CONFLUENTES MATHEMATICI
Alexandre MARTIN
On the Limiting absorption principlefor a new class of Schrödinger Hamiltonians Tome 10, no1 (2018), p. 63-94.
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10, 1 (2018) 63-94
ON THE LIMITING ABSORPTION PRINCIPLE FOR A NEW CLASS OF SCHRÖDINGER HAMILTONIANS
ALEXANDRE MARTIN
Abstract. We prove the limiting absorption principle and discuss the continuity properties of the boundary values of the resolvent for a class of form bounded perturbations of the Euclidean Laplacian∆that covers both short and long range potentials with an essentially optimal behaviour at infinity. For this, we give an extension of Nakamura’s results (see [16]).
CONTENTS
1. Introduction 63
2. Notation and basic notions 69
2.1. Notation 69
2.2. Regularity 70
3. Nakamura’s ideas in a more general setting:
Boundary values of the resolvent 71
4. An extension of Nakamura’s results 73
5. Concrete potentials 78
5.1. A non Laplacian-compact potential 79
5.2. A class of oscillating potential 84
5.3. An unbounded potential with high oscillations 86
5.4. A short range potential in a weak sense 89
6. Flow 93
References 93
1. INTRODUCTION
The purpose of this article is to prove a limiting absorption principle for a certain class of Schrödinger operator with real potential and to study their essential spectrum. Because this operators are self-adjoint, we already know that their spectrum is in the real axis. We also know that the non negative Laplacian operator∆(Schrödinger operator with no potential) has for spectrum the real set[0,+∞)with purely absolutely continuous spectrum on this set. If we add to∆a "small" potential (with compact properties with respect to∆), the essential spectrum of this new operator is the same that ∆ essential spectrum which is continuous. We are interested in the nature of the essential spectrum of the perturbed operator and in the behaviour of the resolvent operator near the essential spectrum.
We will say that a self-adjoint operator hasnormal spectrum in an open real setOif it has no singular continuous spectrum inOand its eigenvalues inOare of finite multiplici- ties and have no accumulation points insideO. Note that if we have a Limiting Absorption Principle for an operatorH onO,Hhas normal spectrum onO.
2010Mathematics Subject Classification:35J10, 35P25, 35Q40, 35S05, 47B15, 47B25, 47F05.
Keywords:Schrödinger operators, Mourre theory, Limiting Absorption Principle.
63
A general technique for proving this property is due to E. Mourre [14] and it involves a local version of the positive commutator method due to C.R. Putnam [17, 18]. For various extensions and applications of these techniques we refer to [1]. Roughly speaking, the idea is to search for a second self-adjoint operatorAsuch thatH is regular in a certain sense with respect toAand such thatHsatisfies the Mourre estimate on a setIin the following sense
E(I)[H, iA]E(I)>c0E(I) +K
whereE(I)is the spectral measure ofH onI,c0 > 0andKa compact operator. Then one says that the operatorAisconjugatetoHonI.
WhenH = ∆ +V is a Schrödinger operator, we usually apply the Mourre theorem with the generator of dilations
AD= 1
2(p·q+q·p),
wherep=−i∇andqis the vector of multiplication byx(see [1, Proposition 7.4.6] and [7, Section 4]). But in the commutator expressions, derivatives ofV appears which can be a problem, if, for example,V has high oscillations at infinity.
In a recent paper S. Nakamura [16] pointed out the interesting fact that a different choice of conjugate operator for H can be used to have a limiting absorption principle. This allows us to avoid imposing conditions on the derivative of the long range part of the potential. More precisely, if the operator of multiplication byV(q)is∆-compact and two other multiplication operators, which include differencies onV and not derivatives, are∆- bounded, thenH has normal spectrum in(0, π2/a2)and the limiting absorption principle holds forH locally on this set, outside the eigenvalues. This fact is a consequence of the Mourre theorem withAN (see (1.1)) as conjugate operator.
Our purpose in this article is to put the results of Nakamura in a more general abstract setting and get a generalisation of his result. Moreover, we will show that this general- isation can be applied to potentials for which the Mourre theorem with the generator of dilations as conjugate operator cannot apply (our potentials are not of long range type).
Furthermore, Nakamura’s result cannot apply to this type of potentials which are not∆- bounded. Finally, as usual, we will derive from the limiting absorption principle an appli- cation of this theory to wave operators.
We denoteX =Rν andH=L2(X). LetH1be the first order Sobolev space onX, denoteH−1its adjoint space and similarly, we denoteH2the second order Sobolev space onXandH−2its adjoint space. All this spaces realised the following
H2⊂ H1⊂ H ⊂ H−1⊂ H−2.
SetB1=B(H1,H−1)andB2=B(H2,H−2). If needed for clarity, ifuis a measurable function onXwe denoteu(q)the operator of multiplication byuwhose domain and range should be obvious from the context. Ifa∈ X letTa be the operator of translation bya, more precisely(Taf)(x) =f(x+a).
We say thatV ∈ B,B = B1orB = B2, is amultiplication operator ifV θ(q) = θ(q)V for anyθ∈Cc∞(X). Note thatV is not necessarily the operator of multiplication by a function, it could be the operator of multiplication by a distribution of strictly positive order. For example, in the one dimensional case V could be equal to the derivative of a bounded measurable function. Anyway, ifV is a multiplication operator then there is a uniquely defined temperate distributionvonXsuch thatV f =vffor allf ∈Cc∞(X)and
thenTaV Ta∗=v(·+a). In general we simplify notations and do not distinguish between the operatorV and the distributionv, so we writeV =V(q)andTaV Ta∗=V(q+a).
We will extend Nakamura’s result in two directions. First, we will use the Mourre theory with a class of potentialV :H2→L2orV :H1→ H−1(cf. [1]) and satisfying a weaker regularity. In particular, this includes potentials with Coulomb singularities, and also short range potentials (see Definition 1.3). Secondly, we will use the Mourre theory with a more general class of conjugate operators includingAN (see (1.5)).
Leta >0and letsin(ap) = sin(−ia∂x1),· · ·,sin(−ia∂xν) . Let AN =1
2 sin(ap)·q+q·sin(ap)
. (1.1)
Fix a real functionξ∈C∞(X)such thatξ(x) = 0if|x|<1andξ(x) = 1if|x|>2.
THEOREM 1.1. — Leta∈RandV :H2 →L2(respectivelyV :H1 → H−1) be a compact symmetric multiplication operator with, for all vectoreof the canonical basis of Rν,
Z ∞
1
kξ(q/r)|q|(V(q+ae)−V(q))kBdr
r <∞ (1.2)
whereB=B2(respectivelyB=B1). Then the self-adjoint operatorH= ∆ +V onH has normal spectrum in(0,(π/a)2)and, for some appropriate Besov spaceK, the limits
(H−λ±i0)−1:= w*-lim
µ↓0 (H−λ±iµ)−1 (1.3)
exist inB(K,K∗), locally uniformly inλ∈(0,(π/a)2)outside the eigenvalues ofH. We make some comments in connection with the Theorem 1.1.
(1) The Besov space is defined in Section 2 by (2.2).
(2) Condition (1.2) is satisfied ifkhqiµ(V(q+a)−V(q))kB <∞for a fixedµ >
1. To satisfy this conditions, it suffices thatkhqiµVkB < ∞. In particular, in dimensionν 6= 2, ifV is a real function onRν and if there isµ > 1such that
h·iµV(·)p
is in the Kato class, withp= 1ifν= 1, andp=ν/2ifν >3, then condition (1.2) is satisfied (see Proposition 4.7). If this is the case for alla ∈R, then the limiting absorption principle is true on(0,+∞).
(3) In the case whereV : H2 → L2 is compact, in [16], V is assumed to satisfy q(V(q+ae)−V(q))andq2(V(q+ae) +V(q−ae)−2V(q))be∆-bounded.
This assumptions implies to theC2(AN,H2, L2)regularity. Observe that, since q(V(q+ae)−V(q))appears in[V, iAN], (1.2) implies theC1,1(AN,H2,H−2) regularity which is implied by theC2(AN,H2, L2)regularity.
(4) In one dimension, letV such that qVc(ξ) =
+∞
X
n=−∞
λnχ(ξ−n), (1.4)
whereb·is the Fourier transform, λn ∈ Randχ is compactly support, then V satisfies assumptions of Theorem 1.1 withB=B1butV is neither∆-bounded nor of classC1(AD,H1,H−1). In particular, we can neither apply the Mourre Theorem with the generator of dilation (see [1, p.258]) nor Nakamura’s Theorem (see lemma 5.3).
As in [16], the limiting absorption principle is limited to(0,(π/a)2). The bound(π/a)2 is artificial and appears with the choice of vector fieldsin(ap). In fact, by a simple com- putation with the Laplacian∆inL2(Rν), we have
[∆, iAN] = [∆, i1
2 sin(ap)·q+q·sin(ap) ]
= 1
2 sin(ap)·[∆, iq] + [∆, iq]·sin(ap)
= 2p·sin(ap)
which implies a loss of positivity on(π/a)2. This is a drawback except if we can apply (1.2) for alla >0. We will use Nakamura’s method in a broader framework that allows the removing of this drawback.
Let us denoteH0 = ∆ =p2. ThenH0is a self-adjoint operator inHwith domainH2 which extends to a linear symmetric operatorH1→ H−1for which we keep the notation H0. LetV :H1→ H−1be a linear symmetric compact operator. ThenH =H0+V is a symmetric operatorH1 → H−1which induces a self-adjoint operator inHfor which we keep the notationH. LetE0andEbe the spectral measures ofH0andH.
Note that for each non realzthe resolventR(z) = (H−z)−1of the self-adjoint operator HinHextends to a continuous operatorR(z) :H−1→ H1which is in fact the inverse of the bijective operatorH−z:H1→ H−1.
AN andAD belongs to a general class of conjugate operator, which appears in [1, Proposition 4.2.3]. This is the class of operator which can be written like
Au= 1
2(u(p)·q+q·u(p)) (1.5)
whereuis aC∞vector field with all the derivates bounded. We will see that this conjugate operator is self-adjoint on some domain (see Section 6). Conjugate operators of this form were already used in Mourre’s paper [14, page 395].
Remark that the commutator of such conjugate operator with a function ofpis quite explicit: denotingh0 =∇hthen
[h(p),iAu] = [h(p),iu(p)q] =u(p)·h0(p) = (u·h0)(p). (1.6) In particular[H0,iAu] = 2p·u(p). We denote by the same notationeiτ Au theC0-group inH1and inH−1.
One says thatAisstrictly conjugate toH0onJif there is a real numbera >0such that E0(J)[H0,iA]E0(J)>aE0(J), which in our case means2k·u(k)>afor eachk∈X such that|k|2∈J. TakingAuin this class, we have the following
THEOREM 1.2. — LetV : H1 → H−1be a compact symmetric operator such that there isuaC∞bounded vector field with all derivatives bounded such thatV is of class C1,1(Au,H1,H−1)in the following sense:
Z 1
0
kVτ+V−τ−2VkB1 dτ
τ2 <∞, whereVτ= eiτ AuVe−iτ Au. (1.7) LetJbe an open real set such that
inf{k·u(k)|k∈X,|k|2∈J}>0. (1.8) ThenHhas normal spectrum inJand the limits
R(λ±i0) := w*-lim
µ↓0 R(λ±iµ) (1.9)
exist inB H−11/2,1,H1−1/2,∞
, locally uniformly inλ∈ J outside the eigenvalues ofH, whereH−11/2,1andH1−1/2,∞are interpolation spaces which are defined on Section 2.
We make some remarks about this Theorem:
(1) To check theC1,1(Au,H1,H−1)property, it is useful to haveeitAuH1 ⊂ H1. For that, we will make a comment in the Section 6 on the flow generated by the vector fielduassociated toAu.
(2) Ifk·u(k)is positive for allk6= 0, Theorem 1.2 applies withJ = (0,+∞).
(3) IfV is the divergence of a short range potential (see Definition 1.3), then Theorem 1.2 applies. A certain class of this type of potential were already studied in [5] and [6].
(4) SinceR(λ±i0)exists inB H−11/2,1,H1−1/2,∞
, forD0 the space of distributions this operator exists inB(Cc∞(Rν), D0).
(5) IfV can be seen as a compact operator fromH2toL2, Theorem 1.2 is still valid if we replace the assumption "V is of class C1,1(Au,H1,H−1)" by the weak assumption "V ∈C1,1(Au,H2,H−2)" with the same proof.
(6) Consider the∆-compact operatorV(q)where V(x) = (1−κ(|x|))sin(|x|α)
|x|β ,
withκ∈Cc∞(R,R)withκ(|x|) = 1if|x|< 1,0 6κ61,α >0andβ > 0.
Note that this type of potential was already studied in [2, 8, 9, 10, 13, 21, 22]. In [13], they proved that if|α−1|+β >1, thenV has the good regularity withAD
but, if|α−1|+β < 1,H /∈ C1(AD). In the latter case, we cannot apply the Mourre theory with the generator of dilation. Here, we prove that, with a certain choice ofu,V ∈C1,1(Au,H2,H−2)if2α+β >3(see lemma 5.4). In that case, Theorem 1.2 applies. In particular, in the region2α+β >3andα+β 62, we have the limiting absorption principle butHis not of classC1(AD). In Section 5, we will see that Theorem 1.2 also applies ifβ 60under certain condition onα.
(7) Letκ∈Cc∞(R,R)such thatκ= 1on[−1,1]and06κ61. Let V(x) = (1−κ(|x|)) exp(3|x|/4) sin(exp(|x|)).
We can show that, for allubounded, V ∈ C∞(Au,H1,H−1)and Theorem 1.2 applies (see Lemma 5.6). Moreover, this implies good regularity properties on the boundary values of the resolvent. SinceV is not∆-bounded, we cannot use theC1(AD,H2,H−2)(see [1, Theorem 6.3.4]).
We can also prove that V /∈ C1(AD,H1,H−1). In particular, Theorem 1.2 does not apply withAu=AD.
(8) IfV : H1 → H−1is compact and if there isµ >0such thatx7→ hxi1+µV(x) is inH−1 (V is assumed to be short range in a quite weak sense), then we can apply Theorem 1.2 with an appropriateu(see lemma 5.8). We will provide in this class an concrete example which cannot be treated with the generator of dilations or Nakamura’s result (see lemma 5.10).
Now we will see a third result concerning existence of wave operators which are useful in scattering theory (see [20]).
DEFINITION1.3. — A linear operatorS∈B1isshort rangeif it is compact, symmet- ric, and
Z ∞
1
kξ(q/r)SkB1dr <∞. (1.10) Remark that (1.2) is satisfied if
Z ∞
1
kξ(q/r)|q|VkBdr
r <∞ (1.11)
which is a short range type condition.
Note that we do not requireSto be local. Clearly this condition requires less decay than the condition (1.11). Then we have:
THEOREM 1.4. — LetH be as in Theorem 1.1 and letS be a short range operator.
Then the self-adjoint operatorK =H +S has normal spectrum in(0,∞)and the wave operators
Ω±= s-lim
t→±∞eitKe−itHEHc (1.12)
exist and are complete, whereEHc is the projection onto the continuity subspace ofH. Proof. — We haveK= ∆ +V +Sand from [1, 7.5.8] it follows thatS, henceV+S, is of classC1,1(AN,H1,H−1)for alla > 0so that we can use Theorem 1.2 to deduce thatKhas normal spectrum in(0,∞)and that the boundary values of its resolvent exist as in the case ofH. For the existence and completeness of the wave operators we use [1, Proposition 7.5.6] with the following change of notations: H0, H, V from the quoted proposition are ourH, K, Srespectively. It remains only to check thatSsatisfies the last condition required onV in that proposition: but this is a consequence of [12, Theorem
2.14].
We will give on Section 4 more explicit conditions which ensure that the assumptions of Theorem 1.1 and 1.4 are satisfied in the case whereV andSare real functions.
We make two final remarks. First, the assumption of compactness ofV andS as op- eratorsH1 → H−1 is too strong for some applications, for example it is not satisfied if X =R3andV(x)has local singularities of order|x|−2. But compactness can be replaced by a notion of smallness at infinity similar to that used in [12] which covers such singu- larities and the arguments there extend to the present setting. Second, let us mention that we treat only the case whenH0is the Laplacian∆ =p2but an extension to more general functionsh(p)is straightforward with the same class of conjugate operatorAu.
The paper is organized as follows. In Section 2, we will give some notations we will use below and we recall some basic fact about regularity with respect to a conjugate operator.
In Section 3, we will prove Theorem 1.2 and extend Nakamura’s results by geting proper- ties about the boundary values of the resolvent. In Section 4 , we will give an extension of Nakamura’s theorem by using the Mourre theory withC1,1regularity with respect to the conjugate operatorAN. In Section 5, we will give some examples of potentials which satisfies Theorem 1.1 and Theorem 1.2 and which are not covered by Mourre Theorem with the generator of dilation and Nakamura’s Theorem. In Section 6, we will study the flow associated to the unitary group generated byAu.
2. NOTATION AND BASIC NOTIONS
2.1. Notation. LetX =Rν and fors∈RletHsbe the usual Sobolev spaces onX with H0=H=L2(X)whose norm is denotedk · k. We are mainly interested in the spaceH1 defined by the normkfk21=R
|f(x)|2+|∇f(x)|2
dxand its dual spaceH−1.
Recall that we set B1 = B(H1,H−1) andB2 = B(H2,H−2)which are Banach spaces with normk · kB,B=B1,B2. These spaces satisfyB1⊂B2.
We denoteqjthe operator of multiplication by the coordinatexjandpj=−i∂jconsid- ered as operators inH. Fork∈Xwe denotek·q=k1q1+· · ·+kνqν. Ifuis a measurable function onXletu(q)be the operator of multiplication byuinHandu(p) =F−1u(q)F, whereFis the Fourier transformation:
(Ff)(ξ) = (2π)−ν2 Z
e−ix·ξu(x)dx.
If there is no ambiguity we keep the same notation for these operators when considered as acting in other spaces.
Throughout this paperξ∈C∞(X)is a real function such thatξ(x) = 0if|x|<1and ξ(x) = 1if|x|>2. Clearly the operatorξ(q)acts continuously in all the spacesHs.
We are mainly interested in potentialsV which are multiplication operators in the fol- lowing more general sense.
DEFINITION 2.1. — A mapV ∈ Bis called amultiplication operator ifVeik·q = eik·qV for allk∈X. Or, equivalently, ifV θ(q) =θ(q)V for allθ∈Cc∞(X).
For the proof of the equivalence, note first that fromVeik·q = eik·qV,∀k ∈X we get V θ(q) =θ(q)V for any Schwartz test functionθbecause(2π)ν2θ(q) =R
eikq(Fθ)(k)dk and second that ifη ∈C1(X)is bounded with bounded derivative thenη(q)is the strong limit inBof a sequence of operatorsθ(q)withθ∈Cc∞(X).
As we mentioned in the introduction, such aV is necessarily the operator of multiplica- tion by a distribution that we also denoteV and we sometimes write the associated operator V(q). For example, the distributionV could be the divergencedivW of a measurable vec- tor fieldW :X →Xsuch that multiplication by the components ofW sendsH1intoH.
For example,W could be a bounded function and if this function tends to zero at infinity thenV will be a compact operatorH1→ H−1. we say that a multiplication operatorV is
∆-compact ifV :H2→L2is a compact operator.
As usualhxi =p
1 +|x|2. Thenhqiis the operator of multiplication by the function x7→ hxiandhpi=F−1hqiF. For reals, twe denoteHtsthe space defined by the norm
kfkHts =khqisfkHt =khpithqisfk=khqishpitfk. (2.1) Note that the adjoint space ofHtsmay be identified withH−t−s.
A finer Besov type versionH−11/2,1ofH−11/2appears naturally in the theory. To alleviate the writing we denote itK. This space is defined by the norm
kfkK=kθ(q)fkH−1+ Z ∞
1
kτ1/2ψ(q/τ)fkH−1dτ
τ (2.2)
whereθ, ψ∈Cc∞(X)withθ(x) = 1if|x|<1,ψ(x) = 0if|x|<1/2, andψ(x) = 1if 1<|x|<2. The adjoint spaceK∗ofKis the Besov spaceH1−1/2,∞(see [1, Chapter 4]).
We will see in Section 6 that ifu:X →Xis aC∞vector field all of whose derivatives are bounded then the operator
Au= 1
2(u(p)·q+q·u(p)) = 1 2
ν
X
j=1
(uj(p)qj+qjuj(p)) =u(p)·q+ i
2(divu)(p) (2.3) with domain Cc∞(X) is essentially self-adjoint inH; we keep the notation Au for its closure. Remark that the unitary group eiτ Au generated byAu leaves invariant all the spacesHstandK(see Section 6).
Since we will use a lot the case of ubounded, letU be the space of vector fields u bounded with all derivatives bounded such thatx·u(x)>0for allx6= 0.
2.2. Regularity. LetF0, F00be two Banach space andT :F0 →F00a bounded operator.
LetAa self-adjoint operator.
Letk ∈ N. we say thatT ∈ Ck(A, F0, F00)if, for allf ∈ F0, the mapR 3 t → eitAT e−itAf has the usualCkregularity. The following characterisation is available:
PROPOSITION 2.2. — T ∈ C1(A, F0, F00)if and only if[T, A]has an extension in B(F0, F00).
It follows that, fork >1,T ∈Ck(A, F0, F00)if and only if[T, A]∈Ck−1(A, F0, F00).
We can define another class of regularity called theC1,1regularity:
PROPOSITION2.3. — we say thatT ∈C1,1(A, F0, F00)if and only if Z 1
0
kTτ+T−τ−2TkB(F0,F00)
dτ τ2 <∞, whereTτ = eiτ AuTe−iτ Au.
An easier result can be used:
PROPOSITION2.4 (Proposition 7.5.7 from [1]). — Letξ∈C∞(X)such thatξ(x) = 0 if|x|<1andξ(x) = 1if|x|>2. IfT satisfies
Z ∞
1
kξ(q/r)[T, iA]kB(F0,F00)
dr r <∞ thenTis of classC1,1(A, F0, F00).
IfT is not bounded, we say thatT ∈ Ck(A, F0, F00)if forz /∈ σ(T)we have(T − z)−1∈Ck(A, F0, F00).
PROPOSITION2.5. — For allk >1, we have
Ck(A, F0, F00)⊂C1,1(A, F0, F00)⊂C1(A, F0, F00).
IfF0 =F00 = His an Hilbert space, we noteC1(A) =C1(A,H,H∗). IfT is self- adjoint, we have the following:
THEOREM2.6 (Theorem 6.3.4 from [1]). — LetAandT be self-adjoint operator in a Hilbert spaceH. Assume that the unitary group{exp(iAτ)}τ∈Rleaves the domainD(T) ofT invariant. SetG=D(T)endowed with it graph topology. Then
(1) Tis of classC1(A)if and only ifT ∈C1(A,G,G∗).
(2) Tis of classC1,1(A)if and only ifT ∈C1,1(A,G,G∗).
Remark that, ifT :H → His not bounded, sinceT :G → G∗is bounded, in general, it is easier to prove thatT ∈C1(A,G,G∗)thanT ∈C1(A).
IfGis the form domain ofH, we have the following:
PROPOSITION2.7 (see p. 258 of [1]). — LetAandT be self-adjoint operators in a Hilbert spaceH. Assume that the unitary group{exp(iAτ)}τ∈Rleaves the form domain GofTinvariant. Then
(1) Tis of classCk(A)if and only ifT ∈Ck(A,G,G∗), for allk∈N.
(2) Tis of classC1,1(A)if and only ifT ∈C1,1(A,G,G∗).
As previously, sinceT : G → G∗ is always bounded, it is, in general, easier to prove thatT ∈Ck(A,G,G∗)thanT ∈Ck(A).
3. NAKAMURA’S IDEAS IN A MORE GENERAL SETTING: BOUNDARY VALUES OF THE RESOLVENT
In this section, we will prove the Theorem 1.2 and we will see how the regularity of the potential, in relation toAu, can implies a good regularity for the boundary values of the resolvent.
Proof of Theorem 1.2. — By taking into account the equation (1.6) and the statement of Theorem 7.5.6 in [1] we only have to explain how the spaceKintroduced before (2.2) appears into the picture (this is not the space also denotedK in [1]). In fact the quoted theorem gives a more precise result, namely instead of ourKone may take the real inter- polation space D(Au,H−1),H−1
1/2,1, whereD(Au,H−1)is the domain of the closure ofAuinH−1. From (2.3) and sinceuis bounded with all its derivatives bounded it follows immediately thatD(Au,H−1)contains the domain ofhqiinH−1, which isH−11 . Hence
D(Au,H−1),H−1
1/2,1contains H−11 ,H−1
1/2,1which isH1/2,1−1 =K.
We say thatV is of classCk(Au,H1,H−1)for some integerk >1 if the mapτ 7→
Vτ ∈B(H1,H−1)isktimes strongly differentiable. We clearly haveC2(Au,H1,H−1)⊂ C1,1(Au,H1,H−1). In [14] the Limiting Absorption Principle is proved essentially for V ∈C2(A)(see [11] for more details); notice that in [15] the limiting absorption principle is proved in a space better (i.e larger) thanK(see (2.1)), but not of Besov type.
Ifs >1/2thenH−1s ⊂ Kwith a continuous and dense embedding. Hence:
COROLLARY3.1. — For eachs >1/2the limitR(λ±i0) = w*-limµ↓0R(λ±iµ) exists in the spacesB Hs−1,H1−s
, locally uniformly inλ∈J outside the eigenvalues of H.
TheC1,1(Au)regularity condition (1.7) onV is not explicit enough for some applica- tions. We now give a simpler condition which ensures that (1.7) is satisfied.
We recall some easily proven facts concerning the classC1(Au,H1,H−1). First,V is of classC1(Au,H1,H−1)if and only if the functionτ 7→Vτ ∈B(H1,H−1)is (norm or strongly) Lipschitz. Notice that we usedeiτ AH1 ⊂ H1 to prove this. Second, note that for an arbitraryV the expression[V,iAu]is well-defined as symmetric sesquilinear form onCc∞(X)andV is of classC1(Au,H1,H−1)if and only if this form is continuous for the topology induced byH1. In this case we keep the notation[V,iAu]for its continuous extension toH1and for the continuous symmetric operatorH1→ H−1associated to it.
As a consequence of Proposition 7.5.7 from [1] with the choiceΛ =hqiwe get:
PROPOSITION3.2. — LetV :H1 → H−1be a symmetric bounded operator of class C1(Au,H1,H−1)such that
Z ∞
1
kξ(q/r)[V,iAu]kBdr
r <∞ (3.1)
withB=B1. ThenV is of classC1,1(Au,H1,H−1).
If the potentialV is of a higher regularity class with respect toAuthen, by using results from [3], we also get an optimal result on the order of continuity of the boundary values of the resolventR(λ±i0)as functions ofλ. From [15] and the improvements in [4] one may also get a precise description of the propagation properties of the dynamical groupeitH in this context, but we shall not give the details here.
To state this regularity result we recall the definition of theHölder-Zygmund continuity classes of order s ∈ (0,∞). LetE be a Banach space andF : R → E a continuous function. If0< s <1thenF is of classΛsifF is Hölder continuous of orders. Ifs= 1 thenFis of classΛ1if it is of Zygmund class, i.e.kF(t+ε) +F(t−ε)−2F(t)k6Cε for all realtandε >0. Ifs >1, let us writes=k+σwithk>1integer and0< σ61;
thenF is of classΛsifF isktimes continuously differentiable andF(k)is of classΛσ. The corresponding local classes are defined as follows: ifF is defined on an open real set U thenF islocally of classΛsifθF is of classΛsfor anyθ∈Cc∞(U).
We say thatV is of classΛs(Au,H1,H−1)if the functionτ7→Vτ∈B1is of classΛs. We mention that in a more general context this class is denoted byCs,∞(Au,H1,H−1), but this does not matter here. In any case, one may easily check that
Λs(Au,H1,H−1)⊂C1,1(Au,H1,H−1)⊂C1(Au,H1,H−1) ifs >1. Ifs>1is an integer thenCs(Au,H1,H−1)⊂Λs(Au,H1,H−1)strictly.
THEOREM 3.3. — Assume that u andJ are as in Theorem 1.2 and let s be a real number such thats >1/2. IfV :H1 → H−1is a compact symmetric operator of class Λs+1/2(Au,H1,H−1)then the functions
λ7→R(λ±i0)∈B(H−1s ,H1−s) (3.2) are locally of classΛs−1/2onJ outside the eigenvalues ofH.
Proof. — We shall deduce this from the theorem on page 12 of [3]. First, note thatH has a spectral gap becauseH0>0and(H+i)−1−(H0+i)−1is a compact operator hence HandH0have the same essential spectrum. Thus we may use the quoted theorem and we get the assertion of the present theorem but withB(H−1s ,H1−s)replaced byB(Hs,H−s).
Then it suffices to observe that ifzbelongs to the resolvent set ofHthen we have R(z) =R(i) + (z−i)R(i)2+ (z−i)2R(i)R(z)R(i)
and to note thatR(i)sendsH−1s intoH1s.
We state explicitly the particular case corresponding to the Mourre condition V ∈ C2(Au,H1,H−1). We mention that this is equivalent to the fact that the sesquilinear form [[V, Au], Au], which is always well-defined onCc∞(X), extends to a continuous sesquilin- ear form onH1.
COROLLARY3.4. — Assume that we are in the conditions of the Theorem 1.2 but with the condition (1.7)replaced by the stronger oneV ∈ C2(Au,H1,H−1). Then the map (3.2)is Hölder continuous of order s−1/2 for alls such that1/2 < s < 3/2 and if s= 3/2then the map(3.2)is of Zygmund class (but could be nowhere differentiable).
Previously, we saw thatH0verified the Mourre estimate onIif and only ifk·u(k)>0 for allksuch that|k|2∈I. Moreover, we saw that ifH0verified the Mourre estimate on I, because the potentialV isH0-compact or compact onH1, the form domain ofH0, to H−1,H verified the Mourre estimate on the same intervalI.
Because
uN(x) = (sin(axj))j=1,···,ν
we have the Mourre estimate only on Ia = (0,(π/a)2), i.e. where uN(x) 6= 0; this function constructs some artificial thresholds. If we can choose a vector fieldusuch that x·u(x)>0ifx6= 0which satisfied some good conditions of regularity for the potential V, we can extend the intervalIatoI= (0,+∞). For example, we can choose the vector fieldu(x) = (arctan(xj))j=1,···,ν, the function arctanbeing non zero forx 6= 0, or u(x) = x/hxi. In particular, with this type of vector field, ifhpi−1qVhpi−1 is bounded, thenV ∈ C1(Au,H1,H−1)and we have the Mourre estimate on all compact subset of (0,+∞).
4. AN EXTENSION OFNAKAMURA’S RESULTS
In this section, we will prove Theorem 1.1 and we will give some conditions easy to verify which assure that assumptions of Theorem 1.1 and 1.4 are satisfied. Moreover, we will give a stronger version of Nakamura’s Theoerem with estimates on the boundary values of the resolvent.
In all this section,B=B1=B(H1,H−1).
We fix a real numbera > 0 and denoteIa = (0,πa22). We apply the general results from Section 3 with the vector fielduas in [16]:
uN(p) = (sin(ap1), . . . ,sin(apν)). (4.1) Then(divuN)(p) =P
j∂pjsin(apj) =P
jacos(apj)hence 2AN =P
j qjsin(apj) + sin(apj)qj
=P
j 2qjsin(apj)−iacos(apj)
. (4.2) The operatorAN behaves well with respect to the tensor factorizationL2(X) =L2(R)⊗ν and this simplifies the computations. Indeed, if we denote B the operator Aacting in L2(R)we haveAN =A1+· · ·+AνwithA1=B⊗1· · · ⊗1,A2= 1⊗B⊗1· · · ⊗1, etc.
LetTj = eiapj be the operator of translation byain thej direction, i.e.(Tjf)(x) = f(x+aej)wheree1, . . . , eνis the natural basis ofX =Rν. For anyV :H1→ H−1set
δj(V) =TjV Tj∗−V (4.3)
which is also an operatorH1 → H−1hence we may considerδkδj(V), etc. IfV =V(q) is a multiplication operator then
δj(V) =V(q+aej)−V(q).
Remark that, when V is a multiplication operator,δj(V)appears in the first commuta- tor[V, iAN]. The operationδj can also be applied to various unbounded operators, for example we obviously have δj(qk) = aδjk, where δjk is the Kronecker symbol, and δj(u(p)) = 0.
IfS∈Bthen[qj, S]andqjSare well-defined as sesquilinear forms onCc∞(X)and we say that one of these expressions is a bounded operatorH1 → H−1 if the corresponding form is continuous in the topology induced byH1.
THEOREM4.1. — LetV :H1→ H−1be a compact symmetric operator such that for anyjthe forms[qj, V]andqjδj(V)are bounded operatorsH1→ H−1and
Z ∞
1
kξ(q/r)[qj, V]kB+kξ(q/r)qjδj(V)kBdr
r <∞. (4.4)
ThenHhas normal spectrum inIaand the limitsR(λ±i0) = w*-limε↓0R(λ±iε)exist inB(K,K∗), locally uniformly inλ∈Iaoutside the set of eigenvalues ofH.
Remark 4.2. — This Theorem give a stronger result than Theorem 1.1, since, ifV is a multiplication operator,[qj, V] = 0and (4.4) reduces to (1.2).
Proof. — This is a consequence of Theorem 1.2 and Proposition 3.2 once we have checked thatV is of classC1(AN)and the relation (3.1) is satisfied. In order to prove that V ∈C1(AN,H1,H−1)it suffices to show that the sesquilinear form onCc∞(X)defined by
[2AN, V] =P
j 2[qjsin(apj), V]−ia[cos(apj), V]
(4.5) is continuous for theH1topology. This is clear for the second term in the sum and for the first one we use
[qjsin(apj), V] = [qj, V] sin(apj) +qj[sin(apj), V]. (4.6) The first term on the right hand side defines a bounded operatorH1→ H−1by one of the hypotheses of the theorem. For the second one we first note that
[Tj, V] =δj(V)Tj and [Tj∗, V] =−Tj∗δj(V) (4.7) from which we get
2i[sin(apj), V] = [Tj−Tj∗, V] =δj(V)Tj+Tj∗δj(V). (4.8) It follows easily thatqj[sin(apj), V]is bounded. ThusV is of classC1(AN,H1,H−1).
It remains to show that (3.1) is satisfied. We use (4.5) again: the terms withsin(apj) are treated with the help of (4.6) and (4.8). The term withcos(apj)is treated similarly by using
2[cos(apj), V] = [Tj+Tj∗, V] =δj(V)Tj−Tj∗δj(V). (4.9)
Using (4.4), this proves (3.1).
Remark 4.3. — The “usual” version of the preceding theorem involves the derivatives [pj, V] of the potential, instead of the finite differences δj(V), cf. [1, Theorem 7.6.8].
Note that the quoted theorem is a consequence of Theorem 4.1 because kδj(V)kB 6 ak[pj, V]kB.
The condition (4.4) says that the operators[qj, V]andqjδj(V)are not only bounded as mapsH1 → H−1but also tend to zero at infinity in some weak sense. Then it is clear that the mapsλ7→R(λ±i0)∈B(K,K∗)are strongly continuous outside the eigenvalues ofH, but nothing else can be said in general. Stronger conditions on this decay improve the smoothness properties of the boundary valuesR(λ±i0)as mapsH1s → H−s−1. This question is solved in general by using Theorem 3.3 but here we consider only a particular case as an example. One may see in [12, Theorem 1.7] the type of assumptionsV has to satisfy in order to improve the smoothness properties of the boundary values.
Remark that ifV is a multiplication operator, we have
δjδk(V) =V(q+aej+aek)−V(q+aej)−V(q+aek) +V(q)
which appears in the second commutator[[V, iAN], iAN].
The next result is an extension of [16, Theorem 1]: we make the regularity assumption V ∈ C2(AN)butV is not necessarily an operator inHand we give the precise Hölder continuity order of the boundary values.
THEOREM 4.4. — LetV =V(q) : H1 → H−1be a symmetric compact multiplica- tion operator. Assume that there is a real numbera >0such that for allj, k
(H)
( qj V(q+aej)−V(q) and
qjqk V(q+aej+aek)−V(q+aek)−V(q+aek) +V(q)
are bounded operatorsH1→ H−1. ThenHhas normal spectrum in the intervalIaand the limitsR(λ±i0) = w*-limε↓0R(λ±iε)exist inB(K,K∗), locally uniformly inλ∈Ia
outside the eigenvalues ofH. If 12 < s <32then the operatorsR(λ±i0)∈B(H−1s ,H1−s) are locally Hölder continuous functions of orders−12 of the parameterλ∈Iaoutside the eigenvalues ofH.
Proof. — We first show that (1.8) is satisfied for any open intervalJ whose closure is included in Ia, i.e.inf{k·uN(k) | k ∈ X,|k|2 ∈ J} > 0. Sincek 7→ k·uN(k)is a continuous function and J¯is a compact inIa, it suffices to check thatak·uN(k) = Pakjsin(akj) > 0 for all k such that|k|2 ∈ Ia. The last condition may be written 0<|ak|< πand this implies|akj|< πfor alljand|akj|>0for at least onej. Clearly then we getak·uN(k)>0.
For the rest of the proof it suffices to check thatV is of classC2(AN,H1,H−1). Indeed, then we may use Theorems 1.2, Corollary 3.1, and Theorem 3.3 (see also Corollary 3.4).
Thus we have to prove that the commutators [AN, V] and[AN,[AN, V]], which are a priori defined as sesquilinear forms on Cc∞(X), extend to continuous forms on H1. Although the computations are very simple, we give the details for the convenience of the reader.
We have[AN, V] =P
[Aj, V]and[AN,[AN, V]] =P
[Aj,[Ak, V]]and we recall the relations (4.7). Sincesin(apj) = 2i1(Tj−Tj∗)andcos(apj) = 12(Tj+Tj∗)we have
2iAj=qj2i sin(apj) +acos(apj) =qj(Tj−Tj∗) +b(Tj+Tj∗) (4.10) whereb=a/2. Thus by using the relations[Tj, V] =δj(V)Tjand[Tj∗, V] =−Tj∗δj(V) and sinceTjqjTj∗=qj+a, we get
[2iAj, V] = qj[Tj−Tj∗, V] +b[Tj+Tj∗, V]
= (b+qj)δj(V)Tj+Tj∗(−b+qj)δj(V) (4.11) BecauseTj = eiapj andTj∗ = e−iapj are bounded, by the assumption(H)the right hand side of this relation is a bounded operator from H1 to H−1, hence V is of class C1(AN,H1,H−1). It remains to treat the second order commutators.
Since[iAN, S∗] = [iAN, S]∗, we have
[iAj,[iAk, V]] = [iAj,(b+qk)δk(V)Tk] + [iAj, Tk∗(−b+qk)δk(V)]
= [iAj,(b+qk)δk(V)]Tk+ (b+qk)δk(V)[iAj, Tk]
+ [iAj, Tk]∗(−b+qk)δk(V) +Tk∗[iAj,(−b+qk)δk(V)]. (4.12)
