On the dynamics of the mean-field polaron in the weak-coupling limit
aa r X i v : . [ m a t h - ph ] S e p On the dynamics of the mean-field polaronin the weak-coupling limit
Marcel Griesemer, Jochen Schmid, Guido Schneider
Fachbereich Mathematik, Universit¨at Stuttgart, D-70569 Stuttgart, Germanyfi[email protected]
We consider the dynamics of the mean-field polaron in the weak-couplinglimit of vanishing electron-phonon interaction, ε →
0. This is a singularlimit formally leading to a Schr¨odinger–Poisson system that is equivalent tothe nonlinear Choquard equation. By establishing estimates between theapproximation obtained via the Choquard equation and true solutions of theoriginal system we show that the Choquard equation makes correct predic-tions about the dynamics of the polaron mean-field model for small valuesof ε > In Pekar’s model of (large) polarons a single electron interacts with a dielectric polar-izable elastic medium. The polarization of the medium by the charge of the electroncreates an electrostatic potential that, in turn, acts on the electron. In the stationarycase this leads to a self-trapped state of the electron called polaron [DA09, Lie77]. Inthe non-stationary case the electron triggers harmonic oscillation of the elastic mediumand the combined system is described by the coupled equations i∂ t u = − ∆ u + vu, (1) ε ∂ t v = − v + ∆ − | u | , (2)for the wave function u = u ( x, t ) ∈ C of the electron and the electrostatic potential v = v ( x, t ) ∈ R associated with the polarization of the medium. Here x ∈ R , t ∈ R ,and ∆ − | u | = − (4 π | · | ) − ∗ | u | . The parameter ε > ε ∂ t v in (2) leads to retardation in the self-interaction, whichmakes it difficult to predict the evolution of the electron. We are therefore interested1n the question whether (1)-(2) may be solved approximately by dropping ε ∂ t v if ε issmall. This approach leads to the Schr¨odinger–Poisson system i∂ t U = − ∆ U + V U, (3)0 = − V + ∆ − | U | , (4)which is equivalent to the Choquard equation i∂ t U = − ∆ U + (∆ − | U | ) U. (5)Such nonlinear Hartree-type equations admit an interpretation as infinite-dimensionalHamiltonian systems, and stationary points of the associated Hamilton functionals leadto solitary-wave solutions [FTY02, MZ10]. In the present case, assuming the legitimacyof letting ε →
0, solitary-wave solutions describe frictionlessly moving polarons. TheChoquard equation (5) in a different context describes the evolution of coherent states(condensates) of bosons in the mean-field limit, and it has been proposed as a model forgravity-induced decoherence [EY01, Pen98]. Similar Hartree-type nonlinear equationsarise in many further areas of mathematical physics.The goal of the present paper is to prove the following approximation theorem.
Theorem 1.1. If U ∈ C ([0 , T ] , H ( R )) is a solution of (5) and V = ∆ − | U | , thenfor C > there exist C > and ε ′ > such that for all ε ∈ (0 , ε ′ ] and all solutions u ε , v ε of (1) and (2) satisfying k u ε ( · , − U ( · , k H + k v ε ( · , − V ( · , k L ∞ + (cid:13)(cid:13) ∆ (cid:0) v ε ( · , − V ( · , (cid:1)(cid:13)(cid:13) L ∩ L ≤ C ε and k ∂ t v ε ( · , k L ∞ + (cid:13)(cid:13) ∆ (cid:0) ∂ t v ε ( · , (cid:1)(cid:13)(cid:13) L ∩ L ≤ C , we have k u ε ( · , t ) − U ( · , t ) k H + k v ε ( · , t ) − V ( · , t ) k L ∞ + (cid:13)(cid:13) ∆ (cid:0) v ε ( · , t ) − V ( · , t ) (cid:1)(cid:13)(cid:13) L ∩ L ≤ C ε for all t ∈ [0 , T ] . See Section 4.2 for a more general version of this result – and a more precise formula-tion specifying, for instance, the notion of solution employed above and the spaces thesolutions live in.
Remark 1.2.
Such approximation results should not be taken for granted. There arevarious counterexamples showing that formally derived limit equations make wrong pre-dictions about the original system [Sch95, SSZ15].
Remark 1.3.
The approximation result is non-trivial since the Lipschitz constant of theright-hand side of the first-order system, cf. (18), associated to (2) is of order O ( ε − ).Hence, especially the nonlinear terms in (2) in principle can lead to some unwantedgrowth rates O ( e ε − t ) for t = O (1). 2he problem described in Remark 1.3 is overcome by an integration by parts w.r.t. t inthe variation of constants formula associated to (2) and by a well adapted choice of spacesand norms. This allows us to use the highly oscillatory linear semigroup associated to(2) to get rid of the ε − in front of the nonlinear terms, cf. also Section 5. Our estimatesimply, in particular, the existence of an interval [0 , T ], that is independent of ε ∈ (0 , ε ′ ],on which the system has a unique (mild) solution.While it appears natural, mathematically, to study the limit ε → ε → ∞ is even more relevant, because the system (1)-(2) is believed, and partly proven, to describe the strong coupling limit, ε → ∞ , of theFr¨ohlich model of large polarons [FS14, FG]. We remark that the Nelson model, which issimilar to the Fr¨ohlich model, in a classical limit leads to the Schr¨odinger-Klein-Gordonsystem [AF14]. The designation of (1)-(2) as mean-field polaron in the title of our paperis adopted from [BNAS00], where these equation, apparently, have been studied for thefirst time.We conclude this introduction with some remarks on our notation. The intersection X ∩ Y and the product X × Y of two function spaces X, Y on R will always be endowedwith the sum norm k·k X ∩ Y := k·k X + k·k Y and the product norm k ( · , ·· ) k X × Y := k·k X + k··k Y , respectively. The operator norm for bounded operators between X and Y will bedenoted by k·k X,Y . We use C ( R ) to denote the space of continuous functions tendingto 0 at infinity, while compact support will be indicated by the notation C kc ( R ) for k ∈ N ∪ {∞} . Finally, distinct constants will be denoted with the same letter C if theycan be chosen independently of the small perturbation parameter ε ≪ Acknowledgement.
The work of J. Schmid is partially supported by the DeutscheForschungsgemeinschaft DFG through the Graduiertenkolleg GRK 1838 ,,Spectral The-ory and Dynamics of Quantum Systems”.
In this section, we introduce the spaces we will work with and investigate the mappingproperties of the Laplace operator ∆ in theses spaces. In particular, we will discussthe properties of the inverse operator ∆ − appearing in the equations (2) and (5). If X, Y ⊂ L ( R ) are function spaces on R , we will write ∆ : D X,Y ⊂ X → Y to denotethe linear operator D X,Y ∋ u ∆ u ∈ Y with domain D X,Y := { u ∈ X : ∆ u ∈ Y } , where ∆ u = ∂ x u + ∂ x u + ∂ x u denotes the distributional Laplacian of u . We willcontinually use the Sobolev spaces X s := H s ( R ) with k u k X s := (cid:16) Z (1 + | ξ | ) s | b u ( ξ ) | dξ (cid:17) / for s ∈ [0 , ∞ ), whose basic and completely well-known properties are summarized in thefollowing lemma for the sake of easy reference.3 emma 2.1. For s, t ∈ [0 , ∞ ) , the following holds true:(i) X s is a Hilbert space which is continuously embedded in X t for all t ≤ s .(ii) If s ∈ [2 , ∞ ) , then X s · X s ⊂ X s ∩ L ( R ) and there is a constant C = C s suchthat k uv k X s ∩ L ≤ C k u k X s k v k X s for all u, v ∈ X s .(iii) D X s ,X s = X s +2 and ∆ : X s +2 ⊂ X s → X s is a self-adjoint linear operator satisfy-ing k ∆ u k X s ≤ k u k X s +2 for all u ∈ X s +2 . We will also need the following refinement of Lemma 2.1 (ii).
Lemma 2.2. If s ∈ [2 , ∞ ) , then X s − · X s ⊂ X s − ∩ L ( R ) and there is a constant C = C s such that k uv k X s − ∩ L ≤ C k u k X s − k v k X s ( u ∈ X s − , v ∈ X s ) . Proof.
We have only to show the inclusion X s − · X s ⊂ X s − and the estimate for the X s − -norm because the respective inclusion and estimate for L ( R ) are an immediateconsequence of Schwarz’s inequality. It follows from Theorem 9.3.5 in [Fri98] that for u ∈ X s − and v ∈ S ( R ) the product uv belongs to X s − with k uv k X s − ≤ C k u k X s − Z (1 + | ξ | ) ( s − / | b v ( ξ ) | dξ ≤ C (cid:16) Z (1 + | ξ | ) − dξ (cid:17) / k u k X s − k v k X s ≤ C k u k X s − k v k X s . An obvious approximation argument now yields the assertion.In the next lemma, we deal with the invertibility of ∆ and the elementary properties ofthe inverse.
Lemma 2.3. ∆ : D C ,L ∩ L ⊂ C ( R ) → L ( R ) ∩ L ( R ) is an invertible linear operatorwith full range and bounded inverse ∆ − satisfying (cid:13)(cid:13) ∆ − ( w ) (cid:13)(cid:13) C ≤ C (cid:0) k w k L + k w k L (cid:1) and ∆ − ( w ) = γ ∗ w (6) for all w ∈ L ( R ) ∩ L ( R ) , where γ is the fundamental solution of Laplace’s equationin R with γ ( x ) = − / (4 π | x | ) for x ∈ R \ { } , and where C is a constant independentof w .Proof. Injectivity is a simple exercise using the structure theorem for distributions withsupport contained in { } . Surjectivity and the properties of ∆ − are equally simple.Indeed, if w ∈ L ( R ) ∩ L ( R ), then Z | b w ( ξ ) || ξ | dξ ≤ (cid:16) Z | ξ | > | ξ | dξ (cid:17) / k b w k L + (cid:16) Z | ξ |≤ | ξ | dξ (cid:17) k b w k C ≤ C ( k w k L + k w k L ) (7)4y the continuity of the Fourier transform from L ( R ) to C ( R ). So, b w/ | · | ∈ L ( R )and thus v := − q ( b w/ | · | ) ∈ C ( R ) and, of course, ∆ v = w , which proves that ∆ : D C ,L ∩ L ⊂ C ( R ) → L ( R ) ∩ L ( R ) is surjective and that∆ − ( w ) = − q (cid:0) b w/ | · | (cid:1) ( w ∈ L ( R ) ∩ L ( R )) . (8)Combining (7) and (8), we obtain the estimate in (6). Additionally, we obtain from (8)the convolution representation of ∆ − ( w ) in (6) by virtue of the convolution theoremfor tempered distributions (together with a suitable approximation argument).In the following, ∆ − will always denote the operator from the lemma above or a re-striction of that operator. In order to control the nonlinear terms in (2) and (5) weuse: Lemma 2.4. If s ∈ [2 , ∞ ) , then for all u ∈ X s and w ∈ X s − ∩ L ( R ) one has u ∆ − ( w ) ∈ X s and (cid:13)(cid:13) u ∆ − ( w ) (cid:13)(cid:13) X s ≤ C k u k X s (cid:0) k w k X s − + k w k L (cid:1) , (9) where C = C s is a constant independent of u and w .Proof. Clearly, we have to show the assertion only for u ∈ S ( R ). So let u ∈ S ( R ) and w ∈ X s − ∩ L ( R ) and set v := ∆ − ( w ). Also, write ρ s ( ξ ) := (1 + | ξ | ) s/ and σ s ( ξ ) := | ξ | s for ξ ∈ R . Since b v = − b w/ | · | by (8), it follows from (7) that b v ∈ L ( R ) and σ s b v ∈ L ( R ) with k b v k L ≤ C ( k w k X s − + k w k L ) and k σ s b v k L ≤ k w k X s − (10)where C is a constant independent of w (and s ). So, b u ∈ S ( R ) and b v ∈ L ( R ) ⊂S ′ ( R ) are classically convolvable and thus, by the convolution theorem for tempereddistributions, we see that c uv ( x ) = (2 π ) − / ( b u ∗ b v )( x ) = (2 π ) − / Z b u ( x − y ) b v ( y ) dy ( x ∈ R ) . (11)Since ρ s ( x ) ≤ Cρ s ( x − y ) + Cσ s ( y ) for all x, y ∈ R with C = 2 s , it follows that ρ s ( x ) | c uv ( x ) | ≤ C (cid:0) ( ρ s | b u | ) ∗ | b v | (cid:1) ( x ) + C (cid:0) | b u | ∗ ( σ s | b v | ) (cid:1) ( x )for all x ∈ R and therefore (cid:16) Z (cid:0) ρ s ( x ) | c uv ( x ) | (cid:1) dx (cid:17) / ≤ C k ρ s | b u |k L k b v k L + C k b u k L k σ s | b v |k L ≤ C k u k X s (cid:0) k w k X s − + k w k L (cid:1) + C k u k X s k w k X s − (12)by Young’s inequality and by the inequalities (10). So, we have u ∆ − ( w ) ∈ X s and theestimate (9) holds true, as desired. 5n view of the above lemmas, we introduce the spaces Y s := ∆ − ( X s ∩ L ( R )) with k v k Y s := k v k C + k ∆ v k X s + k ∆ v k L for s ∈ [0 , ∞ ), whose basic properties are summarized in the following lemma. Lemma 2.5.
For s, t ∈ [0 , ∞ ) , the following holds true:(i) Y s is a Banach space which is continuously embedded in Y t for all t ≤ s .(ii) If s ∈ [2 , ∞ ) , then X s · Y s − ⊂ X s and there is a constant C = C s such that k uv k X s ≤ C k u k X s k v k Y s − for all u ∈ X s and v ∈ Y s − .(iii) ∆ − : X s ∩ L ( R ) → Y s is a bounded linear operator.Proof. Assertion (i) easily follows by the completeness of C ( R ) and X s ∩ L ( R ) andby the boundedness of ∆ − : X s ∩ L ( R ) → C ( R ) (Lemma 2.3). Assertions (ii) and(iii) are immediate consequences of Lemma 2.4 and Lemma 2.3 respectively. In this section, we discuss the solvability of the equations (1)-(2) and of (5), which is ofcourse the very first thing to do in proving the desired approximation result. We startwith the approximation equation (5) and first show mild and classical solvability of thecorresponding abstract initial value problem U ′ = i ∆ U − iU ∆ − ( | U | ) , with U (0) = U (13)in the sense of [Paz83]. Theorem 3.1. If s ∈ [0 , ∞ ) , then for every U ∈ X s +2 there exists a T > and aunique mild solution U ∈ C ( I, X s +2 ) of (13) on I = [0 , T ] .Proof. Clearly, the linear part i ∆ of the equation (13) is the generator of a stronglycontinuous unitary group in X s +2 by virtue of Lemma 2.1 (iii). Also, the nonlinear part f of the equation (13) given by f ( U ) := − iU ∆ − ( | U | )is a map from X s +2 into itself and Lipschitz continuous on bounded subsets by virtueof Lemma 2.5 (ii) and (iii). So, the standard existence and uniqueness result for mildsolutions (Theorem 6.1.4 in [Paz83]) implies that there is a T > U : I = [0 , T ] → X s +2 of (13). In other words, the integral equation U ( t ) = e i ∆ t U − i Z t e i ∆( t − r ) U ( r )∆ − ( | U ( r ) | ) dr (14)has a unique solution U ∈ C ( I, X s +2 ). 6n the situation of the above theorem, we also obtain classical solvability by Theo-rem 6.1.5 in [Paz83]: Corollary 3.2. If s ∈ [0 , ∞ ) and if U ∈ X s +2 and U ∈ C ( I, X s +2 ) are as in the abovetheorem, then U belongs to C ( I, X s ) and is a classical solution of (13) . We now go on with the original equations (1) and (2) and show mild and classicalsolvability of the corresponding abstract initial value problem uvw ′ = i ∆ uε − w − ε − v + − iuv ε − ∆ − ( | u | ) , with uvw (0) = u v w (15)in the sense of [Paz83]. In the following, we will always abbreviateΛ( v, w ) := ( w, − v ) for ( v, w ) ∈ Y s × Y s . Lemma 3.3. If s ∈ [0 , ∞ ) , then ε − Λ is the generator of a continuous group ( e ε − Λ t ) t ∈ R in Y s × Y s , which is uniformly bounded w.r.t. ε ∈ (0 , ∞ ) , that is, sup ε ∈ (0 , ∞ ) ,t ∈ R (cid:13)(cid:13)(cid:13) e ε − Λ t (cid:13)(cid:13)(cid:13) Y s × Y s ,Y s × Y s < ∞ . Proof.
Since Λ is a bounded operator in Y s × Y s having the matrix representationΛ = (cid:18) − (cid:19) , we have the explicit representation formula e ε − Λ t = ∞ X n =0 (cid:0) ε − Λ t (cid:1) n n ! = (cid:18) cos( ε − t ) sin( ε − t ) − sin( ε − t ) cos( ε − t ) (cid:19) from which the assertion is obvious. Theorem 3.4. If s ∈ [2 , ∞ ) and ε > , then for every ( u , v , w ) = ( u ε , v ε , w ε ) ∈ X s × Y s − × Y s − there exists a unique maximal mild solution ( u, v, w ) = ( u ε , v ε , w ε ) ∈ C ( I ε , X s × Y s − × Y s − ) of (15) with I ε ⊂ I , where I = [0 , T ] is the interval from theprevious theorem.Proof. Clearly, the linear part A = A ε = diag( i ∆ , ε − Λ) of the equation (15) given by A ( u, v, w ) := ( i ∆ u, ε − Λ( v, w )) = ( i ∆ u, ε − w, − ε − v )(( u, v, w ) ∈ X s +2 × Y s − × Y s − )7s the generator of a strongly continuous group ( e A ε t ) t ∈ R = (diag( e i ∆ t , e ε − Λ t )) t ∈ R in X s × Y s − × Y s − by virtue of Lemma 2.1 (iii) and Lemma 3.3. Also, the nonlinear part f = f ε of the equation (15) given by f ( u, v, w ) := ( − iuv, , ε − ∆ − ( | u | ))is a map from X s × Y s − × Y s − into itself and Lipschitz on bounded subsets by virtueof Lemma 2.5 (ii)-(iii) and Lemma 2.1 (ii). So, for I = [0 , T ] as in Theorem 3.1, thestandard existence and uniqueness result for mild solutions (Theorem 6.1.4 in [Paz83])implies that there is a unique maximal mild solution ( u ε , v ε , w ε ) : I ε → X s × Y s − × Y s − of (15) with I ε ⊂ I . In other words, the integral equation uvw ( t ) = e A ε t u v w + Z t e A ε ( t − r ) − iu ( r ) v ( r )0 ε − ∆ − ( | u ( r ) | ) dr (16)has a unique maximal solution ( u, v, w ) = ( u ε , v ε , w ε ) ∈ C ( I ε , X s × Y s − × Y s − ) with I ε ⊂ I .In the situation of the above theorem, we also obtain classical solvability by Theo-rem 6.1.5 in [Paz83]: Corollary 3.5. If s ∈ [2 , ∞ ) and ε > and if ( u , v , w ) and ( u, v, w ) = ( u ε , v ε , w ε ) ∈ C ( I ε , X s × Y s − × Y s − ) are as in the above theorem, then ( u, v, w ) belongs to C ( I ε , X s − × Y s − × Y s − ) and is a classical solution of (15) . For the subsequent estimates, we additionally have to control the second-order timederivatives.
Lemma 3.6.
Suppose U ∈ C ( I, X s +2 ) is as in Theorem 3.1 with s ∈ [2 , ∞ ) and V ( t ) :=∆ − ( | U ( t ) | ) for t ∈ I . Suppose further ( u, v, w ) = ( u ε , v ε , w ε ) ∈ C ( I ε , X s × Y s − × Y s − ) is as in Theorem 3.4 with the same s ∈ [2 , ∞ ) as above. Then V ∈ C ( I, Y s − ) and v ∈ C ( I ε , Y s − ) .Proof. With the help of Lemma 2.1 (ii) and (iii) and of Lemma 2.5 (ii) and (iii), itfollows from Corollary 3.2 and (13) that U ′ ∈ C ( I, X s − ) and hence U ∈ C ( I, X s ) ∩ C ( I, X s − ). We easily conclude from this by Lemma 2.2 that | U | ∈ C ( I, X s − ) andtherefore V = ∆ − ( | U | ) belongs to C ( I, Y s − ) by Lemma 2.5 (iii). That v belongs to C ( I ε , Y s − ) is an immediate consequence of Corollary 3.5 and (15). In this section, we are going to bound the approximation error, that is the differencebetween the solutions u = u ε , v = v ε of the original equations (1)-(2) and the solutions U , V of the approximate equations (3)-(4). We show that this difference – measured inthe right norm – remains of order ε for all times t ∈ [0 , T ] provided it was of order ε atthe initial time 0, thus establishing the desired approximation result.8 .1 Integral equations and estimates for the error We first derive integral equations for the scaled approximation errors R u ( t ) = R u,ε ( t ) := ε − ( u ( t ) − U ( t )) , R v ( t ) = R v,ε ( t ) := ε − ( v ( t ) − V ( t )) ,R w ( t ) = R w,ε ( t ) := εR ′ v ( t ) = v ′ ( t ) − V ′ ( t ) , where U ∈ C ( I, X s +2 ) and ( u, v, w ) = ( u ε , v ε , w ε ) ∈ C ( I ε , X s × Y s − × Y s − ) are mildsolutions of (13) and (15) with s ∈ [2 , ∞ ) and where V = ∆ − ( | U | ). With the help ofCorollary 3.2 and 3.5 and Lemma 3.6, we obtain R u ( t ) = e i ∆ t R u (0) − i Z t e i ∆( t − r ) f u ( r ) dr (17)for all t ∈ I ε , where f u ( r ) = f u,ε ( r ) := R u ( r ) V ( r ) + U ( r ) R v ( r ) + εR u ( r ) R v ( r ) . and (cid:18) R v ( t ) R w ( t ) (cid:19) = e ε − Λ t (cid:18) R v (0) R w (0) (cid:19) + Z t e ε − Λ( t − r ) (cid:18) ε − f v ( r ) − V ′′ ( r ) (cid:19) dr (18)for all t ∈ I ε , where f v ( r ) = f v,ε ( r ) := ∆ − (cid:0) R u ( r ) U ( r ) + R u ( r ) U ( r ) + ε | R u ( r ) | (cid:1) . We now derive from the integral equations (17) and (18) integral inequalities which areimplicit in the sense that the scaled approximation errors R u and ( R v , R w ) – measured inthe norm of X s and Y s − × Y s − respectively – show up on both sides of the inequalities.In order to get rid of the dangerous ε − in front of f v in (18) we perform an integrationby parts. Proposition 4.1.
Set S u,ε ( t ) := sup r ∈ [0 ,t ] k R u ( r ) k X s , S ( v,w ) ,ε ( t ) := sup r ∈ [0 ,t ] k ( R v ( r ) , R w ( r )) k Y s − × Y s − , and S ε ( t ) := S u,ε ( t ) + S ( v,w ) ,ε ( t ) for t ∈ I ε . Then there is a constant C = C s such thatfor all ε ∈ (0 , ∞ ) and all t ∈ I ε S u,ε ( t ) ≤ C (cid:16) S ε (0) + Z t S ε ( r ) + εS ε ( r ) dr (cid:17) ,S ( v,w ) ,ε ( t ) ≤ C (cid:16) S ε (0) + 1 + S u,ε ( t ) + εS u,ε ( t ) + Z t S ε ( r ) + εS ε ( r ) + ε S ε ( r ) dr (cid:17) . roof. It follows from Lemma 2.5 (ii) that f u belongs to C ( I ε , X s ) and satisfies theestimate k f u ( r ) k X s ≤ C (cid:16) k R u ( r ) k X s k V ( r ) k Y s − + k U ( r ) k X s k R v ( r ) k Y s − + ε k R u ( r ) k X s k R v ( r ) k Y s − (cid:17) (19)for all r ∈ I ε . Since sup r ∈ I k U ( r ) k X s < ∞ and sup r ∈ I k V ( r ) k Y s − < ∞ , the assertedestimate for S u,ε now follows from (17) with the help of (19) and Lemma 2.1 (iii). Since R u ∈ C ( I ε , X s ) ∩ C ( I ε , X s − ) and U ∈ C ( I, X s ) by Corollary 3.2 and 3.5, it followsfrom Lemma 2.5 (iii) and Lemma 2.2 that f v belongs to C ( I ε , Y s − ). We can thereforeintegrate by parts in (18) and thus obtain (cid:18) R v ( t ) R w ( t ) (cid:19) = e ε − Λ t (cid:18) R v (0) R w (0) (cid:19) − Z t e ε − Λ( t − r ) (cid:18) V ′′ ( r ) (cid:19) dr − e ε − Λ( t − r ) Λ − (cid:18) f v ( r ) (cid:19) (cid:12)(cid:12)(cid:12) r = tr =0 + Z t e ε − Λ( t − r ) Λ − (cid:18) f ′ v ( r ) (cid:19) dr (20)for all t ∈ I ε , where f ′ v ( r ) = ∆ − (cid:16) R u ( r ) U ′ ( r ) + R ′ u ( r ) (cid:0) U ( r ) + εR u ( r ) (cid:1)(cid:17) + c.c.= ∆ − (cid:16) R u ( r ) U ′ ( r ) (cid:17) + i ∆ − (cid:16)(cid:0) ∆ R u ( r ) − f u ( r ) (cid:1)(cid:0) U ( r ) + εR u ( r ) (cid:1)(cid:17) + c.c. . In the above equation, the symbol c.c. stands for the complex conjugate of the terms onthe left of it. With the help of Lemma 2.5 (iii) and Lemma 2.1 (ii), we can estimate k f v ( r ) k Y s − ≤ C (cid:16) (cid:13)(cid:13)(cid:13) R u ( r ) U ( r ) (cid:13)(cid:13)(cid:13) X s ∩ L + ε (cid:13)(cid:13)(cid:13) R u ( r ) R u ( r ) (cid:13)(cid:13)(cid:13) X s ∩ L (cid:17) ≤ C (cid:16) k R u ( r ) k X s k U ( r ) k X s + ε k R u ( r ) k X s (cid:17) (21)for all r ∈ I ε , and with the help of Lemma 2.5 (iii) and Lemmas 2.2 and 2.1 (ii)-(iii), wecan estimate (cid:13)(cid:13) f ′ v ( r ) (cid:13)(cid:13) Y s − ≤ C (cid:16) (cid:13)(cid:13)(cid:13) R u ( r ) U ′ ( r ) (cid:13)(cid:13)(cid:13) X s ∩ L + (cid:13)(cid:13)(cid:13) (∆ R u ( r ))( U ( r ) + εR u ( r )) (cid:13)(cid:13)(cid:13) X s − ∩ L + (cid:13)(cid:13)(cid:13) f u ( r )( U ( r ) + εR u ( r )) (cid:13)(cid:13)(cid:13) X s ∩ L (cid:17) ≤ C (cid:16) k R u ( r ) k X s (cid:13)(cid:13) U ′ ( r ) (cid:13)(cid:13) X s + k R u ( r ) k X s (cid:0) k U ( r ) k X s + ε k R u ( r ) k X s (cid:1) + k f u ( r ) k X s (cid:0) k U ( r ) k X s + ε k R u ( r ) k X s (cid:1)(cid:17) (22)for all r ∈ I ε . Since sup r ∈ I k U ( r ) k X s + k U ′ ( r ) k X s < ∞ and sup r ∈ I k V ( r ) k Y s − + k V ′′ ( r ) k Y s − < ∞ , the asserted estimate for S ( v,w ) ,ε now follows from (20) with thehelp of (21), (22), (19) and Lemma 3.3. 10 .2 Approximation result With the help of Gronwall’s lemma, we finally turn the implicit estimates for the approx-imation error just established into explicit estimates and thus obtain our approximationtheorem. Choosing s = 2, we obtain the version of the theorem stated in the introduc-tion. Theorem 4.2.
Suppose ( U, V ) and ( u, v, w ) = ( u ε , v ε , w ε ) are as in Lemma 3.6 andsuppose further that the initial values satisfy k u ε (0) − U (0) k X s + k v ε (0) − V (0) k Y s − + ε (cid:13)(cid:13) v ′ ε (0) − V ′ (0) (cid:13)(cid:13) Y s − ≤ C ε (23) for all ε ∈ (0 , ε ] with some ε > and some constant C = C ,s . Then there is an ε ′ ∈ (0 , ε ] and a constant C = C ,s such that I ε = I for all ε ∈ (0 , ε ′ ] and such that k u ε ( t ) − U ( t ) k X s + k v ε ( t ) − V ( t ) k Y s − + ε (cid:13)(cid:13) v ′ ε ( t ) − V ′ ( t ) (cid:13)(cid:13) Y s − ≤ C ε (24) for all t ∈ I and all ε ∈ (0 , ε ′ ] .Proof. We plug in the estimate for S u,ε into the estimate for S ( v,w ) ,ε from Proposition 4.1and, by adding the resulting inequality to the inequality for S u,ε , we obtain the followinginequality for S ε : S ε ( t ) ≤ C (cid:16) S ε (0) + 1 + Z t S ε ( r ) + εS ε ( r ) + ε S ε ( r ) + ε S ε ( r ) dr (cid:17) for all t ∈ I ε and all ε ∈ (0 , ∞ ). Since S ε (0) ≤ C for all ε ∈ (0 , ε ] by assumption, wetherefore have that S ε ( t ) ≤ C + C Z t p ( εS ε ( r )) S ε ( r ) dr (25)for all t ∈ I ε and all ε ∈ (0 , ε ], where p ( ξ ) := 1+ ξ + ξ + ξ for ξ ∈ R . So, by Gronwall’slemma, we obtain S ε ( t ) ≤ Ce C R t p ( εS ε ( r )) dr (26)for all t ∈ I ε and all ε ∈ (0 , ε ]. Set M := Ce T + 2 with C being the constant in (26)and choose ε ′ ∈ (0 , ε ] such that Ce p ( ε ′ M ) T ≤ Ce T + 1 = M − . (27)Also, for ε ∈ (0 , ε ′ ] set b ε := sup (cid:8) t ∈ I ε : S ε ( r ) ≤ M for all r ∈ [0 , t ] (cid:9) . We then have, for all t ∈ [0 , b ε ) and all ε ∈ (0 , ε ′ ], that S ε ( t ) ≤ Ce C R t p ( εS ε ( r )) dr ≤ Ce p ( ε ′ M ) T ≤ M − b ε was strictly less than sup I ε for some ε ∈ (0 , ε ′ ], thenfrom (28) we would obtain, using the continuity of S ε , a contradiction to the definitionof b ε . So, b ε = sup I ε for all ε ∈ (0 , ε ′ ]. It follows from this and from (28) thatsup t ∈ I ε S ε ( t ) = sup t ∈ [0 ,b ε ) S ε ( t ) ≤ M − t ∈ I ε k ( u ε ( t ) , v ε ( t ) , w ε ( t )) k X s × Y s − × Y s − ≤ C + ε ′ sup t ∈ I ε S ε ( t ) < ∞ (30)for all ε ∈ (0 , ε ′ ]. Since ( u ε , v ε , w ε ) by definition is the maximal mild solution of (15)with I ε ⊂ I , it follows from (30) by the standard blow-up result for mild solutions(Theorem 6.1.4 in [Paz83]) that I ε must be equal to I for all ε ∈ (0 , ε ′ ]. So, invoking (29)again, we see that sup t ∈ I S ε ( t ) = sup t ∈ I ε S ε ( t ) ≤ M − C (31)for all ε ∈ (0 , ε ′ ], and this immediately implies (24). We close this paper with some remarks on the connection of the presented approach tonormal form transformations. Instead of the approach pursued above, one can try toget rid of the dangerous term ε − ∆ − ( U R u + U R u ) in the equation (18) for R ( v,w ) :=( R v , R w ) by a near-identity change of coordinates of the form e R ( v,w ) = R ( v,w ) + B ( U, R u )where B is a symmetric bilinear mapping. Inserting this transformation into the R ( v,w ) equation yields ∂ t e R ( v,w ) = ε − Λ e R ( v,w ) − ε − Λ B ( U, R u ) + B ( − i ∆ U, R u ) + B ( U, − i ∆ R u )+ ε − ∆ − (cid:0) U R u + U R u (cid:1) + h.o.t. , where h.o.t. stands for the higher-order terms. So, we have to find a bilinear mapping B such that − ε − Λ B ( U, R u ) + B ( − i ∆ U, R u ) + B ( U, − i ∆ R u ) + ε − ∆ − (cid:0) U R u + U R u (cid:1) = 0 , which is not possible, however, since the non-resonance conditioninf k,l | ± ε − + ( k − l ) − l | > B ( U, R u ) =Λ − ∆ − (cid:0) U R u + U R u (cid:1) = O (1). It allowed us to eliminate the terms of order O ( ε − )such that after the transform we had ∂ t e R ( v,w ) = ε − Λ e R ( v,w ) + B ( − i ∆ U, R u ) + B ( U, − i ∆ R u ) + · · · = ε − Λ e R ( v,w ) + O (1) . We finally remark that the energy approach chosen in [DSS16] for a similar limit in theKlein–Gordon–Zakharov system can only be used for (1)-(2) in space dimensions d ≥ − which maps L ( R d ) ∩ L ( R d ) into L ( R d ) only for spacedimensions d ≥ References [AF14] Zied Ammari and Marco Falconi. Wigner measures approach to the classicallimit of the Nelson model: convergence of dynamics and ground state energy.
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