A mathematical model for the Einstein-Podolsky-Rosen argument

A mathematical mo del for the Einstein-P o dolsky-Rosen argumen t Riccardo Adami 1 , Luigi Barletti 2 , and Alessandro T eta 3 1 Dipartimen to di Scienze Matematiche “G.L. Lagrange”, P olitecnico di T orino, Corso Duca degli Abruzzi 24, 10129 T orino, Italy 2 Dipartimen to di Matematica e Informatica “U. Dini”, Universit` a di Firenze, Viale G.B. Morgagni 67/a, 50134 Firenze, Italy 3 Dipartimen to di Matematica “G. Castelnuo v o”, Univ ersit` a di Roma La Sapienza, P .le A. Moro 2, 00185 Roma, Italy F ebruary 25, 2026 Abstract W e study a nonrelativistic system made of tw o quantum particles constrained to mo ve on a line and a spin lo cated at a fixed p oin t of the line. Initially the tw o particles are in a maximally en tangled state and the spin is do wn. The first particle interacts with the spin while the second particle is free, i.e., it do es not interact neither with the first particle nor with the spin. W e rigorously prov e that there is a correlation b et ween the state of the spin and the state of the second particle. More precisely , we sho w that, in a suitable scaling limit, if the first particle flips the spin, then the second particle p ossesses a definite momen tum in the direction opp osite to the spin. 1 In tro duction The celebrated pap er by Einstein, Podolsky and Rosen (EPR) published in 1935 [3] is surely one of the most influen tial w orks on Quantum Mechanics, not only for its deep conceptual meaning, but also b ecause it has highligh ted p eculiar asp ects of the theory that during the y ears hav e led to fundamental applications in man y fields, suc h as quan tum information theory . The aim of this pap er is to form ulate a simplified but mathematically w ell-defined version of the mo del in tro duced b y EPR and to rigorously deriv e its physical consequences. The original EPR w ork aims to show that Quan tum Mec hanics is not complete and it is based on the analysis of a mo del with contin uous v ariables describing tw o particles in dimension one. As is well known, the argument was then reform ulated by Bohm [2] in terms of spin v ariables, which is simpler from a mathematical point of view. Suc h reform ulation is the one commonly discussed and analysed in the literature, while the original EPR formulation is less known. Ho w ev er, since w e are interested in the latter case, for the conv enience of the reader w e summarise the original argument in App endix A. 1 The mo del w e consider consists of a pair of quantum particles, referred to as particle 1 and particle 2, and a spin. The Hilb ert space of the states of the system is then K = L 2 ( R ) ⊗ L 2 ( R ) ⊗ C 2 , so a state of the system at time t is represen ted by a t wo-component v ector function, namely Ψ t ( x 1 , x 2 ) =  Ψ u t ( x 1 , x 2 ) Ψ d t ( x 1 , x 2 )  , where Ψ u t , Ψ d t ∈ L 2 ( R ) ⊗ L 2 ( R ) and the v ariable x j , j = 1 , 2, denotes the p osition v ariable of the j − th particle of the system. The upp er comp onen t Ψ u t ( x 1 , x 2 ) is asso ciated with the v alue ℏ / 2 of the spin, i.e., with the spin up, while the lo wer Ψ d t ( x 1 , x 2 ) refers to the v alue − ℏ / 2, i.e., with the spin do wn. P article 2 is free, while particle 1 can interact with the spin through a lo calized p oten tial γ V , where γ ∈ R is a coupling constant and V is smo oth and rapidly decaying at infinit y . F urthermore, the in teraction can make the spin flip, so that the full interaction reads as γ V σ 2 , where σ 2 =  0 − i i 0  is the second P auli matrix. The Hamiltonian of the system reads then H = H 0 + γ V  x 1 − a δ  σ 2 H 0 = − ℏ 2 2 m 1 I ∆ 1 − ℏ 2 2 m 2 I ∆ 2 + ℏ ω 2 σ 3 (1.1) where: m 1 and m 2 are the masses of the particles; ω is the characteristic frequency of the spin and therefore the free evolution of the spin is p eriodic with p erio d T s = 2 π /ω ; I is the 2 × 2 iden tity matrix; σ 3 is the P auli matrix σ 3 =  1 0 0 − 1  ; γ is the coupling constan t of the p oten tial V that describ es the in teraction b et w een particle 1 and the spin; δ > 0 is the spatial scale of the range of V ; a > 0 is the p osition of the spin, that is fixed. W e study the time ev olution of the system under the assumption that the initial datum has the form Ψ 0 =  0 Ψ d 0  , (1.2) so that the spin equals − ℏ / 2 and the asso ciated energy is − ℏ ω / 2. Moreov er, the pair of particles is in a maximally entangled state concen trated in position around the origin, namely Ψ d 0 ( x 1 , x 2 ) = N ( ϕ P ( x 1 ) ϕ − P ( x 2 ) + ϕ − P ( x 1 ) ϕ P ( x 2 )) , (1.3) 2 where ϕ P ( x ) = 1 √ σ f  x σ  e i P ℏ x , σ > 0 , P > 0 . Here f is smo oth, rapidly deca ying, with f ev en, real and ∥ f ∥ L 2 ( R ) = 1. The normalization factor N is given b y N =  2 + 2 Z dy f ( y ) 2 cos 2 σ P ℏ y  − 1 / 2 W e stress that (1.3) is a sup erposition of the state ϕ P ( x 1 ) ϕ − P ( x 2 ), where particle 1 has mean momentum P and particle 2 mean momentum − P , and the state ϕ − P ( x 1 ) ϕ P ( x 2 ), where mean momen ta are exchanged. In order to let the EPR phenomenon emerge from the dynamics we need to characterize the parameters of the mo del. More precisely , the ph ysical situation we wan t to describ e is the follo wing. Let us define T coll := a m 1 P (1.4) i.e., the classical collision time of particle 1 with momen tum P starting from the origin with the spin placed in a . F or t < T coll w e require that, with high probability , the system is describ ed b y the free evolu- tion. This means that the spin remains down and the particles are describ ed b y a sup erp o- sition giv en by the free evolution of ϕ P ( x 1 ) ϕ − P ( x 2 ) and the free ev olution of ϕ − P ( x 1 ) ϕ P ( x 2 ). In particular this means that for t < T coll eac h particle has p ositiv e momen tum with proba- bilit y approximately 1 / 2 and negative momen tum with probabilit y approximately 1 / 2. A t t ≃ T coll particle 1 in teracts with the spin whic h, with some probabilit y , can flip to state up. More precisely , the interaction o ccurs for the particles describ ed b y the time evolution of the comp onen t ϕ P ( x 1 ) ϕ − P ( x 2 ) of (1.3). Recall that particle 2 do es not in teract with the spin and, moreov er, in the free evolution of the component ϕ − P ( x 1 ) ϕ P ( x 2 ) particle 1 remains far from the p osition of the spin. F or t > T coll w e exp ect that the system is described b y a superp osition represen ting the tw o p ossible situations (apart for some small errors): ( i ) the spin is down and the tw o particles are still describ ed by the free evolution of the initial state (1.3), ( ii ) the spin is up and the t wo particles are describ ed by a pro duct state, with particle 1 ev olving freely with (appro ximate) mean momentum P and mean p osition on the righ t of the spin p osition a , while particle 2 evolv es freely with mean momentum − P and mean p osition on the left of the origin. Notice that situation ( ii ) is the one of in terest for the EPR argumen t, in the sense that if the spin is up then particle 1 has mean momen tum P and therefore particle 2 has mean momen- tum − P . W e shall come back to this p oin t with more details after the precise formulation of our result. The physical situation describ ed ab ov e cannot b e realized for arbitrary v alues of the pa- rameters present in the mo del. It should b e clear that we need a semiclassical regime for the t w o particles, a small and short range interaction p oten tial and excitation energy of the spin muc h smaller than the initial kinetic energy of particle 1, i.e., a quasi-elastic regime for 3 particle 1. In order to imp ose these conditions, it is con venien t to introduce the follo wing scaling (analogous to the one used in [4, 8]): ℏ = ε 2 , ω = ε − 1 , γ = ε 2 , δ = ε σ = ε (1.5) where ε is a small positive parameter, while P and a are of order 1. F or notational simplicity , w e also set m 1 = m 2 = 1. Let us briefly comment on the meaning of our scaling for ε → 0. W e first note that the quan tity ∆ p P is roughly given by ℏ P σ = O ( ε ) , which means that the momentum of the particles is w ell concentrated around the mean v alues P or − P . Moreo ver, σ a = O ( ε ) , δ a = O ( ε ) i.e. the lo calization in position of the particles and the effective range of the in teraction are m uch smaller than the distance a . W e also stress that the spacing of the energy lev els of the spin is ℏ ω = O ( ε ) while the kinetic energy of particle 1 is O (1) (quasi-elastic regime for particle 1). Moreov er, the c hoice of the coupling constan t γ = ε 2 guaran tees applicabilit y of p erturbativ e metho ds. F urthermore, it is in teresting to compare the characteristic times of our system. W e hav e T coll = a P = O (1) while the p eriod of the “spin motion” is T s = 2 π ω = O ( ε ) and the time necessary to particle 1 to cross the in teraction region is T int = δ P = O ( ε ) . Therefore the condition T coll ≫ T s , T int allo ws us to interpret T coll as the effectiv e collision time in the quantum description. Moreov er, the condition T s /T int = O (1) pro ves crucial to ha ve a non trivial transition probability for the spin. According to the scaling (1.5), the Hamiltonian of the system reads H ε = H ε 0 + ε 2 V ε σ 2 H ε 0 = − ε 4 2 I ∆ 1 − ε 4 2 I ∆ 2 + ε 2 σ 3 (1.6) where V ε denotes the multiplication op erator by V  x 1 − a ε  . The initial state has the form (1.2), with Ψ − 0 replaced b y Ψ − ,ε 0 ( x 1 , x 2 ) = N ε ( ϕ ε P ( x 1 ) ϕ ε − P ( x 2 ) + ϕ ε − P ( x 1 ) ϕ ε P ( x 2 )) , (1.7) where ϕ ε P ( x ) = 1 √ ε f  x ε  e i P ε 2 x 4 and N ε is giv en by N ε =  2 + 2 Z dy f ( y ) 2 cos 2 P ε y  − 1 / 2 . Notice that N ε = 1 / √ 2 + O ( ε n ) for all n . With an abuse of notation, from now on w e shall drop the dep endence on ε of the initial state, the Hamiltonians and the corresp onding unitary propagators. In order to form ulate the result we define the interacting propagator U ( t ) = e − i t ε 2 H (1.8) and the free propagator U 0 ( t ) = e − i t ε 2 H 0 (1.9) b oth acting in K . Note that the action of U 0 ( t ) is explicitly giv en by U 0 ( t )Ψ =  e − i t 2 ε U 0 ( t ) ⊗ U 0 ( t )Ψ u e i t 2 ε U 0 ( t ) ⊗ U 0 ( t )Ψ d  for an y Ψ =  Ψ u Ψ d  ∈ K . (1.10) where U 0 ( t ) is the free propagator in L 2 ( R ) U 0 ( t ) f ( x ) = 1 √ 2 π iε 2 t Z R e i ( x − y ) 2 2 ε 2 t f ( y ) dy . (1.11) Our main result is summarized in the follo wing theorem. Theorem 1.1. Assume that V and f b elong to the Schwartz sp ac e S ( R ) , with f r e al, even and ∥ f ∥ L 2 ( R ) = 1 , and let us fix t > T coll . Then for ε → 0 we have Ψ t := U ( t )Ψ 0 = U 0 ( t )Ψ 0 + ε U 0 ( t )Ψ ( I ) + R ( t ) (1.12) wher e Ψ ( I ) ( x 1 , x 2 ) = N  A ( x 1 ) ϕ − P ( x 2 ) 0  , (1.13) A ( x ) = − √ 2 π P e i a εP ( 1+ ε 2 P 2 ) ˆ V ( P − 1 ) 1 √ ε f  x − aP − 2 ε ε  e i ε 2 ( P − εP − 1 ) x , (1.14) ∥R ( t ) ∥ K < C ε 2 (1.15) and the p ositive c onstant C is indep endent of ε and dep ends on t and on the p ar ameters of the mo del. The proof will b e given in sev eral steps in the following sections. Here w e mak e some commen ts on the result. The theorem provides precisely the state of the system w e exp ected for t > T coll in a p ertur- bativ e form. In particular, the zero-th order term describ es the unp erturb ed situation, with spin do wn and the tw o particles describ ed b y the free evolution of the initial state (1.3). The 5 first order term describ es the situation with spin up, particle 1 mo ving freely tow ards the righ t with a slightly reduced momentum P − εP − 1 and lo calized in position around x 1 ( t ) = aP − 2 ε + ( P − εP − 1 ) t = a + ( P − εP − 1 )( t − T coll ) while particle 2 mov es freely tow ards the left with unp erturb ed momentum − P and is lo cal- ized in p osition around x 2 ( t ) = − P t = − a − P ( t − T coll ) . Finally , the rest is sho wn to b e O ( ε 2 ). On the basis of Theorem 1.1 and Born’s rule w e can compute approximate formulas for the probabilities of outcomes of measurements on the system, p erformed for t > T coll . In particular, w e compute the probability to find the spin up or down, denoted by P u , P d resp ectiv ely , and the probability to find momentum − P for particle 2 and the spin up or do wn, denoted P − ,u , P − ,d resp ectiv ely . By (1.13), (1.14), (1.15) we find P u = α ε 2 + O ( ε 3 ) , P d = 1 + O ( ε ) (1.16) P − ,u = α ε 2 + O ( ε 3 ) , P − ,d = 1 2 + O ( ε ) (1.17) where α = π P 2 | ˆ V ( P − 1 ) | 2 . (1.18) This means that, for t > T coll , w e hav e P − ,u P u = 1 + O ( ε ) , (1.19) P − ,d P d = 1 2 + O ( ε ) . (1.20) W e stress that the ab o v e statements hold for any t > T coll and therefore regardless of how large the distance is b et w een the spin and the p osition of particle 2. F orm ula (1.19) can b e considered as a w ay to express the essence of the EPR argument. Indeed, following the line of reasoning of EPR, one should apply the principles of realit y and lo calit y (see also App endix A). Now, F orm ula (1.19) sho ws that, if w e measure the spin and the result is “up”, then we can predict that, almost certainly, particle 2 has momentum − P . This means that one should attribute an element of realit y asso ciated with the momen tum of particle 2 (due to the realit y criterion). Moreov er, as particle 1 and the spin do not in teract with particle 2, such element of realit y would b e existing prior to the measuremen t of the spin (due to the lo calit y principle). On the other hand, b efore the measurement, Quan tum Mec hanics does not pro vide a represen tation for such an elemen t of reality and therefore must b e considered as incomplete. Con trarily to the original EPR argumen t, in this formulation one do es not need to in v oke the uncertaint y principle. Con versely , if the spin remains “do wn” after the collision time T coll , then formula (1.20) says that w e cannot predict with certaint y the momen tum of particle 2. 6 In the framework of non-relativistic Quan tum Mechanics, sev eral models ha ve b een proposed to describ e a pro cess in whic h the momentum of a particle is revealed b y the flip of a spin. In such a situation, the spin acts as the simplest p ossible measurement apparatus [6, 1, 7, 4]. Here w e apply the same approach to the mathematical study of the EPR argument. This allo ws us to treat, as in the EPR pap er but in full mathematical rigour and in a dynamical setting, a contin uous v ariable (the momen tum). More sp ecifically , we define an initial state of the system and compute its time ev olution p erturbativ ely with a quan titative estimate of the error. W e stress that, even though the role of the spin is that of a measuremen t apparatus, we alwa ys consider the whole system “particle 1 + particle 2 + spin”, without ha ving to resort to the wa v e pac ket reduction. As a further feature of our metho d, we remark that dealing rigorously with the Sc hr¨ odinger equation for a contin uous system, brings out the necessit y of a suitable scaling, in order to isolate the phenomenon w e aim at exploring. F or the conv enience of the reader we collect here some further notation that shall b e used in the rest of the pap er. - The Hilb ert space of the sub-system made b y particle 1 and the spin: K 1 ,s = L 2 ( R ) ⊗ C 2 . (1.21) - The free Hamiltonian and the free unitary group in K 1 ,s : H 1 ,s 0 = − ε 4 2 I ∆ 1 + ε 2 σ 3 , U 1 ,s 0 ( t ) = e − i t ε 2 H 1 ,s 0 =  U 0 ( t ) e − i t 2 ε 0 0 U 0 ( t ) e i t 2 ε  . (1.22) - the m ultiplication op erator V ε : V ε ( x 1 ) = V  x 1 − a ε  , x 1 ∈ R . (1.23) - the in teracting Hamiltonian and the interacting unitary group in K 1 ,s : H 1 ,s = − ε 4 2 I ∆ 1 + ε 2 σ 3 + ε 2 V ε σ 2 , U 1 ,s ( t ) = e − i t ε 2 H 1 ,s . (1.24) - ( f ⊗ g )( x 1 , x 2 ) = f ( x 1 ) g ( x 2 ) where g ∈ L 2 ( R ) and f ∈ L 2 ( R ) or f ∈ K 1 ,s . - [( S ⊗ T )( f ⊗ g )]( x 1 , x 2 ) = ( S f )( x 1 )( T g )( x 2 ) where T is a linear op erator in L 2 ( R ) and S is a linear op erator in L 2 ( R ) or in K 1 ,s . - The action of the free propagator: U 0 ( t ), giv en explicitly in (1.10). - The norms: ∥ f ∥ 2 = Z dx | f ( x ) | 2 (1.25) (ho wev er, when necessary , w e shall sp ecify the space as ∥ f ∥ 2 L 2 ( R ) ) 7 ∥ f ∥ 2 L 2 n ( R ) = Z dx | x | n | f ( x ) | 2 (1.26) ∥ f ∥ W 1 ,n n ( R ) = n X m =0 Z dx (1 + | x | m ) | f ( n − m ) ( x ) | . (1.27) 2 Strategy of the pro of Here w e outline the strategy of the pro of. By linearit y , the evolution of the system is given b y Ψ t = U ( t )Ψ 0 = N U ( t )  0 ϕ P ⊗ ϕ − P  + N U ( t )  0 ϕ − P ⊗ ϕ P  (2.1) and, taking in to account that particle 2 is sub ject to a free evolution, w e can write Ψ t = N U 1 ,s ( t )  0 ϕ P  ⊗ U 0 ( t ) ϕ − P + N U 1 ,s ( t )  0 ϕ − P  ⊗ U 0 ( t ) ϕ P (2.2) The analysis is then reduced to the study of the evolution of the particle 1 and the spin. Using Duhamel form ula, we hav e U 1 ,s ( t )  0 ϕ ± P  = U 1 ,s 0 ( t )  0 ϕ ± P  − i Z t 0 dτ U 1 ,s ( t − τ ) V ε σ 2 U 1 ,s 0 ( τ )  0 ϕ ± P  (2.3) W e will sho w that the last term in (2.3) is negligible when the particle 1 has a negative momen tum (recall that the p osition a of the spin is assumed to b e p ositiv e). Therefore we write the w av e function of the system as Ψ t = N U 1 ,s ( t )  0 ϕ P  ⊗ U 0 ( t ) ϕ − P + N e i t 2 ε U 0 ( t ) ϕ − P ⊗ U 0 ( t ) ϕ P  0 1  + N L ( t ) (2.4) where L ( t ) = − i Z t 0 dτ U 1 ,s ( t − τ ) V ε σ 2 U 1 ,s 0 ( τ )  0 ϕ − P  ⊗ U 0 ( t ) ϕ P (2.5) Let us no w consider the first term in the r.h.s. of (2.4). Iterating (2.3), we find U 1 ,s ( t )  0 ϕ P  = U 1 ,s 0 ( t )  0 ϕ P  − i Z t 0 dτ U 1 ,s 0 ( t − τ ) V ε σ 2 U 1 ,s 0 ( τ )  0 ϕ P  − Z t 0 dτ 1 U 1 ,s ( t − τ ) V ε σ 2 U 1 ,s 0 ( τ 1 ) Z τ 1 0 dτ 2 U 1 ,s 0 ( − τ 2 ) V ε σ 2 U 1 ,s 0 ( τ 2 )  0 ϕ P  8 = e i t 2 ε  0 U 0 ( t ) ϕ P  − e − i t 2 ε  U 0 ( t ) I ( t ) ϕ P 0  + J ( t ) (2.6) where I ( t ) = Z t 0 dτ e i τ ε U 0 ( − τ ) V ε U 0 ( τ ) (2.7) J ( t ) = − Z t 0 dτ 1 U 1 ,s ( t − τ 1 ) V ε σ 2 U 1 ,s 0 ( τ 1 ) Z τ 1 0 dτ 2 U 1 ,s 0 ( − τ 2 ) V ε σ 2 U 1 ,s 0 ( τ 2 )  0 ϕ P  . (2.8) T aking in to account (2.4) and (2.6), the wa v e function of the system reads Ψ t = U 0 ( t )Ψ 0 − N e − i t 2 ε  U 0 ( t ) I ( t ) ϕ P 0  ⊗ U 0 ( t ) ϕ − P + N J ( t ) ⊗ U 0 ( t ) ϕ − P + N L ( t ) . (2.9) Let us consider the op erator I ( t ). Its action on the function f X,K ( x ) = 1 √ ε f  x ε − X  e i K ε 2 x , X , K > 0 , f ∈ S ( R ) (2.10) can b e explicitly computed and w e find (see App endix B) I ( t ) f X,K ( x ) = e i K ε 2 x √ 2 π ε Z t 0 dτ Z dξ F ( τ , ξ ) e i ε Φ( τ ,ξ ) (2.11) where F ( τ , ξ ) = b V ( ξ ) f  τ ξ + x ε − X  e i 2 τ ξ 2 + i ε xξ , (2.12) Φ( τ , ξ ) = τ − a ξ + K τ ξ . (2.13) F or notational conv enience, w e hav e dropp ed the dep endence on x, ε of the function F . Notice that (2.11) has the form of a highly oscillating integral for ε → 0 and therefore its asymptotic b eha viour is c haracterized by the critical p oints of the phase ( τ , ξ ) = ( τ c , ξ c ) =  a K , − 1 K  . (2.14) Notice that τ c reduces to T coll when K = P . In order to iden tify the leading order in the asymptotic expansion for ε → 0 w e write Φ( τ , ξ ) = a K + K ( τ − τ c )( ξ − ξ c ) , (2.15) w e define the new in tegration v ariable z = K ε ( τ − τ c ) and w e denote Ω ε =  − a ε , K ε ( t − τ c )  . Then w e find I ( t ) f X,K ( x ) = e i K ε 2 x + i ε a K √ 2 π K √ ε Z Ω ε dz e − iξ c z Z dξ e iz ξ F  τ c + ε K z , ξ  9 = e i K ε 2 x + i ε a K √ 2 π K √ ε Z dz e − iξ c z Z dξ e iz ξ F  τ c , ξ  − e i K ε 2 x + i ε a K √ 2 π K √ ε Z R \ Ω ε dz e − iξ c z Z dξ e iz ξ F  τ c , ξ  + e i K ε 2 x + i ε a K √ 2 π K √ ε Z Ω ε dz e − iξ c z Z dξ e iz ξ  F  τ c + ε K z , ξ  − F  τ c , ξ   = e i K ε 2 x + i ε a K √ 2 π K √ ε F ( τ c , ξ c ) + Q 1 ( x ) + Q 2 ( x ) (2.16) where Q 1 ( t, x ) = − e i K ε 2 x + i ε a K √ 2 π K √ ε Z R \ Ω ε dz e − iξ c z Z dξ e iz ξ F  τ c , ξ  (2.17) Q 2 ( t, x ) = e i K ε 2 x + i ε a K √ 2 π K √ ε Z Ω ε dz e − iξ c z Z dξ e iz ξ  F  τ c + ε K z , ξ  − F  τ c , ξ   . (2.18) Using form ula (2.16), with X = 0 and K = P we find I ( t )Φ P ( x ) = − ε A ( x ) + Q 0 1 ( t, x ) + Q 0 2 ( t, x ) (2.19) where A is defined in (1.14) and Q 0 i = Q i for X = 0 and K = P , i = 1 , 2. Replacing (2.19) in (2.6) w e obtain U 1 ,s ( t )  0 ϕ P  =  ε e − i t 2 ε U 0 ( t ) A e i t 2 ε U 0 ( t ) ϕ P  − e − i t 2 ε  U 0 ( t )( Q 0 1 ( t ) + Q 0 2 ( t )) 0  + J ( t ) (2.20) In conclusion, using formula (2.20) in (2.4), w e obtain the following expression for the w a ve function of the system Ψ t = N  ε e − i t 2 ε U 0 ( t ) A e i t 2 ε U 0 ( t ) ϕ P  ⊗ U 0 ( t ) ϕ − P + N e i t 2 ε U 0 ( t ) ϕ − P ⊗ U 0 ( t ) ϕ P  0 1  + N R ( t ) =  ε N e − i t 2 ε U 0 ( t ) A ⊗ U 0 ( t ) ϕ − P N e i t 2 ε  U 0 ( t ) ϕ P ⊗ U 0 ( t ) ϕ − P + U 0 ( t ) ϕ − P ⊗ U 0 ( t ) ϕ P   + N R ( t ) ≡ U 0 ( t )Ψ 0 + ε U 0 ( t )Ψ ( I ) + N R ( t ) (2.21) where Ψ ( I ) is giv en in (1.13) and R ( t ) = − e − i t 2 ε  U 0 ( t )( Q 0 1 ( t ) + Q 0 2 ( t )) 0  ⊗ U 0 ( t ) ϕ − P + J ( t ) ⊗ U 0 ( t ) ϕ − P + L ( t ) (2.22) 10 The pro of of our result is therefore reduced to the estimate of the norm in K of the three terms in the r.h.s. of (2.22). These estimates will b e giv en in the next sections. 3 Estimate of R ( t ) 3.1 Estimate of Q 1 and Q 2 Lemma 3.1. F or the function F define d in (2.12) the fol lowing identities hold ∂ N ξ F ( τ , ξ ) = X 0 ≤ k,l,m ≤ N k + l + m = N C k,l ,m τ l b V ( k ) ( ξ ) f ( l )  τ ξ + x ε − X  P m  τ ξ + x ε  e i Ξ( τ ,ξ ) (3.1) and ∂ φ ∂ N ξ F ( τ c + φ, ξ ) = X 0 ≤ k,l,m ≤ N k + l + m = N C k,l ,m h l ( τ c + φ ) l − 1 b V ( k ) ( ξ ) f ( l )  ( τ c + φ ) ξ + x ε − X  P m  ( τ c + φ ) ξ + x ε  +( τ c + φ ) l ξ b V ( k ) ( ξ ) f ( l +1)  ( τ c + φ ) ξ + x ε − X  P m  ( τ c + φ ) ξ + x ε  +( τ c + φ ) l ξ b V ( k ) ( ξ ) f ( l )  ( τ c + φ ) ξ + x ε − X  P ′ m  ( τ c + φ ) ξ + x ε  + i 2 ( τ c + φ ) l ξ 2 b V ( k ) ( ξ ) f ( l )  ( τ c + φ ) ξ + x ε − X  P m  ( τ c + φ ) ξ + x ε   e i Ξ( τ c + φ,ξ ) (3.2) wher e C k,l ,m = N ! k ! l ! m ! , P m is a c omplex p olynomial of de gr e e m and Ξ( τ , ξ ) := 1 2 τ ξ 2 + 1 ε xξ . Pr o of. The result can b e prov en by direct computation. By applying Leibniz’s formula to the expression for Q 1 in tro duced in (2.16) one gets ∂ N ξ F ( τ , ξ ) = X 0 ≤ k,l,m ≤ N k + l + m = N C k,l ,m ( ∂ k ξ b V ( ξ ))( ∂ l ξ f  τ ξ + x ε − X  )( ∂ m ξ e i Ξ( τ ,ξ ) ) = X 0 ≤ k,l,m ≤ N k + l + m = N C k,l ,m τ l b V ( k ) ( ξ ) f ( l )  τ ξ + x ε − X  ( ∂ m ξ e i Ξ( τ ,ξ ) ) , where C k,l ,m = n ! k ! l ! m ! , as results from iterating Newton’s binomial formula. Using iden tity 0.430-2 in [5] on iterate deriv ativ es of comp osed functions, and since ∂ ξ Ξ( τ , ξ ) = τ ξ + x/ε , ∂ 2 ξ Ξ( τ , ξ ) = τ and ∂ 3 ξ Ξ( τ , ξ ) = 0, one has ∂ m ξ e i Ξ( τ ,ξ ) = X m 1 +2 m 2 = m m ! m 1 ! m 2 ! i m 1 + m 2 τ m 2 2 m 2  τ ξ + x ε  m 1 e i Ξ( τ ,ξ ) , (3.3) 11 Denoting P m ( ζ ) := X m 1 +2 m 2 = m m ! m 1 ! m 2 ! i m 1 + m 2 τ m 2 2 m 2 ζ m 1 (3.4) one obtains (3.1). Iden tity (3.2) is obtained by replacing τ b y τ c + φ in (3.1) and differen tiating in the v ariable φ . W e can no w state the main result of the section. Theorem 3.2 (Estimate of Q 1 and Q 2 ) . F or the quantities Q 1 and Q 2 define d in (2.16) the fol lowing estimates hold: ∥ Q 1 ( · , t ) ∥ ≤ C ε N , ∀ N ∈ N , ∥ Q 2 ( · , t ) ∥ ≤ C ε 2 , (3.5) wher e the c onstants C ar e indep endent of ε . Pr o of. By F ubini’s theorem, in tegrating N times by parts in the v ariable ξ yields ∥ Q 1 ( t, · ) ∥ 2 = ε 2 π K 2 Z R \ Ω ε dz e − iξ c z z N Z R \ Ω ε dz ′ e iξ c z ′ ( z ′ ) N Z dξ e iz ξ Z dξ ′ e − iz ξ ′ Z dx ∂ N ξ F ( τ c , ξ ) ∂ N ξ ′ F ( τ c , ξ ′ ) (3.6) W e fo cus on the in tegral in the v ariable x . By Lemma 3.1 it rewrites as X 0 ≤ k,l,m ≤ N k + l + m = N X 0 ≤ k ′ ,l ′ ,m ′ ≤ N k ′ + l ′ + m ′ = N C k,l ,m C k ′ ,l ′ ,m ′ τ l c τ l ′ c b V ( k ) ( ξ ) b V ( k ′ ) ( ξ ′ ) Z dx f ( l )  τ c ξ + x ε − X  f ( l ′ )  τ c ξ ′ + x ε − X  P m  τ c ξ + x ε  P m ′  τ c ξ ′ + x ε  e i Ξ( τ c ,ξ ) − i Ξ( τ c ,ξ ′ ) . F or the sake of estimating (3.6) we pass to the mo dulus and, with the change of v ariable y = x ε , w e write        X 0 ≤ k,l,m ≤ N k + l + m = N X 0 ≤ k ′ ,l ′ ,m ′ ≤ N k ′ + l ′ + m ′ = N C k,l ,m C k ′ ,l ′ ,m ′ τ l c τ l ′ c b V ( k ) ( ξ ) b V ( k ′ ) ( ξ ′ ) Z dx f ( l )  τ c ξ + x ε − X  f ( l ′ )  τ c ξ ′ + x ε − X  P m  τ c ξ + x ε  P m ′  τ c ξ ′ + x ε  e i Ξ( τ c ,ξ ) − i Ξ( τ c ,ξ ′ )     ≤ ε X 0 ≤ k,l,m ≤ N k + l + m = N X 0 ≤ k ′ ,l ′ ,m ′ ≤ N k ′ + l ′ + m ′ = N C k,l ,m C k ′ ,l ′ ,m ′ τ l c τ l ′ c | b V ( k ) ( ξ ) || b V ( k ′ ) ( ξ ′ ) | Z dy | f ( l ) ( τ c ξ + y − X ) || f ( l ′ ) ( τ c ξ ′ + y − X ) || P m ( τ c ξ + y ) || P m ′ ( τ c ξ ′ + y ) | ≤ ε X 0 ≤ k,l,m ≤ N k + l + m = N X 0 ≤ k ′ ,l ′ ,m ′ ≤ N k ′ + l ′ + m ′ = N C k,l ,m C k ′ ,l ′ ,m ′ τ l c τ l ′ c ∥ f ( l ) ( · − X ) P m ∥∥ f ( l ′ ) ( · − X ) P m ′ ∥| b V ( k ) ( ξ ) || b V ( k ′ ) ( ξ ′ ) | , 12 where w e used Cauch y-Sc h warz inequality . Then, from (3.6) one has ∥ Q 1 ( t, · ) ∥ 2 ≤ C ε 2 X 0 ≤ k,l,m ≤ N k + l + m = N X 0 ≤ k ′ ,l ′ ,m ′ ≤ N k ′ + l ′ + m ′ = N C k,l ,m C k ′ ,l ′ ,m ′ τ l c τ l ′ c ∥ f ( l ) ( · − X ) P m ∥∥ f ( l ′ ) ( · − X ) P m ′ ∥ Z R \ Ω ε dz z N Z R \ Ω ε dz ′ ( z ′ ) N Z dξ | b V ( k ) ( ξ ) | Z dξ ′ | b V ( k ′ ) ( ξ ′ ) | ≤ C ε 2 N N − 1  1 a N − 1 + 1 ( k ( t − τ c )) N − 1  2 X 0 ≤ k,l,m ≤ N k + l + m = N X 0 ≤ k ′ ,l ′ ,m ′ ≤ N k ′ + l ′ + m ′ = N C k,l ,m C k ′ ,l ′ ,m ′ τ l c τ l ′ c ∥ f ( l ) ( · − X ) P m ∥∥ f ( l ′ ) ( · − X ) P m ′ ∥∥ b V ( k ) ∥ 1 ∥ b V ( k ′ ) ∥ 1 = C ε 2 N , (3.7) so the first inequalit y in (3.5) is prov en. In order to get the second inequality , w e write the quan tity Q 2 in a differen t w ay . First, from (2.16) and (3.2), Q 2 ( t, x ) = e i K ε 2 x + i ε a K √ 2 π K √ ε Z Ω ε dz e − iξ c z Z dξ e iz ξ  F  τ c + ε K z , ξ  − F  τ c , ξ   = e i K ε 2 x + i ε a K √ 2 π K √ ε Z Ω ε dz e − iξ c z 1 + z 4 Z dξ Z z K ε 0 dφ e iz ξ ∂ φ (1 + ∂ 4 ξ ) F  τ c + φ, ξ  = e i K ε 2 x + i ε aK √ 2 π K √ ε Z Ω ε dz e − iξ c z 1 + z 4 Z dξ Z z K ε 0 dφ e iz ξ e i Ξ( τ c + φ,ξ )  ξ b V ( ξ ) f ′  ( τ c + φ ) ξ + x ε − X  + i 2 ξ 2 b V ( ξ ) f  ( τ c + φ ) ξ + x ε − X  X 0 ≤ k,l,m ≤ 4 k + l + m =4 e C k,l ,m  l ( τ c + φ ) l − 1 b V ( k ) ( ξ ) f ( l )  ( τ c + φ ) ξ + x ε − X  P m  ( τ c + φ ) ξ + x ε  + ( τ c + φ ) l ξ b V ( k ) ( ξ ) f ( l +1)  ( τ c + φ ) ξ + x ε − X  P m  ( τ c + φ ) ξ + x ε  + ( τ c + φ ) l ξ b V ( k ) ( ξ ) f ( l )  ( τ c + φ ) ξ + x ε − X  P ′ m  ( τ c + φ ) ξ + x ε  + i 2 ( τ c + φ ) l ξ 2 b V ( k ) ( ξ ) f ( l )  ( τ c + φ ) ξ + x ε − X  P m  ( τ c + φ ) ξ + x ε   , where e C k,l ,m = 24 k ! l ! m ! . In order to estimate the L 2 − norm of Q 2 , we follow the line traced for estimating Q 1 , but noticing that the integration in z is on the domain Ω ε instead of its complemen tary . F urthermore, w e will omit the details of the computation since for each 13 term they are analogous to those used for estimating Q 1 . Then ∥ Q 2 ( t, · ) ∥ ≤ ε √ 2 π K Z Ω ε dz 1 + z 4 Z dξ Z z K ε 0     dφ  | ξ b V ( ξ ) |∥ f ′ ∥ + 1 2 | ξ 2 b V ( ξ ) |∥ f ∥ + X 0 ≤ k,l,m ≤ 4 k + l + m =4 e C k,l ,m  l ( τ c + φ ) l − 1 | b V ( k ) ( ξ ) |∥ f ( l ) ( · − X ) P m ∥ +( τ c + φ ) l | ξ b V ( k ) ( ξ ) |∥ f ( l +1) ( · − X ) P m ∥ + ( τ c + φ ) l | ξ b V ( k ) ( ξ ) |∥ f ( l ) ( · − X ) P ′ m ∥ + 1 2 ( τ c + φ ) l | ξ 2 b V ( k ) ( ξ ) |∥ f ( l ) ( · − X ) P m ∥      . F or each term, the integral in φ gives a con tribution of order ε , the integral in ξ is finite since V is in the Sc hw artz class, and the in tegral in z b ecomes Z Ω ε dz | z | 1 + z 4 ≤ C . Then w e conclude ∥ Q 2 ( t, · ) ∥ ≤ C ε 2 and the pro of is complete. 3.2 Estimate of J ( t ) Prop osition 3.3. L et us fix t > T coll . Then ∥J ( t ) ∥ K ≤ C ε 2 , wher e the c onstant C dep ends on t , a, P , f , V . Pr o of. F rom the definition (2.8) of J ( t ) one has J ( t ) = J 0 ( t ) + M ( t ) (3.8) where J 0 ( t ) = − Z t 0 dτ 1 U 1 ,s 0 ( t − τ 1 ) V ε σ 2 U 1 ,s 0 ( τ 1 ) Z τ 1 0 dτ 2 U 1 ,s 0 ( − τ 2 ) V ε σ 2 U 1 ,s 0 ( τ 2 )  0 ϕ P  (3.9) and M ( t ) = i Z t 0 dτ 1 U 1 ,s ( t − τ 1 ) V ε σ 2 U 1 ,s 0 ( τ 1 ) Z τ 1 0 dτ 2 U 1 ,s 0 ( − τ 2 ) V ε σ 2 U 1 ,s 0 ( τ 2 ) Z τ 2 0 dτ 3 U 1 ,s 0 ( − τ 3 ) V ε σ 2 U 1 ,s 0 ( τ 3 )  0 ϕ P  . (3.10) First w e estimate ∥J 0 ( t ) ∥ K . W e preliminarily observe that, since the matrix σ 2 acts t wice, the only non-trivial comp onen t of J ( t ) is the second one, so the K -norm of J ( t ) equals the norm of suc h comp onen t, denoted by J (2) 0 , in the space L 2 ( R ). 14 By a straigh tforward computation one finds J (2) 0 ( t ) = − U 0 ( t ) Z t 0 dτ 1 e − i τ 1 ε U 0 ( − τ 1 ) V ε U 0 ( τ 1 ) Z τ 1 0 dτ 2 e i τ 2 ε U 0 ( − τ 2 ) V ε U 1 ,s 0 ( τ 2 ) ϕ P . (3.11) F or con venience we in tro duce the notation J 0 ( t, x ) =  Z t 0 dτ 1 e − i τ 1 ε U 0 ( − τ 1 ) V ε U 0 ( τ 1 ) Z τ 1 0 dτ 2 e i τ 2 ε U 0 ( − τ 2 ) V ε U 1 ,s 0 ( τ 2 ) ϕ P  ( x ) (3.12) and notice that ∥J 0 ( t ) ∥ K = ∥ J 0 ( t ) ∥ L 2 ( R ) . (3.13) Using the explicit expression for the propagators and pro ceeding as in App endix B, one gets J 0 ( t, x ) = Z R 6 dξ 1 dξ 2 dy dz dη dz Z t 0 dτ 1 Z τ 1 0 dτ 2 e i Υ( x,ξ 1 ,ξ 2 ,y ,z,η ,ζ ,τ 1 ,τ 2 ) /ε 8 π 3 τ 1 τ 2 ε 4 b V ( ξ 1 ) b V ( ξ 2 ) ϕ P ( ζ ) (3.14) where Υ( x, ξ 1 , ξ 2 , y , z , η , ζ , τ 1 , τ 2 ) = τ 2 − τ 1 ε + z 2 − x 2 + 2 xy − 2 y z 2 τ 1 ε 2 + ζ 2 − z 2 + 2 z η − 2 η ζ 2 τ 2 ε 2 + + y ξ 1 + η ξ 2 − a ( ξ 1 + ξ 2 ) ε . (3.15) Using Z R dη e iη  z − ζ τ 2 ε 2 + ξ 2 ε  = 2 π τ 2 ε 2 δ ( z − ζ + ετ 2 ξ 2 ) Z R dy e iy  x − z τ 1 ε 2 + ξ 1 ε  = 2 π τ 1 ε 2 δ ( x − z + ετ 1 ξ 1 ) (3.16) and then in tegrating in the v ariables z and ζ , one obtains J 0 ( t, x ) = e iP x/ε 2 2 π √ ε Z R 2 dξ 1 dξ 2 Z t 0 dτ 1 Z τ 1 0 dτ 2 b V ( ξ 1 ) b V ( ξ 2 ) f  x ε + τ 1 ξ 1 + τ 2 ξ 2  × e i Υ ε 1 ( x,ξ 1 ,ξ 2 ,τ 1 ,τ 2 ) e i Υ ε 2 ( x,ξ 1 ,ξ 2 ,τ 1 ,τ 2 ) /ε (3.17) where Υ ε 1 ( x, ξ 1 , ξ 2 , τ 1 , τ 2 ) = x ε ( ξ 1 + ξ 2 ) + 1 2 ( τ 1 ξ 2 1 + τ 2 ξ 2 2 ) + τ 1 ξ 1 ξ 2 Υ ε 2 ( x, ξ 1 , ξ 2 , τ 1 , τ 2 ) = τ 2 − τ 1 − a ( ξ 1 + ξ 2 ) + P ( τ 1 ξ 1 + τ 2 ξ 2 ) = P ( τ 1 − τ c )( ξ 1 + ξ c ) + P ( τ 2 − τ c )( ξ 2 − ξ c ) , (3.18) where, as in Section 2, we set τ c = a P , ξ c = − 1 P , p erform the c hanges of v ariable z i = P ε ( τ i − τ c ) , i = 1 , 2, and obtain J 0 ( t, x ) = e iP x/ε 2 2 π P 2 ε 3 2 Z Ω 1 ,ε dz 1 e iz 1 ξ c Z Ω 2 ,ε dz 2 e − iz 2 ξ c Z R dξ 1 e iz 1 ξ 1 Z R dξ 2 e iz 2 ξ 2 × b V ( ξ 1 ) b V ( ξ 2 ) f  x ε + τ ( z 1 ) ξ 1 + τ ( z 2 ) ξ 2  e i Λ ε ( x/ε,z 1 ,z 2 ,ξ 1 ,ξ 2 ) (3.19) 15 where we in tro duced the notation τ ( z ) = εz P + τ c , Ω 1 = [ − a/ε, P ( t − τ c ) /ε ], Ω 2 = [ − a/ε, P ( τ ( z 1 ) − τ c ) /ε ], and Λ ε ( y , z 1 , z 2 , ξ 1 , ξ 2 ) = 1 2 τ ( z 1 )( ξ 2 1 + ξ 1 ξ 2 ) + y ( ξ 1 + ξ 2 ) + 1 2 τ ( z 2 ) ξ 2 2 . (3.20) Pro ceeding analogously to (3.6), w e use the iden tity e iz ξ = 1 1 + z 2 (1 − ∂ 2 ξ ) e iz ξ for the v ariables ξ 1 and ξ 2 , and finally obtain J 0 ( t, x ) = e iP x/ε 2 2 π P 2 ε 3 2 Z Ω 1 ,ε dz 1 e iz 1 ξ c 1 + z 2 1 Z Ω 2 ,ε dz 2 e − iz 2 ξ c 1 + z 2 2 Z R dξ 1 e iz 1 ξ 1 Z R dξ 2 e iz 2 ξ 2 × (1 − ∂ 2 ξ 1 )(1 − ∂ 2 ξ 2 ) b V ( ξ 1 ) b V ( ξ 2 ) f  x ε + τ ( z 1 ) ξ 1 + τ ( z 2 ) ξ 2  e i Λ ε ( x,z 1 ,z 2 ,ξ 1 ,ξ 2 ) = e iP x/ε 2 2 π P 2 ε 3 2 Z Ω 1 ,ε dz 1 e iz 1 ξ c 1 + z 2 1 Z Ω 2 ,ε dz 2 e − iz 2 ξ c 1 + z 2 2 Z R dξ 1 e iz 1 ξ 1 Z R dξ 2 e iz 2 ξ 2 × b V ( ξ 1 ) b V ( ξ 2 ) f  x ε + τ ( z 1 ) ξ 1 + τ ( z 2 ) ξ 2  e i Λ ε ( x,z 1 ,z 2 ,ξ 1 ,ξ 2 ) − e iP x/ε 2 2 π P 2 ε 3 2 Z Ω 1 ,ε dz 1 e iz 1 ξ c 1 + z 2 1 Z Ω 2 ,ε dz 2 e − iz 2 ξ c 1 + z 2 2 Z R dξ 1 e iz 1 ξ 1 Z R dξ 2 e iz 2 ξ 2 × ∂ 2 ξ 1 b V ( ξ 1 ) b V ( ξ 2 ) f  x ε + τ ( z 1 ) ξ 1 + τ ( z 2 ) ξ 2  e i Λ ε ( x,z 1 ,z 2 ,ξ 1 ,ξ 2 ) − e iP x/ε 2 2 π P 2 ε 3 2 Z Ω 1 ,ε dz 1 e iz 1 ξ c 1 + z 2 1 Z Ω 2 ,ε dz 2 e − iz 2 ξ c 1 + z 2 2 Z R dξ 1 e iz 1 ξ 1 Z R dξ 2 e iz 2 ξ 2 × ∂ 2 ξ 2 b V ( ξ 1 ) b V ( ξ 2 ) f  x ε + τ ( z 1 ) ξ 1 + τ ( z 2 ) ξ 2  e i Λ ε ( x,z 1 ,z 2 ,ξ 1 ,ξ 2 ) + e iP x/ε 2 2 π P 2 ε 3 2 Z Ω 1 ,ε dz 1 e iz 1 ξ c 1 + z 2 1 Z Ω 2 ,ε dz 2 e − iz 2 ξ c 1 + z 2 2 Z R dξ 1 e iz 1 ξ 1 Z R dξ 2 e iz 2 ξ 2 × ∂ 2 ξ 1 ∂ 2 ξ 2 b V ( ξ 1 ) b V ( ξ 2 ) f  x ε + τ ( z 1 ) ξ 1 + τ ( z 2 ) ξ 2  e i Λ ε ( x,z 1 ,z 2 ,ξ 1 ,ξ 2 ) = J 0 , 1 ( t, x ) + J 0 , 2 ( t, x ) + J 0 , 3 ( t, x ) + + J 0 , 4 ( t, x ) , (3.21) F or the conv enience of the reader, w e fo cus on the estimate of J 0 , 1 ( t ) and J 0 , 4 ( t ). The tw o remaining terms can b e treated in the same w a y . ∥ J 0 , 1 ( t, · ) ∥ ≤ 1 2 π P 2 ε 3 2 Z Ω 1 ,ε dz 1 1 + z 2 1 Z Ω 2 ,ε dz 2 1 + z 2 2 Z R dξ 1    b V ( ξ 1 )    Z R dξ 2    b V ( ξ 1 )    ×    f  x ε + τ ( z 1 ) ξ 1 + τ ( z 2 ) ξ 2     L 2 ( R x ) = C ε 2 . (3.22) 16 Pro ceeding with J 0 , 4 , one has to distribute tw o deriv ativ es in ξ 1 and tw o in ξ 2 among the factors in the in tegrand. Then ∥ J 0 , 4 ( t, · ) ∥ ≤ ε 2 2 π P 2 X 0 ≤ k 1 ,ℓ 1 ,m 1 ≤ 2 k 1 + ℓ 1 + m 1 =2 X 0 ≤ k 2 ,ℓ 2 ,m 2 ≤ 2 k 2 + ℓ 2 + m 2 =2 Z Ω 1 ,ε dz 1 1 + z 2 1 Z Ω 2 ,ε dz 2 1 + z 2 2 Z R dξ 1    b V ( k 1 ) ( ξ 1 )    Z R dξ 2    b V ( k 2 ) ( ξ 1 )    τ ( z 1 ) ℓ 1 ( z 1 ) τ ( z 2 ) ℓ 2 ( z 2 ) ∥ f ( ℓ 1 + ℓ 2 ) ( y + τ ( z 1 ) ξ 1 + τ ( z 2 ) ξ 2 ) ∂ m 1 ξ 1 ∂ m 2 ξ 2 Λ ε ( y , z 1 , z 2 , ξ 1 , ξ 2 ) ∥ L 2 ( R y ) (3.23) By (3.20) ∂ m 1 ξ 1 ∂ m 2 ξ 2 Λ ε ( y , z 1 , z 2 , ξ 1 , ξ 2 ) = P ( y , εz 1 , εz 2 , ξ 1 , ξ 2 ) , where P is a p olynomial whose maximum degree is one in εz 1 , εz 2 and y , and t w o in ξ 1 and ξ 2 . So, one has ∥ f ( ℓ 1 + ℓ 2 ) ( y + τ ( z 1 ) ξ 1 + τ ( z 2 ) ξ 2 ) ∂ m 1 ξ 1 ∂ m 2 ξ 2 Λ ε ( y , z 1 , z 2 , ξ 1 , ξ 2 ) ∥ L 2 ( R y ) ≤ C (1 + ε | z 1 | + ε | z 2 | + ξ 2 1 + ξ 2 2 ) ∥ (1 + | y | ) f ( ℓ 1 + ℓ 2 ) ( y + τ ( z 1 ) ξ 1 + τ ( z 2 ) ξ 2 ) ∥ L 2 ( R y ) ≤ C (1 + ε | z 1 | + ε | z 2 | + ξ 2 1 + ξ 2 2 ) ∥ (1 + | y | + | ξ 1 | + | ξ 2 | + ε | z 1 | + ε | z 2 | ) f ( ℓ 1 + ℓ 2 ) ( y ) ∥ L 2 ( R y ) ≤ C (1 + ε 2 z 2 1 + ε 2 z 2 2 + | ξ 1 | 3 + | ξ 2 | 3 ) ∥ (1 + | y | ) f ( ℓ 1 + ℓ 2 ) ( y ) ∥ L 2 ( R y ) ≤ C (1 + ε 2 | z 1 | 2 + ε 2 | z 2 | 2 + | ξ 1 | 3 + | ξ 2 | 3 ) . (3.24) Inserting (3.24) in (3.23) one finally gets ∥ J 0 , 4 ( t ) ∥ ≤ C ε 2 2 π P 2 X 0 ≤ k 1 ,ℓ 1 ,m 1 ≤ 2 k 1 + ℓ 1 + m 1 =2 X 0 ≤ k 2 ,ℓ 2 ,m 2 ≤ 2 k 2 + ℓ 2 + m 2 =2 Z Ω 1 ,ε dz 1 1 + z 2 1 Z Ω 2 ,ε dz 2 1 + z 2 2 Z R dξ 1    b V ( k 1 ) ( ξ 1 )    Z R dξ 2    b V ( k 2 ) ( ξ 2 )    (1 + ε 2 z 2 1 + ε 2 z 2 2 + | ξ 1 | 3 + | ξ 2 | 3 ) . (3.25) The integrals in ξ 1 and ξ 2 are finite due to the rapid deca y of b V . The in tegrals in z 1 and z 2 pro ve finite to o, o wing to the elementary estimate Z Ω ε ( ε | z | ) β 1 + z 2 dz ≤ ε β P β ε β t β Z R dz 1 + z 2 ≤ C , where w e used sup | z | ≤ P ε t in the in tergation domain. Then, from (3.25), ∥J 0 ( t ) ∥ K =     Z t 0 dτ 1 e − i τ 1 ε U 0 ( − τ 1 ) V ε U 0 ( τ 1 ) Z τ 1 0 dτ 2 e i τ 2 ε U 0 ( − τ 2 ) V ε U 1 ,s 0 ( τ 2 ) ϕ P     L 2 ( R ) ≤ C ε 2 . (3.26) T o estimate M ( t ), from (3.10), using the unitarit y of U 1 ,s ( t − τ 1 ), ∥M ( t ) ∥ K ≤ ∥ V ∥ ∞ Z t 0 dτ 1 ∥J 0 ( τ 1 ) ∥ K ≤ C ε 2 . (3.27) 17 4 Estimate of L ( t ) Prop osition 4.1. L et us fix t > 0 . Then ∥L ( t ) ∥ K ≤ C n ( t ) ε n ∀ n ∈ N (4.1) wher e the dep endenc e of the c onstant C n ( t ) on t and on the other p ar ameters a, P , f , V is given during the pr o of. Pr o of. F rom the definition of L ( t ) (see (2.5)) w e hav e ∥L ( t ) ∥ K =     Z t 0 dτ U 1 ,s ( t − τ ) V ε σ 2 U 1 ,s 0 ( τ )  0 ϕ − P  ⊗ U 0 ( t ) ϕ P     K ≤ Z t 0 dτ     V ε σ 2 U 1 ,s 0 ( τ )  0 ϕ − P      K 1 ,s ∥ U 0 ( t ) ϕ P ∥ = Z t 0 dτ ∥ V ε U 0 ( τ ) ϕ − P ∥ . (4.2) The estimate of the L 2 -norm of V ε U 0 ( τ ) ϕ − P is elementary if one takes in to account that for ε small the function V ε is strongly concen trated around x = a > 0 while the function U 0 ( τ ) ϕ − P is strongly concen trated around x = − P τ ≤ 0. Then w e write ∥ V ε U 0 ( τ ) ϕ − P ∥ 2 = Z a/ 2 −∞ dx V  x − a ε  2 | U 0 ( τ ) ϕ − P ( x ) | 2 + Z + ∞ a/ 2 dx V  x − a ε  2 | U 0 ( τ ) ϕ − P ( x ) | 2 ≤ ∥ U 0 ( τ ) ϕ − P ∥ 2 L ∞ ( R ) Z a/ 2 −∞ dx V  x − a ε  2 + ∥ V ∥ 2 L ∞ ( R ) Z + ∞ a/ 2 dx | U 0 ( τ ) ϕ − P ( x ) | 2 . (4.3) Let us recall that U 0 ( τ ) ϕ − P ( x ) = 1 √ 2 π iετ Z dy e i ( x − y ) 2 2 ε 2 τ ϕ − P ( y ) = e − i P 2 2 ε 2 τ − i xP ε 2 U 0 ( τ ) f  x + P τ ε  = e − i P 2 2 ε 2 τ − i xP ε 2 1 √ 2 π Z dk e i x + P τ ε k e − ik 2 τ ˆ f ( k ) . (4.4) Hence, from (4.3) w e hav e 18 ∥ V ε U 0 ( τ ) ϕ − P ∥ 2 ≤ ∥ ˆ f ∥ 2 L 1 ( R ) 2 π Z a/ 2 −∞ dx V  x − a ε  2 + ∥ V ∥ 2 L ∞ ( R ) Z + ∞ a/ 2 dx     U 0 ( τ ) f  x + P τ ε      2 = ∥ ˆ f ∥ 2 L 1 ( R ) 2 π ε Z − ε − 1 a/ 2 −∞ dy V ( y ) 2 + ∥ V ∥ 2 L ∞ ( R ) ε Z + ∞ ε − 1 ( P τ + a/ 2) dy | U 0 ( τ ) f ( y ) | 2 . (4.5) It remains to estimate the t wo integrals in the last line of (4.5). F or the first one we ha v e Z − ε − 1 a/ 2 −∞ dy V ( y ) 2 ≤ ε 2 n − 1  2 a  2 n − 1 Z − ε − 1 a/ 2 −∞ dy | y | 2 n − 1 V ( y ) 2 ≤ ε 2 n − 1  2 a  2 n − 1 ∥ V ∥ 2 L 2 2 n − 1 ( R ) (4.6) for an y n ∈ N , where ∥ · ∥ L 2 n ( R ) denotes the w eighted L 2 -norm defined in (1.26). F or the second second in tegral in (4.5), a rep eated in tegration by parts yields Z + ∞ ε − 1 ( P τ + a/ 2) dy | U 0 ( τ ) f ( y ) | 2 = 1 2 π Z + ∞ ε − 1 ( P τ + a/ 2) dy     Z dk e iy k e − ik 2 τ ˆ f ( k )     2 = 1 2 π Z + ∞ ε − 1 ( P τ + a/ 2) dy 1 y 2 n     Z dk  d n dk n e iy k  e − ik 2 τ ˆ f ( k )     2 = 1 2 π Z + ∞ ε − 1 ( P τ + a/ 2) dy 1 y 2 n     Z dk e iy k d n dk n  e − ik 2 τ ˆ f ( k )      2 ≤ 1 2 π  Z dk     d n dk n  e − ik 2 τ ˆ f ( k )       2 Z + ∞ ε − 1 ( P τ + a/ 2) dy 1 y 2 n = ε 2 n − 1  2 a + 2 P τ  2 n − 1 1 2 π (2 n − 1)  Z dk     d n dk n  e − ik 2 τ ˆ f ( k )       2 . (4.7) Let us notice that     d n dk n  e − ik 2 τ ˆ f ( k )      ≤ n X m =0  n m      d n dk n e − ik 2 τ        ˆ f ( n − m ) ( k )    = n X m =0  n m  τ m/ 2       d m dq m e − iq 2  q = k √ τ         ˆ f ( n − m ) ( k )    ≤ n X m =0  n m  τ m/ 2 c m (1 + τ m/ 2 | k | m )    ˆ f ( n − m ) ( k )    19 ≤ n X m =0  n m  c m (1 + τ m )(1 + | k | m )    ˆ f ( n − m ) ( k )    ≤ c n (1 + τ n ) n X m =0 (1 + | k | m )    ˆ f ( n − m ) ( k )    (4.8) where c l denotes a n umerical constant dep ending on l ∈ N . Using (4.7) in (4.8), we find Z + ∞ ε − 1 ( P τ + a/ 2) dy | U 0 ( τ ) f ( y ) | 2 ≤ ε 2 n − 1 (1 + τ n ) 2 ( a + 2 P τ ) 2 n − 1 c n ∥ ˆ f ∥ 2 W 1 ,n n ( R ) (4.9) where ∥ · ∥ W 1 ,n n ( R ) denotes the w eighted Sob olev norm defined in (1.27). T aking in to account (4.2), (4.5), (4.6), (4.9) we obtain ∥L ( t ) ∥ K ≤ ε n c n  t a − n +1 / 2 ∥ ˆ f ∥ L 1 ( R ) ∥ V ∥ L 2 2 n − 1 ( R ) + γ ( t ) ∥ V ∥ L ∞ ( R ) ∥ ˆ f ∥ W 1 ,n n ( R )  (4.10) where γ ( t ) = Z t 0 dτ 1 + τ n ( a + 2 P τ ) n − 1 / 2 (4.11) is a con tinuous function satisfying lim t → 0 γ ( t ) t = a − n +1 / 2 , lim t →∞ γ ( t ) t 3 / 2 = 2 3 / 2 − n 3 P − n +1 / 2 . (4.12) This concludes the pro of of the proposition. App endix A Here, w e briefly recall the original EPR argument [3]. The argument relies on the follo wing t wo fundamental assumptions: - reality criterion (RC): if, without in any wa y disturbing a system, w e can predict with cer- tain ty the v alue of a physical quan tity , then there exists an elemen t of realit y corresponding to the ph ysical quantit y; - lo calit y principle (LP): the elements of physical reality of a system lo calized in a giv en place cannot b e instan taneously influenced b y a ph ysical pro cess inv olving a system, at any distance, that do es not in teract with the first one. It is w orth noticing that, while RC is explicitly stated by the authors, LP is not, since it is considered as ob vious. Under these assumptions, the goal of the argument is to prov e that Quan tum Mec hanics is not a complete theory , i.e., there are elements of reality without a coun terpart in the theory . They consider a simple mo del made of tw o quantum particles 20 in dimension one (set ℏ = 1 for simplicity) and in tro duce the corresp onding p osition and momen tum observ ables P ( j ) f ( x j ) = 1 i ∂ ∂ x j f ( x j ) , X ( j ) f ( x j ) = x j f ( x j ) , j = 1 , 2 . Neglecting some mathematical inconsistencies, they assume that the system is describ ed at some instan t of time by the entangled (i.e., non factorised) state Ψ( x 1 , x 2 ) = Z dp u p ( x 1 ) ψ − p ( x 2 ) =: Z dp e ipx 1 e − ip ( x 2 − x 0 ) where x 0 is an arbitrary fixed p oin t. Notice that u p ( x 1 ) = e ipx 1 is “eigen vector” of P (1) with “eigenv alue” p and ψ − p ( x 2 ) = e − ip ( x 2 − x 0 ) is “eigenv ector” of P (2) with “eigenv alue” − p . Moreo ver, using the identit y δ ( x ) = (2 π ) − 1 R dp e − ipx , w e hav e Ψ( x 1 , x 2 ) = 2 π δ ( x 2 − x 1 − x 0 ) and b Ψ( p 1 , p 2 ) = 2 π δ ( p 1 + p 2 ) . Hence the state Ψ describ es the physical situation where the t wo particles are at (a p os- sibly large) distance x 0 , with opp osite momentum. F urthermore, the state Ψ can also b e represen ted in the form Ψ( x 1 , x 2 ) = 2 π Z dx v x ( x 1 ) φ x + x 0 ( x 2 ) =: 2 π Z dx δ ( x 1 − x ) δ ( x 2 − x − x 0 ) where v x ( x 1 ) = δ ( x 1 − x ) is “eigenv ector” of X (1) with “eigenv alue” x and φ x + x 0 ( x 2 ) = δ ( x 2 − x − x 0 ) is “eigen vector” of X (2) with “eigen v alue” x + x 0 . No w the proof proceeds as follows. Let us p erform a measurement of the momentum P (1) of the first particle and let ¯ p b e the result. By the wa v e pack et collapse, after the measuremen t the state of the system is u ¯ p ( x 1 ) ψ − ¯ p ( x 2 ) Hence, after the measurement on the first particle, w e can predict with certain ty that the second particle has momen tum − ¯ p . By R C this means that there is an element of reality corresp onding to the observ able P (2) of the second particle. By LC this elemen t of reality cannot ha ve b een created by the measurement on the first particle and therefore it existed from the b eginning. Pro ceeding in a similar w ay , starting from the same state Ψ one measures the p osition of the first particle and let ¯ x b e the result. By the wa v e pack et collapse, after the measuremen t the state of the system is v ¯ x ( x 1 ) φ ¯ x + x 0 ( x 2 ) Hence, after the measurement on the first particle, w e can predict with certain ty that the second particle has p osition ¯ x + x 0 and so there exists an elemen t of realit y corresp onding to the p osition of the second particle. Th us, given the state Ψ of the system and p erforming measurements on the first particle, without in an y w ay disturbing the second particle one can attribute elemen ts of reality corresp onding to both momentum and position of the second particle. 21 On the other hand, in Quantum Mec hanics the uncertaint y principle implies that p osition and momen tum cannot b e b oth predicted with certaint y . Therefore, there are elements of realit y that do not hav e counterparts in the theory . It follows that Quantum Mechanics is not complete. App endix B Here w e derive formula (2.11). F rom (2.7) and (2.10) we hav e  I ( t ) f X,K  ( x ) = 1 2 π ε 5 / 2 Z t 0 dτ e i τ ε τ Z dy e − i ( x − y ) 2 2 ε 2 τ V  y − a ε  Z dz e i ( y − z ) 2 2 ε 2 τ f  z ε − X  e i K ε 2 z W e write V  y − a ε  = 1 √ 2 π Z dξ ˆ V ( ξ ) e i y − a ε ξ and then  I ( t ) f X,K  ( x ) = 1 (2 π ) 3 / 2 ε 5 / 2 Z t 0 dτ e i τ ε τ Z dy e − i ( x − y ) 2 2 ε 2 τ Z dξ ˆ V ( ξ ) e i y − a ε ξ Z dz e i ( y − z ) 2 2 ε 2 τ f  z ε − X  e i K ε 2 z = e − i x 2 2 ε 2 τ (2 π ) 3 / 2 ε 5 / 2 Z t 0 dτ e i τ ε τ Z dy e i xy ε 2 τ Z dξ ˆ V ( ξ ) e i y − a ε ξ Z dz f  z ε − X  e i z 2 2 ε 2 τ − i yz ε 2 τ + i K ε 2 z = e − i x 2 2 ε 2 τ (2 π ) 3 / 2 √ ε Z t 0 dτ e i τ ε Z dv e ixv Z dξ ˆ V ( ξ ) e − i a ε ξ + iετ v ξ Z dz f  z ε − X  e i z 2 2 ε 2 τ − iv z + i K ε 2 z No w we exc hange the order of in tegration and by a formal delta-function computation w e find  I ( t ) f X,K  ( x ) = e − i x 2 2 ε 2 τ (2 π ) 3 / 2 √ ε Z t 0 dτ e i τ ε Z dξ ˆ V ( ξ ) e − i a ε ξ Z dz f  z ε − X  e i z 2 2 ε 2 τ + i K ε 2 z Z dv e − iv  z − x − ετ ξ  = e − i x 2 2 ε 2 τ √ 2 π ε Z t 0 dτ e i τ ε Z dξ ˆ V ( ξ ) e − i a ε ξ Z dz f  z ε − X  e i z 2 2 ε 2 τ + i K ε 2 z δ  z − x − ετ ξ  = e i K ε 2 x √ 2 π ε Z t 0 dτ Z dξ ˆ V ( ξ ) f  x ε + τ ξ − X  e i  τ ξ 2 2 + x ε ξ  e i ε ( τ − aξ + K τ ξ ) whic h coincides with (2.11). 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