,3,SAIDAKHMATN.LAKAEV4,5,ANDJANIKULI.ABDULLAEV5
Asaconsequenceoftheseresultsweobtainanestimateforthenumberofeigenvalues
ofthetwo-particlediscreteSchro¨dingeroperatorH(k) = H (k)−V viathe numberof
0
eigenvaluesof the interactionoperatorV which are biggerthan the width w (H (k)) of
b 0
thebandspectrum(band)ofH (k)(Theorem4.2).
0
InthecaseofcontinuousSchro¨dingeroperatorsoneobservestheemergenceofnegative
eigenvaluesfromthebottomofthecontinuousspectrum.
Thisphenomenoniscloselyrelatedtotheexistenceofzero-energyresonancesandzero
eigenvaluesatthebottomofcontinuousspectrumofthetwo-particleSchro¨dingeroperators
(so-calledcriticalHamiltonians).
The appearance of negative eigenvalues for critical (non-negative)Schro¨dinger oper-
ators under infinitesimally small negative perturbationsis remarkable: the presence of a
zero-energy resonances in at least two of the two-particle operators (subsystems) leads
to the existence of infinitely many eigenvalues accumulating to the bottom of the three-
particlecontinuumforthecorrespondingthree-particleSchro¨dingeroperator(theEfimov
effect)(see,e.g.,[1,3,8,12,14,15,16,17]).
Thepresenceofeigenvaluesbelowthebottomofthecontinuousspectrumisespecially
remarkableforthethree-particlelatticeSchro¨dingeroperators(Hamiltonians)H (K),K ∈
3
T3: thepresenceinatleastoneofthetwo-particleoperators(subhamiltonians)impliesfor
H (K)thefinitenessofthenumberofeigenvaluesforallsufficientlysmallnonzerovalues
3
of the three-particle quasi-momentum K ∈ T3 even in the case, where all two-particle
subhamiltonianshaveazeroenergyresonance(see[3]).
If we comparethe results in the lattice case with those in the continuouscase we see
thatintheformercasethereisamechanism,whichcausestheemergenceofeigenvalues
fromthethresholdoftheHamiltonians,thismechanismhasnothingtodowithadditional
(effectivelynegative)perturbationsofthepotentialterm.Infactthemechanismisprovided
bythenontrivialdependenceoftheHamiltoniansH(k)onthequasi-momentumk ∈ T3.
We prove this fact using the lattice analogue of Birman-Schwinger principle (Theorem
6.6).
Thenumberofeigenvaluesoutsideofthebandofthetwo-particleSchro¨dingeroperators
essentiallydependsonthe widthw (·)ofthebandinthedirectionj = 1,2,3,(ej being
jb
theunitvectoralongthej-thdirectioninZ3).
WeestablishthattheoperatorH(k)hasinfinitelymanypositiveeigenvalueslyingout-
side of the band spectrum if width w (·) of the band in direction j vanishes for some
jb
j = 1,2,3 (Theorem 6.10). This corresponds to having an effective mass in direction
j,j =1,2,3,whichincreasesto+∞
Thepaperisorganizedasfollows.
InSection2wedescribethetwo-particleHamiltoniansinthecoordinaterepresentation
andtwo-particlediscreteSchro¨dingeroperatorsH(k)inthemomentumrepresentation.In
Section3weobtainanestimateforthenumberofeigenvaluesoftheperturbedoperatorA−
V lyingoutsideitsessentialspectrumbythenumberofV. InSection4weintroducethe
notions”widthoftheband”and”widthofthebandinthedirectionj”forthetwo-particle
discreteSchro¨dingeroperatorsH(k)=H (k)−V andproveTheorem4.2.InSection5we
0
provetheBirman-Schwingerprincipleforthetwo-particlediscreteSchro¨dingeroperators
H(k). In Section 6 we introduce the notion of a zero-energyresonance for two-particle
discreteSchro¨dingeroperatorsH(k)andproveTheorems6.6and6.10.
ESTIMATESONTHENUMBEROFEIGENVALUESOFTWO-PARTICLEDISCRETESCHRO¨DINGER... 3
2. ENERGYOPERATORS OF TWO PARTICLESON A LATTICEIN THECOORDINATE
REPRESENTATIONAND THE TWO-PARTICLEDISCRETE SCHRO¨DINGER OPERATORS
ThefreeHamiltonianHˆ ofthesystemoftwoquantumparticlesonthethree-dimensional
0
latticeZ3 isdefinedbythefollowing(bounded)self-adjointoperatorontheHilbertspace
ℓ2((Z3)2)
1 1
Hˆ = ∆ + ∆ ,
0 2m x1 2m x2
1 2
with∆ = ∆⊗I and∆ = I ⊗∆,wherem > 0denotesthemassoftheparticleα
x1 x2 α
andI istheidentityoperatoronℓ2(Z3).
The Laplacian ∆ is a difference operator which describes the transport of a particle
fromasitetothenearestneighboringsite,i.e.,
(∆ψˆ)(x)= [ψˆ(x)−ψˆ(x+s)], ψˆ∈ℓ2(Z3).
X
|s|=1
Thetotaltwo-particleHamiltonianHˆ ontheHilbertspaceℓ2((Z3)2)describesthein-
teractionbetweentwoparticles
Hˆ =Hˆ −Vˆ,
0
where
(Vˆψˆ)(x ,x )=vˆ(x −x )ψˆ(x ,x ), ψˆ∈ℓ2((Z3)2),
1 2 1 2 1 2
andvˆisarealboundedfunctiononZ3.
Assumption2.1. Assumethatvˆ(s)isevenfunctiononZ3verifying
lim |s|3+κvˆ(s)=0, κ>1/2.
|s|→∞
Under above assumptions the two-particle Hamiltonian Hˆ is a well defined bounded
self-adjointoperatoronℓ2((Z3)2).
Recall that the study of spectral properties of the two-particle Hamiltonian Hˆ in the
momentum representation reduces to the spectral analysis of the two-particle discrete
Schro¨dingeroperators H(k), k ∈T3,actingintheHilbertspaceL2(T3)(see[3,8,4]
(2.1) H(k)=H (k)−V.
0
Thenon-perturbedoperatorH (k)isthemultiplicationoperatorbythefunctionE (q)
0 k
onL2(T3)
(H (k)f)(q)=E (q)f(q), f ∈L2(T3),
0 k
where
1 1 1 1
E (q)= ε( k+q)+ ε( k−q),
k
m 2 m 2
1 2
3
ε(q)= (1−cosq(j)), q =(q(1),q(2),q(3))∈T3.
X
j=1
Theinteraction(perturbation)operatorV actsontheHilbertspaceL2(T3)by
(Vf)(q)=(2π)−3/2 v(q−s)f(s)ds, f ∈L2(T3).
Z
T3
HerethekernelfunctionvisgivenbytheFourierseries
v(q)=(2π)−3/2 vˆ(s)ei(q,s),
sX∈Z3
4 SERGIOALBEVERIO1,2,3,SAIDAKHMATN.LAKAEV4,5,ANDJANIKULI.ABDULLAEV5
with
3
(q,s)= q(j)s(j), q =(q(1),q(2),q(3))∈T3, s=(s(1),s(2),s(3))∈Z3.
Xj=1
WeseethattheoperatorH(k),k ∈ T3,doesnotsplitintothesumofacenter-of-mass
term and a relative kinetic energy term contrary to the continuum case, where one as is
center-of-mass.
3. ON THENUMBER OF EIGENVALUESOF BOUNDED SELF-ADJOINTOPERATORS
Weestablishsomeauxiliaryresultsonthenumberofeigenvaluesforanabstractopera-
toronaHilbertspaceH.
Foraboundedself-adjointoperatorAinaHilbertspaceH,wedefinen (µ,A)(resp.
+
n (µ,A))as
−
n (µ,A)=sup{dimL:L⊂H;(Af,f)>µ,f ∈L,||f||=1}
+
resp.
n (µ,A)=sup{dimL:L⊂H;(Af,f)<µ,f ∈L,||f||=1}
−
Thevaluen (µ,A)(resp.n (µ,A))isequaltotheinfinityifµisintheessentialspectrum
+ −
andn (µ,A)(resp. n (µ,A))isfinite,itisequaltothenumberoftheeigenvaluesofA
+ −
bigger(resp. smaller)thanµ.
Nowweintroducethenotionof”widthofthespectrumoftheself-adjointoperatorA”.
Definition3.1. ThewidthoftheoperatorA,denotedbyw (A),isdefinedtobe
s
w (A)=M(A)−m(A),
s
whereM(A)=sup (Af,f)andm(A)=inf (Af,f).
kfk=1 kfk=1
Theorem3.2. LetAbeaboundedself-adjointoperatorandV beacompactself-adjoint
operatorinaHilbertspaceH.Then
n (m(A),A−V)≥n (w (A),V) resp. n (M(A),A−V)≥n (−w (A),V) .
− + s + − s
(cid:0) (cid:1)
Proof. LetAbeascalaroperator,i.e.,A = µI,µ ∈ R1.Thenw (A) = 0andhencewe
s
have
n (µ,A−V)=n (0,V).
− +
NowletAbeanon-scalaroperator,i.e.,A 6=µI.Thenw (A) >0.SinceV isacom-
s
pactoperatorthe numbern (w (A),V) is finite. Let H (V) be the subspace
+ s (ws(A),+∞)
generatedbytheeigenfunctionsofV associatedwiththeeigenvaluesbiggerthanw (A).
s
Thenn (w (A),V)=dimH (V)andforanyf ∈H (V),||f||=1
+ s (ws(A),+∞) (ws(A),+∞)
thefollowingrelations
(Af,f)−(Vf,f)<M(A)−w (A)=m(A)
s
hold.Thisimpliesthat
n (m(A),A−V)≥dimH (V)=n (w (A),V).
− (ws(A),+∞) + s
Similarlyweget
n (M(A),A−V)≥n (−w (A),V).
+ − s
(cid:3)
Corollary3.3. Let A bea boundedself-adjointoperatorandV be compactself-adjoint
operatorinH.Then
n (m(A),A−V)+n (M(A),A−V)≥n (w (A),|V|).
− + + s
ESTIMATESONTHENUMBEROFEIGENVALUESOFTWO-PARTICLEDISCRETESCHRO¨DINGER... 5
Proof. Itissufficienttonoticethatthesingularvaluesofa compactself-adjointoperator
V arepreciselytheeigenvaluesof|V|. (cid:3)
4. ON THENUMBEROF EIGENVALUESOF THETWO-PARTICLE DISCRETE
SCHRO¨DINGER OPERATORSH(k),k ∈T3
UnderAssumption2.1theperturbationV oftheoperatorH (k)isatraceclassoperator
0
andthereforeinaccordancewiththewellknowninvarianceofthe absolutelycontinuous
spectrumundertraceclassperturbationstheabsolutelycontinuousspectrumoftheoperator
H(k)definedby(2.1)fillsinthefollowingintervalontherealaxis:
σ (H(k))=[E (k),E (k)],
ac min max
where
E (k)= minE (q), E (k)=maxE (q).
min k max k
q∈T3 q∈T3
NowwewillapplytheresultsobtainedinSection3tothetwo-particlediscreteSchro¨dinger
H(k). WerecallthattheoperatorH (k)hasaband.
0
Letusintroducethenotion”widthofthebandofH (k).”
0
Definition4.1. ThewidthofthebandofH (k),denotedbyw (H (k)),isdefinedtobe
0 b 0
w (H (k))≡maxE (q)−minE (q).
b 0 k k
q∈T3 q∈T3
Theorem4.2. (i)Foranyk ∈T3 thefollowinginequalityholds
n (E (k),H(k))≥n (w (H (k)),V).
− min + b 0
(ii)Letm =m andk =π˜ =(π,π,π). Thentheequalitieshold
1 2
6 6
n ( ,H(π˜))=n (0,V) and n ( ,H(π˜))=n (0,V).
− + + −
m m
Proof. Thepart(i)ofTheorem4.2isaconsequenceofTheorem3.2. Fromthecondition
(ii)ofTheorem4.2weconcludethatE (p) = 6(m)−1,hencethatw (H (π˜)) = 0,and
k b 0
finallythatH (π˜)=6(m)−1I isascalaroperator.Thusthenumbers{6 −vˆ(x),x∈Z3}
0 m
areeigenvaluesoftheoperatorH(π˜). (cid:3)
TheCorollary3.3yieldsimmediatelythefollowing
Corollary4.3. Foranyk ∈T3 thefollowinginequality
n (E (k),H(k))+n (E (k),H(k))≥n (w (H (k)),|V|)
− min + max + b 0
holds.
(cid:3)
Now we introduce the notion ” width of the band spectrum of H (k) in the direction
0
j =1,2,3”.
Definition4.4. Letk = (k(1),k(2),k(3)) ∈ T3.Bythewidthw (k(j))ofthebandspec-
jb
truminthedirectionj =1,2,3ofH (k)wemean
0
w (k(j))≡ max E (q)− min E (q).
jb k k
q(j)∈(−π,π] q(j)∈(−π,π]
6 SERGIOALBEVERIO1,2,3,SAIDAKHMATN.LAKAEV4,5,ANDJANIKULI.ABDULLAEV5
Lemma 4.5. The width of the band spectrum in the direction w (k(j)), j = 1,2,3,
jb
dependsonlyonk(j) ∈(−π,π]andtheequality
3
w (H (k))= w (k(j))
b 0 jb
X
j=1
holds.
Proof. ThefunctionE (p)canberewrittenas
k
3
1 1
(4.1) E (p)=3( + )− (a(k(j))cosp(j)+b(k(j))sinp(j)),
k
m m
1 2 X
j=1
wherethecoefficientsa(k(j))andb(k(j))aregivenby
1 1 k(j) 1 1 k(j)
a(k(j))=( + )cos , b(k(j))=( − )sin .
m m 2 m m 2
1 2 1 2
Theequality(4.1)impliesthefollowingrepresentationforE (p)
k
3
1 1
(4.2) E (p)=3( + )− r(k(j))cos(p(j)−p(k(j))),
k
m m
1 2 X
j=1
where
b(k(j))
r(k(j))= a2(k(j))+b2(k(j)), p(k(j))=arcsin , k(j) ∈(−π,π].
q r(k(j))
FromDefinition4.4ofthewidthw (k(j))ofthebandinthedirectionj =1,2,3andthe
jb
representation(4.2)forE (p)weconcludethat
k
3
w (k(j))=2r(k(j)) and w (H (k))= 2r(k(j)).
jb b 0
X
j=1
(cid:3)
TheproofofLemma4.5impliesthefollowing
Corollary4.6. (i)Thefunctionsw (k(j))≡2r(k(j)),j =1,2,3,arerealanalytic,even
jb
andpositiveontheinterval(−π,π).
(ii)Theequality
w (k(j))=0
jb
holdsifandonlyifm =m =mandk(j) =πforsomej =1,2,3.
1 2
5. THEBIRMAN-SCHWINGER PRINCIPLE FOR THETWO-PARTICLEDISCRETE
SCHRO¨DINGER OPERATORSH(k),k ∈T3
Assumption5.1. AssumethattheinteractionoperatorV ispositive.
Since the operator Vˆ is the multiplication operator by the positive function vˆ(s) its
positiverootVˆ12 is the multiplicationoperatorby vˆ21(s). Hence the positiverootV 12 of
thepositiveoperatorV hasform
1
(V 21f)(p)= v12(p−p′)f(p′)dp′,
(2π)23 ZT3
ESTIMATESONTHENUMBEROFEIGENVALUESOFTWO-PARTICLEDISCRETESCHRO¨DINGER... 7
wherethekernelfunctionv12(p)istheinverseFouriertransformofthefunctionvˆ12(s),i.e.,
1
v21(p)= vˆ12(s)ei(p,s).
(2π)32 sX∈Z3
We define for any k ∈ (−π,π)3 and z ≤ E (k) and also for any k ∈ T3 \
min
(−π,π)3 and z < E (k) the integral operator G(k,z)(cid:0)resp. G (k,z) with the ker-
min 1
2
nelG(k,z;p,q)resp. G (k,(cid:1)z;p,q)
1
2
1
(5.1) G(k,z;p,q)= (2π)3 Z v21(p−t)(Ek(t)−z)−1v12(t−q)dt
T3
resp.
1
G21(k,z;p,q)= (2π)32v21(p−q)(Ek(q)−z)−21.
TheproofofthefollowingvariantoftheBirman-Schwingerprinciplefortwo-particle
discreteSchro¨dungeroperatorH(k)followssimilarlinesasinthecaseofquantumparti-
clesmovingonR3.
Lemma 5.2. Forany k ∈ T3 andz < E (k) the operatorG(k,z)actson L2(T3), is
min
positiveandtheequality
(5.2) n (z,H(k))=n (1,G(k,z))
− +
holds.
NowwearegoingtoobtainageneralizationoftheBirman-Schwingerprincipleforthe
two-particleSchro¨dingeroperatorsonthelatticeZ3.
Theorem5.3. Foranyk ∈(−π,π)3 theoperatorG(k,E (k))actsonL2(T3),isposi-
min
tive,traceclassandtheequality
n (E (k),H(k))=n (1,G(k,E (k)))
− min + min
holds.
Proof. Foranyz ≤E (k)theoperatorG (k,z)isHilbert-Schmidt.Sincetheequality
min 1
2
G(k,z)=G (k,z)(G (k,z))∗
1 1
2 2
holdstheoperatorG(k,z)ispositiveandbelongstothetraceclass. Itiseasytoshowthat
thereexistsC >0sothattheinequality
||G(k,E (k))−G(k,z)||≤C E (k)−z
min min
p
holds.
Letusshowthatn (E (k),H(k)) = n (1,G(k,E (k))).SinceG(k,E (k))is
− min + min min
acompactoperatorthe numbern (1,G(k,E (k))) isfinite. Foranyψ ∈ L2(T3) and
+ min
z <E (k)thefollowingrelations
min
|V1/2ψ(p)|2dp |V1/2ψ(p)|2dp
(G(k,z)ψ,ψ)= ≤ =(G(k,E (k))ψ,ψ)
Z E (p)−z Z E (p)−E (k) min
k k min
T3 T3
hold.
Consequentlywehave
(5.3) n (1,G(k,z))≤n (1,G(k,E (k))).
+ + min
8 SERGIOALBEVERIO1,2,3,SAIDAKHMATN.LAKAEV4,5,ANDJANIKULI.ABDULLAEV5
Let λ ≥ λ ≥ ... ≥ λ > 1 be the eigenvalues and ψ ,ψ ,...,ψ the correspond-
1 2 n 1 2 n
ingeigenfunctionsof the operatorG(k,E (k)). Denoteby H the subspacegenerated
min n
by the eigenfunctions ψ ,ψ ,...,ψ . Since for all z ∈ U (E (k)) = (E (k) −
1 2 n δ min min
δ,E (k)],δ >0sufficientlysmall,theinequality
min
kG(k,z)−G(k,E (k))k<λ −1
min n
holdsforanyψ ∈H ,wehave
n
(5.4)
(G(k,z)ψ,ψ)=(G(k,E (k))ψ,ψ)−([G(k,E (k))−G(k,z)]ψ,ψ)>(ψ,ψ).
min min
From(5.4)weobtainthat
(5.5) n (1,G(k,z))≥dimH =n (1,G(k,E (k)))
+ n + min
foranyz ∈U (E (k)),δ >0sufficientlysmall.
δ min
Combining(5.3)and(5.5)weobtain
(5.6) n (1,G(k,z))=n (1,G(k,E (k)))
+ + min
forallz ∈U (E (k)).
δ min
ByBirman-Schwinger’sprinciple(Lemma5.2)andtheequality(5.6)wehave
(5.7) lim n (z,H(k))=n (E (k),H(k)).
− − min
z→Emin(k)−0
Takingintoaccount(5.6)and(5.7)weobtaintheequality(5.2). (cid:3)
6. THRESHOLDANALYSIS OFH(k),k ∈T3
Proposition6.1. AssumeAssumption2.1. ThentheoperatorH(0)hasanontrivialkernel
ifandonlyiftheintegraloperatorG(0,0)hastheeigenvalueλ=1andthecorresponding
eigenfunctionψ ∈L2(T3)satisfiesthecondition
1
v2(t)ψ(t)dt =0.
Z
T3
Proof. See[4]. (cid:3)
Definition 6.2. Let Assumption 2.1 be fulfilled. The operator H(0) is said to have a
zero energyresonanceifthe integraloperatorG(0,0)hasthe eigenvalueλ = 1 andthe
associatedeigenfunctionψsatisfiesthecondition
(v1/2,ψ)= v1/2(p)ψ(p)dp6=0.
Z
T3
Remark6.3. Ourdefinition6.2ofazeroenergyresonanceisadirectanalogueofthatin
thecontinuouscase(see,e.g.,[2],[14],[16]and[18]andreferencestherein).
Remark6.4. IftheHamiltonianH(0)hasazeroenergyresonancethenthefunction
(V 21ψ)(p)
f(p)=
E (p)
0
obeystheequation
H(0)f =0
andf belongstoLr(T3),1≤r <3/2.
Thefollowinglemmaexpressesanimportantfeatofthetwo-particleSchro¨dingeroper-
ators.
ESTIMATESONTHENUMBEROFEIGENVALUESOFTWO-PARTICLEDISCRETESCHRO¨DINGER... 9
Lemma6.5. Letm =m =mandletH(0)bepositive. Thenforallk ∈T3,k 6=0the
1 2
operatorH(k)ispositive.
Proof. Since
1 k k
E (p)= ε( −p)+ε( +p)
k
m 2 2
(cid:0) (cid:1)
and v(p) are even functions, the subspace L2(T3) resp. L2(T3) of the even resp. odd
e o
functionsofL2(T3)isaninvariantsubspacefortheoperatorH(k).
Thenforanyf ∈L2(T3)wehave
(6.1) (H(k)f,f)=(H(k)f ,f )+(H(k)f ,f ),
e e o o
where
f(p)+f(−p) f(p)−f(−p)
f (p)= , f (p)= .
e o
2 2
Weremarkthatf ∈L2(T3)andf ∈L2(T3).Thereforemakingachangeofvariables
e e o o
onther.h.sof(6.1)wehave
(H(k)f,f)=(H(0)f ,f )+(H(0)f ,f )≥0 for f 6=0,
ek ek ok ok
where
k k
f (p)=f ( −p), f (p)=f ( −p).
ek e ok o
2 2
Wenotethatf , f ∈L2(T3). (cid:3)
ek ok
Ourmainnonperturbativeresultisthefollowing
Theorem6.6. Letm =m =mandassumethattheoperatorH(0)ispositive.Assume
1 2
moreoverthatH(0)hasazeroeigenvalueofmultiplicitynandazeroenergyresonance.
Then for all nonzero k ∈ (−π,π)3 the operator H(k) has at least n + 1 nonnegative
eigenvalueslyingbelowthebottomE (k)ofthebandspectrumofH(k).
min
Proof. UndertheassumptionsofTheorem6.6theequationG(0,0)ψ =ψhasn+1solu-
tionsψ ,ψ ,...,ψ intheHilbertspaceL2(T3). Letψ ∈ L2(T3)beoneofthesolutions.
0 1 n
Then
(E (p))−1|(V1/2ψ)(p)|2dp=(G(0,0)ψ,ψ)=(ψ,ψ).
Z 0
T3
Foranynonzerok ∈(−π,π)3 andp6=0theinequalities
0<E (p)−E (k)<E (p)
k min 0
hold.Therefore,bythedefinition(5.1)oftheoperatorG(k,E (k))wehave
min
|(V1/2ψ)(p)|2dp |(V1/2ψ)(p)|2dp
(ψ,ψ)= < =(G(k,E (k))ψ,ψ)
Z E (p) Z E (p)−E (k) min
0 k min
T3 T3
forallnonzerok ∈(−π,π)3.Thismeans
n (1,G(k,E (k)))≥n+1.
+ min
ApplyingtheBirman-Schwingerprincipleweconcludethatforallnonzerok ∈ (−π,π)3
theoperatorH(k)hasatleastn+1eigenvalueslyingbelowthebottomoftheessential
spectrum. SincebyLemma6.5theoperatorH(k)ispositivewehavethatalleigenvalues
ofH(k)arenonnegative.
(cid:3)
10 SERGIOALBEVERIO1,2,3,SAIDAKHMATN.LAKAEV4,5,ANDJANIKULI.ABDULLAEV5
Corollary6.7. Letm =m =mandassumethattheoperatorH(0)ispositive.Assume
1 2
thatH(0)hasazeroeigenvalueofmultiplicityn. Thenforallnonzerok ∈ (−π,π)3 the
operatorH(k)hasatleastnnonnegativeeigenvalueslyingbelowthebottomE (k)of
min
thebandspectrumofH(k).
Remark6.8. WenotethattheinteractionoperatorV ispositiveandhenceH(k)hasno
eigenvaluebiggerthanE (k).
max
Remark6.9. AresultrelatedtoTheorem6.6hasbeenobtainedin[4]. Inthelatterrefer-
enceonlyexistenceofeigenvaluesbelowE (k)hasbeeninfactproven,usingadifferent
min
method,howeverit wasnotproventhatthese eigenvaluesare nonnegative. See also [4]
forexamples,whichinparticularshowtheexistenceofazero-energyresonanceandzero-
eigenvalueofmultiplicity2or3.
Theorem6.10. Assumem = m andk(j) = π forsomej = 1,2,3.Thenw (π) = 0
1 2 jb
andthefollowinginequality
n (E (k),H(k))≥card{s∈Z1 :vˆ(sej)>0}
− min
holds.
Proof. Under the assumptions of Theorem 6.10 the function E (p) does not depend on
k
p(j) and w (π) = 0. Note that for any x ∈ Z3 the value vˆ(x) is an eigenvalue of the
jb
operatorV.Letℵ ={s∈Z1 :vˆ(sej)>0}.Thentheassociatedeigenfunctions
j
ψ (p(j))=(2π)−3/2eisp(j),s∈ℵ ,
s j
of the operatorH(k) dependonly on p(j). Let H be the n(n ≤ card ℵ )-dimensional
n j
subspaceoftheHilbertspaceL2(T3)generatedbytheeigenfunctions{ψ (p(j))}n ,s ∈
si i=1 i
ℵ .ThenthesubspaceH isinvariantwithrespecttotheoperatorV1/2andtheequality
j n
inf kV 21ψk2 = inf (V 12ψ,V 21ψ)= inf (Vψ,ψ)= min vˆ(siej)kψk2
ψ∈Hn ψ∈Hn ψ∈Hn 1≤i≤n
holds.Nowwewillprovethatforanynaturaln≤cardℵ thereisz <E (k)suchthat
j n min
(6.2) inf (G(k,z )ψ,ψ)>1.
n
ψ∈Hn,kψk=1
ByBirman-Schwingerprinciplethismeans
n (E (k),H(k))≥n (1,G(k,z ))≥cardℵ .
− min + n j
Indeed
inf (G(k,z)ψ,ψ)=
ψ∈Hn,kψk=1
dp
||ψ||=i1n,fψ∈HnZT1|(V 12ψ)(p(j))|2dp(j)ZT2(Ek(p)−z)−1dp(j) =
1 dp(1)dp(2)dp(3)
= min vˆ(s ej) (E (p)−z)−1 .
(2π)2 1≤i≤n i ZT2 k dp(j)
Onecancheck
dp(i)dp(l) 2πlog(E (k)−z)
min
lim = lim (C (k)− )=+∞,
z→Emin(k)−0ZT2 Ek(p)−z z→Emin(k)−0 0 cosk(i) cosk(l)
q 2 2
wherei,l = 1,2,3,i 6= l,j 6= l,i 6= j andC (k)isa positivenumber. Thereforethere
0
existsanumberz <E (k)satisfyinginequality(6.2). (cid:3)
n min