aa r X i v : . [ m a t h - ph ] O c t On long-time decay for modified Klein-Gordon equation
E. A. Kopylova Institute for Information Transmission Problems RASB.Karetnyi 19, Moscow 101447,GSP-4, Russia e-mail: [email protected]
Abstract
We obtain a dispersive long-time decay in weighted energy norms for solutions of theKlein-Gordon equation in a moving frame. The decay extends the results of Jensen, Katoand Murata for the equations of the Schr¨odinger type. We modify the approach to makeit applicable to relativistic equations.
Keywords : Klein-Gordon equation, relativistic equations, resolvent, spectral representa-tion, weighted spaces, Born series, convolution. : 35L10, 34L25, 47A40, 81U05. Supported partly by FWF, DFG and RFBR grants. n long-time decay for Klein-Gordon equation In this paper, we establish a dispersive long time decay in weighted energy norms for thesolutions to 1D Klein-Gordon equation in a moving frame with the velocity v ˙Ψ( t ) = A Ψ( t ) (1.1)where Ψ( t ) = (cid:18) ψ ( t ) π ( t ) (cid:19) , A = (cid:18) v ∇ − m − V v ∇ (cid:19) , ∇ = ddx , ∆ = d dx with m >
0, and | v | <
1. For s, σ ∈ R , we denote by H sσ = H sσ ( R ) the weighted Agmon-Sobolevspaces [1], with the finite norms k ψ k H sσ = kh x i σ h∇i s ψ k L ( R ) < ∞ , h x i = (1 + | x | ) / Denote L σ = H σ . We assume that V ( x ) is a real function, and | V ( x ) | + | V ′ ( x ) | ≤ C h x i − β , x ∈ R (1.2)for some β >
5. Then the multiplication by V ( x ) is bounded operator H s → H s + β for any s ∈ R .We consider the “nonsingular case” in the terminology of [9], when the truncated resolventof the operator − ∆ + γ V ( x ), γ = 1 / √ − v is bounded at the edge point ζ = 0 of thecontinuous spectrum. In other words, the point ζ = 0 is neither eigenvalue nor resonance for the operator − ∆ + γ V ( x ) (1.3)By definition (see [9, page 18]) the point ζ = 0 is the resonance if there exists a nonzero solution ψ ∈ L − / − \ L to the equation ( − ∆ + γ V ( x )) ψ = 0. Definition 1.1. F σ is the complex Hilbert space H σ ⊕ H σ of vector-functions Ψ = ( ψ, π ) withthe norm k Ψ k F σ = k ψ k H σ + k π k H σ < ∞ Our main result is the following long time decay of the solutions to (1.1): in the nonsingularcase, the asymptotics hold kP c Ψ( t ) k F − σ = O ( | t | − / ) , t → ±∞ (1.4)for initial data Ψ = Ψ(0) ∈ F σ with σ > /
2, where P c is a Riesz projection onto the continuousspectrum of the operator A . The decay is desirable for the study of asymptotic stability andscattering for the solutions to nonlinear hyperbolic equations.Let us comment on previous results in this direction. The decay of type (1.4) in weightednorms has been established first by Jensen and Kato [6] for the Schr¨odinger equation in thedimension n = 3. The result has been extended to all other dimensions by Jensen and Nenciu[4, 5, 7], and to more general PDEs of the Schr¨odinger type by Murata [9].The Jensen-Kato-Murata approach is not applicable directly to the relativistic equations.The difference reflects distinct character of wave propagation in the relativistic and nonrela-tivistic equations (see the discussion in [8, Introduction]). n long-time decay for Klein-Gordon equation v = 0. The approach develops the Jensen-Kato-Murata techniques tomake it applicable to the relativistic equations. Namely, we apply the finite Born series andconvolution. Here we extend the result [8] to the case v = 0.Our paper is organized as follows. In Section 2 we obtain the time decay for the solution tothe free modified Klein-Gordon equation and state the spectral properties of the free resolvent..In Section 3 we obtain spectral properties of the perturbed resolvent and prove the decay (1.4). Here we consider the free equation with zero potential V ( x ) = 0:˙Ψ( t ) = A Ψ( t ) (2.1)where A = (cid:18) v ∇ − m v ∇ (cid:19) For t > = Ψ(0) ∈ F , the solution Ψ( t ) to (2.1) admits the spectral Fourier-Laplacerepresentation θ ( t )Ψ( t ) = − π Z R e ( iω + ε ) t R ( iω + ε )Ψ dω, t ∈ R (2.2)with any ε >
0, where θ ( t ) is the Heaviside function, R ( λ ) = ( A − λ ) − for Re λ > A . The representation follows from the stationary equation λ ˜Ψ + ( λ ) = A ˜Ψ + ( λ ) + Ψ for the Fourier-Laplace transform ˜Ψ + ( λ ) := Z R θ ( t ) e − λt Ψ( t ) dt , Re λ >
0. The solution Ψ( t ) is continuous bounded function of t ∈ R with the values in F by theenergy conservation for the equation (2.1). Hence, ˜Ψ + ( λ ) = −R ( λ )Ψ is analytic functionin Re λ > F , and bounded for Re λ > ε . Therefore, the integral (2.2)converges in the sense of distributions of t ∈ R with the values in F . Similarly to (2.2), θ ( − t )Ψ( t ) = 12 π Z R e ( iω − ε ) t R ( iω − ε )Ψ dω, t ∈ R (2.3)Let us calculate the resolvent R ( λ ). We have R ( λ ) = ( A − λ ) − = (cid:18) v ∇ − λ − m v ∇ − λ (cid:19) − , Re λ > (cid:18) − ( ivk + λ ) 1 − ( k + m ) − ( ivk + λ ) (cid:19) − = [( ivk + λ ) + k + m ] − (cid:18) − ( ivk + λ ) − k + m − ( ivk + λ ) (cid:19) n long-time decay for Klein-Gordon equation R ( λ ) = (cid:18) v ∇ − λ − − ∆ + m v ∇ − λ (cid:19) R ( λ ) = (cid:18) ( v ∇ − λ ) R ( λ ) − R ( λ )1 − ( v ∇ − λ ) R ( λ ) ( v ∇ − λ ) R ( λ ) (cid:19) (2.4)where R ( λ ) is the operator with the integral kernel R ( λ, x, y ) = F − k → x − y k + m + ( ivk + λ ) , x, y ∈ R (2.5)which is well defined since the denominator in (2.5) does not vanish for Re λ >
0. Denote H = − (1 − v )∆ + m = − γ ∆ + m . Since( H + λ − vλ ∇ ) ψ ( x ) = e − γ vλx ( H + γ λ ) e γ vλx ψ ( x ) (2.6)we have R ( λ ) = ( H + λ − vλ ∇ ) − = e − γ vλx γ ˜ R ( γ m + γ λ ) e γ vλy (2.7)where ˜ R ( ζ ) == ( − ∆ + ζ ) − = Op h e −√ ζ | z | √ ζ i is the Schr¨odinger resolvent. Finally, R ( λ, x, y ) = e − γ ( √ λ − µ | x − y | + vλ ( x − y )) p λ − µ , µ = imγ (2.8)Denote Γ := ( − i ∞ , − µ, ) ∪ ( µ, i ∞ ). We choose Re p λ − µ > λ ∈ C \ Γ. Then0 < Re ( vλ ) < Re p λ − µ , λ ∈ C \ Γ (2.9)Denote by L ( B , B ) the Banach space of bounded linear operators from a Banach space B toa Banach space B . Formulas (2.8) implies the following properties of R ( λ ): Lemma 2.1. (cf. [1, 9])i) The operator R ( λ ) is analytic function of λ ∈ C \ Γ with the values in L ( H , H ) .ii) For λ ∈ Γ , the convergence (limiting absorption principle) holds R ( λ ± ε ) → R ( λ ± , ε →
0+ (2.10) in L ( H σ , H − σ ) with σ > / , uniformly in | λ | ≥ | µ | + r for any r > .iii) The asymptotics hold R ( λ ) = B ± √ ν + B ± + O ( | ν | / ) , ν = λ ∓ µ → , λ ∈ C \ Γ (2.11) in L ( H σ , H − σ ) with σ > / , where B ± = Op h e ∓ γ vµ ( x − y ) √± µ i ∈ L ( H σ , H − σ ) , σ > / B ± = Op h − γ e ∓ γ vµ ( x − y ) | x − y | i ∈ L ( H σ , H − σ ) , σ > / n long-time decay for Klein-Gordon equation iv) The asymptotics ( ) can be differentiated two times: R ′ ( λ ) = − B ± ν √ ν + O (cid:0) | ν | − / (cid:1) , R ′′ ( λ ) = O (cid:0) | ν | − / (cid:1) , ν = λ ∓ µ → , λ ∈ C \ Γ (2.13) in L ( H σ , H − σ ) with σ > / .v) For s ∈ R , l = − , , , , k = 0 , , , ... and σ > / k the decay holds k R ( k )0 ( λ ) k L ( H sσ ,H s + l − σ ) = O ( | λ | − (1 − l ) ) , | λ | → ∞ , λ ∈ C \ Γ (2.14)
Proof.
We prove the properties ii) and v) since other properties follow directly from (2.8).
Step i)
First, we prove the convergence (2.10). The norm of the operator R ( λ ) : H σ → H − σ isequivalent to the norm of the operator h x i − σ R ( λ ) h y i − σ : L → H The norm of the latter operator does not exceed the sum in k , k = 0 ,
1, of the norms of operators ∂ kx [ h x i − σ R ( λ, x, y ) h y i − σ ] : L → L (2.15)According (2.8) and (2.9), | ∂ kx R ( λ, x, y ) | ≤ C ( λ ) , k = 0 , , x, y ∈ R , λ ∈ C \ ΓHence for σ > / X k Z | ∂ kx [ h x i − σ R ( λ, x, y ) h y i − σ ] | dxdy ≤ C ( λ ) Z h x i − σ h y i − σ dxdy ≤ C ( λ )The estimate implies that Hilbert-Schmidt norms of operators (2.15) is finite. For λ ∈ Γ and x, y ∈ R , there exists the pointwise limit R ( λ ± ε, x, y ) → R ( λ ± , x, y ) , ε → X k Z | ∂ kx [ h x i − σ R ( λ ± ε, x, y ) h y i − σ − h x i − σ R ( λ ± ε, x, y ) h y i − σ ] | dxdy → , ε → Step ii)
Now we prove the decay (2.14). It suffices to verify the case s = 0 since R ( λ ) commuteswith the operators h∇i s with arbitrary s ∈ R . For k = 0 and l = 0 , , k = 0 and l = − R ( λ ) = 1 m + λ (cid:0) R ( λ ) γ + 2 vλ ∇ R ( λ ) (cid:1) (2.16)Namely, using (2.14) with l = 0 and l = 1, we obtain k∇ R ( λ ) k L ( H σ ,H − − σ ) = O ( | λ | − ) , k ∆ R ( λ ) k L ( H σ ,H − − σ ) = O (1)hence (2.16) implies k R ( λ ) k L ( H σ ,H − − σ ) = O ( | λ | − )In the case k = 0 the bounds (2.14) follow similarly by differentiating (2.8). n long-time decay for Klein-Gordon equation Corollary 2.2. i) The resolvent R ( λ ) is analytic function of λ ∈ C \ Γ with the values in L ( F , F ) .ii) For λ ∈ Γ , the convergence (limiting absorption principle) holds R ( λ ± ε ) → R ( λ ± , ε →
0+ (2.17) in L ( F σ , F − σ ) with σ > / .iii) The asymptotics hold R ( λ ) = B ± √ ν + B ± + O ( | ν | / ) , ν = λ ∓ µ → , λ ∈ C \ Γ (2.18) in L ( F σ , F − σ ) with σ > / , where B ± = B ± (cid:18) ∓ iγm − γ m ∓ iγm (cid:19) ∈ L ( F σ , F − σ ) with σ > / and B ± ∈ L ( F σ , F − σ ) with σ > / .iv) The asymptotics ( ) can be differentiated two times: R ′ ( λ ) = −B ± ν √ ν + O ( | ν | − / ) , R ′′ ( λ ) = O ( | ν | − / ) , ν = λ ∓ µ → , λ ∈ C \ Γ (2.20) in L ( F σ , F − σ ) with σ > / .v) For k = 0 , , , ... and σ > / k the asymptotics hold kR ( k )0 ( λ ) k L ( F σ , F − σ ) = O (1) , | λ | → ∞ , λ ∈ C \ Γ (2.21)Denote by G v ( t ) the dynamical group of equation (2.1). Corollary 2.3.
For t ∈ R and Ψ ∈ F σ with σ > / , the group G v ( t ) admits the integralrepresentation G v ( t )Ψ = 12 πi Z Γ e λt h R ( λ − − R ( λ + 0) i Ψ dλ (2.22) where the integral converges in the sense of distributions of t ∈ R with the values in F − σ .Proof. Summing up the representations (2.2) and (2.3), and sending ε → For the integral kernel of the operator G v ( t ) we have G v ( x − y, t ) = G ( x − y − vt, t ) , x, y ∈ R , t ∈ R (2.23) n long-time decay for Klein-Gordon equation G ( z, t ) = (cid:18) ˙ G ( z, t ) G ( z, t )¨ G ( z, t ) ˙ G ( z, t ) (cid:19) , G ( z, t ) = 12 θ ( t − | z | ) J ( m √ t − z ) , z = x − y (2.24)where J is the Bessel function. The relation (2.23) implies the Huygen’s principle for the group G v ( t ), i.e. G v ( x − y, t ) = 0 , | x − y − vt | > t Also, the relation (2.23) implies the energy conservation for the group G v ( t ). Namely, forΨ( t ) = ( ψ ( · , t ) , π ( · , t )) = G v ( t )Ψ we have Z [ | π ( x, t ) + v · ∇ ψ ( x, t ) | + |∇ ψ ( x, t ) | + m | ψ ( x, t ) | ] dx = const , t ∈ R In particular, this gives that k Ψ( t ) k F ≤ C k Ψ k F , t ∈ R We represent G v ( z, t ) as G v ( z, t ) = G b ( z, t ) + G r ( z, t ) , z ∈ R , t ≥ G b ( z, t ) := 1 p mπt/γ − mγ sin[ m ( tγ + γvz ) − π ] cos[ m ( tγ + γvz ) − π ] − m γ cos[ m ( tγ + γvz ) − π ] − mγ sin[ m ( tγ + γvz ) − π ] (2.25)The entries of the matrix G b ( z, t ) admit the bounds |G ijb ( z, t ) | ≤ C ( v ) / √ t, i, j = 1 , , z ∈ R , t ≥ G v ( t ) slow decays, like t − / . We will show that G b ( t ) = Op[ G b ( x − y, t )] is only termresponsible for the slow decay. More exactly, in the next section we will prove the followingbasic proposition Proposition 2.4.
The decay holds G r ( t ) = Op[ G r ( x − y, t )] = O ( t − / ) , t → ∞ (2.27) in the norm of L ( F σ , F − σ ) with σ > / . The following key observation is that (2.25) contains just two frequencies ± µ which are theedge points of the continuous spectrum. This suggests that the term G b ( t ) with “bad decay” t − / should not contribute to the high energy component of the group G v ( t ) and the highenergy component of the group G v ( t ) decays like t − / .More precisely, let us introduce the low energy and high energy components of G v ( t ): G l ( t ) = 12 πi Z Γ e λt l ( iλ ) h R ( λ − − R ( λ + 0) i dλ (2.28) G h ( t ) = 12 πi Z Γ e λt h ( iλ ) h R ( λ − − R ( λ + 0) i dλ (2.29)where l ( ω ) ∈ C ∞ ( R ) is an even function, l ( ω ) = 0 if | ω | > | µ | + 2 ε , and l ( ω ) = 1 if | ω | ≤ | µ | + ε with an ε >
0, and h ( ω ) = 1 − l ( ω ). n long-time decay for Klein-Gordon equation Theorem 2.5. In L ( F σ , F − σ ) with σ > / the decay holds G h ( t ) = O ( t − / ) , t → ∞ (2.30) Proof.
We deduce asymptotics (2.30) from Proposition 2.4.
Step i)
Let Ψ ∈ F σ . DenoteΨ + ( t ) = θ ( t ) G v ( t )Ψ , Ψ + b ( t ) = θ ( t ) G b ( t )Ψ , Ψ + h ( t ) = θ ( t ) G h ( t )Ψ , Ψ + r ( t ) = θ ( t ) G r ( t )Ψ Then Ψ + h ( t ) = − π Z R e iωt h ( ω ) R ( iω + 0)Ψ dω = 12 π Z R e iωt h ( ω ) ˜Ψ + ( iω ) dω = 12 π Z R e iωt h ( ω ) h ˜Ψ + b ( iω ) + ˜Ψ + r ( iω ) i dω = Ψ + r ( t ) + 12 π Z R e iωt h ( ω ) ˜Ψ + b ( iω ) dω − π Z R e iωt l ( ω ) ˜Ψ + r ( iω ) dω (2.31)where ˜Ψ + ( λ ) = ∞ R e − λt Ψ + ( t ) dt and so on. By (2.27) k Ψ + r ( t ) k F − σ = O ( t − / ) , t → ∞ (2.32) Step ii)
Let us consider the second summand in the last line of (2.31). By (2.25) the vectorfunction ˜Ψ + b ( iω ) is a smooth function for | ω | > | µ | + ε , and ∂ kω ˜Ψ + b ( iω ) = O ( | ω | − / − k ), k =0 , , ... , ω → ∞ . Hence partial integration implies that (cid:13)(cid:13)(cid:13) Z R e iωt h ( ω ) ˜Ψ + b ( iω ) dω (cid:13)(cid:13)(cid:13) F − σ = O ( t − N ) , ∀ N ∈ N , t → ∞ (2.33) Step iii)
Finally, let us consider the third summand in the last line of (2.31). Introducing thefunction L ( t ) such that ˜ L ( λ ) = l ( iλ ), we obtain12 π Z R e iωt l ( ω ) ˜Ψ + r ( iω ) dω = [ L ⋆ Ψ + r ]( t ) = O ( t − / ) , t → ∞ (2.34)in the norm of F − σ , since L ( t ) = O ( t − N ), t → ∞ for any N ∈ N , and k Ψ + r ( t ) k F − σ = O ( t − / )by (2.27). Finally, (2.31)- (2.34) imply (2.30). Let us fix an arbitrary ε ∈ ( | v | , ε = ε − | v | . For any t ≥ ∈ F σ in two terms, Ψ = Ψ ′ ,t + Ψ ′′ ,t , Ψ ′ ,t = ( ψ ′ ,t , π ′ ,t ), Ψ ′′ ,t = ( ψ ′′ ,t , π ′′ ,t ), such that k Ψ ′ ,t k F σ + k Ψ ′′ ,t k F σ ≤ C k Ψ k F σ , t ≥ n long-time decay for Klein-Gordon equation ′ ,t ( x ) = 0 for | x | > ε t , and Ψ ′′ ,t ( x ) = 0 for | x | < ε t G r ( t )Ψ ′ ,t and G r ( t )Ψ ′′ ,t separately. Step i)
First we consider G r ( t )Ψ ′′ ,t = G v ( t )Ψ ′′ ,t − G b ( t )Ψ ′′ ,t . Using energy conservation andproperties (2.35)- (2.36) we obtain kG v ( t )Ψ ′′ ,t k F − σ ≤ kG v ( t )Ψ ′′ ,t k F ≤ C k Ψ ′′ ,t k F ≤ C ( ε ) t − σ k Ψ ′′ ,t k F σ ≤ C ( ε ) t − / k Ψ k F σ , t ≥ σ > /
2. Further, (2.26) and the Cauchy inequality imply | ( G b ( t ) π ′′ ,t )( y ) | ≤ C √ t (cid:12)(cid:12)(cid:12) Z π ′′ ,t ( x ) dx (cid:12)(cid:12)(cid:12) ≤ C √ t (cid:16) Z | π ′′ ,t ( x ) | (1 + x ) σ dx (cid:17) / (cid:16) ∞ Z ε t/ dx (1 + x ) σ (cid:17) / ≤ C ( ε ) √ t t − σ +1 / k π ′′ ,t k H σ ≤ C ( ε ) t − / k π ′′ ,t k H σ , t ≥ kG b ( t ) π ′′ ,t k H − σ ≤ C ( ε ) t − / k π ′′ ,t k H σ . The functions G b ( t ) π ′′ ,t and G i b ( t ) ψ ′′ ,t , i = 1 , kG b ( t )Ψ ′′ ,t k F − σ ≤ C ( ε ) t − / k Ψ k F σ , t ≥ kG r ( t )Ψ ′′ ,t k F − σ ≤ C ( ε ) t − / k Ψ k F σ , t ≥ Step ii)
Denote by ζ the operator of multiplication by the function ζ ( | x | /t ), where ζ = ζ ( s ) ∈ C ∞ ( R ), ζ ( s ) = 1 for | s | < ε / ζ ( s ) = 0 for | s | > ε /
2. Obviously, for any k , we have | ∂ kx ζ ( | x | /t ) | ≤ C ( ε ) < ∞ , t ≥ − ζ ( | x | /t ) = 0 for | x | < ε t/
4, then by the energy conservation and (2.35), we obtain || (1 − ζ ) G v ( t )Ψ ′ ,t || F − σ ≤ C ( ε ) t − σ ||G v ( t )Ψ ′ ,t || F ≤ C ( ε ) t − σ || Ψ ′ ,t || F ≤ C ( ε ) t − / || Ψ || F σ , t ≥ | ( G b ( t ) π ′ ,t )( y ) | ≤ C √ t (cid:12)(cid:12)(cid:12) Z π ′ ,t ( x ) dx (cid:12)(cid:12)(cid:12) ≤ C √ t k π ′ ,t k H σ Hence, we obtain k (1 − ζ ) G b ( t ) π ′ ,t k H − σ ≤ C √ t k π ′ ,t k H σ (cid:16) ∞ Z ε t/ dy (1 + y ) σ (cid:17) / ≤ C ( ε ) t − / k π ′ ,t k H σ The functions (1 − ζ ) G b ( t ) π ′ ,t and (1 − ζ ) G i b ( t ) ψ ′ ,t , i = 1 , || (1 − ζ ) G b ( t )Ψ ′ ,t || F − σ ≤ C ( ε ) t − / || Ψ || F σ , t ≥ n long-time decay for Klein-Gordon equation || (1 − ζ ) G r ( t )Ψ ′ ,t || F − σ ≤ C ( ε ) t − / || Ψ || F σ , t ≥ Step iii)
Finally, let us estimate ζ G r ( t )Ψ ′ ,t . Let χ t be the characteristic function of the ball | x | ≤ ε t/
2. We will use the same notation for the operator of multiplication by this characteristicfunction. By (2.36), we have ζ G r ( t )Ψ ′ ,t = ζ G r ( t ) χ t Ψ ′ ,t (2.44)The matrix kernel of the operator ζ G r ( t ) χ t is equal to G ′ r ( x − y, t ) = ζ ( | x | /t ) G r ( x − y, t ) χ t ( y )Well known asymptotics of the Bessel function [10] imply the following lemma, which we provein Appendix. Lemma 2.6.
For any ε ∈ ( | v | , the bounds hold | ∂ kz G r ( z, t ) | ≤ C ( ε )(1 + z ) t − / , | z | ≤ ( ε − | v | ) t, t ≥ , k = 0 , ζ ( | x | /t ) = 0 for | x | > ε t/ χ t ( y ) = 0 for | y | > ε t/ G ′ r ( x − y, t ) = 0 for | x − y | > ε t = ( ε − | v | ) t . Hence, (2.45) imply that | ∂ kx G ′ r ( x − y, t ) | ≤ C ( ε )(1 + ( x − y ) ) t − / , k = 0 , , t ≥ ζ G r ( t ) χ t : F σ → F − σ is equivalent to the norm of the operator h x i − σ ζ G r ( t ) χ t ( y ) h y i − σ : F → F The norm of the later operator does not exceed the sum in k , k = 0 , ∂ kx [ h x i − σ ζ G r ( t ) χ t ( y ) h y i − σ ] : L ( R ) ⊕ L ( R ) → L ( R ) ⊕ L ( R ) (2.47)The bounds (2.46) imply that the Hilbert-Schmidt norms of operators (2.47) do not exceed C ( ε ) t − / since σ > /
2. Hence, (2.35) and (2.44) imply that || ζ G r ( t )Ψ ′ ,t || F − σ ≤ C ( ε ) t − / || Ψ ′ ,t || F σ ≤ C ( ε ) t − / || Ψ || F σ , t ≥ ||G r ( t )Ψ ′ ,t || F − σ ≤ C ( ε ) t − / || Ψ || F σ , t ≥ n long-time decay for Klein-Gordon equation Now we consider the resolvent of the perturbed equation. We use the formula R ( λ ) = (1 + R ( λ ) V ) − R ( λ ) , V = (cid:18) − V (cid:19) (3.1)By (2.4) we have(1 + R ( λ ) V ) − = R ( λ ) V − ( v ∇− λ ) R ( λ ) V − = (1 + R ( λ ) V ) − v ∇− λ ) (cid:16) − (1 + R ( λ ) V ) − (cid:17) (3.2)Let us denote H = − (1 − v )∆ + m + V, R ( λ ) = ( H + λ − vλ ∇ ) − = (1 + R ( λ ) V ) − R ( λ )Substituting (3.2) into (3.1) we obtain R ( λ ) = (1 + R ( λ ) V ) − v ∇ − λ ) (cid:16) − (1 + R ( λ ) V ) − (cid:17) ( v ∇ − λ ) R ( λ ) − R ( λ )( − ∆ + m ) R ( λ ) ( v ∇ − λ ) R ( λ ) = R ( λ )( v ∇ − λ ) − R ( λ )1 − ( v ∇ − λ ) R ( λ )( v ∇ − λ ) ( v ∇ − λ ) R ( λ ) (3.3)Similarly (2.6)-(2.7), we obtain( H + λ − vλ ∇ ) ψ ( x ) = e − γ vλx ( H + γ λ ) e γ vλx ψ ( x ) (3.4) R ( λ ) = e − γ vλx γ ˜ R ( γ m + γ λ ) e γ vλy (3.5)where ˜ R ( ζ ) = ( − ∆ + ζ + V γ ) − is the resolvent of the Schr¨odinger operator − ∆ + V γ . To prove the long time decay for the perturbed equation, we first establish the spectral prop-erties of the generator.
Let the potential V satisfy ( ). Theni) R ( λ ) is meromorphic function of λ ∈ C \ Γ with the values in L ( H , H ) ;ii) For λ ∈ Γ , the convergence holds R ( λ ± ε ) → R ( λ ± , ε →
0+ (3.6) in L ( H σ , H − σ ) with σ > / , uniformly in | λ | ≥ | µ | + r for any r > . n long-time decay for Klein-Gordon equation Proof. Step i)
The statement i) follows from Lemma 2.1-i), the Born splitting R ( λ ) = R ( λ )(1 + V R ( λ )) − (3.7)and the Gohberg-Bleher theorem [2, 3] since V R ( λ ) is a compact operator in L for λ ∈ C \ Γ. Step ii)
The convergence (3.6) follow from (2.10) by the Born splitting (3.7) if[1 +
V R ( λ ± ε )] − → [1 + V R ( λ ± − , ε → +0 , λ ∈ Γin L ( H σ ; H σ ). This convergence holds if and only if both limit operators 1 + V R ( λ ±
0) areinvertible in H σ for λ ∈ Γ. The operators are invertible according to the reversibility of theoperators 1 + γ V ˜ R ( ζ ± i
0) in H σ for ζ < V R ( λ ±
0) = e − γ vλx (cid:0) γ V ˜ R ( γ m + γ ( λ ± i ) (cid:1) e γ vλy which follows from (2.7).Formula (3.3) and Proposition 3.1 imply Corollary 3.2.
Let the conditions ( ) holds. Theni) R ( λ ) is meromorphic function of λ ∈ C \ Γ with the values in L ( F , F ) ;ii) For λ ∈ Γ , the convergence holds R ( λ ± ε ) → R ( λ ± , ε →
0+ (3.8) in L ( F σ , F − σ ) with σ > / . For k = 0 , , , s = 0 , and l = − , , with s + l ∈ { , } the asymptotics hold k R ( k ) ( λ ± k L ( H sσ ,H s + l − σ ) = O ( | λ | − (1 − l + k ) ) , | λ | → ∞ , λ ∈ Γ (3.9) with σ > / k .Proof. The decay follows from formula (3.5) and the known decay of Schr¨odinger resolvent ˜ R ( ζ )(see [1, 6, 9, 8]). Corollary 3.4.
For k = 0 , , and σ > / k the asymptotics hold kR ( k ) ( λ ± k L ( F σ , F − σ ) = O (1) , | λ | → ∞ , λ ∈ Γ (3.10)The resolvents R ( λ ) and R ( λ ) are related by the Born perturbation series R ( λ ) = R ( λ ) − R ( λ ) VR ( λ ) + R ( λ ) VR ( λ ) VR ( λ ) , λ ∈ C \ [Γ ∪ Σ] (3.11)where Σ is the set of eigenvalues of the operator A . An important role in (3.11) plays theproduct W ( λ ) := VR ( λ ) V . Now we obtain the asymptotics of W ( λ ) for large λ . Lemma 3.5.
Let k = 0 , , , and the potential V satisfy ( ) with β > / k + σ where σ > . Then the asymptotics hold kW ( k ) ( λ ) k L ( F − σ , F σ ) = O ( | λ | − ) , | λ | → ∞ , λ ∈ C \ Γ (3.12) n long-time decay for Klein-Gordon equation Proof.
Asymptotics (3.12) follow from the algebraic structure of the matrix W ( k ) ( λ ) = VR ( k )0 ( λ ) V = (cid:18) − V R ( k )0 ( λ ) V (cid:19) since (2.14) with s = 1 and l = − k V R ( k )0 ( λ ) V f k H σ ≤ C k R ( k )0 ( λ ) V f k H σ − β = O ( | λ | − ) k V f k H β − σ = O ( | λ | − ) k f k H − σ since β − σ > / k . The asymptotics hold R ( λ ) = B ± + O ( ν / ) R ′ ( λ ) = O ( ν − / ) R ′′ ( λ ) = O ( ν − / ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν := λ ∓ µ → , λ ∈ C \ Γ (3.13) in the norm L ( F σ , F − σ ) with σ > / , where B ± ∈ L ( F σ , F − σ ) does not depend on λ . First we prove the boundedness of the resolvent near the points ± µ . Lemma 3.7.
Let the conditions ( ) and ( ) hold. Then the families {R ( ± µ + ε ) : ± µ + ε ∈ C \ Γ , | ε | < δ } are bounded in the operator norm of L ( F σ , F − σ ) for any σ > / and sufficientlysmall δ .Proof. Let us consider the equation for eigenfunctions of operator A with eigenvalues λ = ± µ : (cid:18) v ∇ − m − V v ∇ (cid:19) (cid:18) ψπ (cid:19) = ± µ (cid:18) ψπ (cid:19) , Ψ = (cid:18) ψπ (cid:19) ∈ F From the first equation we have π = − ( v ∇ ∓ µ ) ψ . Then the second equation becomes( H + µ ∓ vµ ∇ ) ψ = e ∓ iγvmx ( − γ ∆ + V ) e ± iγvmx ψ = 0 (3.14)Hence, the condition (1.3) implies that Ψ = 0. Similarly, (1.3) implies that the equation A Ψ = ± µ Ψ has no nonzero solutions Ψ ∈ F − / − . Then the required boundedness of theresolvent near the points ± µ follows similarly to [9, Theorem 7.2 ].This lemma implies that the operators (1 + R ( λ ) V ) − = 1 − R ( λ ) V and (1 + VR ( λ )) − =1 − VR ( λ ) are bounded in L ( F − σ , F − σ ) and in L ( F σ , F σ ) respectively for | λ ∓ µ | < δ , λ ∈ C \ Γ.Now we prove more detailed asymptotics
Lemma 3.8.
The asymptotics hold (1+ R ( λ ) V ) − B ± = O ( √ ν ) , B ± (1+ VR ( λ )) − = O ( √ ν ) , ν = λ ∓ µ → , λ ∈ C \ Γ (3.15) in L ( F σ , F − σ ) with σ > / . n long-time decay for Klein-Gordon equation Proof.
The asymptotics (2.18) implies R ( λ ) = (cid:0) R ( λ ) V (cid:1) − R ( λ ) = (cid:0) R ( λ ) V (cid:1) − (cid:0) B ± √ ν + O (1) (cid:1) R ( λ ) = R ( λ ) (cid:0) VR ( λ ) (cid:1) − = (cid:0) B ± √ ν + O (1) (cid:1)(cid:0) VR ( λ ) (cid:1) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν = λ ∓ µ → , λ ∈ C \ ΓHence, the boundedness R ( λ ), (1 + R ( λ ) V ) − and (1 + VR ( λ )) − at the points λ = ± µ incorresponding norms imply the asymptotics (3.15). Corollary 3.9. i) The asymptotics hold k (1 + R ( λ ) V ) − [ e ∓ iγvmx ] k F − σ = O ( √ ν ) , ν = λ ∓ µ → , λ ∈ C \ Γ , σ > / ii) For any f ∈ F σ with σ > / Z e ± iγvmx [(1 + VR ( λ )) − f ]( x ) dx = O ( √ ν ) , ν = λ ∓ µ → , λ ∈ C \ Γ (3.17)
Proof of Proposition 3.13 . Taking into account the identities R ′ = (1 + R V ) − R ′ (1 + VR ) − , R ′′ = h (1 + R V ) − R ′′ − R ′ VR ′ i (1 + VR ) − we obtain from (2.20) and (3.16)-(3.17) the asymptotics (3.13) for the derivatives. The asymp-totics (3.13) for R ( λ ) follows by integration of asymptotics for R ′ ( λ ). Proposition 3.13 isproved. Corollary 3.10.
Let the conditions ( ) and ( ) hold. Then the set Σ of eigenvalues of theoperator A is finite, i.e. Σ = { λ j , j = 1 , ..., N } . Our main result is
Theorem 3.11.
Let conditions ( ) and ( ) hold. Then k e t A − X ω j ∈ Σ e λ j t P j k L ( F σ , F − σ ) = O ( | t | − / ) , t → ±∞ (3.18) with σ > / , where P j are the Riesz projections onto the corresponding eigenspaces.Proof. Corollaries 3.2 and 3.4 and Proposition 3.6 imply similarly to (2.22), thatΨ( t ) − X λ j ∈ Σ e λ j t P j Ψ = 12 πi Z Γ e λt h R ( λ − − R ( λ + 0) i Ψ dλ = Ψ l ( t ) + Ψ h ( t )where P j Ψ := 12 πi Z | λ − λ j | = δ R ( λ )Ψ dλ with a small δ >
0, and low and high energy componentsare defined by Ψ l ( t ) = 12 πi Z Γ l ( iλ ) e λt h R ( λ − − R ( λ + 0) i Ψ dλ (3.19)Ψ h ( t ) = 12 πi Z Γ h ( iλ ) e λt h R ( λ − − R ( λ + 0) i Ψ dλ (3.20)where l ( iλ ) and h ( iλ ) are defined in Section 2.2. We analyze Ψ l ( t ) and Ψ h ( t ) separately. n long-time decay for Klein-Gordon equation We prove the desired decay of Ψ l ( t ) using a special case of Lemma 10.2 from [6]. We consideronly the integral (3.19) over ( µ, µ + 2 iε ). The integral over ( − µ − iε, − µ ) is dealt with in thesame way. Denote by B a Banach space with the norm k · k . Lemma 3.12.
Let F ∈ C ([ a, b ] , B ) , satisfy F ( a ) = F ( b ) = 0 , k F ′′ ( ω ) k = O ( | ω − a | − / ) , ω → a Then b Z a e − itω F ( ω ) dω = O ( t − / ) , t → ∞ Due to (3.13), we can apply Lemma 3.12 with ω = − iλ , F = l ( ω ) (cid:0) R ( iω − − R ( iω + 0) (cid:1) , B = L ( F σ , F − σ ), a = | µ | , b = | µ | + 2 ε and σ > /
2, to get k Ψ l ( t ) k F − σ ≤ C (1 + | t | ) − / k Ψ k F σ , t ∈ R , σ > / Let us substitute the series (3.11) into the spectral representation (3.20) for Ψ h ( t ):Ψ h ( t ) = 12 πi Z Γ e λt h ( iλ ) h R ( λ − − R ( λ + 0) i Ψ dλ + 12 πi Z Γ e λt h ( iλ ) h R ( λ − VR ( λ − − R ( λ + 0) VR ( λ + 0) i Ψ dλ + 12 πi Z Γ e λt h ( iλ ) h R VR VR ( λ − − R VR VR ( λ + 0) i Ψ dλ = Ψ h ( t ) + Ψ h ( t ) + Ψ h ( t ) , t ∈ R We analyze each term Ψ hk , k = 1 , , Step i)
The first term Ψ h ( t ) = G h ( t )Ψ by (2.29). Hence, Theorem 2.5 implies that k Ψ h ( t ) k F − σ ≤ C (1 + | t | ) − / k Ψ k F σ , t ∈ R , σ > / Step ii)
Now we consider the second term Ψ h ( t ). Denote h ( ω ) = p h ( ω ) (we can assume that h ( ω ) ≥ h ∈ C ∞ ( R )). We setΦ h = 12 πi Z Γ e λt h ( iλ ) h R ( λ − − R ( λ + 0) i Ψ dλ It is obvious that for Φ h the inequality (3.21) also holds. Namely, k Φ h ( t ) k F − σ ≤ C (1 + | t | ) − / k Ψ k F σ , t ∈ R , σ > / h ( t ) can be written as a convolution. n long-time decay for Klein-Gordon equation Lemma 3.13. (cf. [8, Lemma 3.11]) The convolution representation holds Ψ h ( t ) = t Z G h ( t − τ ) V Φ h ( τ ) dτ, t ∈ R (3.22) where the integral converges in F − σ with σ > / . Applying Theorem 2.5 with h instead of h to the integrand in (3.22), we obtain that kG h ( t − τ ) V Φ h ( τ ) k F − σ ≤ C kV Φ h ( τ ) k F σ ′ (1 + | t − τ | ) / ≤ C k Φ h ( τ ) k F σ ′− β (1 + | t − τ | ) / ≤ C k Ψ k F σ (1 + | t − τ | ) / (1 + | τ | ) / where σ ′ ∈ (5 / , β − / τ , we obtain by (3.22) that k Ψ h ( t ) k F − σ ≤ C (1 + | t | ) − / k Ψ k F σ , t ∈ R , σ > / Step iii)
Let us rewrite the last term Ψ h ( t ) asΨ h ( t ) = 12 πi Z Γ e λt h ( iλ ) N ( λ )Ψ dλ, where N ( λ ) := M ( λ − − M ( λ + 0) for λ ∈ Γ, and M ( λ ±
0) := R ( λ ± VR ( λ ± VR ( λ ±
0) = R ( λ ± W ( λ ± R ( λ ± , λ ∈ ΓThe asymptotics (2.21), (3.10) and (3.12) for R ( k )0 ( λ ± R ( k ) ( λ ±
0) and W ( k ) ( λ ±
0) imply
Lemma 3.14. (cf.[8, Lemma 3.12]) For k = 0 , , the asymptotics hold kM ( k ) ( λ ± k L ( F σ , F − σ ) = O ( | λ | − ) , | λ | → ∞ , λ ∈ Γ , σ > / k Finally, we prove the decay of Ψ h ( t ). By Lemma 3.14( h N ) ′′ ∈ L (( − i ∞ , − µ − iε ) ∪ ( µ + iε, i ∞ ); L ( F σ , F − σ ))with σ > /
2. Hence, two times partial integration implies that k Ψ h ( t ) k F − σ ≤ C (1 + | t | ) − k Ψ k F σ , t ∈ R This completes the proof of Theorem 3.11.
Corollary 3.15.
The asymptotics ( ) imply ( ) with the projection P c = 1 − P d , P d = X ω j ∈ Σ P j n long-time decay for Klein-Gordon equation A Proof of Lemma 2.6
Formulas (2.23)- (2.24) imply G v ( z, t ) = ˜ G b ( z, t ) + ˜ G r ( z, t )where˜ G b ( z, t ) = θ ( t −| z − vt | ) √ mπ − mt sin( m p t − ( z − vt ) − π ) p ( t − ( z − vt ) ) cos( m p t − ( z − vt ) − π ) p t − ( z − vt ) − m t cos( m p t − ( z − vt ) − π ) p ( t − ( z − vt ) ) − mt sin( m p t − ( z − vt ) − π ) p ( t − ( z − vt ) ) For ε ∈ ( | v | ,
1) and | z | ≤ ( ε − | v | ) t we have | z − vt | ≤ εt . Hence | ∂ kz ˜ G r ( z, t ) | ≤ C ( ε ) t − / , | z | ≤ ( ε − | v | ) t, k = 0 , Q ( z, t ) = ˜ G b ( z, t ) − G b ( z, t ). Let us consider the entry Q ( t, z ): Q ( t, z ) = 1 √ πm h cos( m p t − ( z − vt ) − π ) p t − ( z − vt ) − cos( m ( tγ + γvz ) − π ) p t/γ i For | z | ≤ ( ε − | v | ) t we have (cid:12)(cid:12)(cid:12) p t − ( z − vt ) − p t/γ (cid:12)(cid:12)(cid:12) = | z − vtz | p t − ( z − vt ) p t/γ (cid:0) p t − ( z − vt ) + p t/γ (cid:1)(cid:0)p t − ( z − vt ) + t/γ (cid:1) ≤ C ( ε ) | z | t √ t Further, (cid:12)(cid:12)(cid:12) cos (cid:16) m p t − ( z − vt ) − π (cid:17) − cos (cid:16) mγ ( t + γ vz ) − π (cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) sin (cid:16) m p t − ( z − vt ) − t + γ vzγ (cid:17)(cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12)(cid:12)p t − ( z − vt ) − ( t + γ vz ) /γ (cid:12)(cid:12)(cid:12) ≤ C z (1 + γ v ) | p t − ( z − vt ) + ( t + γ vz ) /γ | ≤ C ( ε ) z t since γ | v || z | ≤ (1 − | v | ) t/ (1 − v ) ≤ t/ (1 + | v | ) ≤ t . Hence, | Q ( t, z ) | ≤ C ( ε )(1 + z ) t − / , | z | ≤ ( ε − | v | ) t (A.23)Differentiating Q ( t, z ), we obtain for | z | ≤ ( ε − | v | ) t∂ z Q ( t, z ) = z − vt √ πm cos( m p t − ( z − vt ) − π )2 p ( t − ( z − vt ) ) + r m π z sin( m p t − ( z − vt ) − π ) p ( t − ( z − vt ) ) + r m π vt h − sin( m p t − ( z − vt ) − π ) p ( t − ( z − vt ) ) + sin( m ( tγ + γvz ) − π ) p ( t/γ ) i Hence, by the arguments above, | ∂ z Q ( t, z ) | ≤ C ( ε )(1 + z ) t − / , | z | ≤ ( ε − | v | ) t (A.24)Other entries Q ij ( t, z ) also admit the estimates of type (A.23) and (A.24). Hence, the lemmafollows since G r ( t ) = ˜ G r ( t ) + Q ( t, z ). n long-time decay for Klein-Gordon equation References [1] Agmon S., Spectral properties of Schr¨odinger operator and scattering theory,
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