Equipartition of Mass in Nonlinear Schrödinger / Gross-Pitaevskii Equations
aa r X i v : . [ m a t h . A P ] N ov Equipartition of Mass in Nonlinear Schr¨odinger /Gross-Pitaevskii Equations
Zhou Gang ∗ , Michael I. Weinstein † November 2, 2018 ∗ Institute for Theoretical Physics, ETH Z¨urich, Switzerland † Department of Applied Physics and Applied Mathematics, Columbia University, New York, U.S.A.
Abstract
We study the infinite time dynamics of a class of nonlinear Schr¨odinger / Gross-Pitaevskii equa-tions. In a previous paper, [6], we prove the asymptotic stability of the nonlinear ground statein a general situation which admits degenerate neutral modes of arbitrary finite multiplicity,a typical situation in systems with symmetry. Neutral modes correspond to purely imaginary(neutrally stable) point spectrum of the linearization of the Hamiltonian PDE about a criticalpoint. In particular, a small perturbation of the nonlinear ground state, which typically excitessuch neutral modes and radiation, will evolve toward an asymptotic nonlinear ground statesoliton plus decaying neutral modes plus decaying radiation. In the present article, we give amuch more detailed, in fact quantitative, picture of the asymptotic evolution. Specificially weprove an equipartition law :The asymptotic soliton which emerges, φ λ ∞ , has a mass which is equal to the initial solitonmass plus one half the mass, | z | , contained in initially perturbing neutral modes: k φ λ ∞ k L = k φ λ k L + 12 | z | + o ( | z | ) In this paper we study the nonlinear Schr¨odinger / Gross-Pitaevskii (NLS/GP) equations in R i∂ t ψ = − ∆ ψ + V ψ − | ψ | σ ψ, (1.1)where σ ≥ V : R → R is a real, smooth function decaying rapidly at spatial infinity. Westudy the large time distribution of mass / energy of solutions with initial data ψ ( x,
0) = ψ , (1.2)which are sufficiently small in the H ( R ) norm Since our results are in the low energy / small amplitude regime, our analysis goes through without changefor the nonlinearities of the form + g | ψ | σ ψ for any fixed real g . Here, we have taken g = 1. N → ∞ , of the linear quantum description of N − weakly interacting bosons. Here, ψ is a collective wave-function and V , a trapping potential, and the nonlinear potential arisesdue to the collective effect of many quantum particles on a representative particle [8, 4]. Inclassical electromagetics and nonlinear optics, NLS/GP arises via the paraxial approximation to Maxwell’s equations, and governs the slowly varying envelope, ψ , of a nearly monochromaticbeam of light, propagating through a waveguide [1, 7]. The waveguide has linear refractiveindex profile, determining the potential, V , and cubic ( σ = 1) nonlinear refractive index, dueto the optical Kerr effect.NLS/GP is a infinite-dimensional Hamiltonian system and a unitary evolution in L ( R ).In the N − body quantum setting the time-invariant L norm corresponds to the conservationof mass. In the electromagnetic setting, it is the conservation of energy (optical power). Inthis paper, we prove an equipartition law (Theorem 3) for the L mass / energy small (weaklynonlinear) solutions. Hence, we may refer to this result equipartition of energy or equipartitionof mass.The mathematical set-up is as follows. We choose a spatially decaying potential V forwhich the Schr¨odinger operator, − ∆ + V , has only two negative eigenvalues e < e < .e is chosen to be closer to the continuous spectrum than to e , (permitting coupling vianonlinearity of discrete and continuum modes at quadratic order in the nonlinear couplingcoefficient, g ): 2 e − e > . The excited state eigenvalue e may be degenerate with multiplicity N . (In Section 5, we allowfor nearly degenerate excited state eigenvalues.) Denote the corresponding eigenvectors by φ lin , ξ lin , · · · , ξ linN . (1.3)For NLS/GP, (1.1), there is a family of nonlinear ground states which bifurcates from thezero solution in the direction of φ lin . The excited state eigenvectors are manifested as neutralmodes (time periodic states with non-zero frequency) of the linearized NLS/GP equation aboutthe ground state family; see Section 2.More specifically, there exists an open interval I , with e as an endpoint, such that for any λ ∈ I , NLS/GP (1.1) has solutions of the form ψ ( x, t ) = e iλt φ λ ( x ) , (1.4)where φ λ is asymptotically collinear to φ lin for small H norm and λ → − e , λ ∈ I .The excited state eigenvalues give rise, in the linear approximation, to neutral modes,( ξ, η ) T , and therefore linearized time-dependent solutions, which are undamped (neutral) os-cillations about φ λ : e iλt (cid:16) φ λ + ( ℜ z ) · ξ + i ( ℑ z ) · η (cid:17) (1.5)where z ∈ C N .In [6], also referred to in this paper as GW1 , we proved the asymptotic stability of theground states. Namely, if the initial condition is of the form ψ = e iγ [ φ λ + R ] (1.6)2or some γ ∈ R and R : R → C satisfying kh x i R k H ≪ k φ λ k , then generically thereexists a λ ∞ ∈ I such that min γ ∈ R k ψ ( t ) − e iγ φ λ ∞ k ∞ → t → ∞ ; (1.7)In particular, the neutral oscillatory modes eventually damp to zero as t → ∞ via the couplingand transfer of their energy to the nonlinear ground state and to continuum radiation modes.When the neutral mode is simple, i.e. N = 1 in (1.3), similar results have been obtained in[11, 12, 2, 9, 5, 3].In the present paper, we seek a more detailed, quantitative description of the large timedynamics. We consider a special class of initial conditions to which the results of GW1 , inparticular, (1.7) apply: ψ = e iγ φ λ + neutral modes + R with k φ λ k ≫ k neutral modes k ≫ kh x i R k H . The main result of this paper, proved by a considerable refinement of the analysis in [6], isthat the emerging asymptotic ground state has, up to high order corrections, a mass equal toits initial mass plus one-half of the mass of the initial excited state mass: k φ λ ∞ k = k φ λ k + 12 k neutral modes k ( 1 + o (1) ) . Thus, half of the excited state mass goes into forming a limiting, more massive, ground state, φ λ ∞ and the other half of the excited state mass is radiated away to infinity. We call this the mass- or energy- equipartition . That this phenomenon is expected, was discussed in [11, 9, 10].The main achievement of the present work is a rigorous quantification of the asymptotic ( t →∞ ) mass / energy distribution.The paper is organized as follows: In Section 2 we review results on the existence and prop-erties of the ground state manifold, and on the spectral properties of the linearized NLS/GPoperator about the ground state. In Section 3 we state and discuss Theorem 3.1 on equipar-tition. In Section 4 we present the proofs, using technical estimates established in the appen-dices, e.g. Sections G- I. In Section 5, we present a generalization of the Theorem 3.1 to thecase of nearly degenerate case, and an outline of its proof. A more extensive list of referencesand a discussion of related work on NLS/GP appears in
GW1 . Acknowledgments
ZG was supported, in part, by a Natural Sciences and Engineering Research Council of Canada(NSERC) Postdoctoral Fellowship and NSF Grant DMS-04-12305 . MIW was supported, inpart, by U.S. NSF Grants DMS-04-12305, DMS-07-07850 and DMS-10-08855. This work wasinitiated while ZG was a visitor at the Department of Applied Physics and Applied Mathe-matics at Columbia University, and was continued while he was a visiting postdoctoral fellowat the Department of Mathematics of Princeton University.3 .1 Notation (1) α + = max { α, } , [ τ ] = max η ∈ Z { η ≤ τ } (2) ℜ z = real part of z , ℑ z = imaginary part of z (3) Multi-indices z = ( z , . . . , z N ) ∈ C N , ¯ z = (¯ z , . . . , ¯ z N ) (1.8) a ∈ N N , z a = z a · · · z a N N | a | = | a | + . . . + | a N | (4) Q m,n denotes an expression of the form Q m,n = X | a | = m, | b | = n q a,b z a ¯ z b = X | a | = m, | b | = n q a,b N Y k =1 z a k k ¯ z kb k (5) J = (cid:18) − (cid:19) , H = (cid:18) L + L − (cid:19) , L = J H = (cid:18) L − − L + (cid:19) (6) σ ess ( L ) = σ c ( L ) is the essential (continuous) spectrum of L , σ disc ( L ) is the discrete spectrum of L .(7) Riesz projections: P disc ( L ) and P c ( L ) = I − P disc ( L ) P disc ( L ) projects onto the discrete spectral part of LP c ( L ) projects onto the continuous spectral part of L (8) h f, g i = R f ( x ) g ( x ) dx (9) k f k pp = R R | f ( x ) | p dx, ≤ p ≤ ∞ (10) k f k H s ( R ) = R (cid:12)(cid:12)(cid:12) ( I − ∆ x ) s f ( x ) (cid:12)(cid:12)(cid:12) dx (11) k f k H s,ν = R R (cid:12)(cid:12)(cid:12) h x i ν ( I − ∆) s f ( x ) (cid:12)(cid:12)(cid:12) dx In this section we review the setting presented in detail in [6]. V ( x ) We assume that the Schr¨odinger operator − ∆ + V has the following properties:(V1) V is real valued and decays sufficiently rapidly, e.g. exponentially, as | x | tends to infinity.(V2) − ∆ + V has two eigenvalues e < e < e is the lowest eigenvalue with ground state φ lin >
0, the eigenvalue e is degeneratewith multiplicity N and eigenvectors ξ lin , ξ lin , · · · , ξ linN . .2 Bifurcation of ground states from e Proposition 2.1.
Suppose that the linear operator − ∆ + V satisfies the conditions above insubsection 2.1. Then there exists a constant δ > and a nonempty interval I ⊂ [ e − δ , e ) such that for any λ ∈ I , NLS/GP ( 1.1) has solutions of the form ψ ( x, t ) = e iλt φ λ ∈ L with φ λ = δ (cid:0) φ lin + O ( δ σ ) (cid:1) and δ = δ ( λ ) = | e + λ | σ (cid:18)Z φ σ +2 lin (cid:19) − σ . (2.1) If we write ψ ( x, t ) = e iλt (cid:0) φ λ + u + iv (cid:1) , then we find the linearized perturbation equationto be: ∂∂t (cid:18) uv (cid:19) = L ( λ ) (cid:18) uv (cid:19) = J H ( λ ) (cid:18) uv (cid:19) , (2.2)where L ( λ ) := (cid:18) L − ( λ ) − L + ( λ ) 0 (cid:19) = (cid:18) − (cid:19) (cid:18) L + ( λ ) 00 L − ( λ ) (cid:19) ≡ J H ( λ ) . (2.3)Here, L + and L − are given by: L − ( λ ) := − ∆ + λ + V − ( φ λ ) σ L + ( λ ) := − ∆ + λ + V − (2 σ + 1)( φ λ ) σ (2.4)The following results on the point spectrum of L ( λ ) appear in [6]; see Proposition 4.1, p. 275and Propositions 5.1-5.2, p. 277: Lemma 2.1.
Let L ( λ ) , or more explicitly, L ( λ ( δ ) , δ ) denote the linearized operator about thethe bifurcating state φ λ , λ = λ ( δ ) . Note that λ (0) = − e . Corresponding to the degeneratee-value, e , of − ∆ + V , the matrix operator L ( λ = − e , δ = 0) has degenerate eigenvalues ± iE ( − e ) = ± i ( e − e ) , each of multiplicity N . For δ > and small these bifurcate to(possibly degenerate) eigenvalues ± iE ( λ ) , . . . , ± iE N ( λ ) with neutral modes (cid:18) ξ ± iη (cid:19) , (cid:18) ξ ± iη (cid:19) , · · · , (cid:18) ξ N ± iη N (cid:19) satisfying the estimates h ξ m , η n i = δ m,n , h ξ m , φ λ i = h η m , ∂ λ φ λ i = 0 (2.5) and = lim λ → e ξ n = lim λ → e η n ∈ span { ξ linn , n = 1 , , · · · , N } in H k spaces for any k > . (2.6) Remark 2.1.
Since E ( − e ) = e − e , it follows that if e − e > , then for sufficientlysmall δ , E n ( λ ) > λ, n = 1 , , ··· , N . This ensures nonlinear coupling of discrete to continuousspectrum at second order (in the nonlinearity coefficient, g ). Thus, to ensure such coupling,we assume:(V3) e − e > . (2.7)5 emma 2.2. Assume the potential V = V ( | x | ) and the functions ξ linn admit the form ξ linn = x n | x | ξ lin ( | x | ) for some function ξ lin , then φ λ , hence ∂ λ φ λ , is spherically symmetric, E n = E forany n = 1 , , · · · , N = d and we can choose ξ n , η n such that ξ n = x n | x | ξ ( | x | ) and η n = x n | x | η ( | x | ) for some real functions ξ and η. In this paper we make the following assumptions on the spectrum of the operator L ( λ ) : (SA) The linearized operator L ( λ ) has discrete spectrum given by:- an eigenvalue 0 with generalized eigenspace spanned by (cid:26) (cid:18) φ λ (cid:19) , (cid:18) ∂ λ φ λ (cid:19) (cid:27) - neutral eigenvalues ± iE ( λ ) , E ( λ ) > E ( λ ) > λ and with corresponding eigenvectors (cid:12)(cid:12)(cid:12)(cid:12) ξ n ± iη n (cid:29) , n =1 , . . . , N .For the non self-adjoint operator L ( λ ) the (Riesz) projection onto the discrete spectrum sub-space of L ( λ ), P d = P d ( L ( λ )) = P λd , is given explicitly in [6], Proposition 5.6, p. 280: P d ≡ ∂ λ k φ λ k (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) φ λ (cid:29) (cid:28) ∂ λ φ λ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∂ λ φ λ (cid:29) (cid:28) φ λ (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) + i N X n =1 (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) ξ n iη n (cid:29) (cid:28) − iη n ξ n (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) ξ n − iη n (cid:29) (cid:28) iη n ξ n (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) . (2.8)and the projection onto the essential spectrum by P c ≡ − P d . The large time analysis of NLS/GP requires good decay estimates on the linearized evolu-tion operator, e L ( λ ) t P λc . An obstruction to such estimates are, so-called, threshold resonances(see [6] and references therein), which we preclude with the following hypothesis.( Thresh λ ) Assume L ( λ ) has no resonances at ± iλ For small solitons, δ sufficiently small, ( Thresh λ ) follows from the absence of a zero energyresonance for − ∆ + V . In this subsection we review the definitions and constructions presented in detail in [6] pp.281-282. The amplitudes and phases of the neutral modes are governed by the complex-valuedvector parameter z : R + → C N , first arising in the linear approximation of solution ψ , see e.g. (1.5). Its precise definition is seen in the decomposition of the solution ψ in (3.1), under thecondition (3.6), below, from which it follows that ∂ t z = − iE ( λ ) z − Γ( z, ¯ z ) z + Λ( z, ¯ z ) z + O (cid:16) (1 + t ) − − δ (cid:17) , δ > ± iE ( λ ) are complex conjugate N − f old degenerate neutral eigenfrequencies of L ( λ ), Γis non-negative symmetric and Λ is skew symmetric.6n what follows we define the non-negative, Fermi Golden Rule matrix, Γ. Define vectorfunctions G k , k = 1 , , · · · , N , as G k ( z, x ) := (cid:18) B ( k ) D ( k ) (cid:19) (2.10)with the functions B ( k ) and D ( k ) defined as B ( k ) := − iσ ( φ λ ) σ − [ ( z · ξ ) η k + ( z · η ) ξ k ] ,D ( k ) := − σ ( φ λ ) σ − [ 3( z · ξ ) ξ k − ( z · η ) η k ] − σ ( σ − φ λ ) σ − ( z · ξ ) ξ k , where z · ξ := N X n =1 z n ξ n , z · η := N X n =1 z n η n . In terms of the column 2-vector, G k , we define a N × N matrix Z ( z, ¯ z ) as Z ( z, ¯ z ) = ( Z ( k,l ) ( z, ¯ z )) , ≤ k, l ≤ N (2.11)and Z ( k,l ) = − (cid:10) ( L ( λ ) + 2 iE ( λ ) − − P c G l , iJ P c G k (cid:11) (2.12)Finally, we define Γ( z, ¯ z ) as follows:Γ( z, ¯ z ) := 12 [ Z ( z, ¯ z ) + Z ∗ ( z, ¯ z )] . (2.13)Thus, [ Γ( z, ¯ z ) ] kl = − ℜ (cid:10) ( L ( λ ) + 2 iE ( λ ) − − P c G l , iJ P c G k (cid:11) . (2.14)By (2.9) and (2.14) we find ∂ t | z ( t ) | = − z ∗ Γ( z, ¯ z ) z + . . . . (2.15)In GW1
Γ was shown to be non-negative and we require it to be positive definite. Inparticular, we shall require the following Fermi Golden Rule hypothesis.Let P linc denote the spectral projection onto the essential spectrum of − ∆ + V . Then (FGR) We assume there exists a constant
C > −ℜh i [ − ∆ + V + λ − E ( λ ) − i − P linc ( φ lin ) σ − ( z · ξ lin ) , ( φ lin ) σ − ( z · ξ lin ) i ≥ C | z | for any z ∈ C N . The assumption FGR implies that there exists a constant C > z ∈ C N z ∗ Γ( z, ¯ z ) z ≥ C k φ λ k σ − ∞ | z | . (2.16)Note that for each fixed z , smallness of | λ + e | together with (2.1) and (2.6) imply thatthe leading term in z ∗ Γ( z, ¯ z ) z is z ∗ Γ ( z, ¯ z ) z ≡ − σ ( σ + 1) δ σ − ( λ ) ×ℜh i [ − ∆ + V + λ − E ( λ ) − i − P linc ( φ lin ) σ − ( z · ξ lin ) , ( φ lin ) σ − ( z · ξ lin ) i . (2.17)7 Main Theorem
In this section we state our main results, Theorems 3.1 and 3.2.
Theorem 3.1.
Assume a cubic nonlinearity, σ = 1 , in (1.1) . If the spectral conditions (SA)(Thres λ ) and (FGR) are satisfied, then there exists a constant δ such that if the initialcondition ψ satisfies the condition ψ ( x ) = e iγ [ φ λ + α · ξ + iβ · η + R ] for some real constants γ , λ , real N vectors α and β , function R : R → C , such that for ǫ ≤ δ : | λ − | e || ≤ ǫ, | α | + | β | . ǫ k φ λ k , kh x i R k H . | α | + | β | = O ( ǫ ) , for ǫ ≤ δ , thenthere exist smooth functions λ ( t ) : R + → I , γ ( t ) : R + → R , z ( t ) : R + → C N ,R ( x, t ) : R × R + → C such that the solution of NLS evolves in the form: ψ ( x, t ) = e i R t λ ( s ) ds e iγ ( t ) × [ φ λ + a ( z, ¯ z ) ∂ λ φ λ + ia ( z, ¯ z ) φ λ + ( Re ˜ z ) · ξ + i ( Im ˜ z ) · η + R ] , (3.1) where lim t →∞ λ ( t ) = λ ∞ , for some λ ∞ ∈ I .Here, a ( z, ¯ z ) , a ( z, ¯ z ) : C N × C N → R and ˜ z − z : C N × C N → C N are some polynomials of z and ¯ z , beginning with terms of order | z | .(A) The dynamics of mass/energy transfer is captured by the following reduced dynamicalsystem for the key modulating parameters, λ ( t ) and z ( t ) : ddt k φ λ ( t ) k = z ∗ Γ ( z, ¯ z ) z + S λ ( t ) , (3.2) ddt | z ( t ) | = − z ∗ Γ ( z, ¯ z ) z + S z ( t ) , (3.3) where z ∗ Γ ( z, ¯ z ) z is given in (2.17) , and S λ ( t ) . (1 + t ) − , and S z ( t ) . (1 + t ) − . (3.4) Furthermore, Z ∞ S λ ( τ ) dτ, Z ∞ S z ( τ ) dτ = o ( | z | ) . (3.5) (B) ~R ( t ) = ( Re R ( t ) , Im R ( t )) T lies in the essential spectral part of L ( λ ( t )) . Equivalently, R ( · , t ) satisfies the symplectic orthogonality conditions: ω h R, iφ λ i = ω h R, ∂ λ φ λ i = 0 ,ω h R, iη n i = ω h R, ξ n i = 0 , n = 1 , , · · · , N, (3.6) where ω h X, Y i := Im R XY . C) Decay estimates:
For any time t ≥ k (1 + x ) − ν ~R ( t ) k ≤ C ( kh x i ψ k H )(1 + t ) − , (3.7) k ~R ( t ) k H ≤ ǫ ∞ , (3.8) | z ( t ) | ≤ C ( kh x i ψ k H )(1 + t ) − . (3.9) (D) Mass / Energy equipartition:
Half of the mass of the neutral modes contributes toforming a more massive asymptotic ground state and half is radiated away k φ λ ∞ k = k φ λ k + 12 (cid:2) | α | + | β | (cid:3) + o (cid:0) (cid:2) | α | + | β | (cid:3) (cid:1) . (3.10)The following result applies to the case where σ > Theorem 3.2.
Assume the general nonlinearity σ > . Then statements (A)-(D) of Theorem3.1 hold provided, in addition to the assumptions of Theorem 3.1, we assume:(1) in the case where the neutral modes are degenerate ( N > ), the potential V is sphericallysymmetric and the eigenvectors ξ linn , n = 1 , , · · · , N = d, admit the form ξ linn = x n | x | ξ ( | x | ) for some function ξ. (2) | α | + | β | ≤ [ k φ λ k ] C ( σ ) for some sufficiently large constant C ( σ ) . The statements (B) and (C) are obtained in [6]: all except (3.8) are taken from The-orem 7.1, p. 284. Equation (3.8) is from the proof (Line 18, p. 306) that R ( T ) :=max ≤ t ≤ T k ~R ( t ) k H ≪ R is defined in (11.2) .The bounds on S λ ( t ) and S z ( t ), (3.4), of statement (A) were proved in [6]; see equations(8-9) and (8-11) p. 286. For the estimate | Remainder | . (1 + t ) − , see line 9, p. 306. Theremaining assertions in (A) will be reformulated as Theorems 4.1 and 4.2, and proved inSection 4. Statement (D) is proved just below. Remark 3.1. Mass equipartition:
It is straightforward to interpret (3.10) as implyingequipartition of the neutral mode mass. Indeed, since φ λ is orthongal to ξ m (see (2.5) ) andsince mass is conserved for NLS/GP, i.e. k ψ ( t ) k = k ψ k , we have k ψ ( · , t ) k = k ψ k = k φ λ k + | α | + | β | + o ( | α | + | β | ) , for all t. The theorem implies that ψ ( t, · ) has a weak- L limit, φ λ ∞ , whose mass is given by (3.10) .Thus, half of the mass of the neutral modes is transferred to the ground states while the otherhalf is radiated to infinity. We now use Statement (A) of Theorem 3.1 to prove Statement (D).
Proof of Mass equipartition:
Twice the first plus the second equation in (3.4) yields: ddt (cid:18) (cid:13)(cid:13)(cid:13) φ λ ( t ) (cid:13)(cid:13)(cid:13) + | z ( t ) | (cid:19) = 2 S λ ( t ) + S z ( t ) . (3.11)9ntegration of (3.11) with respect to t from zero to infinity and use of the decay of z ( t ), (3.9),imply 2 (cid:13)(cid:13)(cid:13) φ λ ( ∞ ) (cid:13)(cid:13)(cid:13) = 2 (cid:13)(cid:13)(cid:13) φ λ (0) (cid:13)(cid:13)(cid:13) + | z (0) | + Z ∞ (cid:0) S λ ( t ′ ) + S z ( t ′ ) (cid:1) dt ′ . Dividing by two and estimating the integral, using (3.5), completes the proof of Statement D.
Remark 3.2. Generic data in a neighborhood of the origin:
For the case of cubic nonlinearity, σ = 1 , the condition | α | + | β | ≪ k φ λ k can be improvedto a state about generic (low energy) initial conditions satisfying | α | + | β | ≈ k φ λ k . We impose the stronger condition in the present paper to simplify the treatment and to applydirectly the results in
GW1 [6]. We refer to [9, 13]. See also our remarks 4.1 and F.1 below.
The generality of the nonlinearity in (1.1)
Our results hold not only for focusing nonlinearity, i.e., −| ψ | σ ψ in (1.1) . In fact all of theresults in the Theorems 3.1 and 3.2 can be transferred to the general cases g | ψ | σ ψ, g ∈ R \{ } without difficulty. We restrict to the present consideration in order not to clutter our argumentsby discussing various constants. Theorems 3.1 and 3.2 are derived from a refinement of the analysis of [6] and a generalizationto arbitrary nonlinearity parameter σ ≥
1. In this subsection we explain this.The overall plan for proofs of asymptotic stability can be broken into two parts, motivatedby a view of the soliton as an interaction between discrete and continuum modes:
Part 1 : a) We seek a natural decomposition of the solution into a component evolving alongthe manifold of solitons and a component which is dispersive. However, since the lin-earization about the soliton may have neutral modes, non-decaying time periodic states,we incorporate these degrees of freedom among the discrete degrees of freedom in theAnsatz. The dispersive components of the evolution lie in the subspace bi-orthogonal,in fact symplectic-orthogonal, to the discrete modes. The result is a strongly-coupled system governing the discrete degrees of freedom and dispersive dispersive wave field, R ( t ). Mathematically we decomposed the solution ψ as in (A.1), and by the orthogonalconditions (2.5) and (3.6) we derive equations for ˙ λ , ˙ γ , z and ~R . These are taken from[6] and displayed in Appendix A.b) We solve explicitly for the leading order components of R ( t ), which arise due toresonant forcing by new, nonlinearity-generated, discrete mode frequencies. To achievethis we find the leading order, that is second order in z and ¯ z contributions to R ( x, t ).This is presented in Appendix (B.7).c) This leading order behavior is substituted into the equations governing the discretemodes, leading to a (to leading order) closed equation for the discrete modes, implyingestimates for ˙ λ and ˙ γ . This is Proposition F.2.d) The latter is put into a normal form, via a finite sequence of near-identity changes of10ariables, in which the energy transfer mechanisms are made explicit. This is achievedvia the introduction of z a ( z, ¯ z ) , a ( z, ¯ z ) , p ( z, ¯ z ) and q ( z, ¯ z ) in Appendix B. Part 2 : The full coupled system is now in a form of:a finite dimensional system of (normal form) ODEs, with non-resonant terms removed bynear identity changes of variables, with rapidly time- decaying corrections, determinedby the dispersive part, weakly-coupled to a dispersive PDE, with rapidly decaying and/oroscillating source terms, coming from the discrete components of the solution. The latteris essentially treatable by low-energy scattering methods.In
GW1 [6] we proved that the neutral mode mass and λ ( t ), which through k φ λ ( t ) k controls the ground state mass, is governed by ddt λ ( t ) = Rem λ ( t ) , (3.12) ddt | z ( t ) | = − z ∗ Γ( z, ¯ z ) z + Rem z ( t ) , (3.13)where Rem λ ( t ) and Rem z ( t ) satisfy an estimate of the form: | Rem λ ( t ) | . | z ( t ) | + kh x i − R ( t ) k H + k R ( t ) k ∞ + | z ( t ) | kh x i − ˜ R ( t ) k | Rem z ( t ) | . | z ( t ) | + | z ( t ) | kh x i − R ( t ) k H + | z ( t ) | k R ( t ) k ∞ + | z ( t ) | kh x i − ˜ R ( t ) k (3.14)where, ˜ R is defined in (B.8), and for t ≫ | z ( t ) | ∼ t − , kh x i − R ( t ) k H ∼ t − , k R ( t ) k ∞ ∼ t − , kh x i − ˜ R ( t ) k ∼ t − . Since Rem z ( t ) = O ( t − − τ ) , τ >
0, Rem z ( t ) is dominated by the first term on the right handside of (3.13), which is O ( t − ) and strictly negative, by the Fermi Golden Rule resonancehypothesis (FGR) . Furthermore, Rem λ ( t ) is integrable in t , λ ( t ) has a limit, λ ∞ . In view of the results of
GW1 , we focus on the refinements required. These concern the terms S λ and S z in (3.2) and (3.3) and their estimation in (3.5), for the proofs of Theorems 3.1and 3.2. In this section we derive S λ and S z and estimate them.Technically the main effort in the present paper is to improve the estimates for the variousterms on the right hand side of (3.12) and (3.13). It is relatively easier to improve the estimatefor ∂ t | z | , since the term − z ∗ Γ( z, ¯ z ) z already measures the decreasing of | z | . What is left isto prove the term Rem z is indeed a small correction in certain sense.To improve the estimates of the terms on the right hand side of (3.12) is more involved.From (3.12) we can not tell the increasing or decreasing of the parameter λ. For that purposewe expand the right hand side of (3.12) to fourth order in z and ¯ z to find some sign. This inturn is achieved by expansion of the function R or ˜ R further to third order in z and ¯ z . For thatpurpose we define the third order terms in (B.9) and introduce remainder by R ≥ in (B.13).11e next present some precise estimates on R ≥ , z and ˙ λ , which are defined in AppendicesA- B. To facilitate later discussions we define the constant δ ∞ by: δ ∞ := k φ λ ∞ k L = O ( | λ ∞ + e | − σ ) = O ( δ ( λ ( t ))) for any time t (4.1)where the last estimate follows from the fact the soliton manifold is stable (see [6]). Recall theconstant δ ( λ ) ≡ δ defined and estimated in (2.1), and recall lim t →∞ λ ( t ) = λ ∞ in Theorem 3.1.We have: Proposition 4.1.
Suppose that | z | δ ∞ ≪ for σ = 1 and | z | ≤ δ C ( σ ) ∞ for σ > and C ( σ ) is asufficiently large constant. Then the following results hold: there exists a constant C > suchthat for any time t ≥ | z ( t ) | ≤ C ( | z | − + δ σ − ∞ t ) − ; (4.2) if σ = 1 then kh x i − R ≥ k . | z | (1 + t ) − + δ ∞ | z | | z ( t ) | ; (4.3) | ˙ λ | . δ ∞ | z ( t ) | + δ ∞ | z | | z ( t ) | + δ ∞ | z | (1 + t ) − + δ ∞ | z | | z ( t ) | (1 + t ) − , (4.4) and if σ > then kh x i − R ≥ k . | z | (1 + t ) − + [ | z | + | z | σ − ] | z ( t ) | , (4.5) | ˙ λ | . | z | σ +1 + | z ( t ) | + δ ∞ | z | | z ( t ) | + | z | (1 + t ) − + | z | | z ( t ) | (1 + t ) − . (4.6)This proposition will be formulated as different parts of Propositions F.1 and F.2 inAppendix F.In the next two subsections we find and estimate the functions S z and S λ of (3.2) and(3.3). S z and its estimate In this part we define and estimate the function S z in (3.3).It was proved in [6], p. 293 (and can also be derived from (A.7) and (A.8)) that z satisfiesthe equation ∂ t z + iE ( λ ) z = − Γ( z, ¯ z ) z + Λ( z, ¯ z ) z + K (4.7)where Γ( z, ¯ z ) is positive definite and Λ( z, ¯ z ) is skew symmetric, K = ( K , · · · , K N ) T is definedas K n := − [ ∂ t p n + iE ( λ ) X k + l =2 , ( k − l ) P ( n ) k,l ] − i [ ∂ t q n + iE ( λ ) X k + l =2 , ( k − l ) Q ( n ) k,l ] − * J N ( ~R, p, z ) − X m + n =2 , J N m,n , (cid:18) η n − iξ n (cid:19)+ + Υ , [ h q · η, η n i − h ip · ξ, ξ n i ]+[ ˙ γ − Υ , ][ h ( β + q ) · η, η n i − i h ( α + p ) · ξ, ξ n i ] − ˙ λ [ a h ∂ λ φ λ , η n i + h ( α + p ) · ∂ λ ξ, η n i + ia h ∂ λ φ λ , ξ n i + i h ( β + q ) · ∂ λ η, ξ n i ]+ (cid:28) ~R, ˙ λ (cid:18) ∂ λ η n − i∂ λ ξ n (cid:19) − ˙ γ (cid:18) iξ n η n (cid:19)(cid:29) . | z | measures the neutral mode mass. By direct computation we find ddt | z | = − z ∗ Γ( z, ¯ z ) z + 2 Rez ∗ · K = − z ∗ Γ ( z, ¯ z ) z + S z with the function S z defined by S z := − z ∗ Γ( z, ¯ z ) z + 2 z ∗ Γ ( z, ¯ z ) z + 2 Rez ∗ · K . (4.8)We now estimate different terms on the right hand side of (4.8). Lemma 4.1.
For σ ≥ z ∗ Γ( z, ¯ z ) z = z ∗ Γ ( z, ¯ z ) z + O ( δ σ − ∞ | z | ) . (4.9) If σ = 1 then |K| . δ ∞ | z ( t ) | + δ ∞ kh x i − R ≥ k + δ ∞ | z ( t ) |kh x i − R ≥ k ; (4.10) if σ > then |K| . | z ( t ) | σ +1 + | z ( t ) | + kh x i − R ≥ k + | z ( t ) |kh x i − R ≥ k . (4.11)Equation (4.9) will be proved in Appendix G, (4.10) and (4.11) will be incorporatedinto Proposition F.2. By above estimates we have Theorem 4.1. Z ∞ S z ( s ) ds = o ( | z | ) . (4.12) Proof.
The following two estimates together with Lemma 4.1 are sufficient to prove the theo-rem: Z ∞ | z | ( s ) |K ( s ) | ds = o ( | z | ) (4.13)and Z ∞ δ σ − ∞ | z | ( s ) ds ≤ C | z | . (4.14)We next focus on proving the two inequalities (4.13) and (4.14). The proof of (4.14) isrelatively easy; it follows applying the estimate of z in (4.2) and direct computation.We now turn to (4.13). For σ = 1 we use (4.10) and (4.3) to obtain |K| . δ ∞ | z | + δ ∞ | z | | z | (1 + t ) − + δ ∞ | z | | z | + | z | δ ∞ (1 + t ) − . Together with the assumption on the initial condition | z | ≪ δ ∞ = O ( δ ) (see (4.1)) and (4.2)we have Z ∞ | z ||K| ( s ) ds = o ( | z | ) . (4.15)For the case, σ >
1, the estimate is easier to obtain by applying the stronger condition | z | ≤ O ( δ C ( σ )0 ) = O ( δ C ( σ ) ∞ ) with C ( σ ) being sufficiently large. This completes the proof.13 .2 Definition of S λ and its estimate After expanding the dispersive part ~R into the third order in z and ¯ z , we derive in AppendixC an equation for ddt k φ λ k : ddt k φ λ k = − z ∗ Γ ( z, ¯ z ) z + S λ (4.16)with S λ defined as S λ := 2Ψ + ( 2Π , + z ∗ Γ ( z, ¯ z ) z ) + 2 X m + n =4 , m = n Π m,n here X m + n =4 , Π m,n is a collection of fourth and fifth order terms X m + n =4 , Π m,n := − * X m + n =4 J N m,n , (cid:18) φ λ (cid:19)+ + Υ , * X m + n =2 , R m,n , (cid:18) φ λ (cid:19)+ + Υ , (cid:10) q · η, φ λ (cid:11) + h φ λ , ∂ λ φ λ i (cid:2) ∂ z a · Z , + ∂ ¯ z a · Z , (cid:3) , and Z , := − Γ( z, ¯ z ) z + Λ( z, ¯ z ) z with the latter defined in (2.9); and Ψ is defined asΨ := ( ˙ γ − Υ , ) (cid:20) (cid:28) ~R, (cid:18) φ λ (cid:19)(cid:29) + (cid:10) ( β + q ) · η, φ λ (cid:11) (cid:21) + Υ , (cid:28) R ≥ , (cid:18) φ λ (cid:19)(cid:29) + ˙ λ (cid:28) ~R, (cid:18) ∂ λ φ λ (cid:19)(cid:29) − ˙ λa h ∂ λ φ λ , φ λ i − ˙ λ ( α + p ) h ∂ λ ξ, φ λ i + h φ λ , ∂ λ φ λ i [ ∂ t a + iE ( λ ) X m + n =2 , ( m − n ) A (1) m,n − ∂ z a · Z , − ∂ ¯ z a · Z , ]+ * J N − X m + n =2 J N m,n , (cid:18) φ λ (cid:19)+ , Here we used the convention made in (B.1) and the definitions of Appendix B.To control these terms in S λ we use the following results: (Recall δ ∞ = k φ λ ∞ k , defined in(4.1).) Lemma 4.2. | Ψ | . | z | δ σ − ∞ kh x i − R ≥ k + kh x i − R ≥ k + δ σ − ∞ | z | (4.17)2Π , + z ∗ Γ ( z, ¯ z ) z = O ( δ σ − ∞ | z | ) , (4.18) X m + n =4 , ,m = n Z ∞ Π m,n ( s ) ds . X m + n =4 , ,m = n Z ∞ | ∂ λ Π m,n || ˙ λ | ( s ) + | ∂ z Π m,n || ˙ z + iE ( λ ) z | ( s ) ds (4.19)+ o ( | z | )The bound (4.17) will be proved in Appendix F, (4.19) in Section H and (4.18) in SectionI. We now briefly present the ideas in the proof.141) Ψ is defined in term of functions ˙ λ, ˙ γ , z and ~R . They satisfy a coupled system. Thissystem must be put in matrix form and decoupled. In the end, we bound the functions˙ λ and ˙ γ by the functions of ~R (or R ≥ ) and z .(2) All the integrands in (4.18) are of order | z | in z and ¯ z. What makes the terms differentis the sizes of the coefficients. These depend smoothly on the functions φ λ , ∂ λ φ λ , ξ, η ,which in turn depend smoothly on the small parameter δ ( λ ) = O ( δ ∞ ); see PropositionD.1. The estimate (4.18) follows from a perturbation expansion in the parameter δ ( λ ) . (3) For (4.19) the important observation is that, if m = n , then function Π m,n is a sum ofthe functions of the form C ( λ ) z m ¯ z n = C ( λ ) Q k z m k k Q l ¯ z n l l with m = P k m k , n = P l n l .These are “almost periodic” with period 2 π ( E ( λ )( m − n )) − = 0 since z satisfies theequation ˙ z = − iE ( λ ) z + · · · . This non-trivial oscillation enables us to integrate by partsin the variable s to derive smallness. The term o ( | z | ) in (4.19) is due to a boundaryterm obtained in this way.Based on the estimates in Lemma 4.2 we will prove Theorem 4.2. S λ satisfying the estimate in (3.2) , i.e. Z ∞ S λ ( s ) ds = o ( | z | ) . Proof.
The result follows directly from Lemma 4.2 and the following two estimates: Z ∞ | Ψ | ( s ) ds = o ( | z | ); (4.20) X m + n =4 , ,m = n Z ∞ | ∂ λ Π m,n || ˙ λ | ( s ) + | ∂ z Π m,n || ˙ z + iE ( λ ) z | ( s ) ds = o ( | z | ) . (4.21)We next prove estimates (4.20) and (4.21). In the proof we consider the case σ = 1. Thatof σ > | z | ≤ δ C ( σ ) ∞ for some sufficientlylarge C ( σ ), and hence omit the details.We start with (4.20), by estimating three different terms in the estimate of Ψ in (4.17)on the right hand side. By applying the estimates for z in (4.2) Z ∞ δ ∞ | z | ( s ) ds . Z ∞ δ ∞ ( | z | − + δ ∞ s ) − ds = 23 δ − ∞ | z | = o ( | z | ) (4.22)where the assumption on the initial condition | z | ≪ δ = O ( δ ∞ ) was used.By the estimate of R ≥ in (4.3) and | z ( t ) | in (4.2) R ∞ δ ∞ | z ( s ) |kh x i − R ≥ ( s ) k ds . δ ∞ | z | R ∞ (1 + s ) − ( | z | − + δ ∞ s ) − ds + δ ∞ | z | R ∞ ( | z | − + δ ∞ s ) − ds = o ( | z | ) . The third term can be similarly estimated. Assembling the above estimates yields Z ∞ | Ψ( t ) | dt = o ( | z | ) .
15o prove (4.21) we use the equations for ˙ z and ˙ λ in (4.7) and (4.4) to find that if m + n = 4 , m = n then | ∂ λ Π m,n || ˙ λ ( s ) | + | ∂ z Π m,n || ˙ z + iE ( λ ) z | ( s ) . | z || ˙ λ | + | z | + | z | |K| . Using the estimates in (4.10) and (4.11) for K , and the estimate (4.4) and similar techniquesabove we prove (4.21). This is straightforward, but tedious, hence we omit the details. Remark 4.1.
In the last step of (4.22) we used | z | ≪ δ ∞ to control R ∞ δ ∞ | z | ( s ) ds. If σ = 1 this can be relaxed to | z | ≤ k φ λ k = O ( δ ∞ ) by inspecting closely the terms forming δ ∞ | z | .The term actually is a part of (cid:28) J N ≥ , (cid:18) φ λ (cid:19)(cid:29) , and can be written as X m + n =5 K m,n for someproperly defined K m,n . To evaluate X m + n =5 Z ∞ K m,n ( s ) ds we observe that K m,n , m + n = 5 , are “almost periodic” as Λ m,n of (4.19) . Hence by integrating by parts as in the proof of (4.19) it is easy to obtain the desired estimate X m + n =5 Z ∞ K m,n ( s ) ds = o ( | z | ) . Note that the terms K m,n , m + n = 5 , may not be well defined if σ N . In [6] and the main part of the present paper we have proved that if the neutral modes aredegenerate and their eigenvalues are sufficiently close to the essential spectrum then the groundstate is asymptotically stable and its mass will grow by half of that of the neutral modes.In what follows we extend the results to the cases where the neutral modes are nearlydegenerate, i.e. a cluster of approximately equal eigenfrequencies. For technical simplicity, weconsider the case of cubic nonlinearity, σ = 1. The main result is Theorem 5.1 below. Thekey ideas of the proof will be presented after its statement. As in Subsection 2.1 we assume that the linear operator − ∆ + V has the following properties:(V1) V is real valued and decays sufficiently rapidly, e.g. exponentially, as | x | tends to infinity.(V2) The linear operator − ∆ + V has N + 1 (counting multiplicity if degenerate) eigenvalues e , e k , k = 1 , , · , N, with e < e k ,e is the lowest eigenvalue with ground state φ lin >
0, the eigenvalues { e k } Nk =1 are possiblydegenerate with eigenvectors ξ lin , ξ lin , · · · , ξ linN . (V3) Moreover, for any k = 1 , , · · · , N we assume2 e k − e > . (5.1)16hen the nonlinear equation (1.1) admits a family of ground states solution e iλt φ λ withproperties as described in Proposition 2.1. The linearized operators about the ground states, L ( λ ), takes the same form as in (2.3). The excited states of − ∆ + V bifurcate to the neutralmodes (cid:18) ξ k ± iη k (cid:19) of L ( λ ) with eigenvalues ± iE k ( λ ) , k = 1 , · · · , N. The ground states φ λ andneutral modes satisfy all the estimates in Lemma 2.1 and the estimates (D.1)- (D.4).Assumption (SA) on the spectrum of L ( λ ) is generalized, in the case where near-degeneracyis admitted, as (SA) The discrete spectrum of the linearized operator L ( λ ) consists of: the eigenvalue 0with generalized eigenvectors (cid:18) φ λ (cid:19) and (cid:18) ∂ λ φ λ (cid:19) and eigenvalues ± iE k ( λ ) , E k ( λ ) > , k = 1 , , · · · , N .A consequence of nonzero neutral mode frequency-differences is a slightly different systemfor the neutral mode amplitudes, z ( t ). The solution ψ ( t ) is decomposed as in (A.1). Followingthe same procedure as in [6], we derive ∂ t z = − iE ( λ ) z − Γ( z, ¯ z ) z + Λ( z, ¯ z ) z + · · · (5.2)where E ( λ ) = Diag [ E ( λ ) , · · · , E n ( λ )] is a diagonal N × N matrix, Γ is symmetric and Λ isskew symmetric.We now describe the matrix Γ, which takes a different form from the degenerate case:Define vector functions G ( k, m ) , k, m = 1 , , · · · , N, as G ( k, m ) := (cid:18) B ( k, m ) D ( k, m ) (cid:19) (5.3)with the functions B ( k, m ) and D ( k, m ) defined as B ( k, m ) := − iφ λ [ z m ξ m η k + z m η m ξ k ] ,D ( k, m ) := − φ λ [ 3 z m ξ m ξ k − z m η m η k ] . In terms of the column 2-vector, G ( k, m ), we define a N × N matrix Z ( z, ¯ z ) as Z ( z, ¯ z ) = ( Z ( k,l ) ( z, ¯ z )) , ≤ k, l ≤ N (5.4)and Z ( k,l ) = − * N X m =1 ( L ( λ ) + iE l ( λ ) + iE m ( λ ) − − P c G ( l, m ) , iJ P c N X m =1 G ( k, m ) + (5.5)Finally, we define Γ( z, ¯ z ) as follows:Γ( z, ¯ z ) := 12 [ Z ( z, ¯ z ) + Z ∗ ( z, ¯ z )] . (5.6)We shall require the following Fermi Golden Rule hypothesis: Let P linc be the projectiononto the essential spectrum of − ∆ + V then (FGR) We assume there exists a constant
C > − Re h i [ − ∆ + V + λ − E ( λ ) − i − P linc φ lin ( z · ξ lin ) , φ lin ( z · ξ lin ) i ≥ C | z | for any z ∈ C N . C > δ > k,l | E k ( λ ) − E l ( λ ) | ≤ δ then for any z ∈ C N z ∗ Γ( z, ¯ z ) z ≥ C k φ λ k ∞ | z | . (5.7)We now introduce the leading order contribution to Γ( z, ¯ z ). For each fixed z we use thefact | λ + e | being small and use (2.1) and (2.6) to find the leading term in z ∗ Γ( z, ¯ z ) z is z ∗ Γ ( z, ¯ z ) z defined as z ∗ Γ ( z, ¯ z ) z = − δ ( λ ) ℜh i X m,n ≤ N [ − ∆ + V + λ − E m ( λ ) − E n − i − P linc φ lin ( z m ξ linm )( z n ξ linn ) , φ lin ( z · ξ lin ) i . (5.8) Recall that we only consider the case σ = 1, i.e. the cubic nonlinearity. Theorem 5.1.
There exists a constant δ independent of the initial condition ψ of (1.1) suchthat if max k,l | E k ( λ ) − E l ( λ ) | ≤ δ then all the results in Theorem 3.1 hold with z ∗ Γ ( z, ¯ z ) z replaced by the expression in (5.8) . Moreover all the remainder estimates in (3.2) - (3.10) holdand are independent of the size of δ . In the next we show how to recover all the estimates. To simplify the treatment we onlyconsider the case N = 2 with eigenfrequencies E ( λ ) and E ( λ ) . There are some differences between the degenerate and the nearly degenerate cases. Amongthem, the most outstanding one are terms, which previously vanished identically, which nowneed to be estimated. These terms include, for example, h ImN , , φ λ i , which was proved tobe zero in [6] Lemma 9.4, p. 291 which we see below is non-zero in the nearly degeneratecase. To treat such terms, the key observation is that these terms have a factor E ( λ ) − E ( λ )in their coefficient enabling us to re-express [ E ( λ ) − E ( λ )] z ¯ z as − i ddt z ¯ z + o ( | z | ). Thus,these terms can be removed via integration by parts and a redefinition of the normal formtransformation. We decompose the initial condition in exactly the same way as in (A.1). All equations (A.1)-(A.6), (A.9) and (A.10) hold. The equations for ˙ z are slightly different since z j each havedifferent associated frequencies. Consequently instead of (A.7) and (A.8) we have ∂ t ( α n + p n ) − E n ( λ )( β n + q n ) + · · · , ∂ t ( β n + q n ) + E n ( λ )( α n + p n ) + · · · , requiring a different near-indentity / normal form transformation.To illustrate the main difference in the calculation we study the equation for ˙ λ. Recall thatthe function ˙ λ satisfies the equation˙ λ + ∂ t a = − h φ λ , ∂ λ φ λ i h ImN ( ~R, z ) , φ λ i + · · · and we want to remove the second and third order terms in z and ¯ z from the equation bydefining some polynomial a in z and ¯ z : a := X m + n =2 , A (1) m,n .
18n the degenerate case we set A (1)1 , = 0 (see (B.2)) due to the fact h ImN , , φ λ i = 0 . Whenthe latter no longer holds A (1)1 , has to be redefined. Following steps in [6], p. 291, we use thefact (cid:18) ξ n iη n (cid:19) , n = 1 , , are eigenvectors of L ( λ ) to obtain h ImN , , φ λ i := i X n =1 2 X m =1 ¯ z n z m Z ( φ λ ) ( ξ n η m − ξ m η n )= i X n =1 2 X m =1 ¯ z n z m [ h ( L − − L + ) ξ n , η m i − h ( L − − L + ) ξ m , η n i ]= i [ E ( λ ) − E ( λ )][ z ¯ z − z ¯ z ][ h η , η i + h ξ , ξ i ] . (5.9)To remove (5.9) from the equation of ˙ λ we define A (1)1 , := − h φ λ , ∂ λ φ λ i [ z ¯ z + z ¯ z ][ h η , η i + h ξ , ξ i ] = O (cid:16) k φ λ k | z | (cid:17) , (5.10)where in the last step the estimate (D.4) and the fact ξ lin ⊥ ξ lin are used.For the other terms in a we only re-define A (1)2 , to illustrate the differences: Decompose h φ λ ,∂ λ φ λ i h ImN , , φ λ i as K z + K z z + K z , then instead of the definition in (B.3) wedefine A (1)2 , = − i E ( λ ) K z − iE ( λ ) + E ( λ ) K z z − i E ( λ ) K z . The new normal forms enable the proof of asymptotic stability of the ground states to gothrough, as well as all results in Section D, i.e. all the statements in Theorem 3.1 except (A)and (D), which we discuss in the next subsection.
In this subsection we recover the Statements (A) and (D). Most of the arguments proved in thedegenerate regime still hold. As presented above certain newly-nonzero terms enter differentplaces. In what follows we present the strategy to handle such terms.To illustrate the idea we only study one term whose counterpart is H , in (G.7) D := X m + n =2 m ′ + n ′ =2 D ( m, n, m ′ , n ′ )with D ( m, n, m ′ , n ′ ) being a real function: D ( m, n, m ′ , n ′ ) := Re h i ( − ∆+ V + λ + mE ( λ )+ nE ( λ )) − P linc φ λ ( z ξ ) m ( z ξ ) n , φ λ ( z ξ ) m ′ ( z ξ ) n ′ i . If E ( λ ) = E ( λ ) then we use the observation in (G.7) to prove D ( m, n, m ′ , n ′ ) = D ( m, n, m ′ , n ′ ) = − D ( m ′ , n ′ , m, n ) implies D = 0 . When E ( λ ) = E ( λ ) we use the following result to recover the desired estimate Lemma 5.1. Z ∞ D ( s ) ds = o ( | z | ) . (5.11)19 roof. The facts that D ( m ′ , n ′ , m, n ) is real and ( − ∆ + V + λ + m ′ E ( λ ) + n ′ E ( λ )) − isself-adjoint imply D ( m ′ , n ′ , m, n ) = D ( m ′ , n ′ , m, n, )= − Re h i ( − ∆ + V + λ + m ′ E ( λ ) + n ′ E ( λ )) − P linc × φ λ ( z ξ ) m ( z ξ ) n , φ λ ( z ξ ) m ′ ( z ξ ) n ′ i . The crucial step is to find the presence of E ( λ ) − E ( λ ) in the coefficient: D ( m, n, m ′ , n ′ ) + D ( m ′ , n ′ , m, n ) = − [( m − m ′ ) E ( λ ) + ( n − n ′ ) E ( λ )] ReH = − [ m − m ′ ][ E ( λ ) − E ( λ )] ReH (5.12)where H is defined as H := h i [ − ∆ + V + λ + m ′ E ( λ ) + n ′ E ( λ )] − [ − ∆ + V + λ + mE ( λ ) + nE ( λ )] − × P linc φ λ ( z ξ ) m ( z ξ ) n , φ λ ( z ξ ) m ′ ( z ξ ) n ′ i , and in the last step the fact m + n = m ′ + n ′ = 2 is used.(5.12) enables us to use the same trick as in (4.19), namely integration by parts, to obtainthe desired estimate Z ∞ D ( m, n, m ′ , n ′ ) + D ( m ′ , n ′ , m, n ) ds (5.13)= Z ∞ dds ℜ ( iH ) ds + Z ∞ O ( | ˙ λ || z | + k φ λ k | z | ) ds + o ( | z | ) . (5.14)The proof is complete.In summary, as outlined above, all the estimates obtained in the degenerate can be proved inthe nearly degenerate case. A Decomposition of the solution ψ This section is based upon [6], pp. 286-287. As stated in (3.1), for any time t the solution ψ ( x, t ) can be decomposed as ψ ( x, t ) = e iγ ( t ) e i R t λ ( s ) ds [ φ λ ( t ) ( x ) + a ( t ) ∂ λ φ λ ( t ) ( x ) + ia ( t ) φ λ ( t ) ( x )+ N X n =1 [ α n ( t ) + p n ( t )] ξ λ ( t ) n ( x ) + i N X n =1 [ β n ( t ) + q n ( t )] η λ ( t ) n ( x ) + R ( x, t )] (A.1)for some polynomials a , a , p n , q n (will be defined explicitly in Appendix B) and thefunction R satisfies the symplectic orthogonality conditions (3.6). By this ~R := (cid:18) R R (cid:19) := (cid:18) ReRImR (cid:19) ∈ P c ( L ( λ )) L satisfies the equation ddt ~R = L ( λ ( t )) ~R − P λ ( t ) c J ~N ( ~R, z ) + L ( ˙ λ, ˙ γ ) ~R + G . (A.2)20ere, J ~N ( ~R, z ) := ImN ( ~R, z ) − ReN ( ~R, z ) ! , (A.3) ImN ( ~R, z ) := | φ λ + I + iI | σ I − ( φ λ ) σ I ,ReN ( ~R, z ) := [ | φ λ + I + iI | σ − ( φ λ ) σ ]( φ λ + I ) − σ ( φ λ ) σ I ,I := α · ξ + a ∂ λ φ λ + p · ξ + R ,I := β · η + a φ λ + q · η + R . The operator L ( ˙ λ, ˙ γ ) and the vector function G are defined as L ( ˙ λ, ˙ γ ) := ˙ λ ( ∂ λ P λ ( t ) c ) + ˙ γP λ ( t ) c J, (A.4) G := P λ ( t ) c (cid:18) [ ˙ γ − Υ , ]( β + q ) · η − ˙ λa ∂ λ φ λ − ˙ λ ( α + p ) · ∂ λ ξ − [ ˙ γ − Υ , ]( α + p ) · ξ − ˙ λa φ λλ − ˙ λ ( β + q ) · ∂ λ η (cid:19) + Υ , P λ ( t ) c (cid:18) ( β + q ) · η − ( α + p ) · ξ (cid:19) (A.5)where, Υ , is defined asΥ , := h ( φ λ ) σ − [(2 σ + σ ) | z · ξ | + σ | z · η | ] , ∂ λ φ λ i h φ λ , ∂ λ φ λ i (A.6)with z := ( z , · · · , z N ) T , z n := α n + iβ n , n = 1 , · · · , N, and ξ := ( ξ , · · · , ξ N ) T , η := ( η , · · · , η N ) T . By the orthogonality conditions (3.6) and (2.5) we derive equations for ˙ λ, ˙ γ and z n = α n + iβ n , n = 1 , . . . , N, as ∂ t ( α n + p n ) − E ( λ )( β n + q n ) + h ImN ( ~R, z ) , η n i = F n ; (A.7) ∂ t ( β n + q n ) + E ( λ )( α n + p n ) − h ReN ( ~R, z ) , ξ n i = F n ; (A.8)˙ γ + ∂ t a − a − h φ λ , φ λλ i h ReN ( ~R, z ) , ∂ λ φ λ i = F ; (A.9)˙ λ + ∂ t a + 1 h φ λ , φ λλ i h ImN ( ~R, z ) , φ λ i = F (A.10)where the scalar functions F j,n , j = 1 , , n = 1 , , · · · , N, F , F , are defined as F n = ˙ γ h ( β + q ) · η, η n i − ˙ λa h ∂ λ φ λ , η n i − ˙ λ h ( α + p ) · ∂ λ ξ, η n i − ˙ γ h R , η n i + ˙ λ h R , ∂ λ η n i ,F n = − ˙ γ h ( α + p ) · ξ, ξ n i − ˙ λa h φ λλ , ξ n i − ˙ λ h ( β + q ) · ∂ λ η, ξ n i + ˙ γ h R , ξ n i + ˙ λ h R , ∂ λ ξ n i ,F = 1 h φ λ , φ λλ i h ˙ λ h R , φ λλλ i − ˙ γ h R , φ λλ i − h ˙ γ ( α + p ) · ξ + ˙ λa φ λλ + ˙ λ ( β + q ) · ∂ λ η, φ λλ i i ,F = 1 h φ λ , φ λλ i h ˙ λ h R , φ λλ i + ˙ γ h R , φ λ i + h ˙ γ ( β + q ) · η − ˙ λa ∂ λ φ λ − ˙ λ ( α + p ) · ∂ λ ξ, φ λ i i . (A.11)21 The Normal Form Expansion
All the results in this Appendix, except the definitions of R m,n , J N m,n , m + n = 3, are takenfrom [6]. Specifically the definitions of a , a , p k , q k , k = 1 , , · · · , N, are taken from (9-12)and (9-13), p. 288; the definitions of R m,n , J N m,n , m + n = 2 , from (9-18)-(9-21), p. 290.Before defining various functions we introduce the following convention on notations: wealways use z to stand for a complex N -dimensional vector z = ( z , z , · · · , z N ) andan upper case letter or a Greek letter with two subindices, for example Q m,n , torepresent Q m,n ( λ ) = X | a | = m, | b | = n q a,b ( λ ) N Y k =1 z a k k ¯ z b k k , (B.1) where a, b ∈ N N , | a | := N X k =1 a k . We refer to this kind term as ( m, n ) term. In what follows we define R m,n , m + n = 2 , , J N m ′ ,n ′ , m ′ + n ′ = 2 , , , and the polynomials a , a , p k , q k , k = 1 , , · · · , N, by induction. Definitions of Polynomials a , a , p k , q k , k = 1 , , · · · , N We define the polynomials a , a , p k and q k , k = 1 , , · · · , N, in ( A.1) as a k ( z, ¯ z ) := X m + n =2 , ,m = n A ( k ) m,n ( λ ) , k = 1 , ,p k ( z, ¯ z ) := X m + n =2 , P ( k ) m,n ( λ ) , k = 1 , , · · · , N,q k ( z, ¯ z ) := X m + n =2 , Q ( k ) m,n ( λ ) , k = 1 , , · · · , N (B.2)where the terms on the right hand side take the form:2 iE ( λ ) A (1)2 , := h φ λ ,∂ λ φ λ i h N Im , , φ λ i ;3 iE ( λ ) A (1)3 , := h φ λ ,∂ λ φ λ i h N Im , , φ λ i ; iE ( λ ) A (1)2 , := h φ λ ,∂ λ φ λ i [ h N Im , , φ λ i − i Υ , h z · η, φ λ i ]; (B.3) − iE ( λ ) A (2)2 , − A (1)2 , := h φ λ ,∂ λ φ λ i h N Re , , ∂ λ φ λ i ; − iE ( λ ) A (2)3 , − A (1)3 , := h φ λ ,∂ λ φ λ i h N Re , , ∂ λ φ λ i ; − iE ( λ ) A (2)2 , − A (1)2 , := h φ λ ,∂ λ φ λ i [ h N Re , , ∂ λ φ λ i − Υ , h z · ξ, ∂ λ φ λ i ]; (B.4) A ( n ) k,l := A ( n ) l,k for n = 1 , , k + l = 2 , , k = l ;and − iE ( λ ) P ( n )2 , − E ( λ ) Q ( n )2 , := −h N Im , , η n i , − iE ( λ ) Q ( n )2 , + E ( λ ) P ( n )2 , := h N Re , , ξ n i , − iE ( λ ) P ( n )3 , − E ( λ ) Q ( n )3 , := −h N Im , , η n i , − iE ( λ ) Q ( n )3 , + E ( λ ) P ( n )3 , := h N Re , , ξ n i , (B.5)22 iE ( λ ) P ( n )1 , − E ( λ ) Q ( n )1 , := −h N Im , , η n i + i h N Re , , ξ n i + i Υ , d X k =1 ¯ z k [ h η k , η n i − h ξ k , ξ n i ] − E ( λ ) Q ( n )1 , := −h N Im , , η n i ,E ( λ ) P ( n )1 , := h N Re , , ξ n i . (B.6) P ( n ) k,l := P ( n ) l,k , Q ( n ) l,k := Q ( n ) k,l . The functions
J N m,n = (cid:18) N Imm,n − N Rem,n (cid:19) used above will be defined in the next subsection.
Expansion of
J ~N and ~R For m + n = 2 we define R m,n := R (1) m,n R (2) m,n ! := [ L ( λ ) + iE ( λ )( m − n ) − − P c J N m,n . (B.7)We denote the remainder of the second order expansion by ˜ R, i.e.,˜ R = ~R − X m + n =2 R m,n . (B.8)For m + n = 3 we define R m,n := [ L ( λ ) + iE ( λ )( m − n ) − − P c [ J N m,n + X m,n ] (B.9)where X m + n =3 X m,n := Υ , (cid:18) − β · ηα · ξ (cid:19) and recall the definition of Υ , in (A.6).We define the quadratic terms J N m,n , m + n = 2 , as X m + n =2 J N m,n = σ ( φ λ ) σ − (cid:18) α · ξ )( β · η ) − [2 σ + 1]( α · ξ ) − ( β · η ) (cid:19) , (B.10)and define J N m,n , m + n = 3 , by X m + n =3 J N m,n := X m + n =3 (cid:18) N Imm,n − N Rem,n (cid:19) (B.11)where N Imm,n and N Rem,n are defined as X m + n =3 N Imm,n := 2 σ ( φ λ ) σ − ( α · ξ ) X m + n =2 [ A (2) m,n φ λ + Q m,n · η + R (2) m,n ]+2 σ ( φ λ ) σ − ( β · η ) X m + n =2 [ A (1) m,n ∂ λ φ λ + P m,n · ξ + R (1) m,n ]+ σ ( φ λ ) σ − [( α · ξ ) + ( β · η ) ][ β · η ] + 2 σ ( σ − φ λ ) σ − ( α · ξ ) ( β · η ) , and X m + n =3 N Rem,n := 2 σ (2 σ + 1)( φ λ ) σ − ( α · ξ ) X m + n =2 [ A (1) m,n ∂ λ φ λ + P m,n · ξ + R (1) m,n ]+2 σ ( φ λ ) σ − ( β · η ) X m + n =2 [ A (2) m,n φ λ + Q m,n · η + R (2) m,n ]+ σ [2 σ − ( σ − σ − φ λ ) σ − ( α · ξ ) + σ (2 σ − φ λ ) σ − ( α · ξ )( β · η ) . J ~N to fourth order: X m + n =4 J N m,n := X m + n =4 (cid:18) N Imm,n − N Rem,n (cid:19) (B.12)with N Imm,n and N Rem,n defined as X m + n =4 N Imm,n := 2 σ ( φ λ ) σ − ( α · ξ ) X m + n =3 [ A (2) m,n φ λ + Q m,n · η + R (2) m,n ]+2 σ ( φ λ ) σ − ( β · η ) X m + n =3 [ A (1) m,n ∂ λ φ λ + P m,n · ξ + R (1) m,n ]+2 σ ( φ λ ) σ − X m + n =2 [ A (1) m,n ∂ λ φ λ + P m,n · ξ + R (1) m,n ] X m ′ + n ′ =2 [ A (2) m ′ ,n ′ φ λ + Q m ′ ,n ′ · η + R (2) m ′ ,n ′ ]+ σ ( φ λ ) σ − [(2 σ − α · ξ ) + 3( β · η ) ] X m + n =2 [ A (2) m,n φ λ + Q m,n · η + R (2) m,n ]+2 σ (2 σ − φ λ ) σ − ( β · η )( α · ξ ) X m + n =2 [ A (1) m,n ∂ λ φ λ + P m,n · ξ + R (1) m,n ]+2 σ ( σ − φ λ ) σ − [ σ − ( α · ξ ) ( β · η ) + ( α · ξ )( β · η ) ]and X m + n =4 N Rem,n := 2 σ (2 σ + 1)( φ λ ) σ − ( α · ξ ) X m + n =3 [ A (1) m,n ∂ λ φ λ + P m,n · ξ + R (1) m,n ]+2 σ ( φ λ ) σ − ( β · η ) X m + n =3 [ A (2) m,n φ λ + Q m,n · η + R (2) m,n ]+ σ (2 σ + 1)( φ λ ) σ − [ X m + n =3 ( A (1) m,n ∂ λ φ λ + P m,n · ξ + R (1) m,n )] + σ ( φ λ ) σ − [ X m + n =2 ( A (2) m,n φ λ + Q m,n · η + R (2) m,n )] + σ ( φ λ ) σ − [3 C ( σ )( α · ξ ) + C ( σ )( β · η ) ] X m + n =2 [ A (1) m,n ∂ λ φ λ + P m,n · ξ + R (1) m,n ]+2 σ ( φ λ ) σ − C ( σ )( β · η )( α · ξ ) X m + n =2 [ A (2) m,n φ λ + Q m,n · η + R (2) m,n ]+( φ λ ) σ − [ C ( σ )( α · ξ ) + C ( σ )( β · η ) + C ( σ )( α · ξ ) ( β · η ) ]where C k ( σ ) , k = 1 , , · · · , C k (1) = 1 if k = 1 , , , and C l (1) = 0 if l = 4 , , . Remark B.1.
Note that if in (1.1) σ > the function J N m,n , m + n = 4 might NOTbe well defined: in the last lines of definitions of N Imm,n and N Rem,n we have terms of the form ( φ λ ) σ − ( α · ξ ) where the power σ − of φ λ might be negative. Still the definitions are usefulbecause later we will take inner production with J N m,n , m + n = 4 and (cid:18) φ λ (cid:19) , see (E.5) . To facilitate later discussions we define R ≥ := ~R − X m + n =2 , R m,n (B.13)and J N ≥ := J ~N ( ~R, z ) − X m + n =2 J N m,n . (B.14)24 Derivation of Equation (4.16)
By the equation for ˙ λ ( t ) in (A.10) we derive the following modulation equation12 ddt (cid:13)(cid:13)(cid:13) φ λ ( t ) (cid:13)(cid:13)(cid:13) = h φ λ , φ λλ i ˙ λ = −h ImN ( ~R, z ) , φ λ i + h φ λ , φ λλ i F − h φ λ , φ λλ i ∂ t a . (C.1)To see the increasing of the mass on the ground state, or (cid:13)(cid:13) φ λ ( t ) (cid:13)(cid:13) we resort to expand theterms on the right hand side to fourth order in z and ¯ z :(1) The definitions of a in (B.2) and (B.3) imply ∂ t a = ∂ t a + iE ( λ ) X m + n =2 , ( m − n ) A (1) m,n − h φ λ ,∂ λ φ λ i [ h X m + n =2 , J N m,n , (cid:18) φ λ (cid:19) i − Υ , h β · η, φ λ i ] (C.2)where in the second step the fact h N Im , , φ λ i = 0, proved in [6] Lemma 9.4 on p. 291, isused. Extracting the lower order terms, we find ∂ t a = ∂ z a · Z , + ∂ ¯ z a · Z , − h φ λ ,∂ λ φ λ i [ h X m + n =2 , J N m,n , (cid:18) φ λ (cid:29) − Υ , h β · η, φ λ i ]+ ∂ t a + iE ( λ ) X m + n =2 , ( m − n ) A (1) m,n − ∂ z a · Z , − ∂ ¯ z a · Z , (C.3)with Z , = − Γ( z, ¯ z ) z + Λ( z, ¯ z ) z defined in (4.7).(2) Separate the terms of order | z | , | z | and | z | from F and obtain h φ λ , ∂ λ φ λ i F = Υ , X m + n =2 , h R m,n , (cid:18) φ λ (cid:19) i + Υ , h ( β + q ) · η, φ λ i +( ˙ γ − Υ , )[ h ~R, (cid:18) φ λ (cid:19) i + h ( β + q ) · η, φ λ i ] + Υ , h R ≥ , (cid:18) φ λ (cid:19) i + ˙ λ h ~R, (cid:18) ∂ λ φ λ (cid:19) i − ˙ λa h ∂ λ φ λ , φ λ i − ˙ λ ( α + p ) h ∂ λ ξ, φ λ i . (C.4)(3) The definitions of J N m,n , m + n = 2 , , , and J N ≥ in (B.10)- (B.12) and (B.14) implythat h ImN ( ~R, z ) , φ λ i = h X m + n =2 J N m,n , (cid:18) φ λ (cid:19) i + h J N ≥ , (cid:18) φ λ (cid:19) i . (C.5)Equations (C.1)- (C.5) and cancellation of terms in sum leads to (4.16). D Estimates on the Eigenvectors of L ( λ ) and the Parametersof Normal Form Transformation Precise estimations of S z and S λ defined in Theorems 4.1 and 4.2 require control on coefficientsdepending on norms of φ λ , its derivatives as well as such norms of the neutral modes. Recallthe definition of δ ∞ and the fact δ ∞ = O ( δ ( λ ( t ))) in (4.1).The result is 25 roposition D.1. There exist constants C , C , C ∈ R such that in the space h x i − H φ λ = C | e + λ | σ φ lin + O ( | e + λ | σ ) = O ( δ ∞ ) ∂ λ φ λ = C | e + λ | σ − φ lin + O ( | e + λ | σ ) = O ( δ − (2 σ − ∞ ) ,∂ λ φ λ = C | e + λ | σ − φ lin + O ( | e + λ | σ − ) = O ( δ − (4 σ − ∞ ) . (D.1)1 h φ λ , ∂ λ φ λ i . δ σ − ∞ . (D.2) For the neutral modes we have kh x i ∂ λ ξ n k , kh x i ∂ λ η n k .
1; (D.3) kh x i ( η m − ξ linm ) k H , kh x i ( ξ m − ξ linm ) k H , kh x i ( ξ m − η m ) k H = O ( δ σ ∞ ) . (D.4) Recall P linc is the orthogonal project onto the essential spectrum of − ∆ + VP λc = P linc (cid:18) (cid:19) + O ( δ σ ∞ ) . (D.5) The function Υ , in (A.6) satisfies the estimate | Υ , | . δ σ − ∞ | z | . (D.6) In what follows we estimate various functions defined in Appendix B.For m + n = 2 , kh x i J N m,n k , kh x i − R m,n k . δ σ − ∞ | z | , | A (1) m,n | . δ σ − ∞ | z | , | A (2) m,n | . δ σ − ∞ | z | , | P ( k ) m,n | , | Q ( k ) m,n | . δ σ − ∞ | z | , k = 1 , , · · · , N. For m + n = 3 , kh x i J N m,n k , kh x i − R m,n k . δ σ − ∞ | z | . | A (1) m,n | . δ σ − ∞ | z | , | A (2) m,n | . δ σ − ∞ | z | , | P ( k ) m,n | , | Q ( k ) m,n | . δ σ − ∞ | z | , k = 1 , , · · · , N. For m + n = 4 and σ = 1 kh x i J N m,n k . δ ∞ | z | . Proof.
Since all the functions are defined in term of φ λ , ξ, η and their derivatives, we startwith deriving estimates for them, or proving (D.1)- (D.4).The key observation is these functions can be constructed perturbatively, as can be foundin the known results in the spaces H k , k = 1 , , (see e.g.[12]). In what follows we re-do theproof in the desired space. 26e start with (D.1) by decompose φ λ as φ λ = δφ lin + φ Re with h φ lin , φ Re i L = 0 . On the subspace parallelling to φ λ and its orthogonal we derive two equations δ − φ Re = δ σ ( − ∆ + V + λ ) − P c ( φ lin + δ − φ Re ) σ +1 ( λ + e ) = δ σ h φ lin , ( φ lin + δ − φ Re ) σ +1 i L . (D.7)where, λ + e is a fixed small positive constant, and 1 − P c is the projection onto the L { φ lin } .We prove the existence of the solutions in appropriate Sobolev spaces by applying thecontraction mapping theorem. Its applicability is fairly routine except observing the map( − ∆ + V + λ ) − P c = ( − ∆ + λ ) − P c − ( − ∆ + λ ) V ( − ∆ + V + λ ) − P c : h x i − H → h x i − H is bounded. By the contraction mapping theorem it is easy to construct the small solutions δ − φ Re and δ σ and find they are functions of λ + e with differentiability C . (Actually if σ is an integer then the functions are analytic in λ + e .) The dependence on λ + e can bedisplayed by rewriting (D.7) δ − φ Re = δ σ ( − ∆ + V + λ ) − P c [( φ lin ) σ +1 + δ σ κ ](1 + O ( δ σ )) δ σ = ( λ + e )( R φ σlin ) − + C ( λ + e ) + O ( λ + e ) with κ being some function in h x i − H and C , a constant. From these it is easy to derive(D.1). The estimates (D.3) and (D.4) can be proved similarly.(D.2) is implied by (D.1). The estimates (D.5) and (D.6) are implied by their definitions(2.8) and (A.6) and the estimates (D.1)- (D.4).Recall the definitions of J N m,n , m + n = 2 , , , in (B.10)- (B.12). The desired estimatesare simple applications of (D.1)- (D.6). Since all the other functions are defined in terms of J N m,n and the estimates are straightforward, we omitted the details here. This completes theproof.
E The Estimate on
J N ≥ := J ~N − X m + n =2 , , J N m,n
In the section we estimate the remainder of
J N after expanding it to the fourth order. Recallthe definitions of
J N m,n , m + n = 2 , ,
4, in (B.10)- (B.12).
Proposition E.1. If σ = 1 then J N ≥ = Loc + N onLoc (E.1) with
N onLoc := [ R + R ] J ~R and kh x i Loc k L , kh x i Loc k L . | z | + | z | ( δ ∞ + | z | ) kh x i − R ≥ k + ( δ ∞ + | z | ) kh x i − ~R k , (E.2) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:28) J N ≥ , (cid:18) φ λ (cid:19)(cid:29)(cid:12)(cid:12)(cid:12)(cid:12) . δ ∞ | z | + δ ∞ | z |kh x i − R ≥ k + δ ∞ kh x i − R ≥ k . (E.3)27 roof. Recall the decomposition of ψ in (A.1) and the fact that the nonlinearity of (1.1)is cubic if σ = 1. Hence each term in Loc must be product of three terms taken from φ λ , a φ λλ , a φ λ , N X n =1 ( α n + p n ) ξ n , i N X n =1 ( β n + q n ) η n , R m,n and R ≥ . By considering all thepossibilities and using Proposition D.1 we obtain (E.2). The procedure is tedious but not dif-ficult, hence is omitted here. (E.3) follows easily easily from (E.2) and the fact φ λ = O ( δ ∞ ).This completes the proof.Now we study the cases σ > . Proposition E.2. If σ > then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) J ~N − X m + n =2 , J N m,n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) J ~N − X m + n =2 , J N m,n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L . | z | σ +1 + | z | + | z |kh x i − R ≥ k + | z | σ +1 ( kh x i − ~R k σσ + kh x i − ~R k σ σ ) + k ~R k σ +12 σ +1 + k ~R k σ +12(2 σ +1) (E.4) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)* J ~N ( ~R, z ) − X m + n =2 J N m,n , (cid:18) φ λ (cid:19)+(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . | z | σ +2 + | z | + δ σ − ∞ | z |kh x i − R ≥ k + kh x i − R ≥ k . (E.5) Proof.
As in the proof of the case σ = 1 the basic idea in proving (E.4) and (E.5) is toTaylor expand the function J N in z and ¯ z . What makes the present situation different is thatif σ N , then the nonlinearity | ψ | σ ψ is not smooth at ψ = 0. Technically the decompositionof ψ in (A.1) makes | ψ | σ to be of the form | ψ | σ ( x, t ) = | ( φ λ ) ( x ) + ǫ ( x, t ) | σ for some ǫ ∈ H ( R ) . Since the inequality | ǫ | < ( φ λ ) does not hold for all x ∈ R , we find thatafter expanding in ǫ to certain orders some undesired negative powers of φ λ will be encounteredif σ N . To prevent the negative powers from appearing in the final form we have to adoptsome tricks, namely compare the sizes of φ λ and ǫ and discuss several regimes.Now we start proving the proposition. Recall that | ψ | σ = [( φ λ ) + 2 φ λ ( α · ξ ) + I ] σ with I := 2 φ λ ( I − α · ξ ) + I + I , I and I defined in (A.3). To control the remainder of theexpansion around φ λ we consider separately two regimes( φ λ ) ≥ | φ λ ( α · ξ ) | + 2 | I | and ( φ λ ) < | φ λ ( α · ξ ) | + 2 | I | . For the second regime we have | J ~N − X m + n =2 , J N m,n | ≤ |
J ~N | + | X m + n =2 , J N m,n |≤ | z | σ +1 + | z | + | ~R | σ +1 . (E.6)For the first regime i.e. ( φ λ ) ≥ | φ λ ( α · ξ ) | + 2 | I | . (E.7)we only study one term ( φ λ ) σ +1 O ( ǫ ) with ǫ := ( φ λ ) − [2 φ λ ( α · ξ ) + I ] ≤ .
28t is easy to see that this term is the remainder after expanding ǫ to the third order.We claim that | ( φ λ ) σ +1 O ( ǫ ) | . | α · ξ | σ +1 + | α · ξ | + | I | σ + . (E.8)We compute directly to obtain | ( φ λ ) σ +1 O ( | ǫ | ) | . ( φ λ ) σ − I + ( φ λ ) σ − ( α · ξ ) . Apply (E.7) to control the first term on the right hand side( φ λ ) σ − I . | I | σ + . To bound the second term we have two possibilities: 2 σ − ≥ σ − < . For the firstwe have ( φ λ ) σ − | ( α · ξ ) | . ( α · ξ ) , hence (E.8) holds trivially; for the second apply (E.7) to obtain( φ λ ) σ − ( α · ξ ) = | α · ξ | σ +1 ( φ λ ) σ − | α · ξ | − σ . | α · ξ | σ +1 . By collecting the estimates above we prove (E.8).This completes the proof for the first regime, and moreover this together with (E.6)completes the proof for (E.4).Now we turn to (E.5). Here the function
J N has to be expanded to one more order toobtain the desired estimate. This is enabled by the fact the function was taken inner productwith function φ λ . The technique is similar to the proof of (E.4), hence we omit the detailshere. The proof is complete. F Proof of Proposition 4.1, Equations (4.17) , (4.10) and (4.11) Proposition 4.1 and Equations (4.17), (4.10) and (4.11) are included in Propositions F.1and F.2 below. Recall the definitions of δ ∞ , R ≥ in (4.1), (B.13) respectively. Proposition F.1. If | z | δ ∞ ≪ for σ = 1 and for σ > , | z | ≤ δ C ( σ ) ∞ with C ( σ ) sufficientlylarge, then the following results hold: ~R and R ≥ satisfy the estimates k ~R k . | z | , (F.1) k ~R k ∞ . | z | (1 + t ) − + δ σ − ∞ | z ( t ) | , (F.2) kh x i − ~R k H . | z | (1 + t ) − + δ σ − ∞ | z ( t ) | . (F.3) If σ = 1 then kh x i − R ≥ k . | z | (1 + t ) − + δ ∞ | z | | z ( t ) | ; (F.4) and if σ > then kh x i − R ≥ k . | z | (1 + t ) − + [ | z | + | z | σ − ] | z ( t ) | . (F.5)29he proposition will be proved shortly.We prepare for the proof by defining some functions. To bound various functions in theproposition we define the following estimating functions: M ( T ) := max ≤ t ≤ T k ~R ( t ) k ∞ [ | z | (1 + t ) − + δ σ − ∞ | z ( t ) | ] − , M ( T ) := max ≤ t ≤ T kh x i − ~R ( t ) k H [ | z | (1 + t ) − + δ σ − ∞ | z ( t ) | ] − , M ( T ) := max ≤ t ≤ T k ~R k | z | − . (F.6)For σ = 1 we define M ( T ) := max ≤ t ≤ T kh x i − R ≥ ( t ) k [ | z | (1 + t ) − + δ ∞ | z | | z ( t ) | ] − ;for σ > M ( T ) := max ≤ t ≤ T kh x i − R ≥ ( t ) k [ | z | (1 + t ) − + ( | z | + | z | σ − ) | z ( t ) | ] − . In the present paper we use | z | as a gauge to measure sizes of different functions. This makesit necessary to obtain lower and upper bounds for | z | . Recall the definition Γ( z, ¯ z ) ≡ Γ λ ( z, ¯ z )in (2.15). By Equation (D.1) and the assumption (FGR) there exist constants C ± such that C + δ σ − ∞ | z | ≤ z ∗ Γ λ ( z, ¯ z ) z ≤ C − δ σ − ∞ | z | . (F.7)We define the upper and lower bounds z ± by z ± ( t ) := ( | z | − + C ± δ σ − ∞ t ) − . (F.8)Recall the equations for ˙ λ , the definitions for Ψ and K in (4.16) and (4.7), and recall theequation for ˙ γ in (A.9), and the definition of Υ , in (A.6). In the rest of the paper we use Remainder to represent different terms satisfying the estimate | Remainder | . | z ( t ) | + kh x i − ~R k + | z ( t ) |kh x i − R ≥ k . (F.9)Define a constant ǫ ∞ by ǫ ∞ := max ≤ t< ∞ k ~R ( t ) k H . By (3.8) it is a small constant. The result is
Proposition F.2.
Suppose that | z | δ ∞ ≪ for σ = 1 and | z | = δ C ( σ ) ∞ for σ > with C ( σ ) beingsufficiently large. Thenif X k =1 M k ≤ ( δ ∞ + ǫ ∞ ) − and | z + | ≥ | z | ≥ | z − | in the time interval [0 , δ ] , then in the same interval the following results hold:(1) The function | z | admits lower and upper bounds | z − ( t ) | ≤ | z ( t ) | ≤ | z + ( t ) | for any time t. (F.10)30
2) The functions K , Ψ , ˙ λ and ˙ γ − Υ , satisfy the following estimates | Ψ | . | z | δ σ − ∞ kh x i − R ≥ k + kh x i − R ≥ k + δ σ − ∞ | z | , (F.11) if σ = 1 then | ˙ λ | , |K| . δ ∞ Remainder, (F.12) | ˙ γ − Υ , | . Remainder. (F.13) If σ > then | ˙ λ | , | ˙ γ − Υ , | , |K| . | z | σ +1 + Remainder. (F.14) (3) M ( T ) . δ ∞ [ M ( T ) + M ( T )] + ǫ ∞ M ( T ) , (F.15) M ( T ) . δ ∞ [ M ( T ) + M ( T )] + ǫ ∞ M ( T ) , (F.16) M ( T ) . M ( T ) M ( T ) + M ( T ) M ( T ) + δ ∞ ( M ( T ) + M ( T )) , (F.17) M ( T ) . | z | [ M ( T ) + M σ ( T ) M ( T )] . (F.18)The estimates (F.10)- (F.14), (F.17) and (F.18) of the proposition will be proved inlater sections. The proofs of the other two estimates are almost the same to the correspondingestimates of [6], specifically Propositions 11.3 and 11.5, pp. 299 and 300 respectively, henceare omitted here. Proof of Proposition F.1
By the local well-posedness of (1.1) there exists some interval[0 , δ ] such that for any T ∈ [0 , δ ] X k =1 M k ( T ) ≤ ( δ ∞ + ǫ ∞ ) − and 10 | z + | ≥ | z | ≥ | z − | . Then the estimates (F.10)- (F.18) hold in this time interval, which include5 | z + | ≥ | z | ≥ | z − | . (F.19)Now we turn to the estimates on M k , k = 1 , , , . By substituting estimates of M and M of (F.15) and (F.18) into (F.17) we obtain M ( T ) ≤ ǫ ∞ + δ ∞ + | z | ) P ( M ( T ) , · · · , M ( T ))with P ( x , · , x ) being some polynomial of positive coefficient. This together with the otherestimates implies M . ǫ ∞ + δ ∞ + | z | ) P ( M ) (F.20)with M := P k =1 M k and P being some polynomial.By the assumption on the initial condition we find X n =1 M n (0) .
1, which together with(F.20) implies M . , δ ] . (F.21)31F.21), (F.19) and the local wellposedness of (1.1) imply that (F.10)- (F.18) hold in alarger interval [0 , δ ] , so do (F.21) and (F.19).By induction on the time interval we prove (F.21) holds on the interval [0 , K ] for any K >
0, hence (F.21) and (F.10)- (F.14) hold in [0 , ∞ ) . To conclude the proof the fact M . , ∞ ) implies Proposition F.1.The proof is complete. (cid:3) F.1 Proof of (F.11) - (F.14) In this subsection we study the derivatives of the normal form transformation, which appearin K of (4.7) and Ψ of (4.16).Recall the equation for z in (2.9). Define Z , := − Γ( z, ¯ z ) z + Λ( z, ¯ z ) z (F.22)and define A := ∂ t a + iE ( λ ) X m + n =2 , ( m − n ) A (1) m,n + ∂ z a · Z , + ∂ ¯ z a · Z , ,A := ∂ t a + iE ( λ ) X m + n =2 , ( m − n ) A (2) m,n + ∂ z a · Z , + ∂ ¯ z a · Z , ,P k := ∂ t p k + iE ( λ ) X m + n =2 , ( m − n ) P ( k ) m,n + ∂ z p k · Z , + ∂ ¯ z p k · Z , ,Q k := ∂ t q k + iE ( λ ) X m + n =2 , ( m − n ) Q ( k ) m,n + ∂ z q k · Z , + ∂ ¯ z q k · Z , . The result is
Lemma F.1. |K| , | A | , | A | , | P k | , | Q k | . | z | ( | ˙ λ | + | ˙ γ − Υ , | ) + δ C ( σ ) ∞ Remainder (F.23) with C (1) = 1 , C ( σ ) = 0 if σ > , and k = 1 , , · · · , N. Proof.
The definitions of p k in (B.2), the equation for z in (4.7) imply | P k | ≤ | ˙ λ || ∂ λ p k | + | ∂ z p k ||K| ≤ | ˙ λ || z | + | z ||K| and | Q k | ≤ | ˙ λ || ∂ λ q k | + | ∂ z q k ||K| ≤ | ˙ λ || z | + | z ||K| . Apply Propositions D.1, E.1 and E.2 to obtain |K| ≤ ( | z | + kh x i − ~R k )( | ˙ λ | + | ˙ γ − Υ , | ) + δ C ( σ ) ∞ Remainder (F.24)with C (1) = 1 and C ( σ ) = 0 if σ >
1, which together with the estimate above implies theestimates for K , P k and Q k in (F.23).By almost identical arguments we produce the estimates for A and A . The proof is complete. 32 roof of (F.11) - (F.14) Proof.
We start with estimating ˙ λ, ˙ γ − Υ , : as in [6] pp. 291-293, we put the equations intoa matrix form to find[ Id + Π( z, ~R, a, p, q )] (cid:18) ˙ λ ˙ γ − Υ , (cid:19) = Ω + Remainder (cid:18) δ ∞ (cid:19) , (F.25)where, recall the definition of Remainder in (F.9), and the terms on the right hand side wereobtained by the following arguments:(1) the matrix Ω is defined asΩ := − ∂ t a + iE ( λ ) X m + n =2 , ( m − n ) A (1) m,n ,∂ t a + iE ( λ ) X m + n =2 , ( m − n ) A (2) m,n and is controlled by applying the results in Lemma F.1Ω = O [ | z | ( | ˙ λ | + | ˙ γ − Υ , | ) + δ ∞ Remainder ] + (cid:18) O ( δ ∞ | z | ) O ( | z | ) (cid:19) ; (F.26)(3) Id is the 2 × z, ~R, a, p, q ) := Π + Π s where Π s is a matrix depending on z, ~R, a, p and q, and satisfies the estimate k Π s ( z, ~R, a, p, q ) k . | z | + δ ∞ by the conditions X k =1 M k ( t ) ≤ ( δ ∞ + ǫ ∞ ) − of Proposition F.2; the matrix Π is definedand estimated as Π := − h R ,∂ λ φ λ ih φ λ ,∂ λ φ λ i , ! = O ( | z | ) . To prove this we used the observations that R = ReR ⊥ ∂ λ φ λ in (3.6) and ∂ λ φ λ and ∂ λ φ λ are almost colinear to each other, proved in (D.1). By these and (D.2) we obtain h φ λ ,∂ λ φ λ i |h R , ∂ λ φ λ i| ≤ δ − σ ∞ kh x i − ~R k . The assumption M ≤ δ − ∞ in Proposition F.2implies that kh x i − ~R k ≤ | z | . Consequently1 h φ λ , ∂ λ φ λ i |h R , ∂ λ φ λ i| . δ − σ ∞ | z | ≤ | z | . (F.27)(2) the term Remainder (cid:18) δ ∞ (cid:19) is produced by1 h φ λ , ∂ λ φ λ i Υ , h R , φ λ i + Υ , h q · η, φ λ i − h ImN − X m + n =2 , ImN m,n , φ λ i− Υ , h R , ∂ λ φ λ i − Υ , h p · ξ, ∂ λ φ λ i + h ReN − X m + n =2 , ReN m,n , ∂ λ φ λ i .
33o prove this we use the results in Propositions D.1 and E.1 and the fact h ImN , , φ λ i =0. Here the “almost orthogonality” or “almost colinear condition” between functions ξ k and η k , φ λ and ∂ λ φ λ , implied by Proposition D.1 were used to approximate theorthogonal conditions (2.5) and (3.6). An example is in proving (F.27) above. Weomit the details here.Now inverting the matrix [ Id + Π] − = Id − Π + O (Π )in (F.25) we obtain the desired estimates on ˙ λ and ˙ γ − Υ , , which are (F.12)- (F.14) exceptthese on K .The estimate on K are implied by (F.23) and (F.12)- (F.14).By similar arguments we prove (F.11).The proof is complete. F.2 Proof of Equation (F.10)
Proof.
As usual we only prove the case σ = 1, the cases σ > | z | ≤ δ C ( σ ) ∞ for some sufficiently large C ( σ ) . By (4.7) and the estimate for K in (F.23) ddt | z | = − Rez ∗ Γ( z, ¯ z ) z + 2 Rez ∗ · Remainder (F.28)with
Rez ∗ · Remainder satisfying the estimate | Rez ∗ · Remainder | . δ ∞ | z | Remainder . δ ∞ | z | + δ ∞ | z | [ | z | (1 + t ) − + δ ∞ | z | | z ( t ) | ] M + δ ∞ | z | [ | z | (1 + t ) − + δ ∞ | z | | z ( t ) | ] M . Observe that | Rez ∗ · Remainder | is NOT a higher order correction of Rez ∗ Γ( z, ¯ z ) z = O ( δ ∞ | z | ) in a neighborhood of t = 0. This forces us to divide the region t ∈ [0 , ∞ ) into twoparts t ≤ δ − ∞ | z | − and t > δ − ∞ | z | − . In the finite time interval we define | ˜ z | ( t ) := | z | ( t ) e − R t Rez ∗· Remainder | z | s ) ds ≈ | z | ( t ) . The assumptions | z | ≥ | z − ( t ) | & ( | z | − + δ ∞ t ) − and M ≤ δ − ∞ imply that for any time t ≤ δ − ∞ | z | − , 0 ≤ Z t | Rez ∗ · Remainder || z | ds = O ( | z | δ ∞ ) ≪ . (F.29)Moreover by (F.28) and the estimates of Rez ∗ Γ( z, ¯ z ) z in (F.7)34 C − δ ∞ ≤ ddt | ˜ z | − ≤ C + δ ∞ . (F.30)Consequently when t ≤ δ − ∞ | z | − we have the desired estimate2 | z + ( t ) | ≥ | ˜ z ( t ) | ≥ | z − ( t ) | , hence 3 | z + ( t ) | ≥ | z ( t ) | ≥ | z − ( t ) | (F.31)34here, recall the definitions of z ± ( t ) = ( | z | − + C ± δ ∞ t ) − in Equation (F.8).When t ≥ δ − ∞ | z | − we consider (F.28) in the regime [ δ − ∞ | z | − , ∞ ) with the initialcondition satisfying the estimate in (F.31). This is easier by the fact the second term in (F.28)is a true correction to the first: Use (1 + t ) − ≤ | z | − δ − ∞ + t ) − = 2 δ ∞ ( | z | − + δ ∞ t ) − tofind δ ∞ | z | ≫ | Rez ∗ · Remainder | . Thus there exists an 0 ≤ ǫ ≪ − (2 + ǫ ) Rez ∗ Γ( z, ¯ z ) z ≤ ddt | z | ≤ − (2 − ǫ ) Rez ∗ Γ( z, ¯ z ) z which together with the condition in (F.31) at t = δ − ∞ | z | − enables us to obtain (F.10).The proof is complete. Remark F.1.
In the proof above, specifically (F.29) , we used | z | ≪ δ ∞ = O ( k φ λ k ) to show δ ∞ | z | ≪ Rez ∗ Γ( z, ¯ z ) z . Actually this condition can be weaken to be | z | ≤ k φ λ k by refiningnormal form transformation: Namely examine closely the equation of z to find that ∂ t z + iE ( λ ) z = − Γ( z, ¯ z ) z + Λ( z, ¯ z ) z + X m + n =4 Z m,n + Remainder where Remainder satisfies the estimate | Remainder | . δ ∞ | z | + δ ∞ | z |kh x i − R ≥ k + δ ∞ kh x i − R ≥ k . The fourth order term Z m,n can be removed by choosing a new parameter ˜ z by ˜ z = z + X m + n =4 iE ( λ )( m − n − Z m,n ≈ z. By studying the equation for ˜ z we obtain the desired estimate. F.3 The Estimate of k ~R k : Proof of (F.18) We only prove the case σ = 1, the cases σ > | z | ≤ δ C ( σ ) ∞ for some sufficiently large C ( σ ) . By taking time derivative on k ~R k and using the equation for ~R in (A.2) we find ddt h ~R, ~R i = K + K with K n , n = 1 , , defined as K := h ( L ( λ ) + ˙ γJ ) ~R, ~R i + h ~R, ( L ( λ ) + ˙ γJ ) ~R i + ˙ λ h P cλ ~R, ~R i + ˙ λ h ~R, P cλ ~R i ; K := −h P λc J N ( ~R, z ) , ~R i − h ~R, P λc J N ( ~R, z ) i + h P λc G , ~R i + h ~R, P λc Gi . By the observation J ∗ = − J and the fact that J L ( λ ) is self-adjoint we cancel all thenonlocalized terms in K and obtain: | K | . kh x i − ~R k . Recall the definition of P λc G in ( A.2). By various estimates in Proposition D.1 and theestimates in previous subsections we obtain | K | . δ ∞ | z | kh x i − ~R k + kh x i − ~R k + k ~R k . | ddt h ~R, ~R i| . δ ∞ | z | kh x i − ~R k + kh x i − ~R k + k ~R k . δ ∞ | z | + kh x i − ~R k + k ~R k . Integrate the equation from 0 to T and use the fact k ~R (0) k . | z | to obtain k ~R ( T ) k . | z | + Z T δ ∞ | z | ( s ) + kh x i − ~R ( s ) k + k ~R ( s ) k ds. (F.32)Now we estimate the different terms inside integral.We start with the first term. By the assumption 10 | z + | ≥ | z | ≥ | z − | we obtain Z T δ ∞ | z ( s ) | ds . Z T δ ∞ ( | z | − + δ ∞ s ) − ds ≤ | z | . For the second term we use the definition ~R = X m + n =2 , R m,n + R ≥ to obtain kh x i − ~R k ≤ X m + n =2 , kh x i − R m,n k + kh x i − R ≥ k . The two terms on the right hand sides admit the following estimate.(1) By kh x i − R m,n k . δ ∞ | z | proved in Proposition D.1 and the assumption 10 | z + | ≥ | z | ≥ | z − | Z T kh x i − R m,n ( s ) k ds . δ ∞ Z ∞ ( | z | − + δ ∞ s ) − ds = | z | ;(2) By the definition of M Z T kh x i − R ≥ ( s ) k ds ≤ M ( T ) | z | . For the third term R T k ~R ( s ) k ds Z T k ~R ( s ) k ds ≤ Z T k ~R ( s ) k ∞ k ~R ( s ) k ds ≤ M ( T ) M ( T ) | z | . Collecting the estimates above we complete the proof.
F.4 The Estimate of kh x i − R ≥ k : Proof of (F.17) We only prove the case σ = 1, the cases σ > | z | ≤ δ C ( σ ) ∞ for some sufficiently large C ( σ ) . Use the definition of R ≥ in (B.13) and the equation (A.2) to derive an equation for R ≥ ∂ t R ≥ = L ( λ ) R ≥ + L ( ˙ λ, ˙ γ ) R ≥ − P λc J N ≥ + G (F.33)with the terms J N ≥ and G defined as: J N ≥ := J N ( ~R, p, z ) − X m + n =2 , J N m,n , := G − Υ , P λ ( t ) c (cid:18) β · η − α · ξ (cid:19) + P c S > , and P c S > := L ( ˙ λ, ˙ γ ) X m + n =2 , R m,n + X m + n =2 , [ L ( λ ) R m,n − ddt R m,n − P λc J N m,n ] . The functions in G take certain forms Lemma F.2.
There exist some functions φ ( m, n, k ) such that G = X m + n =2 , ,k =0 , , [ L ( λ ) + iE ( λ )( m − n ) − − k φ ( m, n, k ) (F.34) where φ ( m, n, k ) are smooth functions admitting the estimate kh x i φ ( m, n, k ) k . | ˙ λ || z | + | z || ˙ γ − Υ , | + δ ∞ | z | . (F.35) Proof. (F.34) is taken from [6], Theorem 8.1, p. 285.(F.35) is an improvement from [6] by the fact one has to find that the lowest order termin φ ( m, n, k ) is of the order δ ∞ | z | . By Proposition D.1 and direct computation we find itis generated by Υ , (cid:18) q · η − p · ξ (cid:19) , Υ , X m + n =2 J R m,n and part of X m + n =2 [ L ( λ ) R m,n − ddt R m,n − P λc J N m,n ]. The procedure is tedious, but easy. We omit the detail here.The proof is complete.Rewrite the equation for R ≥ in (F.33) as ∂ t R ≥ = L ( λ ) R ≥ + L ( ˙ λ, ˙ γ ) R ≥ − P λc ( Loc + N onLoc ) + P λc G (F.36)where, recall the definitions of Loc and
N onLoc in (E.1).Following the steps in [6], p. 302, we now derive an integral equation for P λ c R ≥ . Rewrite L ( λ ( t )) as L ( λ ( t )) = L ( λ ) + L ( λ ( t )) − L ( λ ) with λ := λ ( T ) for some fixed time T , andrewrite ( F.36) one more time to obtain ddt P λ c R ≥ = L ( λ ) P λ c R ≥ + [ ˙ γ + λ − λ ] i ( P + − P − ) R ≥ + P λ c O R ≥ + P λ c G − P λ c [ Loc + N onLoc ] . (F.37)Here for the terms on the right hand side we have(1) O is the operator defined by O := ˙ λP cλ + L ( λ ) − L ( λ ) + ˙ γP λc J − [ ˙ γ + λ − λ ] i ( P + − P − )and the function O R ≥ satisfies the estimate: when t ≤ T then apply Proposition F.2to obtain kh x i O R ≥ k . [ | ˙ λ | + | ˙ γ | + | λ − λ | ] kh x i − R ≥ k . [ | z | (1 + s ) − + δ ∞ | z | ( | z | − + δ ∞ s ) − [1 + δ ∞ M + δ ∞ M ](F.38)372) Recall that L ( λ ) has two branches of essential spectrum [ iλ, i ∞ ) and ( − i ∞ , − iλ ], we use P + and P − to denote the projection operators onto these two branches of the essentialspectrum of L ( λ ( T )).Then we have Lemma F.3.
For any function h and any large constant ν > we have (cid:13)(cid:13)(cid:13) h x i ν (cid:16) P λ c J − i ( P + − P − ) (cid:17) h (cid:13)(cid:13)(cid:13) ≤ c (cid:13)(cid:13) h x i − h (cid:13)(cid:13) . (F.39)The following estimates are taken from [6], Theorem 5.7, p. 280. Lemma F.4.
There exists a constant c such that if the parameter λ satisfies the estimate | λ + e | ≪ , then for any function h and t ≥ we have kh x i − e tL ( λ ) ( L ( λ ) ± ikE ( λ ) − − n P ± h k ≤ c (1 + t ) − kh x i h k (F.40) with n = 0 , , , k = 2 , ; kh x i − e tL ( λ ) P ± h k . (1 + | t | ) − ( k h k + k h k ) . (F.41)Apply the Duhamel’s principle on Equation ( F.37) and use the observation that the oper-ators P + , P − and L ( λ ) commute with each other to find kh x i − P λ c R ≥ k ≤ X l =1 A l , (F.42)with A := kh x i − e tL ( λ )+ ia ( t, P + − P − ) P λ c R ≥ (0) k where a ( t, s ) := R ts [ ˙ γ ( τ ) + λ ( τ ) − λ ] dτ ∈ R ,A := Z t kh x i − e ( t − s ) L ( λ )+ ia ( t,s )( P + − P − ) P λ c [ O R ≥ − P λc Loc ] k ds,A := Z t kh x i − e ( t − s ) L ( λ )+ ia ( t,s )( P + − P − ) P λ c P λc G ( s ) k ds, and A := Z t kh x i − e ( t − s ) L ( λ )+ ia ( t,s )( P + − P − ) P λ c N onLoc k ds. Now we estimate A k , k = 1 , , , . In what follows we use repeatedly the assumption10 | z + | ≥ | z | ≥ | z − | and the fact e ia ( t ,t )( P + − P − ) = e ia ( t ,t ) P + + e − ia ( t ,t ) P − : h x i k L → h x i k L is uniformly bounded for any k ∈ R . 381) By the propagator estimate (F.40) we have A ≤ kh x i − e tL ( λ ( T )) P ± ~R (0) k + kh x i − e tL ( λ ( T )) P ± X m + n =2 , R m,n (0) k . (1 + t ) − [ kh x i ~R (0) k + | z | ] . (1 + t ) − | z | . (2) Equation (F.43) below and the estimates of O R ≥ in (F.38), Loc in Proposition E.1imply that for any time t ≤ TA . R t (1 + t − s ) − [ | z | (1 + s ) − + δ ∞ | z | ( | z | − + δ ∞ s ) − [1 + δ ∞ M + δ ∞ M ] ds . [ | z | (1 + t ) − + δ ∞ | z | ( | z | − + δ ∞ t ) − ][1 + δ ∞ M + δ ∞ M ] . (3) Equation (F.40) and G of Equation (F.34) imply A . R t (1 + t − s ) − [ δ ∞ | z ( s ) | + Remainder ( s ) | z ( s ) | ] ds . [ | z | (1 + s ) − + δ ∞ | z | ( | z | − + δ ∞ s ) − ][1 + δ ∞ M + δ ∞ M ] . (4) We control A by the form of N onLoc in Proposition E.1 and the propagator estimate(F.41): A . R t (1 + t − s ) − [ k ~R ( s ) k + k ~R ( s ) k ] ds . R t (1 + t − s ) − [ k ~R ( s ) k k ~R k ∞ + k ~R ( s ) k ∞ k ~R k ] ds . | z | R t (1 + t − s ) − [(1 + s ) − / + δ ∞ ( | z | − + δ ∞ s ) − ] ds [ M M + M M ] . Apply (F.43) to obtain A . | z | [(1 + t ) − + δ ∞ ( | z | − + δ ∞ t ) − ][ M M + M M ] . Collecting the estimates above we have kh x i − R ≥ k . [ | z | (1+ t ) − + δ ∞ | z | ( | z | − + δ ∞ t ) − ][1+ M M + M M + δ ∞ ( M + M )] . The proof is complete by the definition of M in Equation (F.17). (cid:3) In the proof we used the following lemma.
Lemma F.5. Z t t − s ) ( | z | − + δ ∞ s ) − ds . ( | z | − + δ ∞ t ) − . (F.43) Proof.
We divide the regime into two parts 0 ≤ s ≤ t and t ≤ s ≤ t. For the latter Z t t t − s ) ( | z | − + δ ∞ s ) − ds ≤ | z | − + δ ∞ t ) − Z t t t − s ) ds . ( | z | − + δ ∞ t ) − . I : = R t t − s ) ( | z | − + δ ∞ s ) − ds ≤ √ t ) − R t ( | z | − + δ ∞ s ) − ds = 2 √ δ − ∞ (1 + t ) − ln [1 + | z | δ ∞ t ] . Change variable u = δ ∞ | z | t to obtain I . δ − ∞ (1 + δ − ∞ | z | − u ) − ln (1 + u ) . | z | u − (1 + u ) − ln (1 + u ) . | z | (1 + u ) − = ( | z | − + δ ∞ t ) − . The proof is complete.
G Proof of (4.9)
Proof.
In the next we study z ∗ Γ( z, ¯ z ) z defined in (2.15). To prepare for the proof we list afew estimates.(1) If n = − , − , , , L ( λ )+ inE ( λ )+0] − P c = [( − ∆+ V + λ ) J + inE ( λ )+0] − P linc (cid:18) (cid:19) + O ( kh x i φ λ k σH )(G.1)as operators mapping from space h x i − L ∞ to space h x i − L ∞ . This is resulted fromthe estimate of P c − P linc in (D.5) and the fact L ( λ ) defined in (2.3) is of the form L ( λ ) = ( − ∆ + V + λ ) J + O (( φ λ ) σ ) . (2) To diagonalize the matrix J we define a unitary 2 × U as U := 1 √ (cid:18) ii (cid:19) (G.2)which makes U ∗ J U = iσ with σ being the Pauli matrix.(3) Apply (D.4) to derive the leading orders from N X k =1 z k G k to obtain N X k =1 z k G k = J N , = σ ( φ λ ) σ − (cid:18) − i ( z · ξ )( z · η ) − (2 σ + 1)( z · ξ ) + ( z · η ) (cid:19) = − σδ σ − ∞ ( φ lin ) σ − ( z · ξ lin ) (cid:18) iσ (cid:19) + O ( δ σ − ∞ | z | ) . (G.3)40ow we begin perturbation-expanding the function z ∗ Γ( z, ¯ z ) z in the variable δ ( λ ) . Use(G.1)- (G.3) to obtain z ∗ Γ( z, ¯ z ) z = − σ δ σ − ( λ ) Re h [( − ∆ + V + λ ) J + 2 iE ( λ ) − − P linc ( φ lin ) σ − ( z · ξ lin ) (cid:18) iσ (cid:19) ,i ( φ lin ) σ − ( z · ξ lin ) J (cid:18) iσ (cid:19) i + O [ δ σ − ∞ | z | ]= − σ δ σ − ( λ ) Re h U ∗ [( − ∆ + V + λ ) J + 2 iE ( λ ) − − U P linc ( φ lin ) σ − ( z · ξ lin ) U ∗ (cid:18) iσ (cid:19) , ( φ lin ) σ − ( z · ξ lin ) U ∗ J i (cid:18) iσ (cid:19) i + O [ δ σ − ∞ | z | ] . In the new expression certain terms can be computed explicitly: U ∗ [( − ∆ + V + λ ) J + 2 iE ( λ ) − − U = (cid:18) − i [ − ∆ + V + λ + 2 E ( λ )] − i [ − ∆ + V + λ − E ( λ ) − i − (cid:19) , (G.4)and iU ∗ J (cid:18) iσ (cid:19) = 1 √ (cid:18) i ( σ − σ + 1 (cid:19) , U ∗ (cid:18) iσ (cid:19) = 1 √ (cid:18) i (1 − σ ) σ + 1 (cid:19) (G.5)Put this back into the expression to find z ∗ Γ( z, ¯ z ) z = z ∗ Γ ( z, ¯ z ) z + H , + O ( δ σ − ∞ | z | ) (G.6)with H , := − δ σ − ( λ ) σ ( σ − × Re h i ( − ∆ + V + λ + 2 E ( λ )) − P linc ( φ lin ) σ − ( z · ξ lin ) , ( φ lin ) σ − ( z · ξ lin ) i . We observe that H , = 0 (G.7)by the fact that the operator ( − ∆ + V + λ + 2 E ( λ )) − P linc is self-adjoint, hence that h ( − ∆ + V + λ + 2 E ( λ )) − P linc Ω , Ω i is real for any Ω . Hence (G.6) is the desired estimate and the proofis complete.
H Proof of Equation (4.19)
Recall the ideas present after Lemma 4.2, which basically is that the functions Π m,n , m = n are almost periodic with period πE ( m − n ) .Compute directly to obtain R ∞ Π m,n ds = R ∞ Π m,n + dds [ iE ( λ )( m − n ) Π m,n ] ds − R ∞ dds [ iE ( λ )( m − n ) Π m,n ] ds = R ∞ ˙ λ∂ λ ( iE ( λ )( m − n ) Π m,n ) + iE ( λ )( m − n ) ∂ z Π m,n · [ ˙ z + iE ( λ ) z ]+ iE ( λ )( m − n ) ∂ ¯ z Π m,n · [ ˙¯ z − iE ( λ )¯ z ] ds + iE ( λ )( m − n ) Π m,n | t =0 . (H.1)41t is easy to see that the last term satisfies the estimate1 iE ( λ )( m − n ) Π m,n | t =0 = o ( | z | ) . Collecting the estimates above we prove (4.19).The proof is complete. (cid:3)
I Proof of Equation (4.18)
To facilitate our discussions we define ρ := z · ξ, ω := z · η. Recall the definition of Π , in (4.16). By direct computation we findΠ , = −h J N , , (cid:18) φ λ (cid:19) i + Υ , h R , , (cid:18) φ λ (cid:19) i + Υ , N X n =1 Q ( n )1 , h φ λ , η n i . (I.1)Further expand the first term to obtain (cid:28) J N , , (cid:18) φ λ (cid:19)(cid:29) = X n =1 (Φ n + Φ n ) + Ω σ> (I.2)with Φ := (cid:28) R , , σ ( φ λ ) σ − (cid:18) − i (2 σ − ρω − ω + (2 σ − ρ (cid:19)(cid:29) , Φ := * A (1)2 , ∂ λ φ λ + N X n =1 P ( n )2 , ξ n A (2)2 , φ λ + N X n =1 Q ( n )2 , η n , σ ( φ λ ) σ − (cid:18) − i (2 σ − ρω − ω + (2 σ − ρ (cid:19)+ , Φ := * N X n =1 P ( n )1 , ξ nN X n =1 Q ( n )1 , η n + R , , σ ( φ λ ) σ − (cid:18) i (2 σ − ρ ¯ ω − ¯ ρω ) ω ¯ ω + (2 σ − ρ ¯ ρ (cid:19)+ , Φ := * N X n =1 P ( n )2 , ξ nN X n =1 Q ( n )2 , η n + A (1)2 , ∂ λ φ λ A (2)2 , φ λ ! , (cid:18) − iσ ( φ λ ) σ ωσ ( φ λ ) σ ρ (cid:19)+ , Φ := (cid:28) R , , (cid:18) − iσ ( φ λ ) σ ωσ ( φ λ ) σ ρ (cid:19)(cid:29) , := * σ ( φ λ ) σ [ A (1)2 , ∂ λ φ λ + N X n =1 P ( n )2 , ξ n + R (1)2 , ] , [ A (2)2 , φ λ + N X n =1 Q ( n )2 , η n + R (2)2 , ] + , Φ := σ * ( φ λ ) σ [ N X n =1 P ( n )1 , ξ n + R (1)1 , ] , N X n =1 Q ( n )1 , η n + R (2)1 , + , Ω σ> only appears in σ > σ> := 18 i σ ( σ − σ − h ( φ λ ) σ − | ρ | , ρ ¯ ω − ¯ ρω i − i σ ( σ − h ( φ λ ) σ − | ω | , ¯ ωρ − ¯ ρω i . The terms defined above satisfy the following estimate:
Lemma I.1. X n =2 | Φ n + Φ n | + | Υ , h R (2)1 , , φ λ i| + | Υ , d X n =1 Q ( n )1 , h φ λ , η n i| + | Ω σ> | . δ σ − ∞ | z | , (I.3)The lemma will be proved in the later part of this section.Now we prove (4.18). Proof of Equation (4.18) Collecting the estimates in Equations (I.1), (I.2) and LemmaI.1 we have Π , = − Re h R , , K , i + O ( δ σ − ∞ | z | ) (I.4)with K , := σ ( φ λ ) σ − (cid:18) − i (2 σ − ρω − ω + (2 σ − ρ (cid:19) = σ ( φ λ ) σ − ρ (cid:18) − i (2 σ − σ − (cid:19) + O [ δ σ − ∞ | z | ]= σ δ σ − ( λ ) φ lin ( z · ξ lin ) (cid:18) − i (2 σ − σ − (cid:19) + O [ δ σ − ∞ | z | ]and recall δ ( λ ) defined in (2.1).Use (G.1) to expand R , in the space h x i L ∞ R , = ( L ( λ ) + 2 iE ( λ ) − − P c J N , = [( − ∆ + V + λ ) J + 2 iE ( λ ) − − P linc J N , + O ( δ σ − ∞ | z | ) . The estimate for
J N , in (G.3) implies − Re h R , , K , i = M , + O ( δ σ − ∞ | z | )with M , := 2 σ δ σ − ( λ ) h [( − ∆ + V + λ ) J + 2 iE ( λ ) − − P linc ( φ lin ) σ − ( z · ξ lin ) (cid:18) iσ (cid:19) ,φ lin ( z · ξ lin ) (cid:18) − i (2 σ − σ − (cid:19) i To make the expression easier we diagonalize the matrix operator [( − ∆+ V + λ ) J +2 iE ( λ ) − − which is essentially to diagonalize the matrix J . Recall the definition of the unitary matrix U in (G.2). Put U U ∗ = Id into appropriate places to obtain M , = 2 σ δ σ − ( λ ) h U ∗ [( − ∆ + V + λ ) J + 2 iE ( λ ) − − U P linc ( φ lin ) σ − ( z · ξ lin ) U ∗ (cid:18) iσ (cid:19) ,φ lin ( z · ξ lin ) U ∗ (cid:18) − i (2 σ − σ − (cid:19) i .
43 few terms in the expression can be computed explicitly: U ∗ [( − ∆ + V + λ ) J + 2 iE ( λ ) − − U was computed in (G.4), U ∗ (cid:18) iσ (cid:19) = 1 √ (cid:18) i (1 − σ )1 + σ (cid:19) , U ∗ (cid:18) − i (2 σ − σ − σ (cid:19) = − √ (cid:18) i ( σ − σ + 1 (cid:19) . Put this back into the expression of M , to obtain M , = 12 z ∗ Γ ( z, ¯ z ) z + ˜ M , (I.5)with ˜ M , := − δ σ − ( λ ) σ ( σ − × Re h i ( − ∆ + V + λ + 2 E ( λ )) − P linc ( φ lin ) σ − ( z · ξ lin ) , ( φ lin ) σ − ( z · ξ lin ) i , where, recall the definition of z ∗ Γ ( z, ¯ z ) z in (2.17). Observe ˜ M , = 0 by the same argumentas in proving (G.7). Hence − Re h R , , K , i = 12 z ∗ Γ ( z, ¯ z ) z + O ( δ σ − ∞ | z | ) . These together with (I.4) imply Equation (4.18).The proof is complete. (cid:3)
In the rest of this section we prove Lemma I.1 by considering the cases σ = 1 and σ > Proof of Lemma I.1 for σ = 1In what follows we only study the term Φ and part of Φ . The estimates on the other termsare similar and easier.We start with analyzing
J N , since the terms are related to it. The definition in (B.11)and the fact k ρ − ω k H = O ( δ ∞ | z | ) imply that in the space h x i − H J N , = − ρ ¯ ρ (cid:18) i (cid:19) + O ( δ ∞ | z | ) . (I.6)Now we turn to Φ . Recall the definition of R , from (B.9). By the facts P c (cid:18) ρ (cid:19) = P c (cid:18) ω (cid:19) = 0 and ρ − ω = O ( δ ∞ | z | ) in (D.4) we obtainΥ , P c (cid:18) iωρ (cid:19) = Υ , P c (cid:18) i ( ω − ρ ) ρ − ω (cid:19) = O ( δ ∞ | z | ) . This together with (I.6), (G.1) implies that R , = − [( − ∆ + V + λ ) J + iE ( λ )] − P linc ρ ¯ ρ (cid:18) i (cid:19) + O ( δ ∞ | z | ) . = Θ , + O ( δ ∞ | z | ) (I.7)with Θ , := − h [( − ∆ + V + λ ) J + iE ( λ )] − P linc ρ ¯ ρ (cid:18) i (cid:19) , ( φ λ ) ρ (cid:18) − i (cid:19) i . Now we claim Θ , = 0which trivially implies the desired estimate on Φ .To prove the claim we diagonalize the matrix J to obtain a convenient form. Recall thedefinition of the unitary matrix U in (G.2). Insert U U ∗ = Id into appropriate places and usethe fact U ∗ J U = iσ to obtainΘ , = h iσ [( − ∆ + V + λ ) Id + σ E ( λ )] − P linc ρ ¯ ρU ∗ (cid:18) i (cid:19) , ( φ λ ) ρU ∗ (cid:18) − i (cid:19) i = h iσ [( − ∆ + V + λ ) Id + σ E ( λ )] − P linc ρ ¯ ρ (cid:18) (cid:19) , ( φ λ ) ρ (cid:18) − i (cid:19) i = 0 . (I.8)This last line follows from the observations that the column vector functions (cid:18) (cid:19) and (cid:18) i (cid:19) are ‘disjoint’, and the operator σ [( − ∆ + V + λ ) Id + σ E ( λ )] − is diagonal.Now we choose a ‘difficult’ term in Φ to study: D , := h A (1)2 , ∂ λ φ λ A (2)2 , φ λ ! , (cid:18) − i ( φ λ ) ω ( φ λ ) ρ (cid:19) i . Put the definitions of A (1)2 , and A (2)2 , in (B.2) into the expression to find D , = iE ( λ ) h φ λ ,∂ λ φ λ i D (1)2 , − E ( λ ) h φ λ ,∂ λ φ λ i D (2)2 , + A (1)2 , iE ( λ ) h φ λ , ( φ λ ) ρ i . with D (1)2 , := h N Im , , φ λ ih ∂ λ φ λ , − i ( φ λ ) ω i − h N Re , , ∂ λ φ λ ih φ λ , ( φ λ ) ρ i and D (2)2 , := Υ , [ h ω, φ λ ih ∂ λ φ λ , − i ( φ λ ) ω i + i h ρ, ∂ λ φ λ ih φ λ , ( φ λ ) ρ i ]= Υ , [ h ω − ρ, φ λ ih ∂ λ φ λ , − i ( φ λ ) ω i + i h ρ − ω, ∂ λ φ λ ih φ λ , ( φ λ ) ρ i ]where in the last step the facts φ λ ⊥ ρ, ∂ λ φ λ ⊥ ω in (2.5) are used.Now we estimate all the three terms in the definition of D , .It is easy to see the third term is of the order O ( δ ∞ | z | ) by the estimate of A (1)2 , in Propo-sition D.1.By the estimate ρ − ω = O ( δ ∞ | z | ) it is not hard to obtain D (2)2 , , − E ( λ ) h φ λ , ∂ λ φ λ i D (2)2 , = O ( δ ∞ | z | ) .
45o estimate D (1)2 , we use the definitions of N Re , and N Im , in (B.11) and various estimatesin Proposition D.1 to find N Im , = − i ρ ¯ ρ + O ( δ ∞ | z | ) , N Re , = − ρ ¯ ρ + O ( δ ∞ | z | ) . Put this into the expression of D (1)2 , and use (2.1) and (D.1) on φ λ and ∂ λ φ λ , after thecancelation of the terms of order δ ∞ | z | we obtain D (1)2 , , iE ( λ ) h φ λ , ∂ λ φ λ i D (1)2 , = O ( δ ∞ | z | ) . Collecting all the estimates we obtain D , = O ( δ ∞ | z | ) . The proof is complete. (cid:3)
Proof of Lemma I.1 for σ > σ > σ = 1, forexample (I.8), do not hold any more. Instead we use an important observation resulted fromTheorem 3.2 whose second statement requires that ξ linn = x n | x | ξ ( | x | ), n = 1 , , · · · , N = d, forsome function ξ lin . By Lemma 2.2 this implies that (cid:18) ξ n iη n (cid:19) = x n | x | (cid:18) ξ ( | x | ) iη ( | x | ) (cid:19) for some real functions ξ and η .This makes Ω σ> = 0 by observing ξ m η n − ξ n η m = 0 . In estimating the other terms on the right hand side of (I.3) we only study Φ , theestimation on the other terms are similar.After some manipulation similar to that in (I.7) we findΦ = iC ( σ ) D + iC ( σ ) D + O ( δ σ − ∞ | z | ) for some C , C ∈ R , (I.9)where D and D are defined as D := h [ − ∆ + V + λ + E ( λ )] − P linc ( φ λ ) σ − ρ ¯ ρ, ( φ λ ) σ ρ i D := h [ − ∆ + V + λ − E ( λ )] − P linc ( φ λ ) σ − ρ ¯ ρ, ( φ λ ) σ ρ i . We claim that D , D ∈ R . If the claim holds then it together with (I.9) yields the desired estimate Re Φ = O ( δ σ − ∞ | z | ) . Now we prove the claim for D , the proof for D is almost the same. The facts ξ k = x k | x | ξ ( | x | )and the potential V and φ λ are spherically symmetric imply D = X k,l ≤ N = d, k = l z k [¯ z l ] D ( k, l ) + X k,l ≤ N = d, k = l | z k | | z l | D ( k, l ) + X k ≤ N | z k | D ( k, k )= D (1 , X k,l ≤ N = d, k = l z k [¯ z l ] + D (1 , X k,l ≤ N = d, k = l | z k | | z l | + D (1 , X k ≤ N | z k | D ( k, l ) := h [ − ∆ + V + λ + E ( λ )] − P linc ( φ λ ) σ − ξ k ξ l , ( φ λ ) σ ξ l i . The last line follows fromthe observations D ( k, l ) = D (1 , ∈ R if k = l and D ( k, k ) = D (1 , ∈ R resulted from thepermutation of coordinates. By observing X k,l ≤ N = d, k = l z k [¯ z l ] = | X k z k | − X l | z l | ∈ R wehave D ∈ R .The proof is complete. (cid:3) References [1] R. Boyd.
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