Korteweg-de Vries waves in peridynamical media

We consider a one-dimensional peridynamical medium and show the existence of solitary waves with small amplitudes and long wavelength. Our proof uses nonlinear Bochner integral operators and characterizes their asymptotic properties in a singular scaling limit.


Introduction
Peridynamics is a nonlocal theory which provides an alternative approach to problems in solid mechanics and replaces the partial differential equations of the classical theory by integro-differential equations that do not involve spatial derivatives, see for instance [Sil00].The internal forces between different material points are described by pairwise interactions similar to nonlinear springs and thus there exists, at least in one space dimension, a close connection to discrete atomistic models such as Fermi-Pasta-Ulam-Tsingou chains (FPUT) with nearest neighbor interactions.
Ever since the seminal paper [FPUT55] there has been an ongoing interest in the propagation of traveling waves within atomistic or related systems.A concise and very readable summary of both the existing literature and the current state of research can be found in the review article [Vai22].Traveling waves in peridynamical media are studied in [DB06,Sil16] numerically and [PV19,HM19] establish the existence of solitary waves with large amplitudes by means of two different but related variational methods.
In this paper we show the existence of Korteweg-deVries (KdV) waves with small amplitudes and generalize similar asymptotic results for various types of lattices.[ZK65] established the existence of KdV waves in FPUT chains using formal asymptotic analysis and the first rigorous existence proof has been given in the first part of the four-part series of papers [FP99,FP02,FP04a,FP04b], while the three other parts deal with the nonlinear orbital stability of those waves.Periodic waves have been studied in [FML15] while [HML16] concerns chains with more than nearest neighbor interactions.
The existence of KdV-type waves has also been proven for dimer chains, in which the masses and/or the spring constants alternate between two values, as well as for mass-in-mass systems, where each particle interacts additionally with an internal resonator.The results in [HW17, FW18, Fav20, FH20, Fav21, FH23] imply under certain generic conditions the existence of wave solutions in which an underlying KdV soliton is superimposed by periodic ripples that are either small (micropterons) or extremely small (nanopterons).See also [KVSGD13,XKS15,KSX16] for more details, [GSWW19, FGW20, FH21] for numerical stability investigations, and [VSWP16] for a discussion of non-generic cases without tail oscillations The KdV equation also governs Cauchy problems in atomistic systems provided that initial values are chosen appopriately.[SW00,KP17] show that the FPUT dynamics can be approximated on large time scales by two KdV solutions traveling in opposite directions and [HW08, HW09, SKSH14] establish similar results that hold for all times but only a special subclass of initial data.Moreover, [GMWZ14] and [MW22] study the KdV limit in chains with periodically varying and random parameters, respectively, while [HW20] concerns mass-in-mass lattices.

Setting of the problem
We consider a spatially one-dimensional continuous and infinitely extended medium whose material points interact pairwise.According to [Sil16], the simplest peridynamical equation of motion is given by the integro-differential equation where ũ denotes the scalar displacement field describing the position of material point ỹ at time t.Moreover, ξ is the bond variable and the interactions are modeled by the force function ∂ r Φ, which stems from the peridynamical potential Φ and is assumed to satisfy Newton's third law of motion via ∂ r Φ(r, ξ) = −∂ r Φ(−r, −ξ) .
Thanks to this identity, we can replace (1) by the formula which is more convenient for our purposes as it involves only positive bond variables ξ > 0.Moreover, it can be viewed as a universal equation for elastic wave propagation in one space dimension and includes many other lattice or PDE models as special or limiting cases, see the discussion in [Sil00, HM19,Kle23].In particular, assuming that all dominant forces originate from the finetely many bonds ξ ∈ {1, . . ., M }, the integral with respect to ξ can be replaced by a sum and the peridynamical wave equation (2) reduces via Φ m (r) = Φ(r, m) to This equation describes that any material point interacts with finitely many other points only and the equivalent lattice model coincides for M = 1 with the well-known FPUT chain.
In this paper, we study traveling waves in peridynamical media.Combining (2) with the rescaled traveling wave ansatz we obtain the nonlinear and nonlocal equation where ε is an additional scaling parameter.The existence of solutions has been proven in [PV19,HM19] by means of constrained optimization techniques (using ε = 1) but here we are interested in effective formulas for the long-wave length regime ε → 0 (KdV limit), in which the waves have small amplitudes and propagate with near sonic speed as in (10).In view of the known results for nonlinar lattices and PDEs, the existence of KdV-type waves in nonlocal media is not surprising and generally expected.The rigorous proof, however, is more complicated in the peridynamical setting as it involves the additional variable ξ and requires asymptotic estimates for the continuum of nonlinear interaction forces.Of particular importance is the ξ-dependence of the linear and the quadratic terms in the Taylor expansion of ∂ r Φ(r, ξ) with respect to r.
Assumption 1.1.The force function ∂ r Φ : R × [0, ∞) → R can be written as where the coefficient functions α and β are piecewise continuous and positive for all ξ ∈ [0, ∞).The function ∂ r ψ is continuously differentiable in r, continuous in ξ, and satisfies ∂ r ψ(0, ξ) = 0 as well as for any ξ and all |r| ≤ 1.Moreover, the integrals are positive and finite, while are well-defined and nonnegative.
The assumptions on (7) 1 , (7) 2 and (8) 1 are essential for the asymptotic problem to be well-defined, while the other integrability conditions simplify the analysis and might be weakened at the price of more technical effort.In mechanics one often postulates a finite interaction horizon H such that ∂ r Φ(r, ξ) = 0 holds for all r and ξ > H but our analysis also allows for H = ∞ provided that α(ξ), β(ξ) and γ(ξ) decay sufficiently fast for ξ → ∞.We further mention that alternative constitutive laws can be found in the literature.For instance, the peridynamical forces in [HM19] are modeled via in terms of a single effective potential.The choice where χ [0, H] denotes the indicator function of the interval [0, H], implies and is compatible with Assumption 1.1.Similar constitutive relations have been proposed and studied in [Sil16].
1.2 Overview on the main result and the proof strategie Our asymptotic analysis generalizes ideas and methods from [FP99] and [HML16], which prove the existence of KdV waves in spatially discrete atomic chains with a single and finitely many bond lenghtes, respectively.However, both the nonlocality and the nondiscreteness of the peridynamical medium necessitate several adjustments, especially the use of Bochner integrals and more careful estimates for the singular limit ε → 0.
x < l a t e x i t s h a 1 _ b a s e 6 4 = " y i s R 4 6 3 d u S 3 y 1 r a r D g r 6 a n J q e e f 3 m 7 W x t b n 6 / y P o m l q 0 4 0 5 k 5 j E Q h t U p l y y q r 5 W F u p x Z 2 n v V / e p X f 1 S P W G q p w 5 / L O 8 6 w e t s p u B < / l a t e x i t >

O(1)
< l a t e x i t s h a 1 _ b a s e 6 4 = " u n 1 p H r w D O G y O r 7 j 0 t 1 P F j S f p 2 n U = " > A A A C 1 3 i c h V F L T 8 J A E B 7 q C / C F e v T S S E z w Q l o 0 0 S O J j 3 g x Y i I P A 2 i 2 Z Y G G v t I u J E i I N + P V m 1 f 9 V / p b P P h 1 L S Z K D N t s Z / a b b 7 6 d 2 T F 8 2 w q F p r 0 n l L n 5 h c W l Z C q 9 v L K 6 t p 7 Z 2 K y E X j 8 w e d n 0 b C + o G S z k t u X y s r C E z W t + w J l j 2 L x q 9 I 6 j e H X A g 9 D y 3 G s A L Y 6 P V R D 8 a s / x 3 q t F M p 5 P X 9 f O H q I F t U 4 4 E n a Z t 2 K I e p H l K R z q m E O k w o v 9 A r v S k 3 y o P y q D x 9 U 5 V E n L N F v 5 b y / A X w R J P 0 < / l a t e x i t > O(" 2 2 ) < l a t e x i t s h a 1 _ b a s e 6 4 = " j L i k r a W h 9 D 9 A X I I / r 1 S 2 / H P g 4 P m I J K 5 z q B r a w j w P W E e E W D / i J X 5 7 y Figure 1: Cartoon of the velocity profile of KdV waves for two different scaling parameters ε 1 (gray) and ε 2 (black) with 0 < ε 1 < ε 2 < 1.The left panel illustrates that the scaled functions W ε converge for ε → 0 to the unique limit W 0 from (13) while the right panel shows how the unscaled counterpart Wε from (16) depends on x, the original space variable in the comoving frame.
As in [HML16], we link the scaling parameter ε to the speed via and regard the scaled velocity profile as the key quantity, while [FP99] works with the distance profile Using a convolution operator A η , which we introduce in (17), we can reformulate (4) as where the Bochner integral operator B ε is given in (26) and collects all terms that are linear with respect to W ε .Moreover, the nonlinear Bochner operators Q ε and P ε are defined in ( 27) and represent all quadratic and higher order terms, respectively.Formal asymptotic arguments applied to (11)see §2 as well as [HML16,Kle23] for more details -yield with formula (31) an analogue to the KdV traveling wave equation for the limit ε → 0. The ODE constants d 1 , d 2 depend on the coefficient functions α, β as described in equation (32) below and the only even and homoclinic solution is given by However, the nonlocal equation ( 11) is not regular but a singular perturbation of (12) and this complicates the analysis for small ε.As in [HML16], we further introduce the predictor-correctoransatz transform the operator equation ( 11) into the equivalent fixed point problem and employ the Contraction Mapping Principle to prove the existence and local uniqueness of V ε for all sufficiently small ε.The key technical problem in this approach is to establish uniform invertibility estimates for a linear but nonlocal operator L ε , which represents the linearization of (11) around W 0 and whose inverse enters the definition of the nonlinear operator F ε , see equations ( 36) and (48).We further mention that [FP99] solves the nonlinar ε-problem for FPUT chains by a variant of the Implicit Function Theorem but also needs careful estimates concerning the inverse of its linearization.
Our main result can be summarized as follows and provides via (3) and a ε-parametrized family of solitary wave solutions to the peridynamical wave equation (2) that is illustrated in Figure 1.
Main result 1.2.Let Assumption 1.1 be satisfied and ε > 0 be sufficiently small.Then there exists a unique solution W ε ∈ L 2 even (R) to the scaled peridynamic equation ( 4) with c ε as in (10) that lies in a small neighborhood of the KdV wave W 0 from (13).In particular, we have The paper is organized as follows: In §2.1 we introduce a family of convolution operators which allows us in §2.2 to transform the rescaled peridynamical equation (4) into the nonlinear integral equation (11) for the velocity profile W ε .The asymptotic properties of the involved operators are discussed in §2.3 and §2.4.In §3.1 we linearize the nonlinear problem (4) around W 0 and prove in §3.2 that the corresponding linear operator L ε is uniformly invertible on the space of all even L 2 -functions.In §3.3 we finally solve the nonlinear problem (11) by applying the Contraction Mapping Principle to the corrector equation (15).Our analysis in §2 and §3 employs Bochner integrals and we refer to the appendix for more details concerning both the general theory and the operators at hand.

Preliminaries
In this section we reformulate the rescaled peridynamical equation (4) as a nonlinear eigenvalue problem and study the properties of the involved integral operators.

The integral operator A η
For η > 0 we denote by A η the integral operator which describes the convolution with the indicator function (2 j+1)! and direct computation we verify for any k ∈ R and conclude that the pseudo-differential operator A η can be regarded as a singular perturbation of the idendity operator Id.In particular, for η := ε ξ 1 we obtain the formal expansion where the error terms contain higher derivatives.The integral operator A η exhibits a number of useful properties which we use throughout the paper.
Lemma 2.1 (properties of A η ).For all η > 0 the operator A η admits the following properties: as well as 4. The convex cone K := W ∈ L 2 (R) : W is even, nonnegativ and unimodal is invariant under A η , where unimodal means that W is monotonically increasing and decreasing for x < 0 and x > 0, respectively.

5.
A η is a pseudo-differential operator and diagonalizes in Fourier space.Its symbol function is given by with sinc (z) = sin (z)/z.
7. For any W ∈ L 1 loc (R), the estimate holds pointwise in x ∈ R.

For any sufficiently regular W we have
and In particular, the convergence is satisfied for any W ∈ L 2 (R).
Proof.All assertions follow from standard arguments and we refer to [Her10, Lemma 2.5] and [HML16, Lemma 2.3] for more details.

Reformulation of the problem
A key observation is that (4) can be reformulated as a nonlinear fixed-point equation that involves the auxiliary operator A η .
In the next step we transform the eigenvalue problem from Lemma 2.2 into the operator equation (11).To this end we insert the Taylor expansion (5) and the speed relation (10) into (25), collect all linear | nonlinear terms on the left | right hand side, and divide by ε 4 .This yields the linear operator while the nonlinear operators Q ε , P ε : L 2 (R) → L 2 (R) are given by All these operators are well-defined in the sense of Bochner integrals and the details are given in the appendix, see Proposition A.3.Of course, (11) admits the trivial solution W ε ≡ 0 but below we show that there also exists another unique solution in a small vicinity of the KdV wave W 0 .

Properties of the operator B ε
The properties of the pseudo-differential operator B ε are determined by its Fourier symbol and imply the existence of B −1 ε thanks to the supersonicity of the wave speed (10).The inverse operator is important for proving that the linearized operator L ε is uniformly invertible, see Proposition 3.4 below.
Lemma 2.3 (properties of B ε ).For any ε > 0, the linear operator B ε is continuous, self-adjoint, uniformly invertible on L 2 (R), and maps the subspace of even functions into itself.Moreover, in Fourier space it corresponds to the multiplication with the symbol function that means we have Proof.The continuity of B ε is a direct consequence of the estimates within the proof of Proposition A.3 and the self-adjointness as well as the invariance of even functions under B ε follow from the corresponding properties of A η , see Lemma 2.1.Moreover, the existence of b ε and the validity of (28) are shown in Proposition A.4.Since 0 ≤ 1 − sinc 2 (y) ≤ 1 holds for all y ∈ R we have This proves the existence of the inverse operator B −1 ε with symbol function b −1 ε as well as the uniform bound for its operator norm.
< l a t e x i t s h a 1 _ b a s e 6 4 = " Y D R p 2 m w n c L t Y + A Z q s 5 y i 1 1 Z J 7 4 w = " > A A A C 6 H i c h V F N S x x B E H 1 O / I 7 R N R 5 F W F w E P W S Z M Y I e x a j k I i q 4 K r g q P W O 7 d r b n g 5 7 e B b P s K X 8 g N / H q L d f k 1 y S / x Y N v 2 j G Q i N h D T 1 W / e v W 6 q i v M t M q t 7 / 8 e 8 N 4 M D g 2 P j I 6 N v 5 < l a t e x i t s h a 1 _ b a s e 6 4 = " t 7 u 8 p + Y 8 s L q X + z 9 < l a t e x i t s h a 1 _ b a s e 6 4 = " g E L 8 m I 7 Figure 2: The symbol functions b ε (black) from (28) and their pointwise limit b 0 (gray).
We now study the asymptotic properties of B ε .Inserting the formal expansion ( 18) into (26) and passing to the limit ε → 0 we obtain for any sufficiently smooth function W as well as However, the operator B 0 is not a regular but a singular limit of B ε since b ε does not converge uniformly to b 0 as ε → 0 as illustrated in Figure 2. In particular, we have Moreover, the positivity and the quadratic growth of b 0 imply that B 0 is defined on the smaller set W 2,2 (R) and admits an inverse with nice smoothing properties while B −1 ε is less regularizing since b ε approaches for k → ±∞ a constant value of order O(ε 2 ).Nonetheless, the operator B ε is still a sufficiently nice counterpart to B 0 as shown by the following three results.
holds pointwise for any fixed k ∈ R. Moroever, for any W ∈ W 2,2 (R) we have Proof.Using the Taylor expansion sinc 2 (y) = 1 for all k ∈ R and hence the claimed pointwise convergence.Moreover, Parseval's Theorem combined with Proposition A.4 implies The auxiliary estimate (see Figure 3 for an illustration) for all k ∈ R and we obtain The second claim is thus a direct consequence of (29), the pointwise convergence b 0 (k) = lim ε→ b ε (k), and the Dominated Convergence Theorem.Finally, let W ∈ W 4,2 (R) be fixed.Proposition A.3 provides and using for all ξ ∈ (0, ∞) thanks to (19) 2 , where we applied (23) 1 to W as well as (23) 2 to W .The third claim now follows in view of (7).
We next show that B −1 ε can be written as the sum of an almost compact operator and a small bounded one.A similar result has been derived for atomic chains in [HML16] but the technical details in the peridynamical setting are more involved due to the continuum of bond variables.We start with an auxiliary result that allows us to replace the function b ε by a simpler one and refer to the right panel in Figure 3 for an illustration.
Lemma 2.5.For all k ∈ R and ε > 0 we have where the positive constants C 0 , C 1 do not depend on ε.
r c b F j I y N y B N a + 5 A R S y j 4 r q K d 8 y + D a j s 8 p h Z X S s q Q N o y C f p q y n z 3 q 4 5 u 7 9 p T 5 U D n u d 7 m q n t / + 2 v f m u W f g s X u I V 3 n C r 6 9 j E e + y x j h C f 8 A 3 f 8 c O 5 d j 4 7 X 5 y v f 0 O d i S Z n G X e O c / M H i M y j 0 Q = = < / l a t e x i t > k < l a t e x i t s h a 1 _ b a s e 6 4 = " + i m P s 7 p 8 X c Y X h O C F + 6 / U P Z G 9 / E 0 = " A C P e Z F B 5 j m t Z X 6 k 7 C q l A Z 2 q b l z U F y t E 9 m n / 8 J z D I o B 5 y q L T h f J 0 w G G p 9 w g T C C G b q E B O e c q g q 4 7 7 k E x J r l j C j J G B T 0 D K 6 a M e r N n 8 u 9 R Z p V W t m E e V a u O 4 X N v L F p 6 n X d q n Q 2 z 1 h G p 0 R X X U Y S P P C 7 3 S m 3 a t C W 2 s T b 5 d t V w W s 0 O / j v b 8 B e O m j t o = < / l a t e x i t > b ",1 < l a t e x i t s h a 1 _ b a s e 6 4 = " F p H L 3 R s h t H h t L P U W o L X 3 j 8 T / w O g = " > A A A C 4 X i c h V F B a x N B F P 6 y V m 1 j 1 W j B S y / B I P Q g Y Z M K e g y 2 F S + F F p q 0 k J Q w u 5 2 s Q 3 Z n l 9 l J I K b 9 A b 2 J V 2 9 e 9 Q / Z 3 + L B b 6 e b g g 2 l s 8 y + N 9 / 7 3 j f v z Q u y W O X W 9 / 9 U v A c r D x 8 9 X l 2 r P l l / + u x 5 7 c X L x e u z H o 6 5 d X u o y 0 6 v 3 W x t N 9 u H 7 x q d j + X A V 7 G J 1 9 j i V N + j g 8 8 4 Y B 0 h z v E T v / D b C 7 1 L 7 5 v 3 / Z r q V c q c D f y 3 v B / / A K q 2 m K I = < / l a t e x i t > O(" 2 ) < l a t e x i t s h a 1 _ b a s e 6 4 = " t 7 u 8 p + Y 8 s L q X + z 9 8 j l G d C 7 X j z Q x n e 6 6 w y Y o j 1 X K S K b 4 7 Z p U b g z g O + Q E z b Y g X 5 K 9 8 p V F 3 H h 7 T C W e l U 4 k J R U M / Q 5 q / P e j j m x v 9 D f e x s N + u N j / X m 1 n J t 5 U s x 8 E m 8 w 3 s s c q q f s I L v 2 G Q d I U 7 w D + e 4 8 H 5 5 p 9 4 f 7 + y W 6 p W K n D k 8 W N 7 f G / 5 T m w A = < / l a t e x i t > O(" 1 ) < l a t e x i t s h a 1 _ b a s e 6 4 = " Y D R p 2 m w n c L t Y + A Z q s 5 y i 1 1 Z J 7 4 w = " > A A A C 6 H i c h V F N S x x B E H 1 O / I 7 R N R 5 F W F w E P W S Z M Y I e x a j k I i q 4 K r g q P W O 7 d r b n g 5 7 e B b P s K X 8 g N / H q L d f k 1 y S / x Y N v 2 j G Q i N h D T 1 W / e v W 6 q i v M t M q t 7 / 8 e 8 N 4 M D g 2 P j I 6 N v 5 m a K g n q K N n 1 9 1 s M x O z + H + t s 5 L J e c 5 V J 5 f 6 W 4 v p E N f B h z m M c i p 7 q K d e y i w j o 8 X O M R T 3 i 2 z q x b 6 8 6 6 / 6 B a u S x n F t + W 9 f A O O 9 G V t w = = < / l a t e x i t > Proof.We choose 0 < h < H < ∞ such that α(ξ) > 0 holds for all ξ with h ≤ ξ ≤ H and estimate Moreover, the properties of the sinus cardinalis (see the left panel in Figure 3) imply and with y = ε k/2 we obtain and setting C 1 = 4/h completes the proof.
Using the constant C 1 from Lemma 2.5 we define the cut-off operator by its symbol function and derive the following result.
Proof.Using Lemma 2.5, Fourier transform, and Parseval's theorem we obtain The desired estimate now follows with D := 2/C 0 .
Although the operator Π ε B −1 ε is much more regular than B −1 ε , it is not yet compact.In the proof of Proposition 3.4 we therefore introduce an additional cut-off in position space.since V (2) n vanishes outside the interval I L .Upper bounds for U (3) n 2 : We split the ξ-integral in the formula for M εn V (3) n , see formula (34), into two parts corresponding to ξ ∈ [0, H] and ξ ∈ (H, ∞).Concerning the first term we observe where we used (22), ε n ξ ≤ H and that and control the first integral contribution to U (3) thanks to (42) and the choice of L in (41).The second contribution can be estimated by due to the choice of H in (40).Together we obtain Derivation of the contradiction: Combining (43) with (44) leads to and (37), (45), ( 46) and (47) imply This, however, contradicts (38) and the proof is complete.
Corollary 3.5 (uniform invertibility of L ε ).The operator L ε is uniformly invertible for all small ε > 0.More precisely, for any sufficiently small ε * > 0 there exists a constant C (which may depend on ε * but not on ε) such that holds for all 0 < ε ≤ ε * and any G ∈ L 2 even (R).Proof.As a direct consequence of Proposition 3.4 we obtain the existence of a constant C * > 0 such that for all V ∈ L 2 even (R) and 0 < ε ≤ ε * .Using this as well as the formula ker L ε = coker L ε (which holds for any continuous and self-adjoint linear operator) we conclude that L ε is both injective and surjective, and that C = 1/C * can be used as continuity constant of the inverse.

The nonlinear fixed point argument
In view of the previous result and based on (35) we define the operator F ε : L 2 even (R) → L 2 even (R) by Theorem 3.6 (nonlinear fixed point argument).There exists 0 < ε * ≤ 1 such that the operator F ε admits for any 0 < ε ≤ ε * a unique fixed point V ε in the set B D = {V ∈ L 2 even (R) : V 2 ≤ D}, where D > 0 is a sufficiently large constant that may depend on ε * but not on ε.
Proof.We show that the operator F ε : B D → B D is a contractive self-mapping for sufficiently large D; the claim is then a direct consequence of the Contraction Mapping Principle.We identify the value of D at the end of this proof and denote by C > 0 any generic constant that is independent of both D and ε.
Estimates for the quadratic terms: Moreover, the choice V 2 := V and V 1 := 0 implies for any V ∈ B D .

A Bochner integrals
Elements of the general theory We summarize two important results of the Bochner theory for functions that are defined on the real semiaxis (0, ∞) and take values in the separable Hilbert space L 2 (R).For the general theory and the proofs we refer to [Růž20, chapter 2] and [HvNVW16, chapter 1].
1. F is Bochner measurable if and only if ξ → F (ξ), G 2 is Lebesgue measurable for all G ∈ L 2 (R), where •, • 2 denotes the inner product on L 2 (R).
2. Let F be Bochner measurable.Then F is Bochner integrable if and only if ξ → F ( ξ ) 2 is Lebesgue integrable.In this case we have

Special results
We next show that the integral operators in (26), (27), and (34) are in fact well-defined in the sense of Bochner integrals.Afterwards we characterize the Fourier transform of B ε .
Proposition A.3 (operators from §2 and §3).The operators B ε , Q ε , P ε , M ε : L 2 (R) → L 2 (R) are well-defined in the sense of Bochner integrals for any ε > 0. They satisfy as well as Moreover, we have Proof.We only discuss B ε and P ε in detail.The statements concerning Q ε and M ε can be established by similar arguments.
Operator B ε : For given V ∈ L 2 (R) and ε > 0 we define F ε : (0, ∞) → L 2 (R) by and observe that the function is Lebesgue measurable for any G ∈ L 2 (R) since ξ → α(ξ) ξ 2 is at least piecewise continuous according to Assumption 1.1 and because Lemma 2.1 ensures that ξ → V, G 2 − A 2 εξ V, G 2 is differentiable with respect to ξ.The function F ε is thus Bochner measurable thanks to Theorem A.1.Moreover, (19) 2 and (7) 1 yield the estimate and this implies the Bochner integrability of F ε as well as the desired bound for B ε V 2 .
Operator P ε : We now define F ε : (0, ∞) → L 2 (R) by fix G ∈ L 2 (R), and write where h ε : (0, ∞) × R → R is given by since A εξ is self-adjoint.This function h ε is continuous by Assumption 1.1 and Lemma 2.1, so Theorem A.1 combined with Fubini's theorem ensures that F ε is Bochner measurable.Moreover, (6), (19), and (22) guarantee that the estimate A εξ |V | holds for any ξ in the sense of functions with variable x.Using also (19) we therefore get and hence This implies both the desired estimate and the claimed convergence result for P ε thanks to (9).
2 / y 3 e K p n a M B z q h m b b J e 3 e N y a m X m s c O 8 Z R Y f s 9 F Z J P 6 Z 9 5 7 4 x W O v P G 3 p G O a 3 w m t a h 4 r h R P C C e 4 J K M / z L 9 j N m v 5 f / M t K s E T W y Z b h T r i w y S 9 u l + 6 e w w o o m 1 T S S P X c N s U c M x 5 y 5 f I K C t s o L 0 l f s K e d P x B a 0 w V h q V I F M U 1 N O 0 6 e u z H o 6 5 9 H O o v 5 2 T c r G 0 X i w f b R Q q S 9 n A x 7 C I Z a x y q p u o Y B + H r M P F F Z 7 w j B e r Y d 1 Z 9 9 b D J 9 U a y H I W 8 G 1 Z j x 8 z 4 5 Z e < / l a t e x i t > b 1 0 (k) < l a t e x i t s h a 1 _ b a s e 6 4 = " K + I N d 1 e u o 4 q Q 7 t p I 0 s I m v w P o Z P w d x x e 0 w l n p V J J S U V D P 0 B a v z 3 o 4 5 u D / o T 5 3 D p f r w c f 6 8 v 5 K b X 2 j H P g o Z j G P R U 5 1 F e v 4 j D 3 W E e E b f u A n f n l f v O / e j X f 7 S P U G y p w Z / L O 8 u w f 7 6 5 r / < / l a t e x i t > O(" 2 ) O 4 p / D p e b o w 0 G l 5 t d 9 t 6 q P n U b h 1F C s z a R y h T Y O k S B E H x E k Y l j 6 G g I Z v z 0 0 4 C M l t o 8 h M U N P u b j E C F P M 7 Z M l y R B E e / x 3 e d o r 0 J j n X D N z 2 S F v 0 d y G m V U s c K 8 5 x Y D s / F Z J P 6 O 9 5 v 7 t s O 6 T N w y d c l 7 h M W 1 A x b J T X C d u c U T G c 5 l R w b y r 5 f n M v C u L D j 6 7 b h T r S x 2 S 9 x n e 6 6 w y Y o j 1 X K S K b 4 7 Z p U b g z g O + Q E z b Y g X 5 K 9 8 p V F 3 H h 7 T C W e l U 4 k J R U M / Q 5 q / Pe j j m x v 9 D f e x s N + u N j / X m 1 n J t 5 U s x 8 E m 8 w 3 s s c q q f s I L v 2 G Q d I U 7 w D + e 4 8 H 5 5 p 9 4 f 7 + y W 6 p W K n D k 8 W N 7 f G / 5 T m w A = < / l a t e x i t > O(" 2 ) 9 6 C B b B U 0 m q o M e C D z w o t G A f U I t s 0 m 0 M e b J J C 7 V 4 8 + Z V f 5 z + F g 9 + u 6 a C F u m G z c x + M / P N y 4 p 9 N 0 k N 4 z 2 n L S w u L a / k C 8 X V t f W N z d L W d i u J h s L m T T v y I 9 G x W M J 9 N + T N 1 E 1 9 3 o k F Z 4 H l 8 7 b l n U l 7 e 8 R F 4 k b h b T q O e S 9 g T u g O X J u l g B r e f a l s V A x 1 9 F n F z J Q y Z a c e l T 7 o j v o U k U 1 D C o h T S C l 0 n x g l + L p k k k E x s B 5 N g A l o r r J z e q I i Y o f w 4 v B g Q D 3 8 H b y 6 G R r i L T k T F W 0 j i 4 8 r E K n T A e 6 l Yr T g L b N y 6 A n k J + 6 j w p x / M 0 w U s 6 x w D G m B s a A Y b 4 C n 9 g T A y z Z R O 4 p / D p e b o w 0 G l 5 t d 9 t 6 q P n U b h 1 F C s z a R y h T Y O k S B E H x E k Y l j 6 G g I Z v z 0 0 4 C M l t o 8 h M U N P u b j E C F P M 7 Z M l y R B E e / x 3 e d o r 0 J j n X D N z 2 S F v 0 d y G m V U s c K 8 5 x Y D s / F Z J P 6 O 9 5 v 7 t s O 6 T N w y d c l 7 h M W 1 A x b J T X C d u c U T G c 5 l R w b y r 5 f n M v C u L D j 6 7 b h T r S x 2 S 9 y C x X O k 3 O e h + C / t J 5 p e b X f b e q z 5 2 g d G o o 1 1 5 a+ Y M m L p A i Q g c x J B J Y + h o C O b 8 T B P C R E T t F j 5 i h p 1 x c o o 9 x 5 n b I k m Q I o m 3 + W z y d l G j C c 6 G Z u + y I t 2 h u w 8 w q F r i 3 n W J I d n G r p J / T 3 n N / d V j r x R t 6 T r m o 8 J o 2 p O K Y U 9 w h b n F F x m u Z c c l 8 q u X 1 z K I r i 0 u s u W 4 U 6 8 s c U v Q Z / d X Z Z M Q Q a 7 t I F V u O 2 a J G 6 M 5 d v k B C 2 2 A F x S s / K V R d x xe 0 w l n p V J J S U V D P 0 B a v z 3 o 4 5 u D / o T 5 3 D p f r w c f 6 8 v 5 K b X 2 j H P g o Z j G P R U 5 1 F e v 4 j D 3 W E e E b f u A n f n l f v O / e j X f 7 S P U G y p w Z / L O 8 u w f 7 6 5 r / < / l a t e x i t > 1 3 y 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " r s 0 p b g o 2 U m l c h z H j 0 s 3 b 2 d 2 3 D j w E 2 3 b L z m r r 3 9 g c G h 4 Z H R s f G J y K j 8 9 c 5 h E b e X J q h c F k T p 2 R S I D P 5 R V 7 e t A H s d K i p Y b y C O 3 u Z n G j 6 6 k S v w o P N C d W J 6 2 R C P 0 6 7 4 n N K H z / H R N 1 5 X w u k 6 v u 9 y r L X X O y u f 5 o l 2 y z S r 8 d p z M K S J b l S j / i h o u E M F D G y 1 I h N D 0 A w g k / E 7 g w E Z M 7 B R d Y o q e b + I S P Y w y t 0 2

Figure 3 :
Figure 3: On the construction of the pointwise bounds for b ε in the proofs of Lemma 2.4 and Lemma 2.5.