On asymptotic stability in energy space of ground states of NLS in 2D
aa r X i v : . [ m a t h . A P ] J a n ON ASYMPTOTIC STABILITY IN ENERGYSPACE OF GROUND STATES OF NLS IN 2D
Scipio Cuccagna and Mirko Tarulli
Abstract.
We transpose work by K.Yajima and by T.Mizumachi to prove dispersiveand smoothing estimates for dispersive solutions of the linearization at a ground stateof a Nonlinear Schr¨odinger equation (NLS) in 2D. As an application we extend todimension 2D a result on asymptotic stability of ground states of NLS proved in theliterature for all dimensions different from 2. § We consider even solutions of a NLS(1 . iu t + ∆ u + β ( | u | ) u = 0 , ( t, x ) ∈ R × R , u (0 , x ) = u ( x ) . We assume:(H1) β (0) = 0, β ∈ C ∞ ( R , R );(H2) there exists a p ∈ (1 , ∞ ) such that for every k = 0 , (cid:12)(cid:12)(cid:12)(cid:12) d k dv k β ( v ) (cid:12)(cid:12)(cid:12)(cid:12) . | v | p − k − if | v | ≥ O such that ∆ u − ωu + β ( u ) u = 0 admits a C -family of ground states φ ω ( x ) for ω ∈ O ;(H4) ddω k φ ω k L ( R ) > ω ∈ O .(H5) Let L + = − ∆+ ω − β ( φ ω ) − β ′ ( φ ω ) φ ω be the operator whose domain is H rad ( R ).We assume that L + has exactly one negative eigenvalue.By [ShS] the ω → φ ω ∈ H ( R ) is C and by [W1,GSS1-2] (H4-5) yields orbitalstability of the ground state e iωt φ ω ( x ). Here we investigate asymptotic stability.We need some additional hypotheses.(H6) For any x ∈ R , u ( x ) = u ( − x ). That is, the initial data u of (1.1) are even. Typeset by
AMS -TEX σ j and the linearization H ω given by:(1 . σ = (cid:20) (cid:21) , σ = (cid:20) i − i (cid:21) , σ = (cid:20) − (cid:21) ; H ω = σ (cid:2) − ∆ + ω − β ( φ ω ) − β ′ ( φ ω ) φ ω (cid:3) + iβ ′ ( φ ω ) φ ω σ . Then we assume:(H7) Let H ω be the linearized operator around e itω φ ω , see (1.2). H ω has a positivesimple eigenvalue λ ( ω ) for ω ∈ O . There exists an N ∈ N such that N λ ( ω ) < ω < ( N + 1) λ ( ω ).(H8) The Fermi Golden Rule (FGR) holds (see Hypothesis 4.2 in Section 4).(H9) The point spectrum of H ω consists of 0 and ± λ ( ω ). The points ± ω are notresonances.Then we prove: Theorem 1.1.
Let ω ∈ O and φ ω ( x ) be a ground state in a family of ground states φ ω . Let u ( t, x ) be a solution to (1.1). Assume (H1)–(H9). In particular assumethe (FGR) in Hypothesis 4.2. Then, if (1.1) is generic, there exist an ǫ > and a C > such that for any ǫ ∈ (0 , ǫ ) and for any u with k u − e iγ φ ω k H < ǫ, thereexist ω + ∈ O , θ ∈ C ( R ; R ) and h ∞ ∈ H with k h ∞ k H + | ω + − ω | ≤ Cǫ such that lim t → + ∞ k u ( t, · ) − e iθ ( t ) φ ω + − e it ∆ h ∞ k H = 0 . Theorem 1.1 is the two dimensional version of Theorem 1.1 [CM]. The one di-mensional version is in [Cu3]. We recall that results of the sort discussed here werepioneered by Soffer & Weinstein [SW1], see also [PW], followed by Buslaev & Perel-man [BP1-2], about 15 years ago. In this decade these early works were followed bya number of results [BS,Cu1-2,GNT,M1,CZ,M2,P,RSS,SW1-3,TY1-3,Wd1]. It washeuristically understood that the rate of the leaking of energy from the so called”internal modes” into radiation, is small and decreasing when N increases, produc-ing technical difficulties in the closure of the nonlinear estimates. For this reasonprior to Gang Zhou & Sigal [GS], the literature treated only the case when N = 1in (H6). [GS] sheds light for N >
1. The results in [GS] deal with all spatial di-mensions different from 2 under the so called Fermi Golden Rule (FGR) hypothesis.[CM,Cu3] strengthen [GS] by considering initial data in H , by showing that the(FGR) hypothesis is a consequence of what looks generic condition, Hypothesis 4.2below, if (H8) is assumed. [CM] treats also the case when there are many eigen-values and Hypothesis 4.2 is replaced by a more stringent hypothesis which is anatural generalization of the (FGR) hypothesis in [GS]. The same result with manyeigenvalues case can be proved also here and in [Cu3], but we skip for simplicity theproof. We recall that Mizumachi [M1], resp. [M2], extends to dimension 1, resp 2,the results in [GNT] valid for small solitons obtained by by bifurcation from ground2tates of a linear equation, while [CZ] extends in 2D the result in [SW2]. [Cu3]transposes [M1] to the case of large solitons, with the generalizations contained in[CM]. Here we consider the case of dimension 2. Thanks to the work by [M2], it isquite clear how to transpose to dimension 2 the higher dimensional arguments in[CM]. The nonlinear arguments in [CM] are not sensitive to the dimension exceptfor the lack in 2D of the endpoint Stricharz estimate. Mizumachi [M2] shows howto replace it with an appropriate smoothing estimate of Kato type. The estimateand its proof are suggested by [M2]. In order to complete the proof of Theorem 1.1we need some dispersive estimates on the linearization H ω which in spatial dimen-sion 2 are not yet proved in the literature. The main technical task of this paperis the transposition to H ω of the proof of of L p boundedness of wave operators ofSchr¨odinger operators in dimension 2 due to Yajima [Y2]. We use the followingnotation. We set H ( ω ) = σ ( − ∆ + ω ); given normed spaces X and Y we denoteby B ( X, Y ) the space of operators from X to Y and given L ∈ B ( X, Y ) we denoteby k L k X,Y or by k L k B ( X,Y ) its norm. We prove: Proposition 1.2.
Assume the hypotheses of Theorem 1.1. The following limits arewell defined isomorphism, inverse of each other:
W u = lim t → + ∞ e itH ω e − itH ( ω ) u for any u ∈ L Zu = lim t → + ∞ e itH ( ω ) e − itH ω for any u ∈ L c ( H ω ) (defined in § p ∈ (1 , ∞ ) and any k the restrictions of W and Z to L ∩ W k,p extend intooperators such that for C ( ω ) < ∞ semicontinuous in ω k W k W k,p ( R ) ,W k,pc ( H ω ) + k Z k W k,pc ( H ω ) ,W k,p ( R ) < C ( ω ) with W k,pc ( H ω ) the closure in W k,p ( R ) of W k,p ( R ) ∩ L c ( H ω ) . We will set L ,s and H m,s k u k L ,s = kh x i s u k L ( R ) and k u k H m,s = kh x i s u k H m ( R ) , where m ∈ N , s ∈ R and h x i = (1 + | x | ) / . For f ( x ) and g ( x ) column vectors,their inner product is h f, g i = R R t f ( x ) · g ( x ) dx . The adjoint H ∗ is defined by h Hf, g i = h f, H ∗ g i . Given an operator H , its resolvent is R H ( z ) = ( H − z ) − . We will write R ( z ) = ( − ∆ − z ) − . We write k g ( t, x ) k L pt L qx = kk g ( t, x ) k L qx k L pt and k g ( t, x ) k L pt L ,sx = kk g ( t, x ) k L ,sx k L pt § We will use the following classical result, [We1,GSS1-2], see also [Cu3]:3 heorem 2.1.
Suppose that e iωt φ ω ( x ) satisfies (H4). Then ∃ ǫ > and a A ( ω ) > such that for any k u (0 , x ) − φ ω k H < ǫ we have for the corresponding solution inf {k u ( t, x ) − e iγ φ ω ( x − x ) k H ( x ∈ R ) : γ ∈ R & x ∈ R } < A ( ω ) ǫ. We can write the ansatz u ( t, x ) = e i Θ( t ) ( φ ω ( t ) ( x ) + r ( t, x )) , Θ( t ) = R t ω ( s ) ds + γ ( t ) . Inserting the ansatz into the equation we get ir t = − r xx + ω ( t ) r − β ( φ ω ( t ) ) r − β ′ ( φ ω ( t ) ) φ ω ( t ) r − β ′ ( φ ω ( t ) ) φ ω ( t ) r + ˙ γ ( t ) φ ω ( t ) − i ˙ ω ( t ) ∂ ω φ ω ( t ) + ˙ γ ( t ) r + O ( r ) . We set t R = ( r, ¯ r ), t Φ = ( φ ω , φ ω ) and we rewrite the above equation as(2 . iR t = H ω R + σ ˙ γR + σ ˙ γ Φ − i ˙ ω∂ ω Φ + O ( R ) . Set H ( ω ) = σ ( − d /dx + ω ) and V ( ω ) = H ω − H ( ω ) . The essential spectrum is σ e = σ e ( H ω ) = σ e ( H ( ω )) = ( −∞ , − ω ] ∪ [ ω, + ∞ ) . L we set N g ( L ) = ∪ j ≥ ker( L j ). [We2]implies that, if {·} means span, N g ( H ∗ ω ) = { Φ , σ ∂ ω Φ } . λ ( ω ) has corresponding realeigenvector ξ ( ω ), which can be normalized so that h ξ, σ ξ i = 1. σ ξ ( ω ) generatesker( H ω + λ ( ω )) . The function ( ω, x ) ∈ O × R → ξ ( ω, x ) is C ; | ξ ( ω, x ) | < ce − a | x | for fixed c > a > ω ∈ K ⊂ O , K compact. ξ ( ω, x ) is even in x since byassumption we are restricting ourselves in the category of such functions. We havethe H ω invariant Jordan block decomposition L = N g ( H ω ) ⊕ (cid:0) ⊕ j, ± ker( H ω ∓ λ ( ω )) (cid:1) ⊕ L c ( H ω ) = N g ( H ω ) ⊕ N ⊥ g ( H ∗ ω )where we set L c ( H ω ) = { N g ( H ∗ ω ) ⊕ ⊕ ± ker( H ∗ ω ∓ λ ( ω )) } ⊥ . We can impose(2 . R ( t ) = ( zξ + ¯ zσ ξ ) + f ( t ) ∈ (cid:2) X ± ker( H ω ( t ) ∓ λ ( ω ( t ))) (cid:3) ⊕ L c ( H ω ( t ) ) . The following claim admits an elementary proof which we skip:
Lemma 2.2.
There is a Taylor expansion at R = 0 of the nonlinearity O ( R ) in(2.1) with R m,n ( ω, x ) and A m,n ( ω, x ) real vectors and matrices rapidly decreasingin x : O ( R ) = X ≤ m + n ≤ N +1 R m,n ( ω ) z m ¯ z n + X ≤ m + n ≤ N z m ¯ z n A m,n ( ω ) f + O ( f + | z | N +2 ) .
4n terms of the frame in (2.2) and the expansion in Lemma 2.2, (2.1) becomes(2 . if t = (cid:0) H ω ( t ) + σ ˙ γ (cid:1) f + σ ˙ γ Φ( ω ) − i ˙ ω∂ ω Φ( t ) + ( zλ ( ω ) − i ˙ z ) ξ ( ω ) − (¯ zλ ( ω ) + i ˙¯ z ) σ ξ ( ω ) + σ ˙ γ ( zξ + ¯ zσ ξ ) − i ˙ ω ( z∂ ω ξ + ¯ zσ ∂ ω ξ )+ X ≤ m + n ≤ N +1 z m ¯ z n R m,n ( ω ) + X ≤ m + n ≤ N z m ¯ z n A m,n ( ω ) f ++ O ( f ) + O loc ( | z N +2 | )where by O loc we mean that the there is a factor χ ( x ) rapidly decaying to 0 as | x | → ∞ . By taking inner product of the equation with generators of N g ( H ∗ ω ) andker( H ∗ ω − λ ) we obtain modulation and discrete modes equations:(2 . i ˙ ω d k φ ω k dω = h σ ˙ γ ( zξ + ¯ zσ ξ ) − i ˙ ω ( z∂ ω ξ + ¯ zσ ∂ ω ξ ) + N +1 X m + n =2 z m ¯ z n R m,n ( ω )+ (cid:0) σ ˙ γ + i ˙ ω∂ ω P c + N X m + n =1 z m ¯ z n A m,n ( ω ) (cid:1) f + O ( f ) + O loc ( | z N +2 | ) , Φ i ˙ γ d k φ ω k dω = h same as above , σ ∂ ω Φ i i ˙ z − λ ( ω ) z = h same as above , σ ξ i . § H ω We need analogues of Lemmas 2.1-3 and Corollary 2.1 in [M2]. We call admissibleall pairs ( p, q ) with 1 /p = 1 / − /q and 2 ≤ q < ∞ . We set ( p ′ , q ′ ) = ( p/ ( p − , q/ ( q − H ω of the form (1.2) forwhich hypotheses (H3-5), (H7) and (H9) hold. Lemma 3.1 (Strichartz estimate).
There exists a positive number C = C ( ω ) upper semicontinuous in ω such that for any k ∈ [0 , :(a) for any f ∈ L c ( ω ) and any admissible all pairs ( p, q ) , k e − itH ω f k L pt W k,qx ≤ C k f k H k . (b) for any g ( t, x ) ∈ S ( R ) and any couple of admissible pairs ( p , q ) ( p , q ) wehave k Z t e − i ( t − s ) H ω P c ( ω ) g ( s, · ) ds k L p t W k,q x ≤ C k g k L p ′ t W k,q ′ x . Lemma 3.1 follows immediately from Proposition 1.2 since W and Z intertwine e − itH ω P c ( H ω ) and e − itH . 5 emma 3.2. Let s > . ∃ C = C ( ω ) upper semicontinuous in ω such that:(a) for any f ∈ S ( R ) , k e − itH ω P c ( ω ) f k L t L , − sx ≤ C k f k L ; (b) for any g ( t, x ) ∈ S ( R ) (cid:13)(cid:13)(cid:13)(cid:13)Z R e itH ω P c ( ω ) g ( t, · ) dt (cid:13)(cid:13)(cid:13)(cid:13) L x ≤ C k g k L t L ,sx . Notice that (b) follows from (a) by duality.
Lemma 3.3.
Let s > . ∃ C = C ( ω ) as above such that ∀ g ( t, x ) ∈ S ( R ) and t ∈ R : (cid:13)(cid:13)(cid:13)(cid:13)Z t e − i ( t − s ) H ω P c ( ω ) g ( s, · ) ds (cid:13)(cid:13)(cid:13)(cid:13) L t L , − sx ≤ C k g k L t L ,sx . As a corollary from Christ and Kiselev [CK], Lemmas 3.2 and 3.3 imply:
Lemma 3.4.
Let ( p, q ) be an admissible pair and let s > . ∃ C = C ( ω ) as abovesuch that ∀ g ( t, x ) ∈ S ( R ) and t ∈ R : (cid:13)(cid:13)(cid:13)(cid:13)Z t e − i ( t − s ) H ω P ( ω ) g ( s, · ) ds (cid:13)(cid:13)(cid:13)(cid:13) L pt L qx ≤ C k g k L t L ,sx . Lemma 3.5.
Consider the diagonal matrices E + = diag (1 , E − = diag (0 , . Set P ± ( ω ) = Z ( ω ) E ± W ( ω ) with Z ( ω ) and W ( ω ) the wave operators associated to H ω .Then we have for u ∈ L c ( H ω )(1) P + ( ω ) u = lim ǫ → + πi lim M → + ∞ Z Mω [ R H ω ( λ + iǫ ) − R H ω ( λ − iǫ )] udλP − ( ω ) u = lim ǫ → + πi lim M → + ∞ Z − ω − M [ R H ω ( λ + iǫ ) − R H ω ( λ − iǫ )] udλ and for any s and s and for C = C ( s , s , ω ) upper semicontinuous in ω , we have (2) k ( P + ( ω ) − P − ( ω ) − P c ( ω ) σ ) f k L ,s ≤ C k f k L ,s . Proof.
Formulas (1) hold with P ± ( ω ) replaced by E ± and H ω replaced by H and for any u ∈ L ( R ). Applying W ( ω ) we get (1) for H ω . Estimate (2) follows bythe proof of inequality (3) in Lemma 5.12 [Cu3] which is valid for all dimensions.6 We restate Theorem 1.1 in a more precise form:
Theorem 4.1.
Under the assumptions of Theorem 1.1 we can express u ( t, x ) = e i Θ( t ) ( φ ω ( t ) ( x ) + N X j =1 p j ( z, ¯ z ) A j ( x, ω ( t )) + h ( t, x )) with p j ( z, ¯ z ) = O ( z ) near 0, with lim t → + ∞ ω ( t ) convergent, with | A j ( x, ω ( t )) | ≤ Ce − a | x | for fixed C > and a > , lim t → + ∞ z ( t ) = 0 , and for fixed C > k z ( t ) k N +1 L N +2 t + k h ( t, x ) k L ∞ t H x ∩ L t W , x < Cǫ. Furthermore, there exists h ∞ ∈ H ( R , C ) such that (2) lim t →∞ k e i R t ω ( s ) ds + iγ ( t ) h ( t ) − e it ∆ h ∞ k H = 0 . The proof of Theorem 4.1 consists in a normal forms expansion and in the closureof some nonlinear estimates. The normal forms expansion is exactly the same of[CM,Cu3], in turn adaptations of [GS]. § We repeat [CM]. We pick k = 1 , , ...N and set f = f k for k = 1. The other f k are defined below. In the ODE’s there will be error terms of the form E ODE ( k ) = O ( | z | N +2 ) + O ( z N +1 f k ) + O ( f k ) + O ( β ( | f k | ) f k ) . In the PDE’s there will be error terms of the form E P DE ( k ) = O loc ( | z | N +2 ) + O loc ( zf k ) + O loc ( f k ) + O ( β ( | f k | ) f k ) . In the right hand sides of the equations (2.3-4) we substitute ˙ γ and ˙ ω using themodulation equations. We repeat the procedure a sufficient number of times untilwe can write for k = 1 and f = fi ˙ ω d k φ ω k dω = h N +1 X m + n =2 z m ¯ z n Λ ( k ) m,n ( ω ) + N X m + n =1 z m ¯ z n A ( k ) m,n ( ω ) f k + E ODE ( k ) , Φ( ω ) i i ˙ z − λz = h same as above , σ ξ ( ω ) i i∂ t f k = ( H ω + σ ˙ γ ) f k + E P DE ( k ) + X k +1 ≤ m + n ≤ N +1 z m ¯ z n R ( k ) m,n ( ω ) , A ( k ) m,n , R ( k ) m,n and Λ ( k ) m,n ( ω, x ) real exponentially decreasing to 0 for | x | → ∞ andcontinuous in ( ω, x ). Exploiting | ( m − n ) λ ( ω ) | < ω for m + n ≤ N , m ≥ n ≥ f k with k ≤ N by f k − = − X m + n = k z m ¯ z n R H ω (( m − n ) λ ( ω )) R ( k − m,n ( ω ) + f k . Notice that if R ( k − m,n ( ω, x ) is real exponentially decreasing to 0 for | x | → ∞ , thesame is true for R H ω (( m − n ) λ ( ω )) R ( k − m,n ( ω ) by | ( m − n ) λ ( ω ) | < ω . By induction f k solves the above equation with the above notifications. Now we manipulate theequation for f N . We fix ω = ω (0). We write(4 . i∂ t P c ( ω ) f N − { H ω + ( ˙ γ + ω − ω )( P + ( ω ) − P − ( ω )) } P c ( ω ) f N =+ P c ( ω ) e E P DE ( N ) + X m + n = N +1 z m ¯ z n P c ( ω ) R ( N ) m,n ( ω )where we split P c ( ω ) = P + ( ω ) + P − ( ω ) with P ± ( ω ), see Lemma 3.5, where P + ( ω ) are the projections in σ c ( H ω ) ∩ { λ : ± λ ≥ ω } and with(4 . e E P DE ( N ) = E P DE ( N ) + X m + n = N +1 z m ¯ z n (cid:16) R ( N ) m,n ( ω ) − R ( N ) m,n ( ω ) (cid:17) + ϕ ( t, x ) f N ϕ ( t, x ) := ( ˙ γ + ω − ω ) ( P c ( ω ) σ − ( P + ( ω ) − P − ( ω ))) f N + ( V ( ω ) − V ( ω )) f N + ( ˙ γ + ω − ω ) ( P c ( ω ) − P c ( ω )) σ f N . By Lemma 3.5 for C N ( ω ) upper semicontinuous in ω , ∀ N we have kh x i N ( P + ( ω ) − P − ( ω ) − P c ( ω ) σ ) f k L x ≤ C N ( ω ) kh x i − N f k L x . The term ϕ ( t, x ) in (4.2) can be treated as a small cutoff function. We write(4 . f N = − X m + n = N +1 z m ¯ z n R H ω (( m − n ) λ ( ω ) + i P c ( ω ) R ( N ) m,n ( ω ) + f N +1 . Then(4 . i∂ t P c ( ω ) f N +1 = ( H ω + ( ˙ γ + ω − ω )( P + ( ω ) − P − ( ω ))) P c ( ω ) f N +1 ++ X ± O ( ǫ | z | N +1 ) R H ω ( ± ( N + 1) λ ( ω ) + i R ± ( ω ) + P c ( ω ) b E P DE ( N )with R + = R ( N ) N +1 , and R − = R ( N )0 ,N +1 and b E P DE ( N ) = e E P DE ( N ) + O loc ( ǫz N +1 ),where we have used that ( ω − ω ) = O ( ǫ ) by Theorem 2.1. Notice that R H ω ( ± ( N +8) λ ( ω ) + i R ± ( ω ) ∈ L ∞ do not decay spatially. In the ODE’s with k = N , bythe standard theory of normal forms and following the idea in Proposition 4.1 [BS],see [CM] for details, it is possible to introduce new unknowns(4 . e ω = ω + q ( ω, z, ¯ z ) + X ≤ m + n ≤ N z m ¯ z n h f N , α mn ( ω ) i , e z = z + p ( ω, z, ¯ z ) + X ≤ m + n ≤ N z m ¯ z n h f N , β mn ( ω ) i , with p ( ω, z, ¯ z ) = P p m,n ( ω ) z m ¯ z n and q ( z, ¯ z ) = P q m,n ( ω ) z m ¯ z n polynomials in( z, ¯ z ) with real coefficients and O ( | z | ) near 0, such that we get(4 . i ˙ e ω = h E P DE ( N ) , Φ i i ˙ e z − λ ( ω ) e z = X ≤ m ≤ N a m ( ω ) | e z m | e z + h E ODE ( N ) , σ ξ i ++ e z N h A ( N )0 ,N ( ω ) f N , σ ξ i . with a m ( ω ) real. Next step is to substitute f N using (4.4). After eliminating bya new change of variables e z = b z + p ( ω, b z, b z ) the resonant terms, with p ( ω, b z, b z ) = P b p m,n ( ω ) z m ¯ z n a polynomial in ( z, ¯ z ) with real coefficients O ( | z | ) near 0, we get(4 . i ˙ b ω = h E P DE ( N ) , Φ i i ˙ b z − λ ( ω ) b z = X ≤ m ≤ N b a m ( ω ) | e z m | b z + h E ODE ( N ) , σ ξ i−− | b z N | b z h b A ( N )0 ,N ( ω ) R H ω (( N + 1) λ ( ω ) + i P c ( ω ) R ( N ) N +1 , ( ω ) , σ ξ i + b z N h b A ( N )0 ,N ( ω ) f N +1 , σ ξ i with b a m , b A ( N )0 ,N and R ( N ) N +1 , real. By x − i = P V x + iπδ ( x ) and by an elementaryuse of the wave operators, we can denote by Γ( ω, ω ) the quantityΓ( ω, ω ) = ℑ (cid:16) h b A ( N )0 ,N ( ω ) R H ω (( N + 1) λ ( ω ) + i P c ( ω ) R ( N ) N +1 , ( ω ) σ ξ ( ω ) i (cid:17) = π h b A ( N )0 ,N ( ω ) δ ( H ω − ( N + 1) λ ( ω )) P c ( ω ) R ( N ) N +1 , ( ω ) σ ξ ( ω ) i . Now we assume the following:
Hypothesis 4.2.
There is a fixed constant Γ > such that | Γ( ω, ω ) | > Γ . By continuity and by Hypothesis 4.2 we can assume | Γ( ω, ω ) | > Γ / . Then wewrite 94 . ddt | b z | − Γ( ω, ω ) | z | N +2 + ℑ (cid:16) h b A ( N )0 ,N ( ω ) f N +1 , σ ξ ( ω ) i b z N +1 (cid:17) + ℑ (cid:16) h E ODE ( N ) , σ ξ ( ω ) i b z (cid:17) . § By an elementary continuation argument, the following a priori estimates implyinequality (1) in Theorem 4.1, so to prove (1) we focus on:
Lemma 4.3.
There are fixed constants C and C and ǫ > such that for any < ǫ ≤ ǫ if we have (4 . k b z k N +1 L N +2 t ≤ C ǫ & k f N k L ∞ t H x ∩ L t W , x ∩ L p p − t W , p x ∩ L t H , − s ≤ C ǫ then we obtain the improved inequalities k f N k L ∞ t H x ∩ L t W , x ∩ L p p − t W , p x ∩ L t H , − s ≤ C ǫ, (4 . k b z k N +1 L N +2 t ≤ C ǫ. (4 . Proof . Set ℓ ( t ) := γ + ω − ω . First of all, we have: Lemma 4.4.
Let g (0 , x ) ∈ H x ∩ L c ( ω ) and let ω ( t ) be a continuous function.Consider ig t = { H ω + ℓ ( t )( P + ( ω ) − P − ( ω )) } g + P c ( ω ) F. Then for a fixed C = C ( ω , s ) upper semicontinuous in ω and s > we have k g k L ∞ t H x ∩ L t W , x ∩ L p p − t W , p x ≤ C ( k g (0 , x ) k H + k F k L t H x + L t H ,sx ) . Lemma 4.4 follows easily from Lemmas 3.1-4 and P ± ( ω ) g ( t ) == e − itH ω e − i R t ℓ ( τ ) dτ P ± ( ω ) g (0) − i Z t e − i ( t − s ) H ω e ± i R ts ℓ ( τ ) dτ P ± ( ω ) F ( s ) ds Lemma 4.5.
Consider equation (4.1) for f N and assume (4.10). Then we cansplit e E P DE ( N ) = X + O ( f N ) + O ( f p N ) such that k X k L t H ,Mx . ǫ for any fixed M and k O ( f N ) + O ( f p N ) k L t H x . ǫ . Proof of Lemma 4.5.
In the error terms for k = N at the beginning of § e E P DE ( N ) = O ( ǫ ) ψ ( x ) f N + O loc ( | z | N +2 ) + O loc ( zf N ) + O loc ( f N ) + O ( f N ) + O ( f p N )10ith ψ ( x ) a rapidly decreasing function, p the exponent in (H2) and with O ( f p N )relevant only for p >
3. Denoting X the sum of all terms except the last one,setting f = f N , by (4.10) we have: :(1) k O ( ǫ ) ψ ( x ) f k L t H ,Mx . ǫ k f k L t H , − Mx . ǫ ;(2) k O loc ( zf ) k L t H ,Mx . k z k ∞ k f k L t H , − Mx . ǫ ;(3) k O loc ( f ) k L t H ,Mx . k f k L t H , − Mx . ǫ . This yields kh x i M X k H x L t . ǫ . To bound the remaining term observe:(4) k| f | f k L t H x . (cid:13)(cid:13)(cid:13) k f k W , x k f k L x (cid:13)(cid:13)(cid:13) L t ≤ k f k L t W , x . ǫ ;(5) k O ( f p ) k L t H x . (cid:13)(cid:13)(cid:13) k f k W , p x k f k p − L p x (cid:13)(cid:13)(cid:13) L t ≤ k f k L p p − t W , p x k f k p − L p p − p t W , p x . ǫ p , where in the last step we use k f k L p p − p t W , p x . k f k αL p p − t L p x k f k − αL ∞ t H x forsome 0 < α < p >
3, interpolation and Sobolev embedding.
Proof of (4.11).
Recall that f N satisfies equation (4.1) whose right hand sideis P c ( ω ) e E P DE ( N ) + O loc ( z N +1 ). In addition to Lemma 4.5 we have the estimate k O loc ( z N +1 ) k L t H ,Mx . k z k N +1 L N +1 t . C ǫ. So by Lemmas 3.1-4, for some fixed c weget schematically k f N k L ∞ t H x ∩ L t W , x ∩ L p p − t W , p x ≤ c C ǫ + ǫ + O ( ǫ )where ǫ comes from initial data, O ( ǫ ) from all the nonlinear terms save for the R ( N ) m,n ( ω ) z m ¯ z n terms which contribute the 2 c C ǫ . Let now f N = g + h with ig t = { H ω + ℓ ( t )( P + ( ω ) − P − ( ω )) } g + X + O loc ( z N +1 ) , g (0) = f N (0) ih t = { H ω + ℓ ( t )( P + ( ω ) − P − ( ω )) } h + O ( f N ) + O ( f p N ) , h (0) = 0in the notation of Lemma 4.5. Then, by Lemmas 3.2 and 3.3 and by the estimatesin Lemma 4.5 we get k g k L t H , − sx . C ǫ + O ( ǫ ) + c ǫ for a fixed c . Finally, Z ∞ k e − i ( t − s ) H ω e ± i R ts ℓ ( τ ) dτ ( O ( f N ) + O ( f p N ))( s ) k L t H , − s . Z ∞ k ( O ( f N ) + O ( f p N ))( s ) k H . ǫ . So if we set C ≈ C + c + 1 we obtain (4.11). We need to bound C . Proof of (4.12).
We first need: 11 emma 4.6.
We can decompose f N +1 = h + h + h + h with for a fixed large M > :(1) k h k L t L ,Mx ≤ O ( ǫ ); (2) k h k L t L ,Mx ≤ O ( ǫ ); (3) k h k L t L ,Mx ≤ O ( ǫ ); (4) k h k L t L ,Mx ≤ c ( ω ) ǫ for a fixed c ( ω ) upper semicontinuous in ω .Proof of Lemma 4.6 . We set i∂ t h = ( H ω + ℓ ( t )( P + − P − )) h h (0) = X m + n = N +1 R H ω (( m − n ) λ ( ω ) + i R ( N ) m,n ( ω ) z m (0)¯ z n (0) . We get k h k L t L , − Mx ≤ c ( ω ) | z (0) | P k R ( N ) m,n ( ω ) k L ,Mx = O ( ǫ ) by the followinglemma: Lemma 4.7.
There is a fixed s such that for s > s , (4 . k e − iH ω t R H ω (Λ + i P c ( ω ) ϕ k L t L , − sx < C s (Λ , ω ) k ϕ ( x ) k L ,sx (cid:13)(cid:13)(cid:13)(cid:13)Z t e − iH ω ( t − τ ) R H ω (Λ + i P c ( ω ) g ( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13) L t L , − sx < C s (Λ , ω ) k g ( t, x ) k L t L ,sx with C s (Λ , ω ) upper semicontinuous in ω and in Λ > ω . Let us assume Lemma 4.7 for the moment, for the proof see §
9. We set h (0) = 0and i∂ t h = ( H ω + ℓ ( t )( P + − P − )) h ++ O ( ǫz N +1 ) R H ω (( N + 1) λ ( ω ) + i R ( N ) N +1 , ( ω )+ O ( ǫz N +1 ) R H ω ( − ( N + 1) λ ( ω ) + i R ( N )0 ,N +1 ( ω ) . Then we have h = h + h with h j = P ± h j ± with h ± ( t ) = Z t e − iH ω ( t − s ) e ± i R ts ℓ ( τ ) dτ P ± O ( ǫz N +1 ) R H ω (( N + 1) λ ( ω ) + i R ( N ) N +1 , ( ω ) ds and h ± defined similarly but with R H ω ( − ( N + 1) λ ( ω ) + i R ( N )0 ,N +1 . Now by(4.13) we get k h j ± ( t ) k L t L , − Mx ≤ Cǫ k z k N +1 L N +2 t and so k h ( t ) k L t L , − Mx = O ( ǫ ) . Let h (0) = 0 and12 ∂ t P c ( ω ) h = ( H ω + ℓ ( t )( P + ( ω ) − P − ( ω ))) P c ( ω ) h + P c ( ω ) e E P DE ( N ) . Then by the argument in the proof of (4.11) we get claim (3). Finally let h (0) = f N (0) and i∂ t P c ( ω ) h = ( H ω + ℓ ( t )( P + ( ω ) − P − ( ω ))) P c ( ω ) h . Then by Lemma 3.2 kh x i − M h k L tx . k f N (0) k L x ≤ c ( ω ) ǫ we get (4). Continuation of proof of Lemma 4.3 . We integrate (4.9) in time. Then byTheorem 2.1 and by Lemma 4.4 we get, for A an upper bound of the constants A ( ω ) of Theorem 2.1, k b z k N +2 L N +2 t ≤ A ǫ + ǫ k b z k N +1 L N +2 t + o ( ǫ ) . Then we can pick C = ( A + 1) and this proves that (4.10) implies (4.12). Fur-thermore b z ( t ) → ddt b z ( t ) = O ( ǫ ) . As in [CM,Cu3] in the above argument we did not use the sign of Γ( ω, ω ). Withthe same argument in [CM,Cu3] one can prove Corollary 4.8.
If Hypothesis 4.2 holds, then Γ( ω, ω ) > Γ . The proof that, for t f N ( t ) = ( h ( t ) , h ( t )), h ( t ) is asymptotically free for t → ∞ ,is similar to the analogous one in [CM] and we skip it. § L theory for H ω In sections § § ± ω :(a) H ω has no eigenvalues in [ ω, + ∞ ) ∪ ( −∞ , − ω ];(b) if g ∈ W , ∞ ( R , C ) satisfies H ω g = ωg or H ω g = − ωg then g = 0.Because of the fact that H ω is not a symmetric operator, we need some prepara-tory work to show that in fact H ω is diagonalizable in the continuous spectrum.This work is done in § W whichis the basis to develop in § q ( x ) areal valued function with: q ( x ) ≥ q ( x ) > q ( x ) ∈ C ∞ ( R ).We set h q = − ∆ + q ( x ). Then we have: 13 emma 5.1. Let C + = { z ∈ C : ℑ z > } . Suppose q ( x ) = 0 for r ≥ r > . Thenwe have the following facts.(1) There exists s > and C > such that for s ≥ s , R h q ( z ) extends into afunction z → R + h q ( z ) which is in ( L ∞ ∩ C )( C + , B ( L ,s , L , − s )) .(2) For any n ∈ N there exists s > such that for any a > there is a choice of C > such that for n ≤ n (cid:13)(cid:13)(cid:13)(cid:13) d n dz n R + h q ( z ) : L ,s ( R ) → L , − s ( R ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ C h z i − (1+ n ) ∀ z ∈ C + ∩ { z : | z | ≥ a } . (3) The same argument can be repeated for C − = { z ∈ C : ℑ z < } and R − h q ( z ) . Claim (2) follows from [Ag] and [JK] and claim (3) follows along the lines ofthe previous two claims. In view of (2), it is enough to prove (1) for z ≈ . For ζ = re iθ with θ ∈ ( − π, π ) let √ ζ = √ re iθ/ . With this convention for z [0 , ∞ )for R ( z ) = ( − ∆ − z ) − we have R ( z ) = 12 π K ( √− z | x | ) ∗ = i H +0 ( i √− z | x | ) ∗ = − i H − ( − i √− z | x | ) ∗ for the Macdonald function K and the Hankel functions H ± . We set G = − π log | x |∗ , P f = R R f dx . We have for M ( z ) = (1 + √ qR ( z ) √ q ) the identity(4) R h q ( z ) = R ( z ) − R ( z ) √ qM − ( z ) √ qR ( z ) . From the expansion at 0 in C + of H +0 and by the argument in Lemma 5 [Sc] wehave in B ( L ,s , L , − s ) , for s sufficiently large,(5) R ( z ) = c ( z ) P − G + O ( − z log √− z ) c ( z ) = i − γ π − π log( √− z/ . Consider the projections in L ( R ), P = √ q h· , √ q i / k q k L and Q = 1 − P . Let T = 1 + √ qG √ q . Then QT Q is invertible in QL ( R ). Denote its inverse in QL ( R ) by D = ( QT Q ) − . Consider the operator in L = P L ⊕ QL defined by S = (cid:20) P − P T QD Q − QD QT P QD QT P T QD Q (cid:21) and h ( z ) = k q k L c ( z ) + trace( P T P − P T QD QT P ). Then by [Sc](6) R h q ( z ) = R ( z ) − h − ( z ) R ( z ) √ qS √ qR ( z ) − R ( z ) √ qQD Q √ qR ( z ) − R ( z ) √ qO ( − z log √− z ) √ qR ( z ) .
14y direct computation h − ( z ) R ( z ) √ qS √ qR ( z ) = c ( z ) h ( z ) h· , i√ qS √ q h· , i + c ( z ) h ( z ) h· , i√ qS √ qG ++ c ( z ) h ( z ) G √ qS √ q h· , i + c ( z ) h ( z ) G √ qS √ qG + O ( − z log √− z ) , where all terms, except the first on the right hand side, admit continuous extensionin C + at 0. We have h· , i√ qS √ q h· , i = k q k L P and so by (5) R ( z ) − c ( z ) h ( z ) k q k L P admits continuous extension in C + at 0. By direct computation R ( z ) √ qQD Q √ qR ( z ) = G √ qQD Q √ qG + O ( − z log √− z )admits continuous extension in C + at 0. So R h q ( z ) admits continuous extension in C + at 0, and so on all C + .A consequence of Lemma 5.1 is the h q smoothness in the sense of Kato [Ka] ofmultiplication operators involving rapidly decreasing functions ψ : Lemma 5.2.
Let ψ ( x ) ∈ L ∞ ( R ) ∩ L ,s ( R ) for s ≫ and q as in Lemma 5.1.Then the multiplication operator ψ is h q smooth, that is, for a fixed C > Z R k ψR h q ( λ + iε ) u k dλ < C k u k for all u ∈ L ( R ) and ε = 0 . This follows from one of the characterizations of H smoothness in the case H isselfadjoint, see Theorem 5.1 [Ka], specifically from the fact that by Lemma 5.1 wehave that for ψ , ψ ∈ L ∞ ∩ L ,s for s ≫ C > z R we have k ψ R h q ( z ) ψ k L ,L < C. We consider now H q = σ ( − ∆ + q + ω ) and consider our linearization H ω . Write H ω = H q + ( V ω − σ q ), and factorize V ω − σ q = B ∗ A with A, B smooth | ∂ βx A ( x ) | + | ∂ βx B ( x ) | < Ce − α | x | ∀ x, for some α, C > | β | ≤ N , N sufficiently large.We have σ H q = − H q σ , σ H ω = − H ω σ . We choose the factorization B ∗ A sothat σ B ∗ = − B ∗ σ , σ A = Aσ . By these equalities σ R H q ( z ) = − R H q ( − z ) σ and σ R H ω ( z ) = − R H ω ( − z ) σ , so in some of the estimates below it is enough toconsider z ∈ C ++ with C ++ = { z : ℑ z > , ℜ z > } . emma 5.3. For z ∈ C + the function R + H q ( z ) is well defined and satisfies thefollowing properties:(1) There exists s > and C > such that for s ≥ s the function z → R + H q ( z ) isin ( L ∞ ∩ C )( C + , B ( L ,s , L , − s )) .(2) For any n ∈ N there exists s > such that for any a > there is a choice of C > such that for n ≤ n and ∀ z ∈ C + ∩ { z : dist ( z, ± ω ) ≥ a } , (cid:13)(cid:13)(cid:13)(cid:13) d n dz n R + H q ( z ) : L ,s ( R ) → L , − s ( R ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ C h z i − (1+ n ) . (3) For any ψ ( x ) ∈ L ∞ ( R ) ∩ L ,s ( R ) for s ≫ the multiplication operator ψ is H q smooth, that is, for a fixed C > Z R k ψR H q ( λ + iε ) u k dλ < C k u k for all u ∈ L ( R ) and ε = 0 . (4) Analogous statements hold for z ∈ C − and the function R − H q ( z ) . Lemma 5.3 is a trivial consequence of Lemmas 5.1-2. The properties in Lemma5.4 are partially inherited by H ω . Let Q + q ( z ) = AR + H q ( z ) B ∗ . Then for z ∈ C + Lemma 5.4.
Fix an exponentially decreasing bounded function ψ . For z ∈ C + thefunction AR H ω ( z ) ψ extends into a function AR + H ω ( z ) ψ for z ∈ C + \ σ d ( H ω ) withthe following properties:(1) ∀ a > ∃ C > such that for X a = C + ∩ { z : dist ( z, σ d ( H ω )) ≥ a } AR + H ω ( z ) ψ ∈ ( L ∞ ∩ C )( X a , B ( L , L )) (2) For any n ∈ N there exists s > such that for any a > there is a choice of C > such that for n ≤ n and ∀ z ∈ X a ∩ { z : dist ( z, ± ω ) ≥ a } , (cid:13)(cid:13)(cid:13)(cid:13) d n dz n AR + H ω ( z ) ψ : L ( R ) → L ( R ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ C h z i − (1+ n ) . (3) There is a constant C > such that Z k AR H ω ( λ + iε ) u k dλ ≤ C k u k for all u ∈ L c ( H ω ) and ε = 0 . (4) Analogous statements hold for z ∈ C − and the function R − H ω ( z ) .Proof. Let us write Q + q ( z ) = AR + H q ( z ) B ∗ and for z ∈ C + (5) AR H ω ( z ) = (1 + Q + q ( z )) − AR H q ( z ) . By Lemma 5.3 we have lim z →∞ k Q + q ( z ) k L ,L = 0. By analytic Fredholm theory1+ Q + q ( z ) is not invertible only at the z ∈ C + where ker(1+ Q + q ( z )) = 0. This set has0 measure in R . By Lemma 2.4 [CPV] if at some z = ± ω we have ker(1+ Q + q ( z )) = 0,then z is an eigenvalue. By hypothesis there are no eigenvalues in σ e ( H ω ) . Hencewe get claim (2). 16 emma 5.5. If ker(1 + Q + q ( ω )) = 0 then there exists g ∈ W , ∞ ( R ) with g = 0 such that H ω g = ωg Let us assume Lemma 5.5. By hypothesis such g does not exist. This yields(1). By (5), claim (4) Lemma 5.4 and Neumann expansion we get (4). Next, apply(5) to u ∈ L c ( H ω ). AR H ω ( z ) u is an analytic function in z with values in L ( R )for z near any isolated eigenvalue z of H ω because the natural projection of u in N g ( H ω − z ) is 0. Away from isolated eigenvalues of H ω , (1 + Q + q ( z )) − is uniformlybounded. Hence (3) in Lemma 5.3 implies (3) in Lemma 5.4. Proof of Lemma 5.5.
Let 0 = e g ∈ ker(1 + Q + q ( ω )). Then B ∗ e g + ( V ω − q ) R H q ( ω ) B ∗ e g = 0 . Set g = R H q ( ω ) B ∗ e g . Then Ag = − e g and so g = 0. By g + R H q ( ω )( V ω − q ) g = 0 wehave g ∈ H loc ( R ) and H ω g = ωg . We want now to show that g ∈ L ∞ ( R ), contraryto the hypotheses. We have t g = ( g , g ) with g = (∆ − q − ω ) − ( B ∗ e g ) , where B ∗ e g ∈ L ,s ( R ) for any s , so g ∈ H ( R ). We have g = R + h q (0)( B ∗ e g ) with g ∈ L , − s ( R ) for sufficiently large s . We split L , ± s = L , ± sr ⊕ (cid:0) L , ∓ sr (cid:1) ⊥ where L , ± sr arethe radial functions and we are considering the standard pairing L ,s × L , − s → C given by R R f ( x ) g ( x ) dx . We decompose g = g r + g nr with g r ∈ L , − sr and g nr ∈ ( L ,sr ) ⊥ . In ( L , − sr ) ⊥ → ( L ,sr ) ⊥ we have R + h q (0) = G − G q (1+ QG qQ ) − G with Q = 1 − P , for P = P q , q = c − q , c = R R qdx , P u = R R udx . Then g nr = G ( B ∗ e g ) nr − G q (1 + QG qQ ) − G ( B ∗ e g ) nr and by asymptotic expansion for | x | → ∞ we conclude that for some constants ∂ αx (cid:18) g nr − a − b x + b x | x | (cid:19) = O ( | x | − − α − ǫ )for some ǫ >
0. Finally we look ar e g r . We can consider solutions φ ( r ) and ψ ( r ) of h q u = 0 with: φ (0) = 1 and φ r (0) = 0; ψ ( r ) = 1 and | ψ ( r ) | bounded for r ≥ r , ψ ( r ) ≈ c log r with c = 0 for r →
0. In terms of these two functions the kernel of R + h q (0) in L ((0 , ∞ ) , dr ) is R + h q (0)( r , r ) = φ ( r ) ψ ( r ) W ( r ) if r < r or = φ ( r ) ψ ( r ) W ( r ) if r > r , with W ( r ) = [ φ ( · ) , ψ ( · )]( r ) = c/r for some c = 0. We have g r ( r ) == c − ψ ( r ) Z r φ ( s )( B ∗ e g ) r ( s ) s ds + c − φ ( r ) Z + ∞ r ψ ( s )( B ∗ e g ) r ( s ) s ds. r ≥ r , | g r ( r ) | ≤| c − ψ ( r ) | Z r | φ ( t )( B ∗ e g ) r ( t ) | t dt + | c − φ ( r ) | Z + ∞ r | ψ ( t )( B ∗ e g ) r ( t ) | t dt . k log h x ik L , − s ( R ) k B ∗ e g k L ,s ( R ) + log(2 + r ) k B ∗ e g k L ,s ( { x ∈ R : | x |≥ r } ) = O (1) . Then we conclude that we have a nonzero g ∈ H loc ( R ) ∩ L ∞ ( R ) such that H ω g = ωg . But this is contrary to the nonresonance hypothesis.Analogous to Lemma 5.4 is: Lemma 5.6.
Fix an exponentially decreasing bounded function ψ . For z ∈ C + thefunction BR H ∗ ω ( z ) ψ extends into a function BR + H ∗ ω ( z ) ψ for z ∈ C + \ σ d ( H ω ) withthe following properties:(1) For any a > there exists C > such that BR + H ∗ ω ( z ) ψ ∈ L ∞ ( X a , B ( L , L )) where X a = C + ∩ { z : dist ( z, σ d ( H ω )) ≥ a } . (2) For any n ∈ N there exists s > such that for any a > there is a choice of C > such that for n ≤ n and ∀ z ∈ X a ∩ { z : dist ( z, ± ω ) ≥ a } , (cid:13)(cid:13)(cid:13)(cid:13) d n dz n BR + H ∗ ω ( z ) ψ : L ( R ) → L ( R ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ C h z i − (1+ n ) . (3) There is a constant C > such that Z k BR H ∗ ω ( λ + iε ) u k dλ ≤ C k u k for all u ∈ L c ( H ∗ ω ) and ε = 0 . (4) Analogous statements hold for z ∈ C − and the function R − H ∗ ω ( z ) . From § Lemma 5.7.
There are isomorphisms f W : L → L c ( H ω ) and e Z : L c ( H ω ) → L ,inverses of each other, defined as follows: for u ∈ L , v ∈ L c ( H ∗ ω ) , h f W u, v i = h u, v i + lim ǫ → + πi Z + ∞−∞ h AR H q ( λ + iǫ ) u, BR H ∗ ω ( λ + iǫ ) v i dλ ; for u ∈ L c ( H ω ) , v ∈ L , h e Zu, v i = h u, v i + lim ǫ → + πi Z + ∞−∞ h AR H ω ( λ + iǫ ) u, BR H q ( λ + iǫ ) v i dλ. e have H ω f W = f W H q and H q e Z = e ZH ω , e itH ω f W = f W e itH q and e itH q e Z = e Ze itH ω P c ( H ω ) . The operators f W and e Z depend continuously on e A and e B ∗ andcan be expressed as f W u = lim t → + ∞ e itH ω e − itH q u for any u ∈ L e Zu = lim t → + ∞ e itH q e − itH ω for any u ∈ L ( H ω ) . In particular we remark:
Lemma 5.8.
We have for C ( ω ) upper semicontinuous in ω and k e − itH ω g k ≤ C ( ω ) k g k for any g ∈ L c ( H ω ) . Having proved that e − itH ω P c ( H ω ) are bounded in L , we want to relate H ω to H = σ ( − ∆ + ω ) . Write H = H + V ω , V ω = B ∗ A . We have σ H = − H σ , σ H ω = − H ω σ . We choose the factorization of V ω so that σ B ∗ = B ∗ σ , σ A = − Aσ . By these equalities σ R H ( z ) = − R H ( − z ) σ and σ R H ω ( z ) = − R H ω ( − z ) σ . We have the following result about existence and completeness ofwave operators: Lemma 5.9.
The following limits are well defined:
W u = lim t → + ∞ e itH ω e − itH u for any u ∈ L (1) Zu = lim t → + ∞ e itH e − itH ω for any u ∈ L c ( H ω ) . (2) W ( L ) = L c ( H ω ) is an isomorphism with inverse Z .Proof. The existence of P c ( H ω ) ◦ W follows from Cook’s method and Lemma 5.8.By an elementary argument W u ∈ L c ( H ω ) for any u ∈ L , so W = P c ( H ω ) ◦ W .We have W = f W ◦ W with W u = lim t → + ∞ e itH q e − itH u for any u ∈ L ( R ) f W u = lim t → + ∞ e itH ω ω e − itH q for any u ∈ L . By standard theory W is an isometric isomorphism of L ( R ) into itself with inverse Z u = lim t → + ∞ e itH e − itH q u and by Lemma 5.7 f W is an isomorphism L ( R ) → L c ( H ω ) with inverse e Z . Then by product rule the limit in (2) exists and we have Z = Z ◦ e Z with Z the inverse of W . 19 emma 5.10. For u ∈ L ,s ( R ) with s > / we have W u = u − πi Z | λ |≥ ω R − H ω ( λ ) V ω (cid:2) R + H ( λ ) − R − H ( λ ) (cid:3) udλ. Proof.
W u ∈ L ( R ) by Lemma 5.9, but the above formula is meaningful in thelarger space L , − s ( R ). For v ∈ L ,s ( R ) ∩ L c ( H ∗ ω ) and for h u, v i = R R u · vdx thestandard L pairing, we have by Plancherel h W u, v i = h u, v i + lim ǫ → + Z + ∞ h V ω e − iH t − ǫt u, e − iH ∗ ω t − ǫt v i dt = h u, v i + lim ǫ → + π Z + ∞−∞ h AR H ( λ + iǫ ) u, BR H ∗ ω ( λ + iǫ ) v i dλ. By the orthogonality in L ( R ) of boundary values of Hardy functions in H ( C + )and in H ( C − ) we have for ǫ > Z + ∞−∞ h AR H ( λ + iǫ ) u, BR H ∗ ω ( λ + iǫ ) v i dλ = Z + ∞−∞ h A [ R H ( λ + iǫ ) − R H ( λ − iǫ )] u, BR H ∗ ω ( λ + iǫ ) v i dλ. By u ∈ L ,s ( R ) and v ∈ L ,s ( R ) ∩ L c ( H ∗ ω ) the limit in the right hand side for ǫ ց h W u, v i = h u, v i +12 π Z + ∞−∞ h A [ R H ( λ + i − R H ( λ − i u, BR H ∗ ω ( λ + i v i dλ = h u, v i + 12 π Z | λ |≥ ω h A [ R H ( λ + i − R H ( λ − i u, BR H ∗ ω ( λ + i v i dλ. This yields Lemma 5.10. The crucial part of our linear theory is the proof of thefollowing analogue of [Y]:
Lemma 5.11.
For any p ∈ (1 , ∞ ) the restrictions of W and Z to L ∩ L p extendinto operators such that for C ( ω ) < ∞ semicontinuous in ω k W k L p ( R ) ,L pc ( H ω ) + k Z k L pc ( H ω ) ,L p ( R ) < C ( ω ) . In the next two sections we will consider W only, since the proof for Z is similar.The argument in the following two sections is a transposition of [Y]. We considerdiagonal matrices E + = diag(1 ,
0) and E − = diag(0 , . σ R ( z ) = − R ( − z ) σ for R ( z ) equal to R H ω ( z ) orto R H ( z ) and σ L c ( H ω ) = L c ( H ω ), it is easy to conclude that the L p boundnessof W is equivalent to L p boundness of U u := Z λ ≥ ω R − H ω ( λ ) V ω (cid:2) R + H ( λ ) − R − H ( λ ) (cid:3) udλ = Z λ ≥ ω R − H ω ( λ ) V ω (cid:2) R +0 ( λ ) − R − ( λ ) (cid:3) E + udλ. As in [Y] we deal separately with high,treated in §
6, and low energies, treated in §
7. We introduce cut-off functions ψ ( x ) ∈ C ∞ ( R ) , and ψ ( x ) ∈ C ∞ ( R ) , with ψ ( x ) + ψ ( x ) = 1 , ψ ( − x ) = ψ ( x ) , ψ ( x ) = 1 for | x | ≤ C and ψ ( x ) = 0 or | x | > C for some C > ω . § L p boundness of U : high energies This part is almost the same of the corresponding part in [Y2]. For ψ ( x ) thecutoff function introduced after Lemma 5.11, ψ ( H ) is a convolution operator withsymbol ψ ( | ξ | + ω ). Both ψ ( H ) and ψ ( H ) are bounded operators in L p ( R )for any p ∈ [1 , ∞ ] . In order to estimate the high frequency part (the so called highenergy)
U ψ ( H ) , we expand R − H ω ( λ ) into the sum of few terms of Born series R − H ω ( λ ) = R − H ( λ ) − R − H ( λ ) V ω R − H ( λ ) + R − H ( λ ) V ω R − H ( λ ) V ω R − H ω ( λ ) , getting by Lemma 5.10 the decomposition U = U + U + U with U u = − πi Z λ ≥ ω R − H ( λ ) V ω R +0 ( λ − ω ) E + udλ,U u = 12 πi Z λ ≥ ω R − H ( λ ) V ω R − H ( λ ) V ω R +0 ( λ − ω ) E + udλ,U u = − πi Z λ ≥ ω R − H ( λ ) V ω R − H ( λ ) V ω R − H ω ( λ ) V ω R +0 ( λ − ω ) E + udλ. Lemma 6.1.
The operator U ψ ( H ) is bounded in L p ( R ) for all < p < ∞ . Specifically for any s > there exists a constant C s > so that for T = U ψ ( H )(1) k T u k L p ≤ C s kh x i s V ω k L k u k L p for all u ∈ L p ( R ) . Proof.
Recall R ( z ) = ( − ∆ − z ) − and R ± H ( z ) = diag( R ± ( z − ω ) , − R ± ( z + ω )) . For u = ( u , u ), and for F the Fourier transform, we are reduced to operators ofschematic form F ( E ± U u )( ξ ) == Z λ ≥ ω dλ Z R | ξ | + ω ∓ λ + i b u ( ξ − η ) δ ( λ − ( | ξ − η | + ω )) b V ( η ) dη, b V the Fourier transform of the generic component of V ω . Then E ± U u = Z R dη b V ( η ) T ± η u η where u η ( x ) = e ix · η u ( x ), T − η u η = π K ( q η + ω | · | ) ∗ u η and by [Y1] T + η u η ( x ) = i | η | Z ∞ e it | η | u η ( x + tη/ | η | ) dt. By [Y2] we have that T = E + U satisfies inequality (1) while for T = E − U we use k T ± η u k L p ≤ π (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) K ( r η ω | x | ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L x k u k L p ≤ C h η i − k u k L p and so k E − U u k L p . k b V ( η ) / h η ik L k u k L p . Lemma 6.2.
The operator U ψ ( H ) is bounded in L p ( R ) for all < p < ∞ , moreover, there exists a constant C s > so that for T = U ψ ( H )(1) k T u k L p ≤ C s kh x i s V ω k L k u k L p for all u ∈ L p ( R ) . is valid, provided s > . Proof.
By [Y1] and with the notation of Lemma 6.1 we are reduced to a combi-nation of operators I ± , ± u = Z R dη T ± η Z R dη b V ( η ) b V ( η − η ) T ± η u η .T f = I − , − u satisfies inequality (1) by Proposition 2.2 [Y2] . The other cases followfrom Lemma 6.1. For example, for K ( η , η ) = b V ( η ) b V ( η − η ) and e K ( x, η ) = R dηe iη · x K ( η, η ), k I ± , ± u k L p = k Z R dη Z R dη K ( η , η ) T − η T + η u η k L p ≤ b C s Z R dη kh x i s e K ( x, η ) k L x k T + η u η k L p ≤ e C s Z R dη kh x i s e K ( x, η ) k L x h η i − k u k L p C s kh x i s V ω k L k u k L p . emma 6.3. Set T = U ψ ( H ) . Then T is bounded in L p ( R ) for all ≤ p ≤ ∞ . Proof.
Schematically E + U ψ ( H ) u = Z k ≥ R − ( k ) V F ( k + ω ) V (cid:2) R +0 ( k ) − R − ( k ) (cid:3) ψ ( λ + ω ) u kdk, with F ( k + ω ) = R − H ( k ) V R − ( k ) and V the generic component of V ω . By (3)Lemma 5.4 for G ± k,y ( x ) = e ∓ ik | y | G ± ( x − y, k ) with G ± ( x, k ) = ± i H ± ( k | x | ) we havethe following analogue of inequality (3.5) [Y2](1) (cid:12)(cid:12)(cid:12) ∂ jk h F ( k + ω ) V G ± k,y , V G + k,x i (cid:12)(cid:12)(cid:12) ≤ C j kh x i s V ω k ∞ k p h x ih y i and by Proposition 3.1 [Y2] this yields the desired result for T = E + U ψ ( H ) . Since (1) continues to hold if we replace G + k,x with e − ik | x | G k,x with G k,x ( y ) = G ( x − y, k ) , where G ( x, k ) = K ( √ k + ω | x | ), we get also the desired result for T = E − U ψ ( H ) . § L p boundness of U : Low energies Set
T u := Z λ ≥ ω R − H ω ( λ ) V ω (cid:2) R +0 ( λ − ω ) − R − ( λ − ω ) (cid:3) ψ ( λ ) E + udλ. We want to prove:
Lemma 7.1.
For any p ∈ (1 , ∞ ) the restriction of T on L ∩ L p extends into anoperator such that k T k L p ( R ) ,L p ( R ) < C ( ω ) for C ( ω ) < ∞ semicontinuous in ω . Let V ω = V = { V ℓj : ℓ, j = 1 , } , W = { W ℓj : ℓ, j = 1 , } with W = W = 0, W = 1 ∈ R and W ( x ) = 1 for V ( x ) ≥ W ( x ) = − V ( x ) < B ∗ = h x i − N for some large N >
0, and A = { A ℓj : ℓ, j = 1 , } with A ( x ) = | V ( x ) | , A ( x ) = W ( x ) V ( x ) and A j ( x ) = V j ( x ) . Then W = 1, B ∗ W A = V .Let k > k = λ − ω and set M ( k ) = W + AR − H ( λ ) B ∗ . Then R − H ω ( λ ) = R − H ( λ ) − R − H ( λ ) B ∗ M − ( k ) AR − H ( λ ) . We have M ( k ) = W + c − ( k ) P + A f G B ∗ + O ( k log k ) where: c − ( k ) = a − + b − log k ; P is a projection in L defined by P = (cid:20) A A (cid:21) h· , B ∗ ik V k L ;23 G = diag (cid:18) − π log | x |∗ , − R ( − ω ) (cid:19) ; k d j /dk j O ( k log k ) k L ,L ≤ Ck − j h log k i j = 0 , , , < k < c. Let Q = 1 − P and let M = W + A f G B ∗ . Then QM Q is invertible in QL if andonly if ω is not a resonance or an eigenvalue for H ω and in that case M − ( k ) = g − ( k )( P − P M QD Q − QD QM P M QD Q + QD Q + O ( k log k ))with g ( k ) = c − log k + d − for c − = 0 and D = ( QM Q ) − by [JN]. We claimnow that QD Q − QW Q is a Hilbert-Schmidt operator. In fact, following the theargument in Lemma 3 [JY], we get that the operator L = P + QM Q is invertiblein QL , and D = QL − Q. We have L = W + [ A f G B ∗ + P + P M P − P M Q − QM P ] . Set L := W (1 + e S ) , the operators P, P M P, P M Q, QM P are of rank one while A f G B ∗ is a Hilbert-Schmidt operator. From the fact that W is invertible, we getthat also (1 + e S ) is invertible. Moreover the identity (1 + e S ) − = 1 − e S (1 + e S ) − yields L − − W = − e S (1 + e S ) − W , that is the product of an Hilbert-Schmidt operator with one in B ( L ( R ) , L ( R )) . Finally, an application of the Theorem VI.22, Chapter VI, in [RS], shows that L − − W is of Hilbert-Schmidt Type.So we are reduced to the following list of operators: T +0 u := Z ∞ R − ( k ) E + V ω E + (cid:2) R +0 ( k ) − R − ( k ) (cid:3) ψ ( λ ) ukdk, and T − defined as above but with R − ( k ) E + replaced by R ( − k − ω ) E − whichare bounded in L p for 1 < p < ∞ by Lemma 6.1; T +1 u := Z ∞ R − ( k ) E + N ( k ) (cid:2) R +0 ( k ) − R − ( k ) (cid:3) ψ ( λ ) E + uk dk with k d j /dk j N ( k log k ) k L , − s ,L ,s ≤ Ck − j h log k i j = 0 , , , < k < c which is bounded in L p for 1 ≤ p ≤ ∞ by Proposition 4.1 [Y]; T +2 u := Z ∞ R − ( k ) E + B ∗ ( d ( k ) F + L + W ) A (cid:2) R +0 ( k ) − R − ( k ) (cid:3) ψ ( λ ) E + uk dk F a rank 3 operator, L a Hilbert Schmidt operator in L , and d ( k ) = g − ( k ).There are also operators T − j , for j = 0 , , , defined as above but with R − ( k ) E + replaced by R ( − k − ω ) E − and bounded in L p . So T ± = T ± , d ( √− ∆) + T ± , + T ± , with T ± ,j for j = 1 , , L p for 1 < p < ∞ because of thefollowing statement proved in [Y2] (the + case is exactly that in [Y2], and the − case can be proved following the same argument):if K is an operator with integral kernel K ( x, y ) such that for some s > k K k s := Z R dy (cid:18)Z R dx h x i s | K ( x, x − y ) | (cid:19) < ∞ then the operators Z + u := Z ∞ R − ( k ) K (cid:2) R +0 ( k ) − R − ( k ) (cid:3) uk dkZ − u := Z ∞ R ( − k + 2 ω ) K (cid:2) R +0 ( k ) − R − ( k ) (cid:3) uk dk are bounded in L p for 1 < p < ∞ with k Z ± k L p ,L p < C s,p k K k s . § We mimic Mizumachi [M2]. By the limiting absorption principle we have P c ( ω ) e − itH ω f = 12 πi Z ∞−∞ e − itλ ( λ ) P c ( ω )[ R + H ω ( λ ) − R − H ω ( λ )] f dλ. We consider a smooth function χ ( x ) satisfying 0 ≤ χ ( x ) ≤ x ∈ R , χ ( x ) = 1 if x ≥ χ ( x ) = 0 if x ≤ χ M ( x ) is an even function satisfying χ M ( x ) = χ ( x − M )for x ≥
0. Let e χ M ( x ) = 1 − χ M ( x ) . We have:
Lemma 8.1.
For any fixed s > there exists a positive C ( ω ) upper semicontinuousin ω, such that for any u ∈ S ( R ) we have k R ± H ω ( λ ) f k L λ ( σ c ( H ω ); L , − sx ) ≤ C k f k L . First, we prove Lemma 3.2 assuming Lemma 8.1.
Proof of Lemma 3.2.
We split P c ( ω ) e − itH ω f = P c ( ω ) e − itH ω χ M ( H ω ) f + P c ( ω ) e − itH ω e χ M ( H ω ) f with P c ( ω ) χ M ( H ω ) e − itH ω f = 12 πi Z ∞−∞ e − itλ χ M ( λ )( R + H ω ( λ ) − R − H ω ( λ )) P c ( ω ) f dλ,P c ( ω ) e − itH ω e χ M ( H ω ) f = 12 πi Z ∞−∞ e − itλ e χ M ( λ )( R + H ω ( λ ) − R − H ω ( λ )) P c ( ω ) f dλ. S ′ x ( R ) for any t = 0 and f ∈ S x ( R ) P c ( ω ) e − itH ω f = ( it ) − j πi Z ∞−∞ dλe − itλ ∂ jλ P c ( ω ) { ( R + H ω ( λ ) − R − H ω ( λ )) χ M ( λ ) } f. Since by (3) Lemma 5.4 for high energies we have k ∂ jλ P c ( ω ) R ± H ω ( λ ) : h x i ( j +1) / L → h x i − ( j +1) / − L ) k . h λ i − ( j +1) / , the above integral absolutely converges in h x i − ( j +1) / − L x for j ≥
2. Let g ( t, x ) ∈ S ( R × R ) . By Fubini and integration by parts, j ≥ h χ M ( H ω ) e − itH ω P c ( ω ) f, g i t,x = 12 πi Z R dt ( it ) − j Z R dλe − itλ ∂ jλ (cid:10) χ M ( λ )( R + H ω ( λ ) − R − H ω ( λ )) f, g (cid:11) x = 12 πi Z R dλ (cid:28) ∂ jλ (cid:8) χ M ( λ )( R + H ω ( λ ) − R − H ω ( λ )) (cid:9) P c ( ω ) f, Z R dt ( − it ) − j g ( t ) e itλ (cid:29) x = 1 √ πi Z R dλ D χ M ( λ )( R + H ω ( λ ) − R − H ω ( λ )) P c ( ω ) f, b g ( λ ) E x . Hence, by Fubini and Plancherel, we have (cid:12)(cid:12) h χ M ( H ω ) e − itH ω P c ( ω ) f, g i t,x (cid:12)(cid:12) ≤≤ (2 π ) − / k χ M ( λ )( R + H ω ( λ ) − R − H ω ( λ )) f k L λ ( σ c ( H ω ); L , − sx ) k b g ( λ, · ) k L λ L ,sx =(2 π ) − / k χ M ( λ )( R + H ω ( λ ) − R − H ω ( λ )) f k L λ ( σ c ( H ω ); L , − sx ) k g k L t L ,sx , In a similar way we have |h e − itH ω e χ M ( H ω ) f, g i t,x | ≤≤ (2 π ) − / ( k e χ M ( H ω )( R + H ω ( λ ) − R − H ω ( λ )) f k L λ ( σ c ( H ω ); L , − sx ) k g k L t L ,sx , therefore we achieve |h e − itH ω P c ( ω ) f, g i t,x | ≤≤ (2 π ) − / (cid:0) k χ M ( λ )( R H ω ( λ + i − R H ω ( λ − i f k L λ ( σ c ( H ω ); L , − sx ) + k e χ M ( λ )( R + H ω ( λ ) − R − H ω ( λ )) f k L λ ( σ c ( H ω ); L , − sx ) k g k L t L ,sx . and by Lemma 8.1 this estimate yields Lemma 3.2.26 roof of Lemma 3.3 By Plancherel’s identity and H¨older inequalities we have k Z t e − i ( t − s ) H ω P c ( ω ) g ( s, · ) ds k L , − sx L t ≤≤ k R + H ω ( λ ) P c ( ω ) b χ [0 , + ∞ ) ∗ λ b g ( λ, x ) k L , − sx L λ ≤≤ (cid:13)(cid:13)(cid:13) k R + H ω ( λ ) P c ( ω ) k L ,sx ,L , − sx k b χ [0 , + ∞ ) ∗ λ b g ( λ, x ) k L ,sx (cid:13)(cid:13)(cid:13) L λ . By Lemma 5.4 sup λ ≥ ω k R + H ω ( λ ) P c ( ω ) k B ( L ,s ,L , − s ) . h λ i − / , and sosup λ ∈ R k R + H ω ( λ ) P c ( ω ) k B ( L ,sx ,L , − sx ) k g k L ,sx L t ≤ C k g k L ,sx L t . The above inequalities yields Lemma 3.3.
Proof of Lemma 3.4
Let ( q, r ) be admissible and let T be an operator defined by T g ( t ) = Z R dse − i ( t − s ) H ω P c ( ω ) g ( s ) . Using Lemmas 3.2 and 3.3 we get f := R R dse isH ω P c ( ω ) g ( s ) ∈ L ( R ) and that thereexists a C > k T g ( t ) k L qt L rx ≤ C k g k L t L ,sx for every g ∈ S ( R × R ). Since q >
2, it follows from Lemma 3.1 in [SmS] (see also[Bq]) and (1) that (cid:13)(cid:13)(cid:13)(cid:13)Z s There exists a positive constant C such that for s > k R ± H ( λ ) f k L , − sx L λ ( ω, ∞ ) ≤ C k f k L . roof. E + R ± H ( λ ) f = R ± ( λ − ω ) E + f and by Lemma 4.2 [M2] we get(1) k R ± ( λ ) E + f k L , − sx L λ (0 , ∞ ) ≤ C sup x k R ± ( λ ) E + f k L λ (0 , ∞ ) ≤ C k E + f k L . We have E − R ± H ( λ ) f = − R ( − ω − λ ) E − f = − − ∆+ ω − λ − ∆+2 ω + λ R +0 ( λ − ω ) E − f . So by (1) k E − R ± H ( λ ) f k L , − sx L λ ( ω, ∞ ) ≤ (cid:13)(cid:13)(cid:13)(cid:13) − ∆ + ω − λ − ∆ + ω + λ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ λ (( ω, ∞ ) ,B ( L , − sx ,L , − sx )) × k R ± ( λ ) E − f k L , − sx L λ (0 , ∞ ) ≤ C k R ± ( λ ) E − f k L , − sx L λ (0 , ∞ ) ≤ C C k E − f k L . Proof of inequality (8.1). We consider the operator h q = − ∆ + q ( x ) introducedin § H q = σ ( h q + ω ). We claim that(1) k R ± H q ( λ ) f k L λ (( ω, ∞ ) ,L , − sx ) ≤ C k f k L . Indeed E + R ± H q ( λ ) f = R ± h q ( λ − ω ) E + f and k R ± h q ( λ ) E + f k L λ (0 , ∞ ) ,L , − sx ) ≤ C k f k L by Lemma 4.1 [M2]. On the other hand E − R ± H q ( λ ) f == − R h q ( − λ − ω ) E − f = − R ( − λ − ω ) E − f + R ( − λ − ω ) qR h q ( − λ − ω ) E − f. The bound for the first term comes from Lemma 8.2 and k R ( − λ − ω ) qR h q ( − λ − ω ) E − f k L , − sx L λ . k R ( − λ − ω ) qR h q ( − λ − ω ) E − f k L ∞ x L λ . k qR h q ( − λ − ω ) E − f k L ∞ λ L x ≤ C k E − f k L x . Armed with inequality (1) we consider the identity(8 . R ± H ω ( λ ) = (1 + R ± H q ( λ )( V ω − σ q )) − R ± H q ( λ ) == R ± H q ( λ ) − R ± H q ( λ )( V ω − σ q )(1 + R ± H q ( λ )( V ω − σ q )) − R ± H q ( λ ) . By (1) it is enough to bound the last term in the last sum. This is bounded by k R ± H q ( λ )( V ω − σ q )(1 + R ± H q ( λ )( V ω − σ q )) − R ± H q ( λ ) f k L λ L , − sx ≤k R ± H q ( λ )( V ω − σ q )(1 + R ± H q ( λ )( V ω − σ q )) − k L ∞ λ B ( L , − sx ,L , − sx ) k R ± H q ( λ ) f k L λ L , − sx . k R ± H q ( λ ) k L ∞ λ ( B ( L ,sx ,L , − sx )) k (1 + R ± H q ( λ )( V ω − σ q )) − k L ∞ λ B ( L , − sx ,L , − sx ) k f k L x . k f k L x by (1) and by the fact that the above L ∞ λ ( ω, ∞ ) norms are bounded byLemmas 5.1 and 5.4. 28 The proof is standard and analogous to Lemma 5.8 [Cu2]. Recall: Lemma 4.7. We have for ϕ ( x ) and ϕ ( t, x ) Schwarz functions, for t ∈ [0 , ∞ ) andfor fixed s > sufficiently large k e − iH ω t R + H ω (Λ) P c ( ω ) ϕ k L t L , − sx < C (Λ , ω ) k ϕ ( x ) k L ,sx (cid:13)(cid:13)(cid:13)(cid:13)Z t e − iH ω ( t − τ ) R + H ω (Λ) P c ( ω ) ϕ ( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13) L t L , − sx < C (Λ , ω ) k ϕ ( t, x ) k L t L ,sx with C (Λ , ω ) upper semicontinuous in ω and in Λ > ω .Proof. We consider ω < a/ < a << Λ < b < ∞ and the partition of unity1 = g + e g with g ∈ C ∞ ( R ) with g = 1 in [ a, b ] and g = 0 in [ a/ , b ]. By Lemma3.2 we get k e − iH ω t R + H ω (Λ) P c ( ω ) e g ( H ω ) ϕ k L t L , − sx ≤ C ( ω ) k R + H ω (Λ) P c ( ω ) e g ( H ω ) ϕ k L x ≤ C ( ω ) c ( a, b, ω ) k ϕ k L x . Similarly by the proof of Lemma 3.3, for any s > k Z t e − i ( t − s ) H ω R + H ω (Λ) P c ( ω ) e g ( H ω ) ϕ ( s, · ) ds k L , − sx L t ≤≤ k R + H ω ( λ ) R + H ω (Λ) e g ( H ω ) P c ( ω ) b χ [0 , + ∞ ) ∗ λ b ϕ ( λ, x ) k L , − sx L λ ≤≤ (cid:13)(cid:13)(cid:13) k R + H ω ( λ ) R + H ω (Λ) e g ( H ω ) P c ( ω ) k L ,sx ,L , − sx k b χ [0 , + ∞ ) ∗ λ b ϕ ( λ, x ) k L ,sx (cid:13)(cid:13)(cid:13) L λ ≤ C ( s, a, b, ω ) k ϕ k L ,sx L t by ( λ − Λ) R + H ω ( λ ) R + H ω (Λ) = R + H ω ( λ ) − R + H ω (Λ), Lemma 5.4 and | λ − Λ | ≥ a ∧ b .We consider now(9 . ǫ ) h x i − γ g ( H ω ) e − iH ω t R H ω (Λ + iǫ ) P c ( H ω ) h y i − γ = e − i Λ t h x i − γ Z + ∞ t e − i ( H ω − Λ − iǫ ) s g ( H ω ) P c ( H ω ) ds h y i − γ . We claim the following: Lemma 9.1. There are functions u ( x, ξ ) defined for x ∈ R and for | ξ | ∈ [ a/ , b ] with values in C such that for any χ ∈ C ∞ ( a/ , b ) we have (for t uσ f the productrow column and t u the transpose of a column vector) . χ ( H ω ) f ( x ) = (2 π ) − Z R u ( x, ξ ) t u ( y, ξ ) σ f ( y ) χ ( | ξ | + ω ) dξdy. There are constants c αβ such that (9 . | ∂ αx ∂ βξ u ( x, ξ ) | ≤ c αβ h x i | β | for all x ∈ R and | ξ | ∈ [ a/ , b ] . Let us assume Lemma 9.1. Then we can write the kernel of operator (9.1) as(9 . h x i − γ g ( H ω ) e − iH ω t R H ω (Λ + iǫ ) h y i − γ = (constant ) ×h x i − γ Z R u ( x, ξ ) e − i ( σ ( ξ + ω ) − Λ − iǫ ) s g ( ξ + ω ) t u ( y, ξ ) dξ h y i − γ . Estimates (9.3) and elementary integration by parts yields | (9 . | ≤ c h x i − γ + r h y i − γ + r s − r e − ǫt and so | (9 . + | ≤ c h x i − γ + r h y i − γ + r h t i − r +1 . For γ > r + 1 and r ≥ 3, we obtain k e − iH ω t R + H ω (Λ) g ( H ω ) P c ( H ω ) ϕ k L t ((0 , ∞ ) ,L , − γ ) ≤ C k ϕ ( x ) k L ,γ . Similarly k Z t e − i ( t − s ) H ω R + H ω (Λ) P c ( ω ) g ( H ω ) ϕ ( s, · ) ds k L t L , − γx ≤≤ (cid:13)(cid:13)(cid:13)(cid:13)Z t h t − s i − k ϕ ( s, · ) ds k L ,γx (cid:13)(cid:13)(cid:13)(cid:13) L t ≤ C k ϕ k L t L ,γx We need now to prove Lemma 9.1. § 10 Proof of Lemma 9.1 First of all we explain how to define the u ( x, ξ ). We set V ω = B ∗ A with A ( x )and B ∗ ( x ) rapidly decreasing and continuous. Then we have Lemma 10.1. For any λ > ω and any ξ ∈ R with λ = ω + | ξ | , in L ( R ) thesystem (1) (cid:0) AR + H ( λ ) B ∗ (cid:1) e u = Ae − iξ · x −→ e admits exactly one solution e u ( x, ξ ) ∈ H such that for any [ a, b ] ⊂ (1 , ∞ ) \ σ p ( H ) there is a fixed C < ∞ such that for any λ ∈ [ a, b ] and any ξ as above we have (2) k e u ( · , ξ ) k H ≤ C. roof. AR + H ( λ ) B ∗ is compact and ker (cid:0) AR + H ( λ ) B ∗ (cid:1) = { } for λ > ω by[CPV], since in that case λ σ p ( H ω ). By Fredholm alternative we get existenceand uniqueness of e u ( x, ξ ) . Regularity theory and continuity of the coefficients ofsystem (1) with respect to ξ yield (2)Let now t e = (1 , 0) and G ( | x | , k ) = diag( i H +0 ( k | x | ) , − π K ( √ k + 2 ω | x | )) for k > 0. We have G ( r, k ) = i √ √ iπkr e ikr e + O ( r − ) and ∂ r G ( r, k ) = − k √ k √ iπr e ikr e + O ( r − ). We set u ( x, ξ ) = e − iξ · x e + v ( x, ξ ) = e − iξ · x e − R + H ( λ ) B ∗ e u ( · , ξ ) . Then ( H ω − λ ) u ( x, ξ ) = B ∗ (cid:0) Ae − iξ · x e − e u − AR + H ( λ ) B ∗ e u (cid:1) = 0 . Notice B ∗ e u = V ω u so v ( x, ξ ) = e − ix · ξ w ( x, ξ ) where w ( x, ξ ) is the unique solution in L − s , s > 1, of theintegral equation(1) w ( x, ξ ) = − F ( x, ξ ) − Z R G ( | x − z | , | ξ | ) e i ( x − z ) · ξ V ω ( z ) w ( z, ξ ) dz, with F ( x, ξ ) = Z R G ( | x − z | , | ξ | ) V ω ( z ) e i ( x − z ) · ξ e dz. It is elementary to show that, for | ξ | ∈ [ a, b ], then | ∂ αx ∂ βξ F ( x, ξ ) | ≤ ˜ c αβ h x i | β |− / . By standard arguments and Lemmas 5.3 and 5.4 we have | ∂ αx ∂ βξ w ( x, ξ ) | ≤ ˜ c αβ h x i | β | . This yields (9.3). To get (9.2) we follow the presentation in Chapter 9 [Ta]. Wedenote by R ± H ω ( x, y, k ) the kernel of R ± H ω ( k + ω ). We set R + H ω ( x, y, k ) = G ( | x − y | , k ) + h ( x, y, k )with h ( · , y, k ) = − R + H ( k + ω ) V ω G ( | · − y | , k ) . Let ( r, Σ) be polar coordinates onthe sphere S , then we claim: Lemma 10.2. Let k > . For r → ∞ we have uniform convergence on compactsets of, with u · (1 , the raw column product between column u and raw (1 , , R + H ω ( x, r Σ , k ) = i √ √ iπkr e ikr u ( x, k Σ) · (1 , 0) + O ( r − )(1) ∂∂r R + H ω ( x, r Σ , k ) = − √ √ iπkr ke ikr u ( x, k Σ) · (1 , 0) + O ( r − ) , (2) R + H ω ( r Σ , y, k ) = i √ √ iπkr e ikr (cid:20) (cid:21) t u ( y, k Σ) σ + O ( r − ) , (3) ∂∂r R + H ω ( r Σ , y, k ) = − √ √ iπkr ke ikr (cid:20) (cid:21) t u ( y, k Σ) σ + O ( r − ) . (4) 31 or R − H ω ( x, y, k ) the asymptotic expansion follows from R − H ω ( x, y, k ) = R + H ω ( x, y, k ) . We write R + H ω ( x, r Σ , k ) = G ( | x − r Σ | , k ) + h ( x, r Σ , k ) with h ( x, r Σ , k ) = − R + H ( k + ω ) V ω G ( | · − r Σ | , k )= − R + H ( k + ω ) " V ω ( x ) i √ √ iπkr e ikr e − ik Σ · x diag(1 , 0) + O ( r − ) ! . We have k V ω ( x ) G ( | x − r Σ | , k ) − V ω ( x ) i √ √ iπkr e ikr e − ik Σ · x diag(1 , k L ,sx = O ( r − / ) . From v ( x, ξ ) = − R + H ( k + ω ) V ω ( x ) e − ik Σ · x e , with t e = (1 , 0) we get v ( x, ξ ) t e = − R + H ( k + ω ) V ω ( x ) e − ik Σ · x diag(1 , . Then we conclude for any s > k h ( x, r Σ , k ) − i √ √ iπkr v ( x, k Σ) t e k L , − s = O ( r − / )and k R + H ω ( x, r Σ , k ) − i √ √ iπkr u ( x, k Σ) t e k L , − s = O ( r − / ) . Then point wise h ( x, r Σ , k + i − i √ √ iπkr v ( x, k Σ) t e = O ( r − / ) and R + H ω ( x, r Σ , k ) − i √ √ iπkr u ( x, k Σ) t e = O ( r − / ) . This yields (1) in Lemma 10.2. (2) can be obtained with a similar argument. (3)and (4) follow from (1) and (2) by σ R ± H ω ( x, y, k ) σ = R ± H ∗ ω ( x, y, k ) = t R ∓ H ω ( y, x, k ) . By Lemma 3.5 for v ∈ L ( H ω ) ∩ C ∞ and for ϕ ∈ C ∞ ( R ) supported in ( ω, ∞ ) wehave ϕ ( H ω ) v ( x ) = 2 π Z ∞ k dk Z R ϕ ( k + ω ) ℑ R + H ω ( x, y, k ) v ( y ) dy. We prove (here u t u is a raw column product between column u and raw t u )(3) ℑ R + H ω ( x, y, k ) = 18 π Z S u ( x, k Σ) t u ( y, k Σ) σ d Σ , d Σ is the standard measure on S . By the Green theorem for S R = { z ∈ R : | z | = R } , | x | < R , | y | < R and r = | z | By Green theorem for S R = { z ∈ R : | z | = R } , | x | < R and | y | < R , ℑ R + H ω ( x, y, k ) = 12 i Z S R I ( x, y, z, k ) dℓ ( z ) I ( x, y, z, k ) := R + H ω ( x, z, k ) σ ∂ | z | R − H ω ( z, y, k ) − ( ∂ | z | R + H ω ( x, z, k )) σ R − H ω ( z, y, k )By Lemma 10.2 (cid:12)(cid:12)(cid:12)(cid:12) ℑ R + H ω ( x, y, k ) − π Z S u ( x, k Σ) t u ( y, k Σ) σ d Σ (cid:12)(cid:12)(cid:12)(cid:12) == (cid:12)(cid:12)(cid:12)(cid:12) R i Z S I ( x, y, r Σ , k ) | r = R d Σ − π Z S u ( x, k Σ) t u ( y, k Σ) σ d Σ (cid:12)(cid:12)(cid:12)(cid:12) ≤ O ( R − ) . Therefore, taking R → + ∞ , we arrive at (3). Moreover, we obtain ϕ ( H ω ) v ( x ) = 2 π Z ∞ k dk Z R ϕ ( k + ω ) ℑ G ( x, y, k ) v ( y ) dy == 14 π Z ∞ k dk Z R Z S u ( x, k Σ) t u ( y, k Σ) σ v ( y ) ϕ ( k + ω ) d Σ dy == (2 π ) − Z R u ( x, ξ ) t u ( y, ξ ) σ v ( y ) ϕ ( | ξ | + ω ) dξdy, that is the integral representation (9.2). 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