Dispersive estimates for Schroedinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

0.44 MB·English

by  Mehmet Burak Erdogan

DISPERSIVE ESTIMATES FOR SCHRO¨DINGER OPERATORS IN THE
PRESENCE OF A RESONANCE AND/OR AN EIGENVALUE AT ZERO
6 ENERGY IN DIMENSION THREE: II
0
0
2 M.BURAKERDOG˘ANANDWILHELMSCHLAG

n
a
J Abstract. We investigate boundedness of the evolution eitH inthe senseof L2(R3)→L2(R3)as
6 wellasL1(R3)→L∞(R3)forthenon-selfadjointoperator

−∆+µ−V1 −V2
] H=
P " V2 ∆−µ+V1 #
A
where µ>0 and V1,V2 are real-valued decaying potentials. Such operators arise when linearizing
.
h afocusingNLSequationaroundastandingwaveandtheaforementionedboundsareneededinthe
t study of nonlinear asymptotic stability of such standing waves. We derive our results under some
a
m naturalspectralassumptions(correspondingtoagroundstatesolitonofNLS),seeA1)–A4)below,
[ but without imposing any restrictions on the edges ±µ of the essential spectrum. Our goal is to

developan“axiomaticapproach”,whichfreesthelineartheoryfromanynonlinearcontextinwhich
2
itmayhavearisen.
v
5
8
5
4
1. The matrix case: Introduction
0
5
0 Considerthe Schr¨odingeroperatorH = ∆+V inR3,where V is areal-valuedpotential. LetPac
−
h/ be the orthogonalprojection onto the absolutely continuous subspace of L2(R3) which is determined
at by H. In Journ´e, Soffer, Sogge [JouSofSog], Yajima [Yaj1], Rodnianski, Schlag [RodSch], Goldberg,
m Schlag [GolSch] and Goldberg [Gol], L1(R3) L (R3) dispersive estimates for the time evolution
∞
→
v: eitHPac were investigated under various decay assumptions on the potential V and the assumption
i
X that zero is neither an eigenvalue nor a resonance of H. Recall that zero energy is a resonance iff
ar there is f ∈ L2,−σ(R3)\L2(R3) for all σ > 12 so that Hf = 0. Here L2,−σ = hxiσL2 are the usual
weighted L2 spaces and x :=(1+ x2)12. For a survey of recent work in this area see [Sch2].
h i | |
In [ErdSch], the authors investigated dispersive estimates when there is a resonance or eigenvalue
atenergyzero. Itiswell-known,seeRauch[Rau],Jensen,Kato[JenKat],andMurata[Mur],thatthe
decay in that case is t−12. Moreover, these authors derived expansions of the evolution into inverse
powers of time in weighted L2(R3) spaces. In [ErdSch], the authors obtained such expansions with
respect to the L1 L∞ norm, albeit only in terms of the powers t−12 and t−23. Independently,
→
Yajima [Yaj2] achieved similar results.
Date:February1,2008.
ThisworkwasinitiatedinJuneof2004,whilethefirstauthorvisitedCaltechandhewishestothankthatinstitution
foritshospitalityandsupport. Thefirstauthor waspartiallysupportedbytheNSFgrantDMS-0303413. Thesecond
authorwaspartiallysupportedbyaSloanfellowshipandtheNSFgrantDMS–0300081. TheauthorsthankAvySoffer
forhisinterestinthiswork.
1
2 M.BURAKERDOG˘ANANDWILHELMSCHLAG
In this paper we obtain analogous expansions for a class of matrix Schro¨dinger operators.
Consider the matrix Schr¨odinger operator
∆+µ 0 V V
1 2
= +V = − + − −
0
H H " 0 ∆ µ # " V2 V1 #
−
on L2(R3) L2(R3). Here µ>0 and V , V are real-valued. It follows from Weyl’s criterion that the
1 2
×
essential spectrum of is ( , µ] [µ, ). The discrete spectrum may intersect C R, and the
H −∞ − ∪ ∞ \
algebraic and geometric multiplicities of eigenvalues may be different (i.e., has a nonzero nilpotent
H
part at these eigenvalues).
Such operators appear naturally as linearizations of a nonlinear Schr¨odinger equation around a
standing wave (or soliton), see below. Dispersive estimates in the context of such linearizations
were obtained in Cuccagna [Cuc], Rodnianski, Schlag, Soffer [RodSchSof1], and [Sch1] under various
decay assumptions on the potential and the assumption that zero is neither an eigenvalue nor a
resonance of . In addition, one always assumes that there are no imbedded eigenvalues in the
H
essential spectrum.
The emphasis of the present paper is to develop an ”abstract” (or ”axiomatic”) approach, which
frees the linear theory from any reference to a nonlinear context in which it may have arisen. More
specifically, our results will require the following assumptions on (in what follows, σ is one of the
3
H
Pauli matrices, see (14)):
Assumptions:
A1) σ V is a positive matrix
3
−
A2) L := ∆+µ V +V 0
1 2
− − − ≥
A3) For some β >0,
(1) V1(x) + V2(x) . x −β
| | | | h i
A4) There are no imbedded eigenvalues in ( , µ) (µ, )
−∞ − ∪ ∞
AssumptionsA1)-A3)holdintheimportantexampleofalinearizednonlinearSchr¨odingerequation,
provided the linearization is performed around the (positive) ground state standing wave. Indeed,
suppose that ψ(t,x)=eitα2φ(x) is a standing wave solution of the NLS
i∂ ψ+∆ψ+ ψ 2βψ =0,
t
| |
where β >0. Here we assume that φ is a ground state, i.e.,
α2φ ∆φ=φ2β+1, φ>0.
−
Is known that such φ exist and that they are radial, smooth, and exponentially decaying, see
Strauss [Str1], Berestycki, Lions [BerLio1] and for uniqueness, see Coffman [Cof], McLeod, Ser-
rin [McLSer], and Kwong [Kwo]. Linearizing around the standing wave solution yields a matrix
potential with V = (β +1)φ2β and V = βφ2β. Hence V > 0 and V > V , which is the same as
1 2 1 1 2
| |
Assumption A1). Moreover,L = ∆+α2 φ2β satisfies L φ=0 and L 0 follows from φ>0.
− − − − − ≥
DISPERSIVE ESTIMATES 3
There is a large body of literature concerning the orbital (or Lyapunov) stability (or instabil-
ity) of this ground state standing wave, see for example Shatah [Sha], Shatah, Strauss [ShaStr],
Weinstein [Wei1], [Wei2], Cazenave, Lions [CazLio], Grillakis, Shatah, Strauss [Gri], [GriShaStr1],
[GriShaStr2], and Comech, Pelinovsky [ComPel]. Reviews of much of this work are in Strauss [Str2],
and Sulem, Sulem [SulSul].
Thequestionofwhenthestrongerpropertyofasymptoticstabilityholdshasreceivedalotofatten-
tion over the past decade. Starting with Soffer and Weinstein [SofWei1], [SofWei2], who studied the
modulationequationsgoverningtheevolutionofsmallsolitons1,therehasbeenmuchworkalsoonthe
case of large solitons, see Buslaev, Perelman [BusPer1], [BusPer2], Cuccagna [Cuc], Perelman [Per1],
[Per2], Rodnianski, Soffer, Schlag [RodSchSof1], [RodSchSof2]. It is for this purpose, rather than for
theaforementionedorbitalstability,thatthedispersiveestimatesofthepresentpaperareofrelevance.
Let us note that for the case of small solitons the potentials V ,V will be small and therefore the
1 2
matrix operator above becomes easier to treat (this is because of dimension three and analogous to
the caseofscalarSchr¨odingeroperatorswith smallpotentials,seee.g., Rodnianski,Schlag[RodSch]).
Only for large V ,V can significant (spectral) difficulties arise on the linear level.
1 2
ItisknownthatAssumptionA2)impliesthatthespectrumspec( )satisfiesspec( ) R iRand
H H ⊂ ∪
thatallpointsofthediscretespectrumotherthanzeroareeigenvalueswhosegeometricandalgebraic
multiplicities coincide. For this see Grillakis [Gri], [BusPer1] or [RodSchSof1], as well as Section 2
below.
UnfortunatelyitisunknownatthispointhowtoguaranteeAssumptionA4),althoughitisbelieved
toholdforsystemsthatarisefromagroundstatesolitonasexplainedabove(in1-dthisisknown,see
Perelman [Per1], due to the explicit form of the ground state in that case). It would be desirable to
havean ”abstract”approachto this question. But sofarthis is unknown, andit is animportantopen
problem to settle this issue (even for radialpotentials). Note that there can be imbedded eigenvalues
for V = 0 and V large and positive. But in that case Assumption A2) does not hold. However,
2 1
Assumptions A2) and A3) alone do not imply A4) by an example2 of Denissov [Den]. Let us remark
that because of these examples where imbedded eigenvalues can exist for our systems even though
the potentials are smooth and decay rapidly, it seems certain that the methods known for the scalar
case (say, commutator methods in the spirit of Mourre theory) alone will not suffice. Some extra
information needs to be used (like A2 plus additional restrictions) to insure the absence of imbedded
eigenvalues.
ForthecaseofscalarSchr¨odingeroperatorsitiswidelyknownthatimbeddedeigenvaluesareunsta-
ble. Infact,undergenericperturbationstheyturnintoresonancesinthecomplexplane(Fermigolden
rule). Hence, one may hope that A4) holds generically in a suitable sense. However, in the matrix
case the situation is more complicated and imbedded eigenvalues can turn into complex eigenvalues
under small perturbations, see Cuccagna, Pelinovsky, and Vougalter [CucPelVou], [CucPel], as well
1Such solitons only arise in an NLS equation with a linear potential. They are are generated by bifurcation off a
boundstateofthelinearSchro¨dingeroperator.
2His example is in one dimension. However, since conditions A1)-A4) are ”abstract” and dimension less, this is
relevanttoourdiscussion.
4 M.BURAKERDOG˘ANANDWILHELMSCHLAG
as Gang, Sigal, Vougalter [GanSigVou]. More precisely, whether or not this happens depends on the
sign of σ f,f where f belongs to the realsubspace associatedwith an imbedded eigenvalue. This
3
h H i
is analogous to Krein’s theorem and the Krein signature in classical mechanics, see MacKay [MacK],
or Avez, Arnold [AveArn].
Unlike the self-adjoint case, for our matrix operators the boundedness of eit as t
H 2 2
H k k → | |→∞
is generally false. Indeed, this is the case in the presence of any complex spectrum. Moreover, even
if there is no complex spectrum, then this operator norm can grow polynomially in t due a nonzero
nilpotent part of the root-space of at zero. Thus, we are lead to consider the boundedness of
H
eit P , where I P is the Riesz projection corresponding to the discrete spectrum. This has
H s 2 2 s
k k → −
been studied before in the case where the thresholds µ are neither eigenvalues nor resonances, see
±
[Cuc, CucPelVou, RodSchSof1]. In fact, the first results on such L2 (or H1)-boundedness are due
to Weinstein [Wei1], [Wei2] who used variational methods. Such an approach is intimately tied up
with the underlying nonlinear problem because it uses the properties of the ground state. For this
reason, Weinstein needs to assume that he is in the stable (L2-subcritical) case. However, the recent
work [Sch1] requires such bounds also in the super-critical case.
OurfirstresultestablishessuchanL2 boundinthefullgeneralityofAssumptionsA1)-A4). Inpar-
ticular,itshowsthatneitherthresholdresonancesnorthresholdeigenvaluesaffecttheL2-boundedness.
Theorem 1. Assume that V satisfies Assumptions A1)–A4) with β >5. Then
sup eit P C
H s 2 2
t Rk k → ≤
∈
with a constant that depends on V.
In this context we would like to mention the work of Gesztesy, Jones, Latushkin, and
Stanislavova [GesJonLatSta]. They prove, for linearized NLS, that σ(ei P )= z : z =1 .
H s
{ | | }
In order to formulate our main dispersive estimate, we need to introduce the analogue of the
projection onto the continuous spectrum from the self-adjoint case. This is done as follows. First,
let P be the Riesz projection corresponding to the discrete spectrum of . Second, let P be the
d µ
H
projection with range equal to ker( µ) and kernel equal to (ker( µ)) . Moreover, P = 0
∗ ⊥ µ
H− H −
if µ is not an eigenvalue of . Similarly with P . We show below, see Lemma 10, that P are
µ µ
H − ±
well-defined, and that P ,P ,P commute. In fact, P P =P P =P P =0. Now, define
d µ µ d µ d µ µ µ
− − −
P =(I P )(I P )(I P )=I P P P .
c d µ µ d µ µ
− − − − − − − −
Clearly, P is the analogue of the continuous spectral projectionin the self-adjointcase. It eliminates
c
all the eigenfunctions, including those at the thresholds (recall that we are assuming absence of
imbedded eigenvalues).
Theorem 2. Assume that V satisfies Assumptions A1)–A4) with β >10. Then there exists a time-
dependent operator F such that
t
sup F < , eit P t 1/2F Ct 3/2.
t k tkL1→L∞ ∞ H c− − t 1 ≤ −
(cid:13) (cid:13) →∞
(cid:13) (cid:13)
(cid:13) (cid:13)
DISPERSIVE ESTIMATES 5
If both µ and µ are not eigenvalues, then F is of rank at most two. Moreover, if µ are neither
t
− ±
eigenvalues nor resonances, then F 0.
t
≡
Inallcases,theoperatorsF canbegivenexplicitly,andtheycanbeextractedfromourproofswith
t
more work. We carry this out explicitly for the case when µ are not eigenvalues, see formula (58)
±
below. ForscalarSchr¨odingeroperators,suchexplicitrepresentationsofthe kernelsofF (intermsof
t
resonancefunctionsandprojectionsontotheeigenspaces)werederivedbyYajima[Yaj2]. Hisformulas
show that F has finite rank in all cases, and the same should be true in Theorem 2. It is important
t
3
to note that the t−2 bound is destroyed by an eigenvalue at zero, even if zero is not a resonance and
even after projecting the zero eigenfunction away (this was discovered by Jensen, Kato [JenKat] for
scalar operators).
Finally,weremarkthatitwasnotourintentiontoobtaintheminimalvalueofβinAssumptionA3).
Our results can surely be improved in that regard. Needless to say, the problem of lowering the
requirementonβ isonlyoneofmanyremainingissues. Morerelevanttononlinearquestionsseemsto
be how to prove A4), and/or how to deal with imbedded eigenvalues when they do occur (in regards
toourtheorems). Inasimilarvein,itwouldofcoursebeinterestingtodevelopthislineartheorywhen
A2)doesnothold. Thisisthecase,forexample,whenlinearizingaroundexcitedstates,see[BerLio2].
2. The matrix case: Generalities
In this section we shall developsome standardand well-knownproperties of the spectra and resol-
vents of under Assumptions A2)-A4). It should be mentioned that Assumption A1) seems to be
H
needed only in order to apply the symmetric resolventidentity, see Section 3 below. However,in this
section we work with the usual resolvent identity and therefore do not need A1).3
Lemma 3. Let β >0 bearbitrary in (1). Then the essential spectrum of equals ( , µ] [µ, ).
H −∞ − ∪ ∞
Moreover, spec( )= spec( )=spec( )=spec( ) and spec( ) R iR. The discrete spectrum
∗
H − H H H H ⊂ ∪
of consists of eigenvalues z N , 0 N , of finite multiplicity. For each z = 0 the
H { j}j=1 ≤ ≤ ∞ j 6
algebraic and geometric multiplicities coincide and Ran( z ) is closed. The zero eigenvalue has
j
H−
finite algebraic multiplicity, i.e., the generalized eigenspace ∞ ker( k) has finite dimension. In
k=1 H
fact, there is a finite m 1 such that ker( k)=ker( k+1) for all k m.
S
≥ H H ≥
Proof. The statement about the essential spectrum follows from Weyl’s criterium. To see this, note
1 i
that conjugation of by the matrix leads to the matrix operator
H " 1 i #
−
0 iL
−
" iL+ 0 #
−
3It seems that one can work with the usual resolvent identity throughout this paper, which would then allow us
to dispense with A1) altogether. However, A1) holds in important applications and we find it convenient to use the
symmetricresolventidentity.
6 M.BURAKERDOG˘ANANDWILHELMSCHLAG
where L is as above and with L = ∆+µ V V . We will again denote this matrix by . Let
+ 1 2
− − − − H
H = ∆+µ and set W = V +V , W = V V ,
0 1 1 2 2 1 2
− − − −
0 iH 0 iW
0 1
(2) = , W = ,
0
H " iH0 0 # " iW2 0 #
− −
0 H +W
0 1
= +W =i .
0
H H " H0 W2 0 #
− −
0 i
By means of the matrix J = one can also write
" i 0 #
−
H 0 H +W 0
0 0 1
= J, = J.
0
H " 0 H0 # H " 0 H0+W2 #
Clearly, is aclosedoperatoronthe domainDom( )=W2,2 W2,2. Since = itfollowsthat
H H × H0∗ H0
spec( ) R. One checks that for z =0
0
H ⊂ ℜ 6
(H2 z2) 1 0
( z) 1 = ( +z) 0 − −
0 − 0
H − − H " 0 (H02−z2)−1 #
(H2 z2) 1 0
(3) = 0 − − ( +z)
0
−" 0 (H02−z2)−1 # H
1
(4) ( z) 1 =( z) 1 ( z) 1U 1+U J( z) 1U −U J( z) 1
− 0 − 0 − 1 2 0 − 1 2 0 −
H− H − − H − H − H −
h i
where (4) also requires the expression in brackets to be invertible, and with
W1 12 0 W1 12sign(W1) 0
U = | | , U = | | .
1 1 2 1
" 0 W2 2 # " 0 W2 2sign(W2) #
| | | |
Itfollowsfrom(3)thatspec( )=( , µ] [µ, ) R. SinceV (x) 0andV (x) 0asx ,
0 1 2
H −∞ − ∪ ∞ ⊂ → → →∞
it follows from Weyl’s theorem, see Theorem XIII.14 in [ReeSim4], and the representation(4) for the
resolvent of , that spec ( ) = spec ( ) = ( , µ] [µ, ) R. Moreover, (4) implies via
ess ess 0
H H H −∞ − ∪ ∞ ⊂
the analyticFredholmalternativethat( z) 1 isameromorphicfunctioninC ( , µ] [µ, ).
−
H− \ −∞ − ∪ ∞
Furthermore, the poles are eigenvalues4 of of finite multiplicity and Ran( z ) is closed at each
j
H H−
pole z .
j
The symmetries of the spectrum are consequences of the commutation properties of with the
H
Pauli matrices
0 1 0 i 1 0
σ = , σ = , σ = .
1 2 3
" 1 0 # " i 0 # " 0 1 #
− −
4NotethatsinceHisnotself-adjoint,itcanhappenthat
ker(H−z)26=ker(H−z)
for some z ∈C. In other words, H can possess generalized eigenspaces. In the NLS applications this does happen at
z=0duetosymmetrieslikemodulation.
DISPERSIVE ESTIMATES 7
Now let us check that the spectrum lies in the union of the real and imaginary axes. Thus, suppose
that
0 iL f f
1 1
− =E
" −iL+ 0 #(cid:18)f2(cid:19) (cid:18)f2(cid:19)
with E 6= 0 and ff12 ∈ L2 \{0}. Then f1 6= 0 and f2 6= 0 and f1 ⊥ ker{L−}. Hence, g = L−−12f1
satisfies
(cid:0) (cid:1)
1 1
L2L L2g =E2g
+
− −
and thus E2 R, as desired. Here we used that L L L with domain W4,2(R3) is a selfadjoint
+
∈ − −
operator. For a proof of this see Lemma 11.10 in [RodSchSof2]. That same lemma also contains a
p p
proof of the fact that for any eigenvalue other than zero the algebraic and geometric multiplicities
coincide.
Let P be the Riesz projection at zero. Then, on the one hand one checks that
0
RanP ker( m) for all m 1.
0
⊃ H ≥
On the other hand, if ( z) 1 C z ν, then
− −
k H− k≤ | |
νP =0.
0
H
Thus RanP ker( ν). See [HisSig] Chapter 6 for these generalstatements about Riesz projections.
0
⊂ H
(cid:3)
It will follow from the next section that N < in Lemma 3 provided β >5 (which can probably
∞
be relaxed). Indeed, in that section we will derive expansions of the resolvent ( z) 1 about the
−
H−
thresholds µwhichwill preclude the eigenvalues fromaccumulatingat these points. Thus there can
±
only be finitely many eigenvalues, i.e., N < .
∞
Next, weneedtodevelopa limiting absorptionprinciple forthe resolvents( z) 1 when z >µ.
−
H− | |
As observed in [CucPelVou] and [Sch1], this can be done along the lines of the classical Agmon
argument [Agm]. We now present some of these arguments.
WebeginbyrecallingsomeweightedL2 estimatesforthefreeresolvent( z) 1 whichgobythe
0 −
H −
name”limitingabsorptionprinciple”. TheweightedL2-spacesherearetheusualonesL2,σ = x σL2.
−
h i
It will be convenient to introduce the space
X :=L2,σ(R3) L2,σ(R3).
σ
×
Clearly, X =X . The statement is that
σ∗ σ
−
(5) |λ|≥sλu0p,0<ǫ|λ|21 k(H0−(λ±iǫ))−1kXσ→Xσ∗ <∞
providedλ >µ and σ > 1 and was provedin this formby Agmon[Agm]. By the explicit expression
0 2
for the kernel of the free resolvent in R3 one obtains the existence of the limit
lim ( (λ iǫ)) 1φ,ψ
0 −
ǫ 0+h H − ± i
→
8 M.BURAKERDOG˘ANANDWILHELMSCHLAG
for any λ R and any pair of Schwartz functions φ,ψ. Hence ( (λ i0)) 1 satisfies the same
0 −
∈ H − ±
bound asin (5) provided λ λ >µ. Thereis a correspondingbound whichis validfor allenergies.
0
| |≥
It takes the form
(6) sup ( z) 1 <
z Ck H0− − kXσ→Xσ∗ ∞
∈
providedσ >1. Itis muchmoreelementarytoobtainthan(5)since itonly usesthatthe convolution
with x 1 is bounded from L2,σ(R3) L2, σ(R3) provided σ > 1. In fact, it is Hilbert-Schmidt in
− −
| | →
these norms. We now state a lemma about absence of imbedded resonances.
Lemma 4. Let β > 1. Then for any λ R, λ > µ the operator ( (λ i0)) 1V is a compact
0 −
∈ | | H − ±
operator on X X and
1 1
−2− → −2−
I+( 0 (λ i0))−1V
H − ±
is invertible on these spaces.
Proof. The compactness is standard and we refer the reader to [Agm] or [ReeSim4]. Let λ > µ. By
the Fredholm alternative, the invertibility statement requires excluding solutions (ψ ,ψ ) X of
1 2 1
∈ −2−
the system
0=ψ R (λ µ+i0)(V ψ +V ψ )
1 0 1 1 2 2
− −
0=ψ R ( λ µ)(V ψ +V ψ ),
2 0 2 1 1 2
− − −
where R (z) is the free, scalar resolvent( ∆ z) 1. Notice that these equations imply that ψ L2
0 − 2
− − ∈
and that
0= ψ ,V ψ + ψ ,V ψ R (λ µ+i0)(V ψ +V ψ ),V ψ +V ψ
1 1 1 1 2 2 0 1 1 2 2 1 1 2 2
h i h i−h − i
0= ψ ,V ψ R ( λ µ)(V ψ +V ψ ),V ψ
2 2 1 0 2 1 1 2 2 1
h i−h − − i
0= ψ ,V ψ R ( λ µ)V ψ ,V ψ R ( λ µ)V ψ ,V ψ .
2 1 2 0 2 1 1 2 0 1 2 1 2
h i−h − − i−h − − i
Since V ,V are real-valued, inspection of these equations reveals that
1 2
R (λ µ+i0)(V ψ +V ψ ),V ψ +V ψ =0.
0 1 1 2 2 1 1 2 2
ℑh − i
So Agmon’s well-known bootstrap lemma (see Theorem 3.2 in [Agm]) can be used to conclude that
ψ L2(R3). But then we have an imbedded eigenvalue at λ, which contradicts Assumption A4). So
1
∈
one can invert
I+( (λ i0)) 1V
0 −
H − ±
on X and we are done. (cid:3)
1
−2−
As usual, one converts the information of the previous lemma into a bound for the perturbed
resolvent by means of the resolvent identity.
Proposition 5. Let β >1 and fix an arbitrary λ >µ. Then
0
(7) sup λ 12 ( (λ iǫ))−1 <
| | k H− ± k ∞
|λ|≥λ0,0<ǫ
where the norm is the one from X X .
21+ → −12−
DISPERSIVE ESTIMATES 9
Proof. Let z = λ+iǫ, λ λ , ǫ = 0. By the resolvent identity and the fact that the spectrum of
0
≥ 6 H
belongs to R iR,
∪
(8) ( z)−1 =(I +( 0 z)−1V)−1( 0 z)−1
H− H − H −
as operatorsonL2(R3). Becauseof the λ−21-decayin (5), there exists a positive radiusrV suchthat
| |
1
( z) 1V <
0 −
k H − k 2
for all z >r in the operator norm of X X . In conjunction with (8) this implies that
V 1 1
| | −2− → −2−
( z)−1 C z −21
k H− k≤ | |
forall z >r intheoperatornormofX X . Nowsuppose(7)fails. Itthenfollowsfrom(8)
| | V 21+ → −12−
and(6)thatthereexistasequencez with (z ) λ andfunctions f X with f =1
n ℜ n ≥ 0 n ∈ −21− k nkX−21−
and such that
(9) [I+( z ) 1V]f 0
k H0− n − nkX 1 →
−2−
as n . Necessarily, the z accumulate at some point λ [λ ,r ]. Without loss of generality,
n 0 V
→ ∞ ∈
z λ and (z )>0 for all n 1. Next, we claim that (9) also holds in the following form:
n n
→ ℑ ≥
(10) [I+( (λ+i0)) 1V]f 0
k H0− − nkX 1 →
−2−
as n . If so, then it would clearly contradict Lemma 4. To prove (10), let
→∞
S :=I+( (λ+i0)) 1V
0 −
H −
for simplicity. Then
I+( z ) 1V =S+(( z ) 1 ( (λ+i0)) 1)V
0 n − 0 n − 0 −
H − H − − H −
(11) = I+(( z ) 1 ( (λ+i0)) 1)VS 1 S.
0 n − 0 − −
H − − H −
Our claim now follows from the fact t(cid:2)hat the expressionin bracketsis aninvertible(cid:3)operatorfor large
n on X . This in turn relies on bounds of the form: Given ǫ > 0, there exists δ > 0 so that for
1
−2−
z >0, and all z close to z,
′
ℜ
(12) ( ∆ z) 1 ( ∆ z ) 1 C z z δ
− − − − − − ′ − L2,21+ǫ L2,−21−ǫ ≤ δ,ǫ| − ′|
→
see [Agm].5 (cid:13) (cid:13) (cid:3)
(cid:13) (cid:13)
As in the case of the free Hamiltonian , it is now possible to define the boundary values of the
0
H
resolvent ( z) 1. More precisely, the following corollary holds.
−
H−
Corollary 6. Let β >1. Define
(13) ( (λ i0)) 1 :=(I +( (λ i0)) 1V) 1( (λ i0)) 1
− 0 − − 0 −
H− ± H − ± H − ±
for all λ >µ. Then as ǫ 0+,
| | →
( (λ iǫ)) 1 ( (λ i0)) 1 0
− −
k H− ± − H− ± k→
5Ofcourseδ→0asǫ→0. Moreover,ifδ=1,thenoneneedsǫ>1.
10 M.BURAKERDOG˘ANANDWILHELMSCHLAG
in the norm of X X and one can extend (7) to ǫ 0.
21+ → −12− ≥
Proof. Definition (13) is legitimate by Lemma 4 andmotivated by (8). Thus, the resolvent( (λ
H− ±
iǫ)) 1 is well-defined for all ǫ 0 and λ >λ . In view of (12),
− 0
≥ | |
( (λ iǫ)) 1 ( (λ i0)) 1 0
0 − 0 −
k H − ± − H − ± k→
as ǫ 0 in the norm of X X . Moreover,by (11) and again (12),
→ 21+ → −21−
[I+( (λ+iǫ)) 1V] 1 [I +( (λ+i0)) 1V] 1
0 − − 0 − −
H − − H −
=S−1 I+(( 0 (λ+iǫ))−1 ( 0 (λ+i0))−1)VS−1 −1 S−1
H − − H − −
= ∞ S(cid:2) 1 (( (λ+iǫ)) 1 ( (λ+i0)) 1)VS(cid:3)1 k
− 0 − 0 − −
− H − − H −
k=1
X (cid:2) (cid:3)
tends to zero in the norm of X as ǫ 0+. (cid:3)
1
−2− →
3. Resolvent expansions at thresholds
In view of Assumption A1), we write
V = σ vv = σ v2 =:v v ,
3 ∗ 3 1 2
− −
where v = σ v, v =v =v,
1 3 2 ∗
−
1 0
(14) σ = .
3
" 0 1 #
−
It follows from (1) that the entries of v ,v are real-valued and decay like x β/2. Let λ = µ+z2,
1 2 −
h i
where Im(z)>0 and z small. We have the symmetric resolvent identity:
| |
(15) R(λ):=( λ) 1 =R (λ) R (λ)v (I +v R (λ)v ) 1v R (λ).
− 0 0 1 2 0 1 − 2 0
H− −
Recall that (see previous section) the essential spectrum of is ( , µ] [µ, ). As in the
H −∞ − ∪ ∞
scalarcase[ErdSch], we obtain resolventexpansionsat thethresholdλ=µ inthe case ofa resonance
and/or eigenvalue. Recall that R (λ) has the kernel
0
1 eiz|x−y| 0
R (λ)(x,y)= .
0 4π|x−y|" 0 −e−√2µ+z2|x−y| #
We have a similar representationof R (λ) for λ around µ. Let
0
−
A(z)=I+v R (λ)v
2 0 1
=:A +zA (z),
0 1
where
A =I+v R (µ)v ,
0 2 0 1
1
A (z)=: v (R (λ) R (µ))v ,
1 2 0 0 1
z −