Spectral deformation for two-body dispersive systems with e.g. the Yukawa potential
aa r X i v : . [ m a t h - ph ] O c t Spectral deformation for two-bodydispersive systems with e.g. theYukawa potential
Matthias EngelmannFachbereich MathematikUniversit¨at StuttgartMorten Grud RasmussenDepartment of Mathematical SciencesAalborg UniversityJuly 17, 2018
Abstract
We find an explicit closed formula for the k ’th iterated commutator ad kA ( H V ( ξ ))of arbitrary order k ≥ H V ( ξ ) = M ω ξ + S ˇ V and a conju-gate operator A = i ( v ξ · ∇ + ∇ · v ξ ), where M ω ξ is the operator of multiplicationwith the real analytic function ω ξ which depends real analytically on the param-eter ξ , and the operator S ˇ V is the operator of convolution with the (sufficientlynice) function ˇ V , and v ξ is some vector field determined by ω ξ . Under certain as-sumptions, which are satisfied for the Yukawa potential, we then prove estimatesof the form k ad kA ( H V ( ξ ))( H ( ξ ) + i ) − k ≤ C kξ k ! where C ξ is some constant whichdepends continuously on ξ . The Hamiltonian is the fixed total momentum fiberHamiltonian of an abstract two-body dispersive system and the work is inspiredby a recent result [3] which, under conditions including estimates of the men-tioned type, opens up for spectral deformation and analytic perturbation theoryof embedded eigenvalues of finite multiplicity. Mathematical Subject Classification:
Keywords: dispersive systems, iterated commutators, spectral deformation, Yukawapotentials 1
Introduction
In this paper we consider a two-body dispersive system where the two particles in-teract via a pair-potential V . We study the iterated commutators ad kA ( H V ( ξ )) =[ad k − A ( H V ( ξ )) , A ] of the fixed total momentum fiber Hamiltonian H V ( ξ ) of this sys-tem with an operator A , which (in the sense of Mourre) is conjugate to H V ( ξ ). Itis well-known from the literature that the nature of the spectrum and regularity ofeigenstates is related to the (iterated) commutators with a conjugate operator, seee.g. [12, 9, 1, 8, 2, 4, 5, 11]. See also [14] for another result where control of iteratedcommutators is needed.Recently, together with Jacob Schach Møller, the authors developed an analyticperturbation theory for embedded eigenvalues in [3], which also contains an exampleof a non-trivial model which satisfies the needed conditions for the abstract theory ofthat paper. We consider a version of this model (introduced in details in Section 2)which has fiber Hamiltonians of the type H V ( ξ ) = ω ξ + S ˇ V , where ω ξ denotes multiplication by a certain analytic function (see Condition 2.1) and S ˇ V denotes convolution by the inverse Fourier transform of the interaction potential V . The abstract conditions in [3] involve the requirement that there exists a constant C ξ > k ∈ N , the iterated commutator ad kA ( H V ( ξ )) exists as a H V ( ξ )-bounded operator and k ad kA ( H V ( ξ ))( H V ( ξ ) + i ) − k ≤ C kξ k !, which serves as amotivation to study these iterated commutators.The result presented in this paper covers the results on the model in [3], but candeal with more singular potentials such as the Yukawa potential, see Proposition 2.7.The main result of this paper, Theorem 2.5, states the existence of a constant C ξ > V satisfies either Condition 2.2 or Condition 2.4. In fact, we prove that this constant iscontinuous as a function of ξ . This allows us to reach the conclusion of [3, Theorem 3.2],if certain further assumptions are satisfied, namely relative boundedness of S ˇ V wrt.the multiplication operator ω ξ and that [ H V ( ξ ) , iA ] satisfies a Mourre estimate. SeeTheorem 2.10 for details. We note that these further assumptions have already beenverified for a large class of potentials in [3].The main obstacle in proving the bound on the iterated commutator is to controlthe commutator with the interaction term. This is done in several steps. First, wefind a closed formula for the k ’th iterated commutator for any k , see Theorem 4.1.Then we find appropriate estimates for every term that appears in the formula. Thesebounds then turn out to have the right behaviour.The paper is organized as follows. In Section 2 the model is introduced, conditionsare stated, and the main results are formulated. To write the closed commutatorformula in a compact, readable form and for use in proofs and various intermediateresults, we introduce some terminology and notation in Section 3. In Section 4, we thenstate and prove the main technical result, Theorem 4.1, which is the closed commutatorformula. In Section 5, a technical lemma is proven which paves the way for an estimate2n the scalar factors in the formula, which is then stated and proved in Section 6. Thenext step is to estimate the number of terms in the sum in the formula, which is donein Section 7. Before stating some sufficient a posteriori assumptions and making thefinal estimates on the interaction commutator, we turn our attention to the iteratedcommutator of the free Hamiltonian in Section 8. This section is essentially a repetitionof known results by the present authors and J. S. Møller, see [3], and is included forthe reader’s convenience. The methods used for the free Hamiltonian can be copied todeal with certain parts of the interaction commutator, which we return to in Section 9,where we identify a posteriori assumptions which are sufficient to prove the right kindof bounds on the iterated commutators with the interaction term. The main result inthis section is Theorem 9.4 in which we show that Conditions 2.2 and 2.4 both lead tothe desired bounds. We conclude with Section 10 where Proposition 2.7, which statesthat in dimension d = 3 Yukawa potentials satisfy Condition 2.2, is proven. We introduce a two-particle Hamiltonian on L ( R d ) by H ′ V = ω ( p ) + ω ( p ) + V ( x − x ) , where p i = − i ∇ x i , x i ∈ R d .We impose the following set of conditions on ω , ω and V : Condition 2.1 (Properties of ω , ω and V ) .
1. The ω i ’s are real-valued, real analytic functions on R d and there exists R > ,such that the ω i ’s extend to analytic functions in the d -dimensional strip S d R := (cid:8) ( z , . . . , z d ) ∈ C d (cid:12)(cid:12) | Im( z i ) | < R, i = 1 , . . . , d (cid:9) . We denote the analytic continuations of these functions by the same symbols.2. There exist real numbers p = s ≥ s , s > and a constant C > such that | ∂ α ω j ( k ) | ≤ C h k i s j , | ω j ( k ) | ≥ C h k i s j − C (1) for every multi-index α ∈ N d , | α | ≤ and all k ∈ S d R .3. The Fourier transform ˆ V of V exists and ˆ V ∈ L ( R d ) . Here and hereafter, h x i = √ x + 1 and ˆ f = f ∧ denotes the Fourier transform of f .Conjugating with the Fourier transform, we see that H ′ V is unitarily equivalent to H V = ω ( k ) + ω ( k ) + t V , where t V is the partial convolution operator( t V f )( k , k ) := Z R d ˆ V ( u ) f ( k − u, k + u ) d u V ( k ) = (2 π ) − d/ Z R d e − i k · x V ( x ) d x. In order to fibrate H V w.r.t. total momentum ξ = k + k , we introduce a unitaryoperator I : L ( R d × R d ) → L ( R d ; L ( R d )) by setting( If )( ξ ) = f ( ξ − · , · ) . Under this transformation, we find that the Hamiltonian takes the form IH V I ∗ = Z ⊕ R d H V ( ξ ) d ξ, where H ( ξ ) = H V ( ξ ) = ω ξ + S ˇ V , and ω ξ ( k ) = ω ( ξ − k ) + ω ( k ) , ( S ˇ V f )( k ) = ( ˇ V ∗ f )( k ) . Here ˇ V ( k ) = ˆ V ( − k ) is the inverse Fourier transform of V and ˇ V ∗ f denotes theconvolution product. Note that H ( ξ ) corresponds to the case where the potential isabsent. For brevity, we will often suppress the subscript V in H V ( ξ ) and just write H ( ξ ). For later use, let T = S ˇ V for a fixed V .Furthermore, we define a self-adjoint operator for every total momentum ξ ∈ R d by A ξ = i (cid:0) v ξ · ∇ k + ∇ k · v ξ (cid:1) . (2)If no confusion can arise, we will just write A = A ξ . The vector field v ξ is given by v ξ ( k ) = e − k − ξ ( ∇ k ω ξ )( k ) . (3)The choice of vector field can be regarded as a generalization of the standard choice A = i x · ∇ + i ∇ · x in Mourre theory for Schr¨odinger operators. Note that x = ∇ x .Thus, the quadratic dispersion relation in the Schr¨odinger case gets substituted bya more general one. The exponential weight in both x and ξ is added to make theuniform estimates in Sections 8 and 9 work.In addition to Condition 2.1, which is always assumed, we will sometimes need oneof the following two conditions. First, we recall the definition of the weak L s ( R d )norm: k f k s,w = sup α> |{ x | | f ( x ) | > α }| s . Here | A | for a subset A of R d denotes theLebesgue measure of A . L sw ( R d ) is then the set of all functions for which k f k s,w < ∞ . Condition 2.2 (L sw –bounds on ˆ V ) . Let s > with max { , − pd } < s ≤ d , where p = s comes from Condition 2.1. There exists c ′ > such that ∀ α ∈ N d : k ˆ V ( α ) k s,w ≤ α ! c ′| α | , where k·k s,w denotes the weak L s ( R d ) norm. Remark 2.3.
Note that s ≤ d is only a restriction for d ≥
3; for lower dimensionsthe condition s > ondition 2.4 (L –bounds on ˆ V ) . There exists c ′ > such that ∀ α ∈ N d : k ˆ V ( α ) k ≤ α ! c ′| α | , Theorem 9.4 can be proven for a potential satisfying Condition 2.2 or 2.4. However itis Condition 2.2 that we show to hold in the case of the Yukawa potential in Proposition2.7.The main result of the paper is now as follows.
Theorem 2.5.
Assume Condition 2.1 and either
Condition 2.2 or Condition 2.4.Then there exists a constant C ( ξ ) which depends continuously on ξ , such that k ad kA ( H ( ξ ))( H ( ξ ) + i ) − k ≤ C ( ξ ) k k ! , for all k ∈ N . Proof.
This follows directly from Proposition 8.1 and Theorem 9.4.
Remark 2.6.
In fact, Theorem 2.5 is true under a weaker assumption which can befound in Theorem 9.4. The logical structure of the argument is that both Condition 2.2and Condition 2.4 imply this weaker assumption.The following Proposition implies that Condition 2.2 is satisfied for the Yukawapotential whenever p > ≤ s < Proposition 2.7.
Let d = 3 , V ( x ) = e −| x | | x | and s ≥ / . Then ˆ V ( k ) = 4 π (1 + k ) − and there exists c > such that ∀ α ∈ N : k ˆ V ( α ) k s,w ≤ α ! c | α | . Remark 2.8.
Some authors use the Fourier transform ˆ V ( k ) = 4 π (1 + k ) − to define the Yukawa potential in other dimensions. In that case, a similar result holds in thesedimensions. Remark 2.9.
In [3], the conclusion of Theorem 2.5 is (indirectly) reached by com-pletely different arguments for the class of Hamiltonians satisfying the following con-dition in addition to Condition 2.1: • Let d ′ = 2[ d/
2] + 2. We suppose that V ∈ C d ′ ( R d ) and there exists a >
0, suchthat for all α ∈ N d with | α | ≤ d ′ , we have sup x ∈ R d e a | x | | ∂ αx V ( x ) | < ∞ .It is easy to see that for dispersion relations and potentials satisfying these conditions,Condition 2.4 is also satisfied. Indeed, one can prove that for some a >
0, ˆ V has ananalytic continuation to the d -dimensional strip S da and ∀ k ∈ S da : (cid:12)(cid:12) b V ( k ) (cid:12)(cid:12) ≤ C V (cid:0) | k | d ′ (cid:1) − , d = 1,2 πn ! k ˆ V ( n ) k ≤ Z (cid:16)Z Γ r | ˆ V ( w ) || w − z | n +1 d w (cid:17) d z ≤ Z (cid:16)Z ˜Γ r C V | r − | z || r n +1 d w (cid:17) d z = 2 C V r n Z d z | r − | z || , where Γ r is a path around z with radius r < a and ˜Γ r is a path around 0. The case d > p >
1, namely relative compactness of the interaction termand the existence of a Mourre estimate.
Theorem 2.10.
Suppose Condition 2.1 and either
Condition 2.2 or Condition 2.4.Assume that T = S ˇ V is relatively bounded wrt. H and that for all ξ ∈ R d and all λ ∈ R , there exists positive constants e, C, κ > and a compact operator K such that [ H ( ξ ) , i A ξ ] ≥ e − CE H ( ξ ) ( R \ [ λ − κ, λ + κ ]) h H ( ξ ) i − K. Let Σ pp be the joint energy-momentum point spectrum Σ pp = (cid:8) ( λ, ξ ) ∈ R × R d (cid:12)(cid:12) λ ∈ Σ pp ( ξ ) (cid:9) , Σ pp ( ξ ) = σ pp ( H ( ξ )) and T the energy-momentum threshold set T = (cid:8) ( λ, ξ ) ∈ R × R d (cid:12)(cid:12) λ ∈ T ( ξ ) (cid:9) , T ( ξ ) = (cid:8) λ ∈ R (cid:12)(cid:12) ∃ k ∈ R d : ω ξ ( k ) = λ and ∇ k ω ξ ( k ) = 0 (cid:9) . Let ( λ , ξ ) ∈ Σ pp \ T . If we fix v ∈ R d with k v k = 1 , then there exist • r, ρ > • natural numbers ≤ m ± ≤ n and n ± , . . . , n ± m ± ≥ with n ± + · · · + n ± m ± ≤ n , • real analytic functions λ ± , . . . , λ ± m ± : I ± → R , I − = ( − r, and I + = (0 , r ) ,such that1. for any j , lim t → ± λ ± j ( t ) = λ ,2. for any t ∈ I ± , we have σ pp ( H ( ξ + tv )) ∩ ( λ − ρ, λ + ρ ) = { λ ± ( t ) , . . . , λ ± m ± ( t ) } ,3. The eigenvalue branches I ± ∋ t → λ ± j ( t ) have constant multiplicity n ± j . This theorem, with the assumptions replaced by just Condition 2.1 and the conditionmentioned in Remark 2.9, is similar to [3, Theorem 3.2]. This result, however, doesn’tcover the Yukawa potential due to the singularity at x = 0.6 Terminology and notation
In this section, we introduce some notation which we use to state and prove Theo-rem 4.1. The main objects are the polyindices which we usually denote α , β , a or b (seebelow), and the two classes of operators indexed by the polyindices; the multiplicationoperators M α,β and the convolution operators T β,b .In the following, C ( N , N d ) will denote the set of non-negative, integer valuedfunctions on N which are zero except on a finite set. Such functions will be referredto as polyindices of dimension d . We will usually use the letters α and a for 1-dimensional polyindices, while β and b are reserved for d -dimensional polyindices. Forany polyindex β ∈ C ( N , N d ) of dimension d , we will call the finite number k β k = d X σ =1 ∞ X i =0 β σ ( i )( i + 1)the order of β , and the finite number | β | = d X σ =1 ∞ X i =0 β σ ( i )will be called the size of β . The order factorial of β will be written as, and defined by, β ¨! = d Y σ =1 ∞ Y i =0 β σ ( i )!(( i + 1)!) β σ ( i ) , and is likewise a finite number for any polyindex β (all factors except a finite numberare 1). The reduced order factorial of β is then β ˙! = d Y σ =1 ∞ Y i =0 β σ ( i )!( i + 1) β σ ( i ) . We will sometimes need the ratio β ¯! := β ¨! β ˙! = Q dσ =1 Q ∞ i =0 ( i !) β σ ( i ) . Remark 3.1.
At a later point in the paper we will need the following result which iseasily checked to be true. |{ α ∈ C ( N , N ) | k α k = k }| = p ( k ) , where p ( k ) is the number theoretic partition function (see e.g. [6]) and | M | denotesthe number of elements of a set M . Indeed, if k α k = k , then α determines a uniquepartition of k in the following way: k = k α k = ∞ X i =0 α ( i ) X j =1 ( i + 1) , and if a partition of k is given, it can be uniquely encoded in an α with k α k = k byletting α ( i −
1) denote the number of i ’s in the partition for all i ≥
1. A simple upperbound for p ( k ) is p ( k ) < e π √ k/ = e c √ k , cf. [6].7he polyindices will be used to index certain operators that appear in the commu-tator formula for the interaction. More specifically, let α and β be a polyindices ofdimension 1 and d , respectively, and let f ∈ C ∞ ( R , R ) and g ∈ C ∞ ( R d , R ). Write D v f α = ∞ Y i =0 ( D iv f ) α ( i ) and D v g β = ∞ Y i =0 d Y σ =1 ( D iv g σ ) β σ ( i ) where D v = i v ξ · ∇ . We define M α,β := M D v w α D v v βξ , where w = i div( v ξ ) and, for a function f , M f denotes the operator of multiplicationwith f . We note that M , = and that M α,β M a,b = M α + a,β + b .The commutator formula also contains some convolution operators which are in-dexed by polyindices. For any pair of d -dimensional polyindices β and b , we now letthe d -dimensional multiindex γ β + b be defined through( γ β + b ) σ = | β σ | + | b σ | . (4)Then T β,b := S (( − | b | ( − x ) γβ + b ¯ V ) ∧ , where S f denotes the operator of convolution with the function f and ¯ V ( x ) = V ( − x ).Note that, although only ˆ V is assumed to be well-defined, T β,b can be interpreted asa form on C ∞ ( R d ) for all β and b . Observe that S ˆ¯ V = T = T , .In the proof of the commutator formula, we will make extensive use of the followingnotation. If γ ∈ N d is a multiindex and σ ∈ { , . . . , d } , then we let γ +( σ ) = γ + δ σ , where δ σ is given by ( δ σ ) σ ′ = δ σ,σ ′ , where δ σ,σ ′ is the Kronecker delta. For a 1-dimensional polyindex α ∈ C ( N , N ), we define α +( i ) = α + δ i , where δ i ( j ) = δ i,j . If β ∈ C ( N , N d ) is a d -dimensional polyindex, then β +( i,σ ) = β + δ ( i,σ ) , where ( δ ( i,σ ) ( j )) σ ′ = δ i,j δ σ,σ ′ . Likewise, we will write α − ( i ) = α − δ i and β − ( i,σ ) = β − δ ( i,σ ) , whenever α ( i ) , β σ ( i ) ≥
1. If α ( i ) or β σ ( i ) is 0, then α − ( i ) respectively β − ( i,σ ) can begiven any (1- respectively d -dimensional polyindex) value; such cases will only appearin expressions that are multiplied by 0. 8 The commutator formula for the interaction term
Theorem 4.1.
Let T = S ˇ V denote the interaction term. Then ad kA ( T ) = X α,β,a,b k α k + k β k + k a k + k b k = k k ! α ¨! β ¨! a ¨! b ¨! M α,β T β,b M a,b (5) as a form identity on C ∞ ( R d ) . Proof.
The proof goes by induction. The case k = 0 is trivially true. Consider[ A, M α,β T β,b M a,b ] = [ A, M α,β ] T β,b M a,b + M α,β [ A, T β,b ] M a,b + M α,β T β,b [ A, M a,b ] , which here and for the rest of this section should be read as form identities on C ∞ ( R d ).We thus need to find [ A, M α,β ] and [
A, T β,b ]:[
A, M α,β ] = ∞ X i =0 (cid:16) α ( i ) M α − ( i )+( i +1) ,β + d X σ =1 β σ ( i ) M α,β − ( i,σ )+( i +1 ,σ ) (cid:17) , while ( AT β,b f )( k ) = ( M w ξ T β,b f )( k ) + d X σ =1 ( M ( v ξ ) σ T β +(0 ,σ ) ,b f )( k )and ( T β,b Af )( k ) = − ( T β,b M w ξ f )( k ) − d X σ =1 ( T β,b +(0 ,σ ) M ( v ξ ) σ f )( k ) , so [ A, T β,b ] = M δ , T β,b + T β,b M δ , + d X σ =1 ( M ,δ (0 ,σ ) T β +(0 ,σ ) ,b + T β,b +(0 ,σ ) M ,δ (0 ,σ ) ) . Putting this together (and noting that T β,b = T β − ( i,σ )+( i +1 ,σ ) ,b = T β,b − ( i,σ )+( i +1 ,σ ) ), weget [ A, M α,β T β,b M a,b ] = ∞ X i =0 (cid:16) α ( i ) M α − ( i )+( i +1) ,β T β,b M a,b + d X σ =1 β σ ( i ) M α,β − ( i,σ )+( i +1 ,σ ) T β − ( i,σ )+( i +1 ,σ ) ,b M a,b (cid:17) + M α +(0) ,β T β,b M a,b + M α,β T β,b M a +(0) ,b + d X σ =1 ( M α,β +(0 ,σ ) T β +(0 ,σ ) ,b M a,b + M α,β T β,b +(0 ,σ ) M a,b +(0 ,σ ) )+ ∞ X i =0 (cid:16) a ( i ) M α,β T β,b M a − ( i )+( i +1) ,b + d X σ =1 b σ ( i ) M α,β T β,b − ( i,σ )+( i +1 ,σ ) M a,b − ( i,σ )+( i +1 ,σ ) (cid:17) . (6)9rom this expression and the induction start, it is clear that the k ’th iterated commu-tator is of the form ad kA ( T ) = X α,β,a,b k α k + k β k + k a k + k b k = k C ( k ) α,β,a,b M α,β T β,b M a,b , (7)where C ( k ) α,β,a,b are some constants to be determined. Assume that the commutatorformula holds true for k . We want to show that it also holds for k + 1. By the abovediscussion, it is enough to let α, β, a, b be arbitrary with k α k + k β k + k a k + k b k = k + 1and show that C ( k ) α,β,a,b = k ! α ¨! β ¨! a ¨! b ¨! , so this is what we do.Using the induction hypothesis we combine (6) and (7) to obtain an expression forad k +1 A ( T ). This enables us to identify those terms in the k ’th iterated commutatorthat through commutation with A contribute to the term C ( k +1) α,β,a,b M α,β T β,b M a,b in the k + 1’st iterated commutator. Before proceeding we illustrate this by an example.Suppose we are given α, β, a, b with k α k + k β k + k a k + k b k = k + 1. One of the termsappearing in the combination of (6) and (7) is C ( k ) α ′ ,β ′ ,a ′ ,b ′ α ′ ( i ) M α ′− ( i )+( i +1) ,β ′ T β ′ ,b ′ M a ′ ,b ′ , where k α ′ k + k β ′ k + k a ′ k + k b ′ k = k. Since the contributing term from the k ’th commutator can only have one polyindexdeviating from α, β, a, b , this term will contribute to C ( k +1) α,β,a,b M α,β T β,b M a,b , if α ′ = α +( i ) − ( i +1) , β ′ = β , a ′ = a and b ′ = b . In the same fashion one easily finds that allpossible contributors have polyindices of one of the following forms:( α − (0) , β, a, b ) , ( α − ( i +1)+( i ) , β, a, b ) , ( α, β − (0 ,σ ) , a, b ) , ( α, β − ( i +1 ,σ )+( i,σ ) , a, b ) , ( α, β, a − (0) , b ) , ( α, β, a − ( i +1)+( i ) , b ) , ( α, β, a, b − (0 ,σ ) ) , ( α, β, a, b − ( i +1 ,σ )+( i,σ ) ) . Appealing again to the induction hypothesis and (6), we see that in our example α ′ ( i ) C ( k ) α ′ ,β ′ ,a ′ ,b ′ = ( α + δ i − δ i +1 )( i ) C ( k ) α + δ i − δ i +1 ,β,a,b = α ( i )( i + 2) k ! α ¨! β ¨! a ¨! b ¨! . Proceeding in the same way we see that the contributions of all the previously listedterms are, respectively α (0) k ! α ¨! β ¨! a ¨! b ¨! , α ( i + 1)( i + 2) k ! α ¨! β ¨! a ¨! b ¨! , β σ (0) k ! α ¨! β ¨! a ¨! b ¨! , β σ ( i + 1)( i + 2) k ! α ¨! β ¨! a ¨! b ¨! ,a (0) k ! α ¨! β ¨! a ¨! b ¨! , a ( i + 1)( i + 2) k ! α ¨! β ¨! a ¨! b ¨! , b σ (0) k ! α ¨! β ¨! a ¨! b ¨! , b σ ( i + 1)( i + 2) k ! α ¨! β ¨! a ¨! b ¨! , times M α,β T β,b M a,b . Finally, summing up all these possible contributions gives us the10alue of C ( k +1) α,β,a,b : C ( k +1) α,β,a,b = k ! α ¨! β ¨! a ¨! b ¨! ∞ X i =0 ( α ( i + 1)( i + 2) + a ( i + 1)( i + 2))+ k ! α ¨! β ¨! a ¨! b ¨! ∞ X i =0 d X σ =1 ( β σ ( i + 1)( i + 2) + b σ ( i + 1)( i + 2))+ k ! α ¨! β ¨! a ¨! b ¨! α (0) + a (0) + d X σ =1 ( β σ (0) + b σ (0)) ! = k ! α ¨! β ¨! a ¨! b ¨! ( k α k + k a k + k β k + k b k ) = ( k + 1)! α ¨! β ¨! a ¨! b ¨! , where we have used that by assumption k α k + k a k + k β k + k b k = k + 1 in the last line.This completes the proof. To be able to use Theorem 4.1 and (5) to estimate ad kA ( T ), we need some control onthe reduced order factorials, α ˙!, β ˙!, etc. In fact, we will just use the trivial estimates a ˙! , α ˙! ≥ M D v w α .For the d -dimensional polyindices β and b , we will make a more careful estimate.More precisely, for each σ ∈ { , , . . . , d } , we want to estimate C k β σ k + k b σ k β σ ˙! b σ ˙! = ∞ Y i =1 (cid:18)(cid:0) β σ ( i ) Y y =1 C i +1 y ( i + 1) (cid:1)(cid:0) b σ ( i ) Y y =1 C i +1 y ( i + 1) (cid:1)(cid:19) (8)from below, where C >
Lemma 5.1.
For any
C > and ℓ ∈ N , let β ℓ : N → N denote the polyindex givenby β ℓ ( i ) = (cid:22) ℓC i ( i + 1) (cid:23) , where ⌊ x ⌋ denotes the integer part of x . For any < ε < , there exists a C suchthat for all C > C and all ℓ , | β ℓ | ≤ (1 + ε ) ℓ (9)11 nd C k β ℓ k ( β ℓ ˙!) ≥ C ′′ | β ℓ | (2 | β ℓ | )! , (10) where C ′′ = C/ (4 e (1 + ε )) . Proof.
The size | β ℓ | of β ℓ clearly depends on C and can be estimated from above(for sufficiently large C ≥ C ) in the following way:1 ≤ ℓ ≤ | β ℓ | = ∞ X i =0 (cid:22) ℓC i ( i + 1) (cid:23) = ℓ + ∞ X i =1 (cid:22) ℓC i ( i + 1) (cid:23) ≤ ℓ + Z ∞ ℓC x ( x + 1) d x = ℓ + (cid:2) ℓC Ei( − ( x + 1) log( C )) (cid:3) ∞ x =0 = ℓ − ℓC Ei( − log( C )) , where Ei denotes the exponential integral function, see e.g. [13] for the definition andproperties. Since x Ei( − log( x )) ∼ − x ) as x → ∞ , for any ε >
0, we can pick C so that for C > C , we have − C Ei( − log( C )) < ε .Putting this together, we get | β ℓ | ≤ (1 + ε ) ℓ for any ℓ and C > C which is (9).Let W denote the Lambert W -function (again, see [13]). Then, using Stirling’sformula, we can estimatelog( C k β ℓ k β ℓ ˙!) = ∞ X i =0 log (cid:16) C (cid:4) ℓCi ( i +1) (cid:5) ( i +1) (cid:4) ℓC i ( i +1) (cid:5) !( i + 1) (cid:4) ℓCi ( i +1) (cid:5)(cid:17) = (cid:4) W ( ℓC log( C ))log( C ) − (cid:5)X i =0 (cid:4) ℓC i ( i +1) (cid:5) ( i + 1) log( C ) + log (cid:0)(cid:4) ℓC i ( i +1) (cid:5) ! (cid:1) + (cid:4) ℓC i ( i +1) (cid:5) log( i + 1) ≥ (cid:4) W ( ℓC log( C ))log( C ) − (cid:5)X i =0 (cid:4) ℓC i ( i +1) (cid:5)(cid:0) ( i + 1) log( C ) + log (cid:0)(cid:4) ℓC i ( i +1) (cid:5)(cid:1) − i + 1) (cid:1) ≥ (cid:4) W ( ℓC log( C ))log( C ) − (cid:5)X i =0 (cid:4) ℓC i ( i +1) (cid:5)(cid:0) log (cid:0) C i +1 ( i + 1) (cid:4) ℓC i ( i +1) (cid:5)(cid:1) − (cid:1) . (11)Since (cid:4) ℓC i ( i +1) (cid:5) = k for some k ≥ i ∈ (cid:8) , . . . , (cid:4) W ( ℓC log( C ))log( C ) − (cid:5)(cid:9) and hence ℓC i ( i +1) < k + 1, we see that Cℓk +1 < C i +1 ( i + 1) and hence, using (9), we can estimate C i +1 ( i + 1) (cid:4) ℓC i ( i +1) (cid:5) > Cℓk +1 k ≥ Cℓ = C ′ ε ) ℓ ≥ C ′ | β ℓ | where C ′ = C/ (4(1 + ε )).Then (11) can be estimated from below by(11) ≥ (cid:4) W ( ℓC log( C ))log( C ) − (cid:5)X i =0 (cid:4) ℓC i ( i +1) (cid:5)(cid:0) log(2 | β ℓ | ) + log( C ′ ) − (cid:1) = | β ℓ | (cid:0) log(2 | β ℓ | ) + log( C ′ ) − (cid:1) . C k β ℓ k ( β ℓ ˙!) ≥ (cid:18) | β ℓ | C ′ e (cid:19) | β ℓ | ≥ C ′′ | β ℓ | (2 | β ℓ | )! , where C ′′ = C ′ /e = C/ (4 e (1 + ε )), which is (10). In this section, we prove a result, Corollary 6.3, which we need to control the orderfactorials in the commutator formula (5) from Theorem 4.1. In Section 5, we took careof a special case in Lemma 5.1. Here, we show that we can split the general case intotwo factors, one which is handled by Lemma 5.1, and another, which can be handledby a simple estimate.
Lemma 6.1.
There exists a constant C > such that for all C > C and all polyin-dices β, b , c | γ β + b | γ β + b ! ≤ C k β k + k b k β ˙! b ˙! , where c can be chosen as C/ (8 e ) . Remark 6.2.
Note that the constant c can be chosen arbitrarily large as long as C is adjusted accordingly. Note also that Lemma 6.1 is “sharp” in the sense that for all k and all multiindices γ with | γ | = k , and β given by β (0) = γ and β σ ( i ) = 0 for i ≥ σ , one has that k β k = k and γ ! = γ β ! = β ˙!. Proof.
We begin by observing that since we can factorize according to dimension c | γ β + b | γ β + b ! = Q dσ =1 c | β σ | + | b σ | ( | β σ | + | b σ | )! and C k β k + k b k β ˙! b ˙! = Q dσ =1 C k β σ k + k b σ k β σ ˙! b σ ˙!,it is enough to prove that for any pair β, b and any σ , 1 ≤ σ ≤ d , we have c | β σ | + | b σ | ( | β σ | + | b σ | )! ≤ C k β σ k + k b σ k β σ ˙! b σ ˙! , (12)for some constants c and C . Let β, b and σ be given, let 0 < ε < C > C with C as in Lemma 5.1, and let ℓ ∈ N be the largest number such that 2 | β ℓ | ≤ | β σ | + | b σ | .Rewrite C k β σ k + k b σ k β σ ˙! b σ ˙! = ∞ Y i =0 C ( β σ ( i )+ b σ ( i ))( i +1) β σ ( i )!( i + 1) β σ ( i ) b σ ( i )!( i + 1) β σ ( i ) = ∞ Y i =0 (cid:18) β σ ( i ) Y y =1 C ( i +1) y ( i + 1) (cid:19)(cid:18) b σ ( i ) Y y =1 C ( i +1) y ( i + 1) (cid:19) = | β σ | + | b σ | Y j =1 p j , (13)for some p j ≤ p j +1 , i.e. a product of | β σ | + | b σ | factors of the form p j = C i +1 y ( i + 1)with either y ≤ β σ ( i ) or y ≤ b σ ( i ). Replacing β σ and b σ by β ℓ in the above identity13ields C k β ℓ k ( β ℓ ˙!) = ∞ Y i =0 (cid:18) (cid:4) ℓCi ( i +1) (cid:5)Y y =1 C i +1 y ( i + 1) (cid:19) , (14)where, for each i , y runs through exactly those integers for which C i +1 y ( i + 1) ≤ Cℓ .Note that (14) contains exactly 2 | β ℓ | (non-trivial) factors. We will now compare thefirst 2 | β ℓ | factors in (13) with the factors appearing in (14). More precisely, we splitthe ordered product in (13) into those which also appear in (14) and a remainder.Note that by the above discussion this splitting corresponds to sorting the relevant p j into those less than or equal to Cℓ and those strictly larger than Cℓ . The first groupcan be written as Y p j ≤ Cℓ p j = ∞ Y i =0 (cid:18) β σ ( i ) ∧ β ℓ ( i ) Y y =1 C ( i +1) y ( i + 1) (cid:19)(cid:18) b σ ( i ) ∧ β ℓ ( i ) Y y =1 C ( i +1) y ( i + 1) (cid:19) , where f ∧ g denotes the minimum of f and g and the whole product can then berewritten as in (15). | β ℓ | Y j =1 p j = ∞ Y i =0 (cid:18) β σ ( i ) ∧ β ℓ ( i ) Y y =1 C ( i +1) y ( i + 1) (cid:19)(cid:18) b σ ( i ) ∧ β ℓ ( i ) Y y =1 C ( i +1) y ( i + 1) (cid:19)! | β ℓ | Y j =1 p j >Cℓ p j (15) ≥ C ′′ | β ℓ | (2 | β ℓ | )! . (16)To obtain the estimate (16) simply note that for each term missing to obtain C k β ℓ k β ℓ ˙!in the first product we find one term in the remainder for which p j > Cℓ . Sincethe missing term’s value must be less than or equal to Cℓ , we may estimate this p j from below by this missing value. The claimed inequality then follows from (10) inLemma 5.1.Since ℓ was chosen as the largest integer such that 2 | β ℓ | ≤ | β σ | + | b σ | , we have that | β σ | + | b σ | < | β ℓ +1 | ≤ ε )( ℓ + 1), where we for the last inequality used (9) fromLemma 5.1. This implies that C ′′ ( | β σ | + | b σ | ) < Cℓ . For the remaining p j , we thennote that | β σ | + | b σ | Y j =2 | β ℓ | +1 p j ≥ ( Cℓ ) | β σ | + | b σ |− | β ℓ | > C ′′| β σ | + | b σ |− | β ℓ | ( | β σ | + | b σ | ) | β σ | + | b σ |− | β ℓ | , (17)where ( x ) n = x ( x − · · · ( x − ( n − c = C ′′ . Corollary 6.3.
With c and C as in Lemma 6.1, we have the following estimate: k ! α ¨! β ¨! a ¨! b ¨! ≤ C k β k + k b k k ! c | γ β + b | γ β + b ! β ¯! b ¯! . Proof.
This follows easily from Lemma 6.1.14
The size of the summation index set
In this section, we show that the number of terms in the commutator formula (5) fromTheorem 4.1 grows in a controllable way.
Proposition 7.1.
The number of terms in the iterated commutator formula for theinteraction term (5) from Theorem 4.1 is bounded by c kd , where c d is some constantwhich only depends on the dimension d . Proof.
For any fixed k , the set { ( α, β, a, b ) : α, a ∈ C ( N , N ) , β, b ∈ C ( N , N d ) , k α k + k β k + k a k + k b k = k } , where C ( N , N ) and C ( N , N d ) denote the sets of 1- and d -dimensional polyindices,respectively, is the index set for the summation formula (5) from Theorem 4.1 for the k ’th iterated commutator of the interaction term T with the conjugate operator A .It can also be written as { ( α, β, a, b ) : α, a, β σ , b σ ∈ C ( N , N ) , β = ( β σ ) dσ =1 , b = ( b σ ) dσ =1 , k α k + k a k + d X σ =1 k β σ k + k b σ k = k } . For any weak composition (see e.g. [7]) P d +2 j =1 k ′ j = k of k with exactly 2 d + 2 terms,there are d +2 Y j =1 { α ∈ C ( N , N ) : k α k = k ′ j } different ways of satisfying the condition: k α k = k ′ , k a k = k ′ , k β σ k = k ′ σ +2 , k b σ k = k ′ σ + d +2 , for σ = 1 , . . . , d. By Remark 3.1 the set { α ∈ C ( N , N ) : k α k = k } has exactly p ( k ) elements.We now want to rewrite the weak composition P d +2 j =1 k ′ j = k of k with exactly 2 d + 2terms in the following way. Let k j = P jn =1 k ′ n for j = 1 , . . . , d + 1. Then k ′ = k , k ′ j = k j − k j − for 2 ≤ j ≤ d + 1 ,k ′ d +2 = k − k d +1 , and 0 ≤ k j ≤ k j +1 ≤ k for any j = 1 , . . . , d . This means that anyweak 2 d + 2-term composition of k is given by a finite, increasing sequence { k j } d +1 j =1 ,i.e. satisfying 0 ≤ k ≤ k ≤ · · · ≤ k d ≤ k d +1 ≤ k . We can now count – and estimate– the number of weak 2 d + 2-term compositions of k by counting the number of wayswe can choose the sequence { k j } d +1 j =1 : X k ,k ,...,k d ,k d +1 ≤ k ≤ k ≤···≤ k d ≤ k d +1 ≤ k (cid:18) k + 2 d + 12 d + 1 (cid:19) = ( k + 1)( k + 2) · · · ( k + 2 d )( k + 2 d + 1)(2 d + 1)! < C kd , (18)15or some sufficiently large constant C d >
0. Note that in the first step of (18) we haveused that the sequence of positive integers k ≤ k ≤ · · · ≤ k d +1 with 0 ≤ k i ≤ k can be identified with a strictly increasing sequence h < h < · · · < h d +1 in a 1 to1 fashion by putting h i = k i + i . However, the number of possible choices for such h i obeying 1 ≤ h i ≤ k + 2 d + 1 is exactly k + 2 d + 1 choose 2 d + 1 as claimed in the firststep of (18).We can now estimate the number of elements in the summation index set from abovein the following way: { ( α, β, a, b ) : α, a ∈ C ( N , N ) , β, b ∈ C ( N , N d ) , k α k + k β k + k a k + k b k = k } , = X ≤ k ≤ k ≤···≤ k d ≤ k d +1 ≤ k p ( k ) p ( k − k ) · · · p ( k d +1 − p d ) p ( k − k d +1 ) < X ≤ k ≤ k ≤···≤ k d ≤ k d +1 ≤ k e c ( √ k + P d +1 j =2 √ k j − k j − + √ k − k d +1 ) ≤ X ≤ k ≤ k ≤···≤ k d ≤ k d +1 ≤ k e ck < ( C d e c ) k , where we used (18) for the last inequality. In this section we assume that V = 0 so that H ( ξ ) = H ( ξ ) is simply given bymultiplication with the function ω ξ . Since A = i div( v ξ ) + i v ξ · ∇ k , it is easy to seethat the commutator form [ A, M f ], where f is some function, is given by the operatorad A ( M f ) = M i v ξ ·∇ k f . If the gradient is finite almost everywhere, this operator is again just a multiplicationoperator by a bounded function and is thus bounded as well. In particular, this istrue for the choice f = ω ξ , see Section 2. Furthermore, we may iterate the preceed-ing calculation and obtain that the n ’th commutator form is given by the boundedmultiplication operator ad nA ( M f ) = M ( i v ξ ·∇ k ) n f , n ≥ n -th derivatives of f remain finite. As noted above the choice f = ω ξ thusyields a bounded operator. Proposition 8.1 (Commutator Bounds in the Free Case) . For all n ∈ N the iteratedcommutator ad nA ( H ( ξ )) is given by a bounded multiplication operator as follows ad nA ( H ( ξ )) = M ( i v ξ ·∇ k ) n ω ξ , (20)16 here M f is the operator given by multiplication with f , and there exists a constant C ξ > independent of n and h , which depends continuously on ξ ∈ R d such that wehave the pointwise estimate | ( i v ξ · ∇ k ) n ω ξ ( h ) | ≤ C nξ n ! h h i s e − h (21) for all h ∈ R d . In particular, there exists a constant c ξ which depends continuously on ξ such that for all k ∈ N , k ad kA ( H ( ξ )) k ≤ c kξ k ! . (22) Proof. (20) follows directly from (19) with the choice f = ω ξ and (22) is implied by(21). It thus suffices to prove (21). Note that( v ξ · i ∇ k ) n ω ξ ( k ) = d n − d n − s u sξ ( k ) | s =0 , u sξ ( k ) := ( v ξ · i ∇ k ω ξ )( γ s ( k )) , where γ s solves the ODE dd s γ ξs ( k ) = v ξ ( γ ξs ( k )) , γ ( k ) = k. By Lemma 3.5 of [3], for all k, ξ ∈ R d the map s γ ξs ( k ) extends analytically to astrip of some width r > k and ξ , such that S r ∋ z γ ξz ( k ) ∈ S dR .Moreover, there exists a constant C ω >
0, which is also independent of k, ξ ∈ R d suchthat | γ ξz ( k ) − k | ≤ C ω | z | , see the Remark 3.8 of [3]. Thus, we may use Cauchy’s integralformula to calculate | ( v ξ · i ∇ k ) n ω ξ ( h ) | = (cid:12)(cid:12)(cid:12)(cid:12) d n − d n − s u sξ ( h ) | s =0 (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( n − π i Z Γ r u ξ ( γ ξz ( h )) z n d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( n − z ∈ Γ r | u ξ ( γ ξz ( h )) | r n − , where Γ r denotes the set {| z | = r } , and, by abuse of notation, a path parametrizingthis set in the counter-clockwise direction. By applying Peetre’s inequality, ∀ q ∈ R , k, h ∈ C d : h k + h i q ≤ | q | h k i | q | h h i q , see e.g. [15, Lemma 1.18], the assumptions, and the estimate | γ ξz ( k ) − k | ≤ C ω | z | , wemay estimate sup z ∈ Γ r | u ξ ( γ ξz ( h )) | ≤ ˜ C ξ h h i s e − h , for some constant ˜ C ξ ≥
1, which can be chosen such that it depends continuously on ξ ∈ R d . Since ˜ C ξ ≥
1, we may define C ξ = ˜ C ξ max { ,r } and conclude the statement.17 Estimates on the interaction commutator
In this section, we make estimates on the commutator from Theorem 4.1, using theestimates established in Sections 5 and 6 on the order factorials, and in Section 7on the number of terms. We also use Proposition 8.1 from Section 8 to control themultiplication operators M α,β . Lemma 9.1.
There exists a constant C ′ ξ which depends continuously on ξ ∈ R d suchthat for all h ∈ R d , all n ∈ N , and all σ ∈ { , , . . . , d } , we have the pointwise estimate | ( i v ξ · ∇ ) n v ξ,σ ( h ) | ≤ C ′ nξ n ! h h i s e − h . Proof.
Mimic the proof of Proposition 8.1 with ω ξ replaced by v ξ,σ . Proposition 9.2 (Estimates on M α,β ) . Let α and β be - and d -dimensional polyin-dices, respectively, and write f α,β = D v w α D v v βξ , such that M α,β = M f α,β . Then wehave the following pointwise estimate | f α,β ( h ) | ≤ C ′′k α k + k β k ξ ( h h i s e − h ) | α | + | β | α ¯! β ¯! , where C ′′ ξ = max { C ξ , C ′ ξ } and C ξ and C ′ ξ are the constants from Proposition 8.1 andLemma 9.1, respectively. Proof.
Note that M α,β = M D v w α M D v v βξ , and that we may write M D v w α = ∞ Y i =0 M ( D iv w ) α ( i ) . Put w ξ ( k ) := w ( k ) and note that (( i v ξ · ∇ k ) n ω ξ )( k ) = (( i v ξ · ∇ k ) n − w ξ )( k ). We nowuse Proposition 8.1 to get the pointwise estimate | D v w α ( h ) | = ∞ Y i =0 | (( D iv w )( h )) α ( i ) | ≤ ∞ Y i =0 ( C iξ i ! h h i s e − h ) α ( i ) ≤ C k α k ξ ( h h i s e − h ) | α | ∞ Y i =0 i ! α ( i ) . Likewise, we note that M D v v βξ = d Y σ =1 ∞ Y i =0 M ( D iv v ξ,σ ) βσ ( i ) which we use to compute the pointwise estimate as before | M D v v βξ ( h ) | ≤ C ′k β k ξ ( h h i s e − h ) | β | d Y σ =1 ∞ Y i =1 i ! β σ ( i ) . Combining these two estimates now gives the proposition.18 roposition 9.3.
If for all d -dimensional polyindices β, b ∈ C ( N , N d ) with totalorder less than k , k β k + k b k ≤ k , the forms M ( h·i s e − ( · )2 ) | β | T β,b M ( h·i s e − ( · )2 ) | β | on C ∞ ( R d ) extend to bounded operators on L ( R d ) , then so do the forms in (5) fromTheorem 4.1 (with M α,β T β,b M a,b interpreted as the bounded operator given by theform). Furthermore, we have the following estimate on ad kA ( T )( H ( ξ ) + i ) − : k ad kA ( T )( H ( ξ ) + i ) − k≤ X α,β,a,b k α k + k β k + k a k + k b k = k C ′′ kξ C k β k + k b k k ! c | γ β + b | γ β + b ! k M ( h·i s e − ( · )2 ) | α | + | β | T β,b M ( ω ξ + i ) − ( h·i s e − ( · )2 ) | a | + | b | k , (23) where C ′′ ξ is the constant from Proposition 9.2, which depends continuously on ξ , and c and C are the constants from Lemma 6.1. Proof.
This follows easily from Theorem 4.1, Theorem 6.3, and Proposition 9.2.
Theorem 9.4.
1. Assume that for some c > and all pairs of polyindices β, b , k M ( h·i s e − ( · )2 ) | β | T β,b M ( ω ξ + i ) − ( h·i s e − ( · )2 ) | b | k ≤ c | β + b | γ β,b ! , (24) where γ β,b is given as in (4) , and M ( h·i s e − ( · )2 ) | β | T β,b M ( ω ξ + i ) − ( h·i s e − ( · )2 ) | b | isinterpreted as the bounded operator given by the form on C ∞ ( R d ) . Then thereexists a constant C > such that k ad kA ( T )( H ( ξ ) + i ) − k ≤ ( CC ′′ ξ c d ) k k ! and the constant C ′′ ξ , which depends continuously on ξ , comes from Proposi-tion 9.2 and c d comes from Proposition 7.1.2. In particular, (24) holds, if there exists a constant c > such that for all β, b , k T β,b ( H ( ξ ) + i ) − k ≤ c | β + b | γ β,b ! . (25)
3. Suppose that V is as in either Condition 2.2 or Condition 2.4. Then (25) issatisfied. Proof.
The first part of the statement follows immediately from Remark 6.2, Propo-sition 7.1, and Proposition 9.3. The second statement follows from observing that if k T β,b ( H ( ξ ) + i ) − k < ∞ , then k M ( h·i s e − ( · )2 ) | β | T β,b M ( ω ξ + i ) − ( h·i s e − ( · )2 ) | b | k ≤ c | β + b | M k T β,b ( H ( ξ ) + i ) − k C M >
0. The third statement can be seen to be correct by the followingargument. Let ˆ V satisfy Condition 2.2. We introduce the shorthand j p ( k ) := (1 + h k i p ) − , where p := s , see Condition 2.1. By the weak Young inequality, see [10, p.107], k T β,b ( H ( ξ ) + i ) − k = sup φ,ψ ∈Hk φ k = k ψ k =1 |h φ, T β,b ( H ( ξ ) + i ) − ψ i . ≤ k ˆ V ( γ β,b ) k s,w sup ψ ∈Hk ψ k =1 k j p ψ k t , where s + t = . Due to s ∈ (1 , t ∈ (1 , k j p ψ k t ≤ k ψ k k j tp k t − t which is finite, if − p t − t + d <
0. This however is equivalent to p > d (1 − s ) which istrue by assumption. Thus, k T β,b ( H ( ξ ) + i ) − k ≤ k ˆ V ( γ β,b ) k s,w k j tp k t − t and we see that the third statement follows if we assume Condition 2.2. If V satisfiesCondition 2.4, the proof is similar and can be carried out directly by applying (theordinary) Young inequality.
10 The Yukawa potential
Before proving Proposition 2.7 we will introduce some notation. Let d ∈ N and D , . . . , D d be discs in C of radius r . We then define D := D × · · · × D d . We denoteby Γ the distinguished boundary of D , that is Γ = ∂D × · · · × ∂D d . Moreover,for any α ∈ N d and z ∈ C d we define z α := Q dj =1 z α j j . For an analytic function f : U ⊂ C d → C d we denote by f ( α ) the iterated partial derivatives of f correspondingto the multi-index α , that is α j derivatives w.r.t. the j -th variable. If we denote by α + ∈ N d the multi-index with whose j -th coordinate is α j + 1, the d -dimensionalgeneralization of Cauchy’s formula is then f ( α ) ( z ) = Z Γ f ( w )( z − w ) α + d d w. Having taken care of these notational issues we can provide a proof of Proposition 2.7.
Proof of Proposition 2.7.
Clearly ˆ V has an extension to an analytic function into the3-dimensional strip S := { z ∈ C | | Im( z j ) | < ˜ r } , where ˜ r <
1. Hence, for r ∈ (0 , ˜ r )20nd k ∈ R the 3-dimensional Cauchy formula allows us to estimate | ˆ V ( α ) ( k ) | ≤ α !(2 π ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z Γ ˆ V ( z )( k − z ) α + d z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α !(2 π ) r | α | π Z π Z π Z π | r − | k || d t d t d t = α ! r | α | π | r − | k || , where Γ is the distinguished boundary of the 3 dimensional polydisc of radius r . Let β >
0. By the above computations we thus have { k ∈ R | | ˆ V ( α ) ( k ) | > β } ⊂ (cid:26) k ∈ R (cid:12)(cid:12)(cid:12)(cid:12) α ! r | α | π | r − | k || > β (cid:27) = (cid:26) k ∈ R (cid:12)(cid:12)(cid:12)(cid:12) r − (cid:18) β − (cid:19) < | k | < (cid:18) β − (cid:19) + r (cid:27) ⊂ (cid:26) k ∈ R (cid:12)(cid:12)(cid:12)(cid:12) | k | < (cid:18) β − (cid:19) + r (cid:27) , where β = r | α | α !4 π β . Due to { k ∈ R | (1 + | r − | k || ) − > β } = ∅ for β ≥
1, we can usethe above inclusions to compute k ˆ V ( α ) k s,w = sup β> β (cid:12)(cid:12)(cid:12) { k ∈ R | | ˆ V ( α ) ( k ) | > β } (cid:12)(cid:12)(cid:12) s ≤ π α ! r | α | sup β ∈ (0 , β (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) k ∈ R (cid:12)(cid:12)(cid:12)(cid:12)
11 + | r − | z || > β (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) s ≤ π α ! r | α | sup β ∈ (0 , β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:26) k ∈ R (cid:12)(cid:12)(cid:12)(cid:12) | k | < (cid:18) β − (cid:19) + r (cid:27)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s = 16 π α ! r | α | M s , where M s = sup β ∈ (0 , β − s (cid:16) (1 − β ) + rβ (cid:17) s < ∞ , due to s > /
2. Choosing c = max { π M s , } r − completes the proof of the state-ment. Acknowledgments
The authors would like to thank Jacob Schach Møller for useful discussions. M. E.acknowledges the support of the Lundbeck Foundation and the German Research Foun-dation (DFG) through the Graduiertenkolleg 1838 and M.G.R. acknowledges supportfrom the Danish Council for Independent Research — Natural Sciences, grant 12-124675, ”Mathematical and Statistical Analysis of Spatial Data”.21 eferences [1] W. O. Amrein, A. Boutet de Monvel, and V. Georgescu, C -groups, commutatormethods and spectral theory of N -body Hamiltonians , Birkh¨auser, 1996.[2] L. Cattaneo, G.M. Graf, and W. Hunziker, A general resonance theory based onMourre’s inequality , Ann. Henri Poincar´e (2006), 583–601.[3] M. Engelmann, J. S. Møller, and M. G. Rasmussen, Local spectral deformation ,arXiv:1508.03474, 2015, Submitted.[4] J. Faupin, J. S. Møller, and E. Skibsted,
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