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A New Class of Schrödinger Operators without Positive Eigenvalues
Alexandre Martin
To cite this version:
Alexandre Martin. A New Class of Schrödinger Operators without Positive Eigenvalues. 2018. �hal-
01636915v2�
EIGENVALUES
October 8, 2018
ALEXANDRE MARTIN
ABSTRACT. Following the proof given by Froese and Herbst in [FH82] with another con- jugate operator, we show for a class of real potential that possible eigenfunction of the Schr¨odinger operator has to decay sub-exponentially. We also show that, for a certain class of potential, this bound can not be satisfied which implies the absence of strictly positive eigenvalues for the Schr¨odinger operator.
CONTENTS
1. Introduction 1
2. Main results 4
3. Notations and basic notions 8
3.1. Notation 8
3.2. Regularity 10
4. Sub-exponential bounds on possible eigenvectors 11
4.1. The operator version 11
4.2. The form version 16
5. Possible eigenvectors can not satisfies sub-exponential bounds 18
6. Concrete potentials 19
6.1. Preliminary results 20
6.2. A class of oscillating potential 22
6.3. A potential with high oscillations 26
Appendix A. The Helffer-Sj¨ostrand formula 27
References 28
1. INTRODUCTION
In this article, we will study the Schr¨odinger operatorH “ ∆`V with a real potential, onL2pRνq, where∆is the non negative Laplacian operator. HereV is a multiplication operator, i.e.V can be the operator of multiplication by a real function or by a distribution of strictly positive order. When V “ 0, we know thatH “ ∆ has a purely absolutly continuous spectrum onr0,`8qwith no embedded eigenvalues. We will try to see what
1
happened if we add to∆a ”small” potentialV, which is compact with respect to∆. In this case, H is a compact perturbation of∆and we already know that the essential spectrum ofHisr0,`8q.
An argument of quantum mechanics can make us believe that our Hamiltonians has no strictly positive eigenvalues, when V is ∆-compact or compact onH1, the first order Sobolev space, to H´1, its dual space. This argument is reinforced by a result of S.
Agmon [Agm70], T. Kato [Kat59], R. Lavine [RS70, Theorem XIII.29] and B. Simon [Sim67]. They proved the absence of positive eigenvalues for the operatorH “∆`V if the potential is a sum of a short range potential and a long range potential, i.e. V can be writtenV “V1`V2with
$
’’
’’
&
’’
’’
% lim
|x|Ñ`8|x|V1pxq “0 lim
|x|Ñ`8V2pxq “0
|x|Ñ`8lim x¨∇V2pxq ď0.
Similarly, L. H¨ormander [H¨or83, Theorem 14.7.2] proved that a possible eigenvector of H, associated to a positive eigenvalue, and its first order derivatives cannot have unlimited polynomial bounds if |x|V is bounded. A.D. Ionescu and D. Jerison [IJ03] proved also this absence of positive eigenvalues for the 1-body Schr¨odinger operator, for a class of potentials with low regularity (V PLν{loc2ifν ě3,V PLrloc,rą1ifν “2).
R. Froese, I. Herbst , M. Hoffman-Ostenhof and T. Hoffman-Ostenhof ([FH82] and [FHHOHO82]) proved a similar result, concerning the N-body Schr¨odinger operator. We will explain below their result for the 1-body Schr¨odinger operator and we will generalize their proof to obtain larger conditions on the potential. More recently, using a similar proof than in [FH82], two other results were proved. T. Jecko and A. Mbarek [JM17]
proved the absence of positive eigenvalues forH “∆`V whereV is the sum of a short range potential, a long range potential and an oscillating potential which are not covered by the previous results. In the case of the discrete Schr¨odinger operator, M.A. Mandich [Man16] proved that under certain assumption on the potential, eigenfunctions decays sub- exponentially and that implies the absence of eigenvalues on a certain subset of the real axis. This three proofs use the generator of dilations AD, or the discrete generator of dilations in [Man16], as conjugate operator. In our case, the continuous case, the generator of dilations has the following expression
AD“1
2pp¨q`q¨pq,
whereqis the multiplication operator byxandp“ ´i∇is the derivative operator with p2“∆.
On the other hand, it is well known that we can construct a potential such that H has positive eigenvalues. For example, in one dimension, the Wigner-von Neuman potential Wpxq “wsinpk|x|q{|x|withką0andw PR›has a positive eigenvalue equal tok2{4 (see [NW29]). Moreover, B. Simon proved in [Sim97] that for all sequencepKnqn“1¨¨¨`8
of distinct positive reals, we can construct a potentialV such thatpK2nqn“1¨¨¨`8are eigen- values ofH. Moreover, B. Simon showed that ifř8
n“1Kn ă 8, then|q|V is bounded, which implies thatV is∆-compact.
In their article [FH82], R. Froese and I. Herbst proved the following
Theorem 1.1([FH82], Theorem 2.1). LetH “∆`V withV a real-valued measurable function. Suppose that
(1) V is∆-bounded with bound less than one,
(2) p∆`1q´1q¨∇Vp∆`1q´1is bounded.
Suppose thatHψ“Eψ. Then SE “sup
"
α2`E; αą0,exppα|x|qψPL2pRνq
*
(1.1) is`8or the Mourre estimate is not valid at this energy withADas conjugate operator.
From this result, they deduce the following
Corollary 1.2([FH82], Theorem 3.1). LetH “∆`V withV a real-valued measurable function. LetEą0. Suppose that
(1) V is∆-compact,
(2) p∆`1q´1q¨∇Vp∆`1q´1is compact,
(3) for someaă2andbPR, we have in the form sense
q¨∇V ďa∆`b. (1.2)
Suppose thatHψ“Eψ. Thenψ“0.
Following their proof, we will extend their result in two directions. First, we will see that for a larger class of∆-compact potential, we can prove that possible eigenvector ofHmust satisfy some sub-exponential bounds in theL2-norm. We will also show that this implies the absence of positive eigenvalue ifV satisfies a condition of type (1.2). Secondly, we will extend their results in the case where the potential is not∆-bounded but compact from H1toH´1. To prove these results we will use another conjugate operator of the form
Au“1
2puppq ¨q`q¨uppqq,
where u is a C8 vector field with all derivatives bounded. Remark that this type of conjugate operator is essentially self-adjoint with the domain DpAuq Ą DpADq (see [ABdMG96, Proposition 4.2.3]). This conjugate operator was also used in [Mar17]. In this paper, it is proved that for a certain choice ofu(ubounded), the commutator between V andAucan avoid us to impose conditions on the derivatives of the potential, which can be useful when V has high oscillations. Moreover, the commutator with the Laplacian, considered as a form with domainH1, is quite explicit:
r∆, iAus “2p¨uppq
which implies that the commutator is bounded fromH1toH´1. Since the unitary group generated byAu leaves invariant the domain and the form domain of the Laplacian (see [ABdMG96, Proposition 4.2.4]), this proves that∆is of class C1pAuqand, similarly if if we add a potential V which is ∆-compact( respectively compact from H1 toH´1), with the regularityC1pAu,H2,H´2q(respectivelyC1pAu,H1,H´1q), since the domain (respectively the form domain) is the same than the domain of the Laplacian, we deduce thatH “∆`V is of classC1pAuq. If we takeusuch thatx¨upxq ą0for allx“0, remark that the Mourre estimate is true withAuas conjugate operator on all compact subset ofp0,`8qfor∆. For this reason, and to follow the proof of [FH82], it will be convenient to chooseuof the formxλpxqwithλ:Rν ÑRa positive function. All differences with [FH82] will be explain in Section4and Section5.
2. MAIN RESULTS
Now we will give our main results. Notice that we will recall the notion of regularity (Ck, CUk,C1,1) with respect to an operator on Section3.2.
To simplify notations, letUbe the set of vector fieldsuwith all derivatives bounded which can be writedupxq “ xλpxqwithλaC8 bounded positive function. In particular,p¨
∇λppqis bounded. We have the following:
Theorem 2.1. LetH “∆`V onL2pRνq, whereV is a symmetric potential such thatV is
∆-bounded with bound less than one. LetEPRandψsuch thatHψ“Eψ. Suppose that there isuPU such thatp∆`1q´1rV, iAusp∆`1q´1is bounded, then, for all0ăβ ă1,
SE “sup
"
α2`E; αą0,exppαxxyβqψPL2pRνq
*
is either`8or inEupHq, the complement of the set of points for which the Mourre esti- mate (see Definition3.2) is satisfied with respect toAu.
We will give some comments about this Theorem:
(a) LetuP U. Since the unitary group generated byAu leaves invariant the Sobolev spaceH2,V P C1pAu,H2,H´2qif and only ifp∆`1q´1rV, iAusp∆`1q´1is bounded. Thus, in this case, we can replace the assumptionp∆`1q´1rV, iAusp∆` 1q´1in Theorem2.1by an assumption of regularity.
(b) Since we do not have an explicit expression for the commutator between an operator of multiplication and the conjugate operatorAu, in the proof of Theorem2.1, it is convenient to chose the functionF, which appears in the proof, with a vanishing gradient at infinity. This is the case if β ă 1 but not ifβ “ 1. Remark that for certain type of potential, by using the interaction between the potential and∆, we can prove the exponential bounds or sub-exponential bounds (β “ 1), even if V RC1pAu,H2,H´2q(see [JM17, Proposition 7.1] and Proposition6.3).
(c) Remark that ifV is∆-compact andV PCu1pAu,H2,H´2q, foruPU,V satisfies assumptions of Theorem2.1and the Mourre estimate is true for allλ P p0,`8q (see [ABdMG96, Theorem 7.2.9]). So, in this case, ifEą0, thenexppαxxyβqψP L2pRνqfor allαą0andβP p0,1q. Moreover, in this case, by the Virial Theorem, we can see that the set of eigenvalues inJ “ p0,`8qhas no accumulation point insideJand are of finite multiplicity.
(d) IfV vanishes at infinity and can be seen as the Laplacian of a short range potential (i.e. V “∆W with lim
|x|Ñ`8xxyW “0), thenV is∆-compact andxqyV : H2 Ñ H´2is compact. In this case, we can apply Theorem2.1toH “∆`V.
(e) ForζPR,θą0,kPR˚andwPR, let
Vpxq “wp1´κp|x|qqsinp|x|ζq
|x|θ ,
with κ P Cc8pR,Rqwith κp|x|q “ 1 if |x| ă 1, 0 ď κ ď 1. Note that this type of potential was already studied in [BAD79,DMR91,DR83a,DR83b,JM17, RT97a,RT97b]. Ifζ ăθor ifθ ą1, we can see thatV is a long range or a short range potential. Moreover, in [JM17], it is proved that ifζ`θ ą 2, thenV has a good regularity with respect toAD. So we can apply Theorem1.1in these two areas. In [JM17], they also showed that if ζ ą 1 andθ ą 1{2, then a possible eigenvector associated with positive energy has unlimited exponential bounds. But, if|ζ´1| `θă1, they proved thatH RC1pADqand so we cannot apply Theorem 1.1with this potential. If2ζ`θ ą3,ζ ą1and0 ăθ ď1{2, thenV is of class
C1,1pAu,H2,H´2q Ă Cu1pAu,H2,H´2qfor allubounded (see [Mar17, Lemma 5.4]). So, Theorem2.1applies if2ζ`θą3withζą1and0ăθď1{2.
Since the Laplacian operator∆can be seen as a form onH1, the first order Sobolev space, to H´1, the dual space of H1, we can also study the case where V : H1 Ñ H´1 is compact. In this case, the difference between the resolvent ofH and the resolvent of∆is compact and the essential spectrum ofHis stillr0,`8q. We have the following
Theorem 2.2. LetH “∆`V onL2pRνq, whereV is a real-valued function such that V : H1 Ñ H´1is bounded with relative bound less than one. LetE P Randψsuch thatHψ “Eψ. If there isuPU such thatxpy´1rV, iAusxpy´1is bounded, then, for all 0ăβă1,
SE “sup
"
α2`E; αą0,exppαxxyβqψPL2pRνq
*
is either`8or inEupHq.
We make some comments about this theorem:
(a) SinceV :H1 ÑH´1is bounded with relative bound less than one, by the KLMN Theorem,Hcan be considered as a form with form domainH1and is associated to a self-adjoint operator.
(b) LetuP U. Since the unitary group generated byAu leaves invariant the Sobolev spaceH1,V PC1pAu,H1,H´1qif and only if
p∆`1q´1{2rV, iAusp∆`1q´1{2
is bounded. Thus, in this case, we can replace the assumption on p∆`1q´1{2rV, iAusp∆`1q´1{2
in Theorem2.2by an assumption of regularity.
(c) IfV :H1ÑH´1is compact and ifV PCu1pAu,H1,H´1q, then p∆`1q´1{2rV, iAusp∆`1q´1{2
is compact. Thus the Mourre estimate is true on all compact subset ofp0,`8q. So, ifEą0, in this case, the sub-exponential bounds are true for allαą0.
(d) ForζPR,θPR,kPR˚andwPR, let
Vpxq “wp1´κp|x|qqsinp|x|ζq
|x|θ ,
withκPCc8pR,Rqwithκp|x|q “1if|x| ă1,0ďκď1. Ifζ`θą2, thenV : H1ÑH´1is compact andV is of classC1,1pAu,H1,H´1q ĂCu1pAu,H1,H´1q for allubounded (see [Mar17, Lemma 5.4]). So, Theorem2.2applies ifζ`θą2, even ifθď0.
(e) Let
Vpxq “wp1´κp|x|qqe3|x|{4sinpe|x|q
withw P R, κ P Cc8pR,Rq,0 ď κ ď 1 andκp|x|q “ 1 if|x| ă 1. Note that this type of potential was already studied in [Com80,CG76]. We can show thatV : H1ÑH´1is compact andV is of classC1,1pAu,H1,H´1q ĂCu1pAu,H1,H´1q for allubounded (see[Mar17, Lemma 5.6]). So, for allwPR, Theorem2.2applies.
Moreover, sinceV is not∆-bounded, we cannot apply Theorem1.1.
(f) Assume thatV :H1ÑH´1is symmetric, bounded with bound less than one and that there isµą 0such thatxxy1`µVpxq P H´1. Then there isu PU such that V P C1,1pAu,H1,H´1q (see [Mar17, Lemma 5.8]). In particular, for this type of potential, Theorem 2.2applies. For example, in dimensionν ě 3, if we take χ : RÑRsuch thatχ PC3,χp|x|q “0if|x| ą1andχ1p0q “ χ2p0q “1, the potential defined by
Vpxq “
`8
ÿ
n“2
np3ν´1q{2χ1pn3ν{2p|x| ´nqq,
is compact on H1 toH´1 and of class C1,1pAu,H1,H´1qfor an appropriateu.
Moreover, we can show that this potential is neither ∆-bounded, neither of class C1pAD,H1,H´1q(see [Mar17, Lemma 5.10]). In particular, Theorems1.1and2.1 do not apply with this potential.
(g) Remark that all examples we gave are central potentials. But it is not necessary to have this property and we gave only examples which are central because it is easier.
In particular, if W satisfies xqy1`ǫW is bounded for one choice ofǫ ą 0, then Theorem2.2applies forV “divpWq.
Since in the proof of Corollary1.2, one use only assumption (1.2) by applying it on certain vectors that are constructed with a possible eigenvector ofH, we can weaken the conditions on the potential. For0ăβ ă1andαą0, letFβpxq “αxxyβ. We have the following Theorem 2.3. Suppose thatV is∆-compact.
Letψsuch thatHψ “EψwithE ą0and such thatψF “exppFβpqqqψPL2pRνqfor allαą0,0ăβă1.
Suppose that there isδ ą ´2,δ1, σ, σ1 P Rsuch thatδ`δ1 ą ´2and, for allα ą 0, 0ăβă1,
pψF,rV, iADsψFq ěδpψF,∆ψFq `δ1pψF,p∇Fβq2ψFq ` pσα`σ1q}ψF}2. (2.1) Thenψ“0.
If we only suppose thatV :H1 ÑH´1is bounded (but not necessarily∆-bounded), we have the following:
Theorem 2.4. Suppose thatV :H1ÑH´1is bounded.
Letψsuch thatHψ “EψwithE ą0. For0ăβ ă1andαą0, letFβpxq “αxxyβ. DenoteψF “exppFβpqqqψ.
Suppose that ψF P L2pRνq for all α ą 0, 0 ă β ă 1, and that there is δ ą ´2, δ1, σ, σ1 PRsuch thatδ` p1` }xpy´1Vxpy´1}qδ1ą ´2and, for allαą0,0ăβă1,
pψF,rV, iADsψFq ěδpψF,∆ψFq `δ1pψF,p∇Fβq2ψFq ` pσα`σ1q}ψF}2. (2.2) Thenψ“0.
We make some comments on the two previous theorems:
(a) Since we suppose thatψhas sub-exponential bounds, forα, βfixed,ψF has sub- exponential bounds too. Moreover, we can remark thatψFis an eigenvector for
HpFq:“eFHe´F “H´ p∇Fq2` pip∇F`i∇F pq.
This makes easier to prove (2.1) and (2.2).
(b) Remark that in (1.2), the inequality is required to be true in the sense of the form. In (2.1) and (2.2), we do not ask to have this inequalities for allφPDpHq XDpAuq, but only for a type of vector with high decrease at infinity.
(c) Assumption (1.2) corresponds to the case whereδ1 “σ“0andδą ´2in (2.1).
In particular, ifV satisfies (1.2), it satisfies (2.1) too.
(d) Remark that ifδ1ě0, conditionsδ`δ1ą ´2andδ`p1`}xpy´1Vxpy´1}qδ1 ą ´2 are always satisfied.
(e) Actually, one only need to require (2.1) and/or (2.2) forβnear1andαlarge enough.
(f) We can replace (2.1) by the similar inequality
pψF,rV, iADsψFq ěδpψF,∆ψFq `δ1pψF,p∇Fβq2ψFq `δ2}g1{2ADψF}2
` pσα`σ1q}ψF}2 (2.1’) withδą ´2,δ`δ1ą ´2andδ2ą ´4. (2.2) may be replaced by
pψF,rV, iADsψFq ěδpψF,∆ψFq `δ1pψF,p∇Fβq2ψFq `δ2}g1{2ADψF}2
` pσα`σ1q}ψF}2 (2.2’) withδ ą ´2,δ` p1` }xpy´1Vxpy´1}q}δ1 ą ´2 and δ2 ą ´4 and the both Theorems remain true. This enlarges the class of admissible potentials (see Section 6).
(g) LetVsrandVlrbe two functions such that there isρsr, ρlr, ρ1lrą0and
|x|1`ρsrVsrpxq,|x|ρlrVlrpxqand|x|ρ1lrx∇Vlrpxqare bounded. Suppose thatV sat- isfies assumptions of Theorem2.3(respectively Theorem2.4).
ThenV˜ “V `Vsr`Vlrsatisfies assumptions of Theorem2.3(respectively Theo- rem2.4) too. To see that, notice thatVlrandVsrare compact onH1and are of class C1pAD,H1,H´1q XCu1pAu,H1,H´1qfor alluPU and that there isσ1, σ2 PR such that
pψF,rVlr, iADsψFq ě σ1}ψF}2
pψF,rVsr, iADsψFq ě ´ǫpψF,∆ψFq `σ2
ǫ }φ}2
for allǫ ą 0. In particular, we can chooseǫ ą 0small enough such that, ifV satisfies (2.1) (respectively (2.2)),V˜ satisfies (2.1) (respectively (2.2)).
(h) IfV can be seen as the derivative of a bounded function (the derivative of a short range potential for example), the conclusion of Theorem2.4is still true if one as- sume (2.2) and if one replaces the conditionδ` p1` }xpy´1Vxpy´1}qδ1ą ´2by the weaker conditionδ`δ1ą ´2.
(i) Forζ, θPR,kPR˚andwPR, let
Vpxq “wp1´κp|x|qqsinp|x|ζq
|x|θ ,
withκ P Cc8pR,Rq withκp|x|q “ 1 if |x| ă 1,0 ď κ ď 1. As for the sub- exponential bounds, we can see that ifθ ą0andζ ăθorθ ą1, then Corollary 1.2applies. In [JM17], they showed that ifζ ą1andθ ą1{2,H “∆`V has no positive eigenvalues. Moreover, they claimed that ifθą0,ζ`θą2and|w|is small enough thenV satisfies (1.2) and so Corollary1.2applies. But their proof is not sufficient ifθď1{2because we need to have the commutator bounded fromH1 toH´1and in this case, it is only bounded fromH2toH´2. Here, we can show a better result: ifζ`θą2,V :H1ÑH´1is compact, of classC1,1pAu,H1,H´1q for all u bounded and satisfies (2.2) for allw. ThereforeV satisfies assumptions of Theorem2.4for allw P R. In particular, ifθ ă 0,V is not bounded. Moreover, ifζ `θ “ 2and1{2 ě θ, thenV satisfies assumptions of Theorem2.3for|w|
sufficiently small. All this results are collected in Proposition6.3.
(j) Let
Vpxq “wp1´κp|x|qqe3|x|{4sinpe|x|q
withwPR,κPCc8pR,Rq,0ďκď1andκp|x|q “1if|x| ă1. For allwPR, we can apply Theorem2.4(see Lemma6.4). Moreover, sinceV is not∆-bounded, we cannot apply Corollary1.2.
Now, we assume that V has more regularity with respect toAu. In this case, we can prove a limiting absorption principle and we can show that the boundary values of the resolvent will be a smooth function outside the eigenvalues. To this end, we need to use the H¨older-Zygmund continuity classesdenotedΛσ. The definition of this particular classes of regularity is recalled on Section3.2. We also need some weighted Sobolev space, denoted Htswhich are defined on Section3.1
Theorem 2.5([Mar17], Theorem (3.3)). LetRpzq “ pH´zq´1be the resolvent operator associate toH. LetV :H1ÑH´1be a compact symmetric operator. Suppose that there isuPUandsą1{2such thatV is of classΛs`1{2pAu,H1,H´1q. Then the limits
Rpλ˘i0q:“w*-lim
µÓ0 Rpλ˘iµq (2.3)
exist, locally uniformly inλP p0,`8qoutside the eigenvalues ofH. Moreover, the func- tions
λÞÑRpλ˘i0q PBpH´s1,H1´sq (2.4) are locally of classΛs´1{2onp0,`8qoutside the eigenvalues ofH.
SinceΛs`1{2pAuq ĂCu1pAuqfor allsą1{2, by combining Theorems2.2,2.4and2.5, we have the following
Corollary 2.6. LetV : H1 Ñ H´1be a compact symmetric potential ands ą 1{2. If there isuPU such thatV is of classΛs`1{2pAu,H1,H´1q, and if (2.2)is satisfied, then the limits
Rpλ˘i0q:“w*-lim
µÓ0 Rpλ˘iµq (2.5)
exist locally uniformly inλP p0,`8qand
λÞÑRpλ˘i0q PBpH´s1,H1´sq (2.6) are of classΛs´1{2onp0,`8q.
The paper is organized as follows. In Section3, we will give some notations and we recall some basic fact about regularity. In Section4, we will prove Theorem2.1and Theorem 2.2. In Section5, we will prove Theorem2.3and Theorem2.4. In Section6, we will give some explicit classes of potential for which we can apply our main results. Finally in AppendixA, we will recall the Helffer-Sj¨ostrand formula and some properties of this formula that we will use in the proof of our main Theorems.
3. NOTATIONS AND BASIC NOTIONS
3.1. Notation. LetX “Rν and forsPRletHsbe the usual Sobolev space onX with H0“H“L2pXqwhose norm is denoted} ¨ }. We are mainly interested in the spaceH1 defined by the norm}f}21“ş `
|fpxq|2` |∇fpxq|2˘
dxand its dual spaceH´1.
We denoteqjthe operator of multiplication by the coordinatexjandpj“ ´iBjconsidered as operators inH. ForkPX we denotek¨q“k1q1` ¨ ¨ ¨ `kνqν. Ifuis a measurable function onXletupqqbe the operator of multiplication byuinHanduppq “F´1upqqF, whereFis the Fourier transformation:
pFfqpξq “ p2πq´ν2 ż
e´ix¨ξupxqdx.
If there is no ambiguity we keep the same notation for these operators when considered as acting in other spaces. Ifuis aC8vector fields with all the derivates bounded, we denote byAuthe symmetric operator:
Au“1
2pq¨uppq `uppq ¨qq “uppq ¨q`i
2pdivuqppq. (3.1) Notice thatAuis essentially self-adjoint (see [ABdMG96, Proposition 4.2.3]). Since we will use vector fieldsuwhich have a particular form, we use the spaceUdefine by Definition 3.1. We defineU the space ofC8 vector fieldsuwith all derivates bounded such that there is a strictly positive bounded function λ : X Ñ R of classC8 with upxq “xλpxqfor allxPX.
LetAD“ 12pp¨q`q¨pqbe the generator of dilations.
As usual, we denotexxy “ p1` |x|2q1{2. Thenxqyis the operator of multiplication by the functionxÞÑ xxyandxpy “F´1xqyF. For reals, twe denoteHtsthe space defined by the norm
}f}Hts “ }xqysf}Ht “ }xpytxqysf}. (3.2) Note that the norm}f}Htsis equivalent to the norm}xqysxpytf}and that the adjoint space ofHtsmay be identified withH´t´s.
We denote∆“p2the non negative Laplacian operator, i.e. for allφPH2, we have
∆φ“ ´
n
ÿ
i“1
B2φ Bx2i.
ForIa Borel subset ofR, we denoteEpIqthe spectral mesure ofHonI.
Definition 3.2. LetA be a self adjoint operator onL2pRνq. Assume thatH is of class C1pAq. We say thatH satisfies the Mourre estimate atλ0with respect to the conjugate operator A if there exists a non-empty open setI containingλ0, a realc0 ą 0 and a compact operatorK0such that
EpIqrH, iAsEpIq ěc0EpIq `K0 (3.3) We denoteEupHqthe complement of the set ofλ0for which the Mourre estimate is satis- fied with respect toAu.
In the Helffer-Sj¨ostrand formula (AppendixA), there is a term of rest which appears. To control it we define the following space of application:
Definition 3.3. ForρPR, letSρbe the class of the functionϕPC8pRν,Cqsuch that
@kPN, Ckpϕq– sup
tPRν
|α|“k
xty´ρ`k|Bαtϕptq| ă 8. (3.4)
Note thatCkdefine a semi-norm for allk.
3.2. Regularity. LetF1, F2be to Banach space andT :F1ÑF2a bounded operator.
LetAa self-adjoint operator such that the unitary group generated byAleavesF1andF2 invariants.
LetkPN. We said thatT PCkpA, F1, F2qif, for allf PF1, the map RQtÞÑeitAT e´itAfhas the usualCkregularity.
We said thatT PCukpA, F1, F2qifT PCkpA, F1, F2qand all the derivatives of the map R Q t ÞÑ eitAT e´itAf are norm-continuous function. The following characterisation is available:
Proposition 3.4 (Proposition 5.1.2, [ABdMG96]). T P C1pA, F1, F2q if and only if rT, As “T A´AThas an extension inBpF1, F2q.
Forką1,T PCkpA, F1, F2qif and only ifT PC1pA, F1, F2qandrT, As PCk´1pA, F1, F2q.
We can defined another class of regularity called theC1,1regularity:
Proposition 3.5. We said thatT PC1,1pA, F1, F2qif and only if ż1
0
}Tτ`T´τ´2T}BpF1,F2q
dτ τ2 ă 8, whereTτ “eiτ AuTe´iτ Au.
An easier result can be used:
Proposition 3.6(Proposition 7.5.7 from [ABdMG96]). LetAbe a self-adjoint operator.
LetGbe a Banach space and letΛbe a closed densely defined operator inG˚with domain included in DpA,G˚qand such that´ir belongs to the resolvent set ofΛ andr}pΛ` irq´1}BpG˚qďCPRfor allrą0. LetξPC8pXqsuch that
ξpxq “ 0 if|x| ă 1 andξpxq “ 1if |x| ą 2. IfT : G Ñ G˚ is symmetric, of class C1pA,G,G˚qand satisfies
ż8 1
}ξpΛ{rqrT, iAs}BpG,G˚q
dr r ă 8 thenTis of classC1,1pA,G,G˚q.
IfT is not bounded, we said thatT PCkpA, F1, F2qif forzRσpTq, pT´zq´1PCkpA, F2, F1q.
Proposition 3.7. For allką1, we have
CkpA, F1, F2q ĂC1,1pA, F1, F2q ĂCu1pA, F1, F2q ĂC1pA, F1, F2q.
IfF1“F2“His an Hilbert space, we noteC1pAq “C1pA,H,H˚q. IfTis self-adjoint, we have the following:
Theorem 3.8(Theorem 6.3.4 from [ABdMG96]). LetA andT be two self-adjoint op- erators in a Hilbert spaceH. Assume that the unitary grouptexppiAτquτPR leaves the domainDpTqofT invariant. SetG“DpTq. Then
(1) Tis of classC1pAqif and only ifT PC1pA,G,G˚q.
(2) Tis of classC1,1pAqif and only ifT PC1,1pA,G,G˚q.
Remark that, ifT :HÑHis not bounded, sinceT :GÑG˚is bounded, in general, it is easier to prove thatT PC1pA,G,G˚qthanT PC1pAq.
IfGis the form domain ofH, we have the following:
Proposition 3.9(see p. 258 of [ABdMG96]). LetAandT be self-adjoint operator in a Hilbert spaceH. Assume that the unitary grouptexppiAτquτPRleaves the form domainG ofTinvariant. Then
(1) Tis of classCkpAqifT PCkpA,G,G˚q, for allkPN. (2) Tis of classC1,1pAqifTPC1,1pA,G,G˚q.
As previously, sinceT :G ÑG˚is always bounded, it is, in general, easier to prove that T PCkpA,G,G˚qthanT PCkpAq.
Now we will recall theH¨older-Zygmund continuity classesof ordersP p0,8q. LetEbe a Banach space andF :RÑE a continuous function. If0ăsă1thenFis of classΛs ifF is H¨older continuous of orders. Ifs“ 1thenF is of classΛ1if it is of Zygmund class, i.e. }Fpt`εq `Fpt´εq ´2Fptq} ďCεfor all realtandεą0. Ifsą1, let us writes“k`σwithkě1integer and0ăσď1; thenFis of classΛsifF isktimes continuously differentiable andFpkqis of classΛσ. We said thatV P ΛspAu,H1,H´1q if the functionτ ÞÑVτ “eiτ AuV e´iτ Au PBpH1,H´1qis of classΛs. Remark that, if sě1is an integer,CspAu,H1,H´1q ĂΛspAu,H1,H´1q.
4. SUB-EXPONENTIAL BOUNDS ON POSSIBLE EIGENVECTORS
In this section we will prove Theorem2.1and Theorem2.2.
4.1. The operator version. Our proof of Theorem2.1closely follows the one of Theorem 2.1 in [FH82]. Therefore, we focus on the main changes. We will use notations of Theorem 2.1.
Forǫą0andτ ą0, define the real valued functionsF andgby Fpxq “τln
ˆ
xxyp1`ǫxxyq´1
˙
and∇Fpxq “xgpxq. (4.1) LetE PRandψ PDpHqsuch thatHψ “Eψ. LetψF “exppFqψ. On the domain of H, we consider the operator
HpFq “eFHe´F “H´ p∇Fq2` pip∇F`i∇F pq. (4.2) As in [FH82],ψF PDp∆q “DpHpFqq,
HpFqψF “EψF (4.3)
andpψF, HψFq “ pψF,pp∇Fq2`EqψFq. (4.4) If we suppose in addition that
xqyβτexppαxqyβqψPL2pRνq (4.5) for allτ and some fixedαě 0,0 ă β ă 1, then (4.3) and (4.4) holds true for the new functionsFandggiven by
Fpxq “αxxyβ`τlnp1`γxxyβτ´1qand∇Fpxq “xgpxq (4.6) for allγą0andτ ą0.
To replace Formula (2.9) in [FH82], we prove the following
Lemma 4.1. Suppose thatV is∆-compact. LetuP U. Assume thatH “∆`V is of classC1pAuq. For both definitions ofFandg, we have
pψF,rH, iAusψFq “ pψF,rp∇Fq2´q¨∇g, iAusψFq
´4
›
›
›
›
λppq1{2g1{2ADψF
›
›
›
›
2
´2ℜ ˆ
gADψF, i∇λppq ¨pψF
˙
`4ℜ ˆ
rg1{2, λppqsg1{2ADψF, ADψF
˙
. (4.7)
We make some remarks about this Lemma:
(a) In the case (4.1), note thatxxyg1{2pxqis bounded. Thus
›
›
›
›
λppq1{2g1{2ADψF
›
›
›
› is well defined.
(b) In the case (4.6), suppose that (4.5) is true for allτand some fixedαě0,0ăβă1, we have
ADψF “ p¨qψF` i 2ψF
“ p¨ q xqyxqy
ˆ
1`γxqyβτ´1
˙τ
exppαxqyβqψ` i 2ψF. Thus,ψF PL2pRνqand
xqy ˆ
1`γxqyβτ´1
˙τ
exppαxqyβqψPL2pRνq.
Moreover, sinceHψ“Eψand∇Fis bounded for allτą0, we can show that xqy
ˆ
1`γxqyβτ´1
˙τ
exppαxqyβqψPH1.
Thus
›
›
›
›
λppq1{2g1{2ADψF
›
›
›
›
is well defined.
(c) IfV :H1ÑH´1is compact andV PC1pAu,H1,H´1q, Lemma4.1is still true with the same proof.
Proof.[Lemma4.1] SinceV is of classC1pAu,G,G˚qwhithG“H2ifV is∆-compact, G “ H1 if V : H1 Ñ H´1 is compact, by a simple computation, we can show that eF∆e´F is of classC1pAu,G,G˚q, which implies thatHpFq “eF∆e´F`V is of class C1pAu,G,G˚q. ForφPDpHq XDpAuq, we have
pφ,rH, iAusφq “ pφ,rpH´HpFqq, iAusφq ` pφ,rHpFq, iAusφq
“ ppH´HpFqqφ, iAuφq ´ pAuφ, ipH´HpFqqφq
`pφ,rHpFq, iAusφq By using (4.2) and (4.3), we have:
pH´HpFqqφ“ pp∇Fq2´ pip∇F`i∇F pqqφ A simple computation gives
pip∇F`i∇F pqφ“ipppqgq ` pqgqpqφ“q¨∇gφ`2igADφ We have
pH´HpFqqφ“ pp∇Fq2´q¨∇g´2igADqφ
thus
pφ,rH, iAusφq “ pφ,rp∇Fq2´q¨∇g, iAusφq
´ p2gADφ, Auφq ´ pAuφ,2gADφq ` pφ,rHpFq, iAusφq (4.8) Sinceupxq “xλpxq,
Au“1
2pλppqp¨q`q¨λppqpq “λppqAD`1
2rq, λppqsp Using the Fourier transform, we see thatrq, λppqs “i∇λppq.
Therefore
Au“λppqAD` i
2∇λppq ¨p which implies
#p2gADφ, Auφq “ p2gADφ, λppqADφq ` p2gADφ,2i∇λppq ¨pφq pAuφ,2gADφq “ pλppqADφ,2gADφq ` p2i∇λppq ¨pφ,2gADφq By sum, we get
p2gADφ, Auφq ` pAuφ,2gADφq “ 2pADφ,pgλppq `λppqgqADφq
`pgADφ, i∇λppq ¨pφq
`pi∇λppq ¨pφ, gADφq.
Sincegandλare positive,
gλppq `λppqg“2g1{2λppq1{2λppq1{2g1{2`g1{2rg1{2, λppqs ` rλppq, g1{2sg1{2. This yields
pADφ,pgλppq `λppqgqADφq “ 2
›
›
›
›
λppq1{2g1{2ADφ
›
›
›
›
2
`2ℜ ˆ
g1{2ADφ,rg1{2, λppqsADφ
˙ . So from (4.8), we obtain
pφ,rH, iAusφq “ pφ,rp∇Fq2´q¨∇g, iAusφq
´4
›
›
›
›
λppq1{2g1{2ADφ
›
›
›
›
2
´2ℜ ˆ
gADφ, i∇λppq ¨pφ
˙
´4ℜ ˆ
g1{2ADφ,rg1{2, λppqsADφ
˙
` pφ,rHpFq, iAusφq.(4.9) Remark that ifF satisfies (4.1), sincexqyg1{2is bounded, all operators which appears on the right hand side of (4.9) are bounded in theH1norm. In particular, this equation can be extended to a similar equation forφPG ĂH1. Thus, sinceψF PG, we obtain a similar equation by replacingφbyψF.
IfF satisfies (4.6), we can see that all operators which appears on the right hand side of (4.9) are bounded in theH11norm. In particular, this equation can be extended to a similar equation forφPH21ifV is∆-compact,φPH11ifV :H1ÑH´1is compact.
In all cases, since, by the Virial theorem and (4.3),pψF,rHpFq, iAusψFq “0, we obtain
(4.7). l
Since in (4.7), we do not know an explicit form for the commutatorrp∇Fq2´q¨∇g, iAus, as in [FH82], we need to control the size of this expression.
Lemma 4.2. Letf :RνÑRbe aC8application such thatf PSρ.
Thenxqy´ρrfpqq, iAusis bounded for allC8vector fieldsuwith bounded derivatives.
Proof. Suppose thatf PSρ. Then
@kPN, sup
tPRν
txty´ρ`k|Bαtfptq|u ă 8 for allαmulti-index such that|α| “k.
Sincerfpqq, qs “0and
Au“q¨uppq ´ i
2divpuqppq, we have
rfpqq, iAus “ rfpqq, iq¨uppq `1
2divpuqppqs “iq¨ rfpqq, uppqs `1
2rfpqq, divpuqppqs (4.10) By using the Helffer-Sj¨ostrand formula onrfpqq, uppqs, withB “q,T “uppqand ϕpxq “fpxq, we have:
rfpqq, uppqs “i∇fpqqdivpuqppq `I2 (4.11) whereI2is the rest of the development of order2in (A.4). Similarly,
rfpqq, divpuqppqs “i∇fpqq∇divpuqppq `I21 (4.12) So, from (4.10), we have
rfpqq, iAus “ ´q¨∇fpqqdivpuqppq ´q¨I2` i
2∇fpqq∇divpuqppq `I21 (4.13) From Proposition A.3, we deduce, sincef P Sρ, thatxqysI2 andxqysI21 are bounded if s ă ´ρ`2. Moreover, since f P Sρ,xxy´ρ`1∇fpxq is bounded, and we conclude thatxqy´ρq¨∇fpqqis bounded. Since, by assumptions,divpuqppqand∇divpuqppqare
bounded, by sum,xqy´ρrfpqq, iAusis bounded. l
Proof.[Theorem2.1] Suppose thatEREupHq.
LetFpxq “τlnpxxyp1`ǫxxyq´1qandΨǫ“ψF{}ψF}.
Following [FH82, equations (2.11) and (2.12)], we can prove that∇Ψǫis bounded and that p∆`1qΨǫconverges weakly to zero asǫÑ0. Thus, for allηą0, sincexqy´ηp∆`1q´1 is compact,}xqy´ηΨǫ}converges to 0 and, similarly,}xqy´η∇Ψǫ}converges to 0.
From Lemma4.1, we deduce that ˆ
Ψǫ,rH, iAusΨǫ
˙ ď
ˆ
Ψǫ,rp∇Fq2´q¨∇g, iAusΨǫ
˙
´2ℜ ˆ
gADψF, i∇λppq ¨pψF
˙
´2ℜ ˆ
gADΨǫ, i∇λppq ¨pΨǫ
˙
. (4.14)
Sincepp∇Fq2´q¨∇gqis inS´2, by Lemma4.2, we havexqy2rp∇Fq2´q¨∇g, iAus is bounded. Thus the first term on right side of (4.14) converges to zero as ǫ Ñ 0. By assumptions,∇λppq ¨pis bounded. Since∇Ψǫis bounded,xqy´1ADΨǫis bounded and , for allµą0,}xqy´1´µADΨǫ}converges to zero asǫÑ0. Thus, sincexqy2gis bounded, the last term on the right side of (4.14) converges to zero asǫÑ0.
Moreover, by the Helffer-Sjostrand formula, we have
rg1{2, λppqs “ ´i∇pg1{2q∇λppq `I
withxqyIxqys1bounded fors1 ă 1. In particular,xqyrg1{2, λppqsxqys1 is bounded for all s1 ă1. Thus,
›
›
›xqyrg1{2, λppqsg1{2ADΨǫ
›
›
›“
›
›
›xqyrg1{2, λppqsg1{2xqy3{2xqy´3{2ADΨǫ
›
›
›, and sincexqyg1{2is bounded, the second term on the right side of (4.14) converges to zero asǫÑ0.
Thus, we deduce that
lim sup
ǫÑ0
ˆ
Ψǫ,rH, iAusΨǫ
˙ ď0.
We follow [FH82, equations (2.16) to (2.19)]to prove that, ifEREupHq, then xxyτψPL2pRνq @τą0.
Suppose now that the Theorem2.1is false so that
SE“α21`E (4.15)
whereα1 ą0andSE REupHq. By definition ofEupHq, we have (3.3) for someδą0, somec0ą0and some compact operatorK0withI“ rSE´δ, SE`δs.
As in [FH82, equations (2.22) and (2.23)], letαP p0, α1qsuch that α2`EP rSE´δ{2, SE`δ{2s.
Let0ăβă1. We have for allτ ą0
xxyβτexppαxxyβqψPL2pRνq. (4.16) Supposeγą0such thatα`γąα1. So we have
}expppα`γqxxyβqψ} “ `8. (4.17) In the following, we suppose thatγis sufficiently small,γP p0,1s. We denote bybj, j “1,2,¨ ¨ ¨ constants which are independant ofα,γandτ.
LetFpxq “αxxyβ`τlnp1`γxxyβτ´1qandψF “exppFqψ,Ψτ “ψF{}ψF}.
By a simple estimate, we have|x∇gpxq| ďb1xxyβ´2and
p∇Fq2pxq ď pα`γq2xxy2β´2ď pα`γq2.
As previously, (4.14) is true. Sincepp∇Fq2 ´q¨∇gqis inS2β´2, by Lemma4.2, we havexqy2´2βrp∇Fq2´q¨∇g, iAusis bounded. Therefore, the first term on right side of (4.14) converges to zero asτ Ñ 8. By assumptions,∇λppq ¨pis bounded. As previously xqy´1ADΨτis bounded and , for allµą0,}xqy´1´µADΨτ}converges to zero asτ Ñ
`8. Thus, sincexqy2´βgis bounded, the last term on the right side of (4.14) converges to zero asτÑ `8.
Moreover, by the Helffer-Sjostrand formula, we have
rg1{2, λppqs “ ´i∇pg1{2q∇λppq `I withxqysIxqys1bounded forsă2,s1ă1ands`s1ă3´β2. In particular,xqy1rg1{2, λppqsxqy1{2is bounded. Thus,
›
›
›xqyrg1{2, λppqsg1{2ADΨǫ›
›
›“›
›
›xqyrg1{2, λppqsxqy1{2g1{2xqy´1{2ADΨǫ›
›
›, and sincexqy1´β2g1{2is bounded, the second term on the right side of (4.14) converges to zero asτÑ `8.
Thus, we deduce that
lim sup
τÑ8
ˆ
Ψτ,rH, iAusΨτ
˙ ď0.
As in [FH82], we have lim sup
τÑ`8
›
›pH´E´ p∇Fq2qΨτ
›
›“lim sup
τÑ`8 }pp¨∇F`∇F¨pqΨτ} and by a simple computation, we have
p¨∇F`∇F¨p“2∇F¨p`i∆F and we have
}p2∇F¨p`i∆FqΨτ} ď2}∇F¨∇Ψτ} ` }∆FΨτ}.
Since|∇F|pxq ďb3xxyβ´1and|∆F|pxq ďb4xxyβ´2, lim sup
τÑ`8 }pH´E´ p∇Fq2qΨτ} “0 which implies that
lim sup
τÑ`8
}pH´E´α2qΨτ} ďb5γ By following [FH82, equations (2.37) to (2.41)], we deduce that
lim inf
τÑ8 pΨτ, EpIqrH, iAusEpIqΨτq ěc0p1´ pb6γq2q. (4.18) Moreover, since
lim sup
τÑ8
pΨτ,rH, iAusΨτq ď0,
we have
lim sup
τÑ8 pΨτ, EpIqrH, iAusEpIqΨτq ďb7γ. (4.19) From (4.18) and (4.19), we have
c0p1´ pb6γq2q ďb7γ.
Sincec0is a fixed positive number, we have a contradiction for all small enoughγ ą0.
Thus the theorem is proved. l
4.2. The form version. If we only suppose thatV :H1 ÑH´1is bounded with bound less than one, we have the following
Proof. [Theorem2.2] Suppose thatEREupHq. We denoteCi ą0constant independant ofǫ.
LetFpxq “τlnpxxyp1`ǫxxyq´1qandΨǫ“ψF{}ψF}. As in [FH82], we can prove that for any bounded setB
ǫÑlim0
ż
B
|Ψǫ|2dnx“0.
By a simple calculus, we have
∇ψF “∇F ψF `eF∇ψ.
