Fredholm properties of nonlocal differential operators via spectral flow
aa r X i v : . [ m a t h . A P ] J un Fredholm properties of nonlocal differential operators via spectral flow
Gr´egory Faye and Arnd Scheel University of Minnesota, School of Mathematics, 206 Church Street S.E., Minneapolis, MN 55455, USA
August 16, 2018
Abstract
We establish Fredholm properties for a class of nonlocal differential operators. Using mild convergenceand localization conditions on the nonlocal terms, we also show how to compute Fredholm indices viaa generalized spectral flow, using crossing numbers of generalized spatial eigenvalues. We illustratepossible applications of the results in a nonlinear and a linear setting. We first prove the existenceof small viscous shock waves in nonlocal conservation laws with small spatially localized source terms.We also show how our results can be used to study edge bifurcations in eigenvalue problems usingLyapunov-Schmidt reduction instead of a Gap Lemma.
Keywords:
Nonlocal operator; Fredholm index; Spectral flow; Nonlocal conservation law; Edge bifurca-tions.
Our aim in this paper is the study of the following class of nonlocal linear operators: T : H ( R , R n ) −→ L ( R , R n ) , U ddξ U − e K ξ ∗ U (1.1)where the matrix convolution kernel e K ξ ( ζ ) = e K ( ζ ; ξ ) acts via e K ξ ∗ U ( ξ ) = Z R e K ( ξ − ξ ′ ; ξ ) U ( ξ ′ ) dξ ′ . Operators such as (1.1) appear when linearizing at coherent structures such as traveling fronts or pulses innonlinear nonlocal differential equations. One is interested in properties of the linearization when analyzingrobustness, stability or interactions of these coherent structures. A prototypical example are neural fieldequations which are used in mathematical neuroscience to model cortical traveling waves. They typicallytake the form [16] ∂ t u ( x, t ) = − u ( x, t ) + Z R K ( | x − x ′ | ) S ( u ( x ′ , t )) dx ′ − γv ( x, t ) (1.2a) ∂ t v ( x, t ) = ǫ ( u ( x, t ) − v ( x, t )) (1.2b)1or x ∈ R and with γ , ǫ positive parameters. The nonlinearity S is the firing rate function and the kernel K is often referred to as the connectivity function. It encodes how neurons located at position x interactwith neurons located at position x ′ across the cortex. The first equation describes the evolution of thesynaptic current u ( x, t ) in the presence of linear adaptation which takes the form of a recovery variable v ( x, t ) evolving according to the second equation. In the moving frame ξ = x − ct , equations (1.2) can bewritten as ∂ t u ( ξ, t ) = c∂ ξ u ( ξ, t ) − u ( ξ, t ) + Z R K ( | ξ − ξ ′ | ) S ( u ( ξ ′ , t )) dξ ′ − γv ( ξ, t ) (1.3a) ∂ t v ( ξ, t ) = c∂ ξ v ( ξ, t ) + ǫ ( u ( ξ, t ) − v ( ξ, t )) , (1.3b)such that stationary solutions ( u ( ξ ) , v ( ξ )) satisfy − c ddξ u ( ξ ) = − u ( ξ ) + Z R K ( | ξ − ξ ′ | ) S ( u ( ξ ′ )) dξ ′ − γv ( ξ ) (1.4a) − c ddξ v ( ξ ) = ǫ ( u ( ξ ) − v ( ξ )) . (1.4b)The linearization of (1.3) at a particular solution ( u ( ξ ) , v ( ξ )) of (1.4) takes the form ∂ t u ( ξ, t ) = c∂ ξ u ( ξ, t ) − u ( ξ, t ) + Z R K ( | ξ − ξ ′ | ) S ′ ( u ( ξ ′ )) u ( ξ ′ , t ) dξ ′ − γv ( ξ, t ) (1.5a) ∂ t v ( ξ, t ) = c∂ ξ v ( ξ, t ) + ǫ ( u ( ξ, t ) − v ( ξ, t )) . (1.5b)Denoting U = ( u, v ) and L the right-hand side of (1.5), the eigenvalue problem associated with thelinearization of (1.3) at ( u , v ) reads λU = L U. (1.6)This eigenvalue problem can be cast as a first-order nonlocal differential equation ddξ U ( ξ ) = e K λξ ∗ U ( ξ ) (1.7)where e K λξ ( ζ ) = − c − (1 + λ ) δ + K ( | ζ | ) S ′ ( u ∗ ( ξ − ζ )) − γδ ǫδ − ( ǫ + λ ) δ ! and δ denotes the Dirac delta at 0.The differential systems (1.4) and (1.7) can be viewed as systems of functional differential equations ofmixed type since the convolutional term introduces both advanced and retarded terms. Such equations arenotoriously difficult to analyze. Our goal here is threefold. First, we establish Fredholm properties of suchoperators. Second we give algorithms for computing Fredholm indices. Last, we show how such Fredholmproperties can be used to analyze perturbation and stability problems.For local differential equations, a variety of techniques is available to study such problems. For example,in the case of the Fitzhugh-Nagumo equations, written in moving frame ξ = x − ct , ∂ t u = c∂ ξ u + ∂ ξξ u + f ( u ) − γv (1.8a) ∂ t v = c∂ ξ v + ǫ ( u − v ) (1.8b)2ith a bistable nonlinearity f , spectral properties of the linearization of (1.8) at a stationary solution( u ∗ ( ξ ) , v ∗ ( ξ )) L ∗ := c∂ ξ + ∂ ξξ + f ′ ( u ∗ ) − γǫ c∂ ξ − ǫ ! , are encoded in exponential dichotomies of the first-order equation [15, 20] ddξ U ( ξ ) = A ( ξ, λ ) U ( ξ ) , A ( ξ, λ ) = λ − f ′ ( u ∗ ) − c − γ − ǫc λ + ǫc . (1.9)In particular, L ∗ − λ is a Fredholm operator if and only if (1.9) has exponential dichotomies on R − and R + . Unfortunately, for nonlocal equations (1.7), neither existence of exponential dichotomies nor Fredholmproperties are known in general. Spectral properties of nonlocal operators such as T in (1.1) are understoodmostly in the cases where T − λ is Fredholm with index zero and U is scalar. We mention the early workof Ermentrout & McLeod [7] who proved that the Fredholm index at a traveling front is zero in the casewhere γ = 0 (no adaptation) for the neural field system (1.2). Using comparison principles, De Masi etal. proved stability results for traveling fronts in nonlocal equations arising in Ising systems with Glauberdynamics and Kac potentials [5]. In a more general setting, yet relying on comparison principles, Chen [3]showed the existence and asymptotic stability of traveling fronts for a class of nonlocal equations, includingthe models studied by Ermentrout & McLeod and De Masi et al. . Bates et al. [2], using monotonicity anda homotopy argument, also studied the existence, uniqueness, and stability of traveling wave solutions ina bistable, nonlinear, nonlocal equation.More general results are available when the interaction kernel is a finite sum of Dirac delta measures. Inparticular, the interaction kernel has finite range in that case. Such interaction kernels arise in the studyof lattice dynamical systems. Mallet-Paret established Fredholm properties and showed how to computethe Fredholm index via a spectral flow [12]. His methods are reminiscent of Robbin & Salamon’s work [19],who established similar results for operators ddξ + A ( ξ ) where A ( ξ ) is self-adjoint but does not necessarilygenerate a semi-group. For the operators studied in [12], Fredholm properties are in fact equivalent to theexistence of exponential dichotomies for an appropriate formulation of (1.1) as an infinite- dimensionalevolution problem [9, 14].Our approach extends Mallet-Paret’s results [12] to infinite-range kernels. We do not know if a dynamicalsystems formulation in the spirit of [9, 14] is possible. Our methods blend some of the tools in [19] withtechniques from [12]. In the remainder of the introduction, we give a precise statement of assumptions andour main results. We are interested in proving Fredholm properties for T : U ddξ U − e K ξ ∗ U. Our main results assume the following properties for e K ξ Exponential localization : the kernel e K ξ is exponentially localized, uniformly in ξ ; see Section 1.4,Hypotheses 1.1 and 1.2. • Asymptotically constant : there exist constant kernels e K ± such that e K ξ −→ ξ →±∞ e K ± ; see Section1.4, Hypotheses 1.1 and 1.2. • Asymptotic hyperbolicity : the asymptotic kernels e K ± are hyperbolic; see Section 1.4, Hypothesis1.3, and Section 2.2. • Asymptotic regularity : the complex extensions of the Fourier transforms of e K ± are bounded andanalytic in a strip containing the imaginary axis; see Section 1.4, Hypothesis 1.4.Our main results can then be summarized as follows. Theorem 1.
Assume that the interaction kernel e K ξ satisfies the following properties: exponential local-ization, asymptotically constant, asymptotic hyperbolicity, and asymptotic regularity. Then the nonlocaloperator T defined in (1.1) is Fredholm on L ( R ) and its index can be computed via its spectral flow. As a first example, we study shocks in nonlocal conservation laws with small localized sources of the form U t = ( K ∗ F ( U ) + G ( U )) x + ǫH ( x, U, U x ) , U ∈ R n . (1.10)Similar types of conservation laws have been studied in [4, 6]. More precisely, using a monotone iterationscheme, Chmaj proved the existence of traveling wave solutions for (1.10) with ǫ = 0, U ∈ R , [4]. Du et al. proposed to study nonlocal conservation laws more systemically and described interesting behavior in theinviscid nonlocal Burgers’ equation [6]. We show how our results can help study properties of shocks insuch systems (1.10). We prove that for small localized external sources there exist small undercompressiveshocks of index −
1, that is, { outgoing characteristics } = { ingoing characteristics } . Shocks can beparametrized by values on ingoing ”characteristics” in the case when characteristic speeds do not vanish.For vanishing characteristic speeds, we show the existence of undercompressive shocks with index −
2, thatis, { outgoing characteristics } = { ingoing characteristics } + 2. Here, we use the term characteristicinformally, a precise definition via the dispersion relation is given in Section 5.1.As a second example, we consider bifurcation of eigenvalues from the edge of the essential spectrum. Ithas been recognized early [23] that localized perturbations of operators can cause eigenvalues to emergefrom the essential spectrum. More recently, spatial dynamics methods have helped to treat a much largerclass of eigenvalue problems using analytic extensions of the Evans function into the essential spectrum,thus tracking eigenvalues into and beyond the essential spectrum; see [8, 11]. This extension, usuallyreferred to as the Gap Lemma, was used to track stability and instability in a conservation law duringspatial homotopies [17, 18], without referring to spatial dynamics but rather to a local tracking functionconstructed via Lyapunov-Schmidt and matching proceedures. In Section 5.2, we will show that such anapproach is possible for nonlocal equations, using the Fredholm properties established in our main results. We are interested in studying linear nonlocal differential equations that can be written as: ddξ U ( ξ ) = Z R K ( ξ − ξ ′ ; ξ ) U ( ξ ′ ) dξ ′ + X j ∈J A j ( ξ ) U ( ξ − ξ j ) + H ( ξ ) . (1.11)4ere U ( ξ ) , H ( ξ ) ∈ C n , and K ( ζ ; ξ ) , A j ( ξ ) ∈ M n ( C ), n ≥
1, the space of n × n complex matrices. The set J is countable and the shits ξ j satisfy (without loss of generality) ξ = 0 , ξ j = ξ k , j = k ∈ J . (1.12)For each ξ ∈ R , we define A ( ξ ) by A ( ξ ) := (cid:16) K ( · ; ξ ) , ( A j ( ξ )) j ∈J (cid:17) , (1.13)such that we may write (1.11) as ddξ U ( ξ ) = N [ A ( ξ )] · U ( ξ ) + H ( ξ ) , (1.14)where N [ A ( ξ )] denotes the linear nonlocal operator N [ A ( ξ )] · U ( ξ ) := Z R K ( ξ − ξ ′ ; ξ ) U ( ξ ′ ) dξ ′ + X j ∈J A j ( ξ ) U ( ξ − ξ j ) . (1.15)We denote K ξ := K ( · ; ξ ) and write (1.15) as a generalized convolution N [ A ( ξ )] · U = K ξ + X j ∈J A j ( ξ ) δ ξ j ∗ U. (1.16)Here ∗ refers to convolution on R ( W ∗ W )( ξ ) = Z R W ( ξ − ξ ′ ) W ( ξ ′ ) dξ ′ , and δ ξ j is the Dirac delta at ξ j ∈ R .Setting H ≡
0, we obtain the homogeneous system ddξ U ( ξ ) = N [ A ( ξ )] · U ( ξ ) . (1.17)A special case of (1.16) are constant coefficient operators A ( ξ ) A ( ξ ) = (cid:16) K ( · ) , (cid:0) A j (cid:1) j ∈J (cid:17) := A , ∀ ξ ∈ R . We have N [ A ] · U = K + X j ∈J A j δ ξ j ∗ U (1.18)and U ′ ( ξ ) = N [ A ] · U ( ξ ) . (1.19)Associated with (1.17), we have the linear operator T A := ddξ − N [ A ( ξ )] . (1.20)5 .4 Notations and hypotheses We denote by H and W the Hilbert spaces L ( R , C n ) and H ( R , C n ) equipped with their usual norm k U k H := max k =1 ··· n k U k k L ( R ) , and k U k W := k U ′ k H + k U k H . For a function K ξ = K ( · ; ξ ) : R → L η ( R , M n ( C )), η >
0, we define its norm as ||K ξ || η := max ( k,l ) ∈ J ,n K kK k,l ( · ; ξ ) e η | · | k L ( R ) . We also introduce the following norm for the kernel
K ∈ C (cid:0) R , L η ( R , M n ( C )) (cid:1) , |||K||| ∞ ,η := sup ξ ∈ R kK ξ k η + sup ξ ∈ R (cid:13)(cid:13)(cid:13)(cid:13) ddξ K ξ (cid:13)(cid:13)(cid:13)(cid:13) η . For a function A ∈ C ( R , M n ( C )) we define its norm as k A k n := sup ξ ∈ R k A ( ξ ) k M n ( C ) + sup ξ ∈ R (cid:13)(cid:13)(cid:13)(cid:13) ddξ A ( ξ ) (cid:13)(cid:13)(cid:13)(cid:13) M n ( C ) . Finally we denote by τ the linear transformation that acts on K ξ as τ · K ξ := K ( · ; · + ξ ) and wenaturally define τ · K : ξ τ · K ξ . We can now give further assumptions on the maps K and ( A j ) j ∈J . Hypothesis 1.1.
There exists η > such that the matrix kernel K satisfies the following properties:1. K belongs to C (cid:0) R , L η ( R , M n ( C )) (cid:1) ;2. K is localized, that is, |||K||| ∞ ,η < ∞ , (1.21a) ||| τ · K||| ∞ ,η < ∞ ; (1.21b)
3. there exist two functions K ± ∈ L ( R , M n ( C )) such that lim ξ →±∞ K ( ζ ; ξ ) = K ± ( ζ ) (1.22) uniformly in ζ ∈ R and lim ξ →±∞ kK ξ − K ± k η = 0 (1.23a)lim ξ →±∞ k τ · K ξ − K ± k η = 0 . (1.23b) Hypothesis 1.2.
The matrices A j satisfy the properties:1. A j ∈ C ( R , M n ( C )) for all j ∈ J ; . with η defined in Hypothesis 1.1, we have, X j ∈J k A j k n e η | ξ j | < ∞ ; (1.24)
3. there exist A ± j ∈ M n ( C ) such that lim ξ →±∞ A j ( ξ ) = A ± j , X j ∈J k A ± j k M n ( C ) e η | ξ j | < ∞ , j ∈ J (1.25) and lim ξ →±∞ X j ∈J k A j ( ξ ) − A ± j k M n ( C ) e η | ξ j | = 0 . (1.26)Note that if we define the map A as A : R −→ L η ( R , M n ( C )) × ℓ η ( M n ( C )) ξ ( ξ ) = (cid:16) K ( · ; ξ ) , ( A j ( ξ )) j ∈J (cid:17) (1.27)then, when Hypotheses 1.1 and 1.2 are satisfied, A ∈ C ( R , L η ( R , M n ( C )) × ℓ η ( M n ( C ))) and is bounded.Here we have implicitly defined ℓ η ( M n ( C )) = ( A j ) j ∈J ∈ M n ( C ) J | X j ∈J k A j k M n ( C ) e η | ξ j | < ∞ . Hypothesis 1.3.
We assume that for all ℓ ∈ R d ± ( iℓ ) := det iℓ I n − c K ± ( iℓ ) − X j ∈J A ± j e − iℓξ j = 0 (1.28) where c K ± are the complex Fourier transforms of K ± defined by c K ± ( iℓ ) = Z R K ± ( ξ ) e − iℓξ dξ. Hypothesis 1.4.
We assume that, with the same η > as in Hypotheses 1.1 and 1.2, the complex Fouriertransforms ν c K ± ( ν ) + X j ∈J A ± j e − νξ j extend to bounded analytic functions in the strip S η := { ν ∈ C | |ℜ ( ν ) | < η } . We can now restate our informal Theorem 1 which we split in two separate theorems. The first theoremstates the Fredholm property of the nonlocal operator T A while the second gives a characterization of theFredholm index via the spectral flow. 7 heorem 2 (The Fredholm Alternative) . Suppose that Hypotheses 1.1, 1.2, and 1.3 are satisfied. Thenthe operator T A : W → H is Fredholm. Furthermore, the Fredholm index of T A depends only on the limitingoperators A ± , the limits of A ( ξ ) as ξ → ±∞ . We denote ι ( A − , A + ) the Fredholm index ind T A . Theorem 3 (Spectral Flow Theorem) . Assume that Hypotheses 1.1, 1.2, 1.3, and 1.4 are satisfied andsuppose, further, that there are only finitely many values of ξ ∈ R for which A ( ξ ) is not hyperbolic. Thenthe Fredholm index of T A ι ( A − , A + ) = − cross( A ) (1.29) is the net number of roots of (1.17) which cross the imaginary axis from left to right as ξ is increased from −∞ to + ∞ ; see Section 4.1 for a precise definition. Remark 1.5.
Similar Fredholm results hold for higher-order differential operators with nonlocal terms.This can be seen by transforming into a system of first-order equations, or, more directly, by following theproof below, which treats the main part of the equation as a generalized operator pencil, thus allowing formore general forms of the equation.
Outline.
This paper is organized as follows. We start in Section 2 by introducing some notation andbasic material needed in the subsequent sections. Section 3 is devoted to the proof of Theorem 2 while inSection 4 we prove Theorem 3. Finally in Section 5, we apply our results to nonlocal conservation laws withspatially localized source term and to nonlocal eigenvalue problems with small spatially localized nonlocalperturbations.
Consider Banach spaces X and Y . We let L ( X , Y ) denote the Banach space of bounded linear operators T : X → Y , and we denote the operator norm by kT k L ( X , Y ) . We write rg T for the range of T and ker T for its kernel, rg T := {T U ∈ Y ; U ∈ X } ⊂ Y , ker T := { U ∈ X ; T U = 0 } ⊂ X . In the proof of Theorem 2, we shall use the following Lemma; see [22] for a proof.
Lemma 2.1 (Abstract Closed Range Lemma) . Suppose that X , Y and Z are Banach spaces, that T : X → Y is a bounded linear operator, and that R : X → Z is a compact linear operator. Assume that thereexists a constant c > such that k U k X ≤ c ( kT U k Y + kR U k Z ) , ∀ U ∈ X . Then T has closed range and finite-dimensional kernel. Let us recall that a bounded operator T : X → Y is a Fredholm operator if(i) its kernel ker T is finite-dimensional;(ii) its range rg T is closed; and 8iii) rg T has finite codimension.For such an operator, the integer ind T := dim (ker T ) − codim (rg T )is called the Fredholm index of T . We introduce the formal adjoint equation of (1.17) as ddξ U ( ξ ) := N [ A ( ξ )] ∗ · U ( ξ ) = − Z R K ∗ ( ξ ′ − ξ ; ξ ′ ) U ( ξ ′ ) dξ ′ − X j ∈J A ∗ j ( ξ + ξ j ) U ( ξ + ξ j ) (2.1)with K ∗ and A ∗ j denoting the conjugate transposes of the matrices K and A j , respectively. Elementarycalculations give that N [ A ( ξ )] ∗ = N [ e A ( ξ )] where e A ( ξ ) = (cid:16) e K ( · ; ξ ) , ( e A j ( ξ )) j ∈J (cid:17) and e K and e A j are defined as e K ( ζ ; ξ ) = −K ∗ ( − ζ ; − ζ + ξ ) ∀ ζ ∈ R , e A j ( ξ ) = − A ∗ j ( ξ + ξ j ) ∀ j ∈ J . Note that e K and e A j also satisfy Hypotheses 1.1 and 1.2.Considering T A as a closed, densely defined operator on H , we find that the adjoint T ∗A : W ⊂ H → H isgiven through T ∗A = − ddξ + N [ A ( ξ )] ∗ . (2.2) Associated to the constant coefficient system (1.19) is the characteristic equation d ( ν ) := det ∆ A ( ν ) = 0 (2.3)where ∆ A ( ν ) = ν I n − c K ( ν ) − X j ∈J A j e − νξ j , ν ∈ C . (2.4)Note that the characteristic equation possesses imaginary roots precisely when there exist solutions of theform e iℓξ to (1.19). More generally, roots of d − ( ν ) detect pure exponential solutions to (1.19). We say thatthis constant coefficient system is hyperbolic when d ( iℓ ) = 0 , ∀ ℓ ∈ R . (2.5)9n the specific case considered here, when c K is a bounded analytic function in the strip S η , there are onlyfinitely many roots of (2.3) in the strip. One can think of roots ν of (2.3) as generalized eigenvalues to thegeneralized eigenvalue problem (1.18).We say that the system (1.17) is asymptotically autonomous at ξ = + ∞ iflim ξ → + ∞ A ( ξ ) = A + where A + is constant. In this case, of course, (1.19) with A = A + is called the limiting equation at+ ∞ . If in addition, the limiting equation is hyperbolic, then we say that (1.17) asymptotically hyperbolicat + ∞ . We analogously define asymptotically autonomous and asymptotically hyperbolic at −∞ . If(1.17) is asymptotically autonomous at both ±∞ , we simply say that (1.17) is asymptotically autonomous,asymptotically hyperbolic if asymptotically hyperbolic at ±∞ .In the case of the constant coefficient system (1.19) it is straightforward to see that we have∆ A ∗ ( ν ) = − ∆ A ( − ¯ ν ) ∗ , so that det ∆ A ∗ ( ν ) = ( − n det ∆ A ( − ν ) . This implies that system (1.19) is hyperbolic if and only if its adjoint is hyperbolic.
For each
T >
0, we define H ( T ) = L ([ − T, T ] , C n ) and W ( T ) = H ([ − T, T ] , C n ). It is easy to see that theinclusion W ( T ) ֒ → H ( T ) defines a compact operator such that the restriction operator R : W → H ( T ) U U [ − T,T ] is a compact linear operator and kR U k H ( T ) = k U k H ( T ) . Lemma 3.1.
There exist constants c > and T > such that k U k W ≤ c (cid:0) k U k H ( T ) + kT A U k H (cid:1) (3.1) for every U ∈ W . Proof.
Following [19], we divide the proof into three steps.
Step - 1
For each U ∈ W , we have kT A U k H = (cid:13)(cid:13)(cid:13)(cid:13) ddξ U ( ξ ) − N [ A ( ξ )] · U (cid:13)(cid:13)(cid:13)(cid:13) H ≥ (cid:13)(cid:13)(cid:13)(cid:13) ddξ U (cid:13)(cid:13)(cid:13)(cid:13) H − C k U k H , where the constant C > C = n q |||K||| ∞ ,η ||| τ · K||| ∞ ,η + X j ∈J k A j k n . k ∈ J , n K , and estimate Z R (cid:12)(cid:12) ( K ξ ∗ U ) k ( ξ ) (cid:12)(cid:12) dξ ≤ n n X l =1 Z R (cid:18)Z R (cid:12)(cid:12) K k,l ( ξ − ξ ′ ; ξ ) U l ( ξ ′ ) (cid:12)(cid:12) dξ ′ (cid:19) dξ ≤ n n X l =1 Z R (cid:18)Z R (cid:12)(cid:12) K k,l ( ξ − ξ ′ ; ξ ) (cid:12)(cid:12) / (cid:12)(cid:12) K k,l ( ξ − ξ ′ ; ξ ) (cid:12)(cid:12) / (cid:12)(cid:12) U l ( ξ ′ ) (cid:12)(cid:12) dξ ′ (cid:19) dξ ≤ n n X l =1 Z R (cid:18)Z R (cid:12)(cid:12) K k,l ( ξ − ξ ′ ; ξ ) (cid:12)(cid:12) dξ ′ (cid:19) (cid:18)Z R (cid:12)(cid:12) K k,l ( ξ − ξ ′ ; ξ ) (cid:12)(cid:12) (cid:12)(cid:12) U l ( ξ ′ ) (cid:12)(cid:12) dξ ′ (cid:19) dξ ≤ n |||K||| ∞ ,η n X l =1 Z R (cid:18)Z R (cid:12)(cid:12) K k,l ( ξ − ξ ′ ; ξ ) (cid:12)(cid:12) dξ (cid:19) (cid:12)(cid:12) U l ( ξ ′ ) (cid:12)(cid:12) dξ ′ ≤ n |||K||| ∞ ,η ||| τ · K||| ∞ ,η k U k H . Similarly, one obtains Z R (cid:12)(cid:12) ( A j ( ξ ) U ( ξ − ξ j )) k (cid:12)(cid:12) dξ ≤ n k A j k n k U k H . This proves the estimate (3.1) with T = ∞ : k U k W ≤ c ( k U k H + kT A U k H ) . (3.2) Step - 2
In the second step, we prove the estimate for a hyperbolic, constant coefficient system (1.19), N [ A ] · U = K + X j ∈J A j δ ξ j ∗ U. Applying Fourier transform to f = T A U gives iℓ I n − c K ( iℓ ) − X j ∈J A j e − iℓξ j b U ( iℓ ) = b f ( iℓ ) ∀ ℓ ∈ R . Using the fact that A is hyperbolic ( d ( iℓ ) = 0), we can invert b U ( iℓ ) = iℓ I n − c K ( iℓ ) − X j ∈J A j e − iℓξ j − b f ( iℓ ) ∀ ℓ ∈ R . This implies that k b U k H ≤ sup ℓ ∈ R iℓ I n − c K ( iℓ ) − X j ∈J A j e − iℓξ j − k b f k H , and, using the Fourier-Plancherel theorem, we obtain k U k H ≤ c kT A U k H ∀ U ∈ W , for some constant c >
0. Using the first step, we finally have the inequality k U k W ≤ c kT A U k H ∀ U ∈ W , (3.3)11ith c > Step - 3
We want to prove that there exist
T > U ( ξ ) = 0 for | ξ | ≤ T − U ∈ W , we have k U k W ≤ c kT A U k H . (3.4)To do so, we first prove that inequality (3.4) is satisfied for functions U ± ∈ W , of the form U + ( ξ ) = 0 for ξ ≤ T − U − ( ξ ) = 0 for ξ ≥ − T + 1 . (3.5)We remark that Hypotheses 1.1 and 1.2 ensure the existence of T > ǫ ( T ) > U ± ∈ W are defined as above, the following estimates are satisfied (cid:13)(cid:13)(cid:0) K ± − K ξ (cid:1) ∗ U ± (cid:13)(cid:13) H ≤ ǫ ( T )2 k U ± k H , (3.6a) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ∈J (cid:16) A ± j − A j ( ξ ) (cid:17) (cid:0) δ ξ j ∗ U ± (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H ≤ ǫ ( T )2 k U ± k H . (3.6b)This ensures that for every U ± ∈ W satisfying (3.5), we have1 c k U ± k W ≤ kT A ± U ± k H ≤ ǫ ( T ) k U ± k H + kT A U ± k H , which proves inequality (3.4) in that case. Here, we have used the implicit notations T A ± = ddξ − N [ A ± ] , N [ A ± ] · U ± = K ± + X j ∈J A ± j δ ξ j ∗ U. Finally, if U ∈ W is such that U ( ξ ) = 0 for | ξ | ≤ T −
1, we decompose U as the sum U + + U − , setting U + ( ξ ) = ( U ( ξ ) , ξ ≥ , ξ < , U − ( ξ ) = ( , ξ > U ( ξ ) , ξ ≤ . Of course, U ± now satisfy (3.5) and we have k U k W = k U + k W + k U − k W ≤ c (cid:0) kT A U + k H + kT A U − k H (cid:1) = c kT A U k H , which gives the desired inequality. Step - 4
Finally, the estimate (3.1) is proved by a patching argument. We choose a smooth cutoff function χ : R → [0 ,
1] such that χ ( ξ ) = 0 for | ξ | ≥ T and χ ( ξ ) = 1 for | ξ | ≤ T −
1. Using estimate (3.2) for χU and(3.4) for (1 − χ ) U , we have k U k W ≤ k χU k W + k (1 − χ ) U k W ≤ c ( k χU k H + kT A ( χU ) k H ) + c kT A [(1 − χ ) U ] k H ≤ c (cid:0) ( k U k H ( T ) + kT A ( U ) k H (cid:1) . T A and its adjoint. Corollary 3.2.
Both, T A and T ∗A , considered as operators from W into H , possess closed range andfinite-dimensional kernel. Proof.
We only need to verify that the Hypotheses 1.1, 1.2 and 1.3 are satisfied for the adjoint operator T ∗A . We recall that in that case we have T ∗A = − ddξ + N [ e A ( ξ )]where e A ( ξ ) = (cid:16) e K ( · ; ξ ) , ( e A j ( ξ )) j ∈J (cid:17) and e K and e A j are defined as e K ( ζ ; ξ ) = −K ∗ ( − ζ ; − ζ + ξ ) ∀ ζ ∈ R , e A j ( ξ ) = − A ∗ j ( ξ + ξ j ) ∀ j ∈ J . As a consequence, Hypotheses 1.1 and 1.2 are satisfied for the adjoint. Hypothesis 1.3 refers to asymptotichyperbolicity of T A . We already noticed that A ± is hyperbolic if and only if its adjoint A ±∗ is hyperbolic,which implies that Hypothesis 1.3 is also satisfied for the adjoint equation. By Lemma 3.1, T ∗A then hasclosed range and finite-dimensional kernel. Proof. [of Theorem 2] The above corollary implies that T A : W → H has finite-dimensional kernel, closedrange, and finite-dimensional co-kernel given by the kernel of its adjoint T ∗A .To prove that the Fredholm index depends only on the limiting operators A ± we consider two families ofoperators A ( ξ ) and A ( ξ ) that satisfy Hypotheses 1.1, 1.2 and 1.3 with coefficients A ( ξ ) = (cid:16) K ( · ; ξ ) , ( A j, ( ξ )) j ∈J (cid:17) , A ( ξ ) = (cid:16) K ( · ; ξ ) , ( A j, ( ξ )) j ∈J (cid:17) and the same shifts ξ j . We assume that the limiting operators at ±∞ are equal, that is, A ± = A ± , where A ± σ = (cid:18) K ± σ , (cid:16) A ± j,σ (cid:17) j ∈J (cid:19) = lim ξ →± ξ A σ ( ξ ) , σ = 0 , . For 0 ≤ σ ≤
1, we define A σ ( ξ ) = (1 − σ ) A ( ξ ) + σ A ( ξ ). Then for each such σ , A σ satisfies Hypotheses1.1, 1.2 and 1.3 and T A σ is a Fredholm operator and T A σ varies continuously in L ( W , H ) with σ . Thus theFredholm index of T A σ is independent of σ and only depends on the limiting operators A ± . Remark 3.3.
The proof immediately generalizes to a set-up where H and W are L p -based, with < p < ∞ ,with the exception of invertibility of the asymptotic, constant-coefficient operators, where we used Fouriertransform as an isomorphism. On the other hand, analyticity of the Fourier multiplier shows that theinverse is in fact represented by a convolution with an exponentially localized kernel, which gives a boundedinverse in L p , so that our theorem holds in L p -based spaces as well. orollary 3.4 (Cocycle property) . Suppose that A , A and A are hyperbolic constant coefficient opera-tors in L η ( R , M n ( C )) × ℓ η ( M n ( C )) , then we have ι ( A , A ) + ι ( A , A ) = ι ( A , A ) . Proof.
We consider, for 0 ≤ σ ≤
1, the system U ′ ( ξ ) = N [ A σ ( ξ )] U ( ξ ) , U ( ξ ) ∈ C n where A σ ( ξ ) = (cid:16) K σ ( · ; ξ ) , ( A j,σ ( ξ )) j ∈J (cid:17) ∈ L η ( R , M n ( C )) × ℓ η ( M n ( C )) K σ ( · ; ξ ) = χ − ( ξ ) K ( · ) 00 K ( · ) ! + χ + ( ξ ) R ( σ ) K ( · ) 00 K ( · ) ! R ( − σ ) ,A j,σ ξ ) = χ − ( ξ ) A j A j ! + χ + ( ξ ) R ( σ ) A j A j ! R ( − σ ) ,R ( σ ) = cos (cid:0) πσ (cid:1) − sin (cid:0) πσ (cid:1) sin (cid:0) πσ (cid:1) cos (cid:0) πσ (cid:1) ! with χ ± ( ξ ) = (1 + tanh( ± ξ )) /
2. For all 0 ≤ σ ≤ A σ ( ξ ) is asymptotically hyperbolic and satisfiesHypotheses 1.1 and 1.2, thus T A σ is Fredholm and the Fredholm index of T A σ is independent of σ . Namely,we have ind T A σ =0 = ind T A σ =1 . At σ = 0 and σ = 1, the equation U ′ ( ξ ) = N [ A σ ( ξ )] U ( ξ ) decouples andone finds that ind T A σ =0 = ι ( A , A ) + ι ( A , A ) , ind T A σ =1 = ι ( A , A ) + ι ( A , A ) = ι ( A , A ) . This concludes the proof.
Throughout this section we fix the shifts ξ j . For ρ ∈ R , we denote by A ρ a continuously varying one-parameter family of constant coefficient operators of the form: A : R −→ L η ( R , M n ( C )) × ℓ η ( M n ( C )) ρ ρ = (cid:18) K ρ ( · ) , (cid:16) A ρj (cid:17) j ∈J (cid:19) . (4.1)For simplicity, we identify the family A ρ with its associated constant nonlocal operator N [ A ρ ]. In thissection we will prove the following result which automatically gives the result of Theorem 3. Theorem 4.
Let A ρ , for ρ ∈ R , a continuously varying one-parameter family of constant coefficientoperators of the form (4.1) . We suppose that:(i) the limit operators A ± are hyperbolic in the sense that ∀ ℓ ∈ R d ± ( iℓ ) = det iℓ I n − c K ± ( iℓ ) − X j ∈J A ± j e − iℓξ j = 0 , ii) ∆ A ρ ( ν ) defined in (2.4) is a bounded analytic function in the strip S η = { λ ∈ C | |ℜ ( λ ) | < η } foreach ρ ∈ R .(iii) there are finitely many values of ρ for which A ρ is not hyperbolic.Then ι ( A − , A + ) = − cross( A ) (4.2) is the net number of roots of (1.17) which cross the imaginary axis from left to right as ρ is increased from −∞ to + ∞ . In our approach to the proof , we approximate the family A ρ of Theorem 4 with a generic family [12, 19].To do so, we need to introduce some notations. We denote by P := P ( R , L η ( R , M n ( C )) × ℓ η ( M n ( C )))the Banach space of all continuous paths for which conditions ( i ) and ( ii ) of Theorem 4 are satisfied. Andfinally, define the open set P := C ( R , L η ( R , M n ( C )) × ℓ η ( M n ( C )) ∩ P . For any continuous path A of the form (4.1), a crossing for A is a real number ρ for which A ρ is nothyperbolic and we letNH( A ) := { ρ ∈ R | equation (1.17) with constant coefficients A ρ is not hyperbolic } , be the set of all crossings of A . Thus A satisfies condition ( iii ) of Theorem 4 if and only if A has finitelymany crossings. In that case, NH( A ) is a finite set that we denote by NH( A ) = { ρ , . . . , ρ m } . Note thatfor all A ∈ P and at any crossing ρ , the equation d ρ ( ν ) := det(∆ A ρ ( ν )) = 0has finitely many zeros in the strip S η , by analyticity and boundedness of ∆ A ρ ( ν ). We define the crossingnumber of A , cross( A ), to be the net number of roots (counted with multiplicity) which cross the imaginaryaxis from left to right as ρ increases from −∞ to + ∞ . More precisely, fix any ρ j ∈ NH( A ) and let ( ν j,l ) k j l =1 denote the roots of d ρ j ( ν ) on the imaginary axis, ℜ ( ν j,l ) = 0. We list multiple roots repeatedly accordingto their multiplicity. Let M j denote the sum of their multiplicities. For ρ near ρ j , with ± ( ρ − ρ j ) > M j roots (counting multiplicity) near the imaginary axis, M L ± j with ℜ ν < M R ± j with ℜ ν >
0, and M j = M L ± j + M R ± j . We definecross( A ) = m X j =1 (cid:16) M R + j − M R − j (cid:17) . For
A ∈ P , we say that a crossing ρ is simple if there is precisely one simple root of d ρ j ( ν ∗ ) locatedon the imaginary axis, and if this root crosses the imaginary axis with non-vanishing speed as ρ passesthrough ρ j . Note that for these simple crossings, we can locally continue the root ν ∗ ∈ i R as a C -functionof ρ as ν ( ρ ). We refer to this root as the crossing root . Non-vanishing speed of crossing can then beexpressed as ℜ ( ˙ ν ( ρ )) = 0. 15ext, suppose that A ∈ P has only simple crossings ρ j ∈ NH( A ). In this case we let ν j ( ρ ) be the complex-valued crossing-value defined near ρ j such that ν j ( ρ ) is a root of d ρ and ℜ ( ν j ( ρ j )) = 0. In this case, thecrossing number is explicitly given throughcross( A ) = m X j =1 sign ( ℜ ( ˙ ν j ( ρ j ))) . (4.3)The following result shows that the set of paths with only simple crossings is dense in P . Lemma 4.1.
Let
A ∈ P be such that NH ( A ) is a finite set. Then given ǫ > , there exists e A ∈ P suchthat:(i) e A ± = A ± ;(ii) | e A ρ − A ρ | < ǫ for all ρ ∈ R ; and(iii) e A has only simple crossings. This lemma is proved in the following section.
Remark 4.2. If ǫ is small enough in Lemma 4.1, then one has cross( A ) = cross( e A ) . The proof follows [12] with some appropriate modifications.We start by introducing submanifolds of M n ( C ). For 0 ≤ k ≤ n we define the sets G k ⊂ M n ( C ) and H ⊂ M n ( C ) × M n ( C ) by G k = { M ∈ M n ( C ) | rank( M ) = k } , H = { ( M , M ) ∈ M n ( C ) × M n ( C ) | rank( M ) = n − ,M is invertible, and rank( M M − M ) = n − (cid:9) . The sets G k and H are analytic submanifolds of M n ( C ) and M n ( C ) × M n ( C ) respectively, of complexdimension dim C G k = n − ( n − k ) , dim C H = 2 n −
2; (4.4)see [12]. We also consider the following maps F , G : (cid:0) L η ( R , M n ( C )) × ℓ η ( M n ( C )) (cid:1) × R → M n ( C ) F × G : (cid:0) L η ( R , M n ( C )) × ℓ η ( M n ( C )) (cid:1) × R → M n ( C ) × M n ( C ) D : (cid:0) L η ( R , M n ( C )) × ℓ η ( M n ( C )) (cid:1) × T → M n ( C ) × M n ( C )16iven by F ( A , ℓ ) = iℓ I n − b K ( iℓ ) − X j ∈J A j e − iℓξ j , (4.5a) G ( A , ℓ ) = I n − b K ′ ( iℓ ) + X j ∈J ξ j A j e − iℓξ j , (4.5b)( F × G )( A , ℓ ) = ( F ( A , ℓ ) , G ( A , ℓ )) , (4.5c) D ( A , ℓ , ℓ ) = ( F ( A , ℓ ) , F ( A , ℓ )) , (4.5d)where A = (cid:16) K , ( A j ) j ∈J (cid:17) ∈ L η ( R , M n ( C )) × ℓ η ( M n ( C )) and T is the set T = (cid:8) ( ℓ , ℓ ) ∈ R | ℓ < ℓ (cid:9) . Proposition 4.3.
Suppose that A = (cid:16) K , ( A j ) j ∈J (cid:17) ∈ L η ( R , M n ( C )) × ℓ η ( M n ( C )) satisfies the conditions ( i ) F ( A , ℓ ) / ∈ G k , ≤ k ≤ n − , ℓ ∈ R ( ii ) ( F × G )( A , ℓ ) / ∈ G n − × G k , ≤ k ≤ n − , ℓ ∈ R ( iii ) ( F × G )( A , ℓ ) / ∈ H , ℓ ∈ R ( iv ) D ( A , ℓ , ℓ ) / ∈ G n − × G n − , ( ℓ , ℓ ) ∈ T (4.6) for all ranges of k, ℓ, ℓ and ℓ . Then the constant coefficient system (1.19) has at most one ℓ ∈ R suchthat ν = iℓ is a root of the characteristic equation det ∆ A ( ν ) = 0 , and the root ν is simple. Proof.
We first note that F ( A , ℓ ) = ∆ A ( iℓ ) as defined in (2.4) and that G ( A , ℓ ) = − i ∆ ′A ( iℓ ). Therefore,condition ( i ) implies that rank(∆ A ( ν )) = n − ν = iℓ . Condition ( ii ) ensures that ∆ ′A ( ν ) isinvertible for such ν . Condition ( iii ) implies that the rank of ∆ A ( ν )∆ ′A ( ν ) − ∆ A ( ν ) is n − ν .Hypothesis 1.4 ensures the existence of η > η − η > f ( ν ) = ∆ A ( ν ) is a holomorphicfunction in a neighborhood of iℓ ∈ S η − η = { ν ∈ C | |ℜ ( ν ) | < η − η } that satisfies: • rank( f ( iℓ )) = n − • f ′ ( iℓ ) is invertible • rank( f ( iℓ ) f ′ ( iℓ ) − f ( iℓ )) = n − g ( ν ) = det f ( ν ) has a simple root at ν = iℓ [12] and ν = iℓ is a simple root of thecharacteristic equation det ∆ A ( ν ) = 0. Finally, the last condition ( iv ) ensures that there is at most onevalue ℓ ∈ R for which det ∆ A ( iℓ ) = 0 which concludes the proof. Proposition 4.4.
The maps F and F × G have surjective derivative with respect to the first argument A at each point ( A , ℓ ) ∈ L ( R , M n ( C )) × ℓ ( M n ( C )) × R . Moreover, if ξ j /ξ k is irrational for some j < k , then the derivative of the map D with respect to the first argument A is surjective at each ( A , ℓ ) ∈ L ( R , M n ( C )) × ℓ ( M n ( C )) × T . roof. From their respective definition, one sees immediately that the derivative of F with respect to A ∈ M n ( C ) is − I n and that the derivative with respect to ( A , A ) ∈ M n ( C ) × M n ( C ) is given by thematrix − I n e − iℓξ I n n − ξ e − iℓξ I n ! which is an isomorphism on M n ( C ) × M n ( C ); in particular, the derivative of both maps is onto.We fix ( ℓ , ℓ ) ∈ T . Then at least one of the quantities ( ℓ − ℓ ) ξ j or ( ℓ − ℓ ) ξ k is irrational. Suppose nowthat ( ℓ − ℓ ) ξ j is irrational. Then the derivative of D with respect to ( A , A j ) is given by − I n e − iℓ ξ j I n I n e − iℓ ξ j I n ! which is an isomorphism. Remark 4.5.
Note that we can always assume that ξ j /ξ k is irrational for some j < k . If it is not thecase, we can enlarge J to J ∪ { ξ ∗ } with an additional constant coefficient A ∗ = 0 in (1.11) so that ξ ∗ /ξ k is irrational for some k ∈ J . In order to complete the proof of Lemma 4.1, we will use the notion of transversality for smooth mapsdefined in manifolds. We say that a smooth map f : X → Y from two manifolds is transverse to asubmanifold
Z ⊂ Y on a subset
S ⊂ X ifrg( Df ( x )) + T f ( x ) Z = T f ( x ) Y whenever x ∈ S and f ( x ) ∈ Z where T p M denotes the tangent space of M at a point p . Theorem 5 (Transversality Density Theorem) . Let V , X , Y be C r manifolds, Ψ :
V → C r ( X , Y ) a repre-sentation and Z ⊂ Y a submanifold and ev Ψ : V × X → Y the evaluation map. Assume that:1. X has finite dimension N and Z has finite codimension Q in Y ;2. V and X are second countable;3. r > max(0 , N − Q ) ;4. ev Ψ is transverse to Z .Then the set { V ∈ V | Ψ V is transverse to Z} is residual in V . The proof of this theorem can be found in [1].
Proposition 4.6.
There exists a residual (and hence dense) subset of P such that for any A in thissubset, all conditions (4.6) are satisfied. Proof.
The idea is to apply the Transversality Density Theorem 5 to exhibit a residual subset of P suchthat all the maps F ( A ρ , ℓ ) , ( F × G )( A ρ , ℓ ) and D ( A ρ , ℓ , ℓ ) are transverse to the manifolds appearing in184.6) on ( ρ, ℓ ) ∈ R and ( ρ, ℓ , ℓ ) ∈ R respectively. For simplicity we only detail the proof for F , the twoother cases being similar.We apply Theorem 5 with manifolds V = P , X = R and Y = M n ( C ) and submanifold Z = G k with0 ≤ k ≤ n −
2. So for any
A ∈ P we define Ψ A : R → M n ( C ) byΨ A ( ρ, ℓ ) = F ( A ρ , ℓ ) , and the evaluation map is simply given by ev Ψ : P × R → M n ( C )ev Ψ ( A , ρ, ℓ ) = F ( A ρ , ℓ ) . We thus have r = 1, N = 2 and Q = 2( n − k ) (the real codimension of G k ). This implies that thethird condition of Theorem 5 is satisfied for all 0 ≤ k ≤ n −
2. Proposition 4.4 ensures that the requiredtransversality hypothesis of the evaluation map is fulfilled.We can then conclude that there exists a residual subset (and hence dense) of P such that for any A inthis subset the composed map F ( A ρ , ℓ ) is transverse to the manifolds appearing in (4.6). Proof. [of Lemma 4.1] We are now ready to prove Lemma 4.1. Let
A ∈ P such that
N H ( A ) is a finiteset. By Proposition 4.6, we may assume that the family A in the statement of Lemma 4.1 is such that allfour conditions (4.6) hold for A ρ for each ρ ∈ R . Thus for each such A ρ , the constant coefficient equation(1.19) has at most one ℓ ∈ R such that ν = iℓ is an root and iℓ is a simple root of the characteristicequation det ∆ A ρ ( ν ) = 0. It is then enough to perturb A to a nearby e A ∈ P with the same endpoints e A ± = A ± such that, by Sard’s Theorem, all the roots of the corresponding family of equations (1.19) crossthe imaginary axis transversely with ρ , that is, e A has only simple crossings. We first introduce the map Σ γ : L η ( R , M n ( C )) × ℓ η ( M n ( C )) → L η ( R , M n ( C )) × ℓ η ( M n ( C )), defined foreach γ ∈ R by Σ γ · A = Σ γ · (cid:16) K , (cid:0) A j (cid:1) j ∈J (cid:17) := (cid:16) K γ , (cid:0) A j,γ (cid:1) j ∈J (cid:17) , where K γ ( ζ ) = K ( ζ ) e γζ , ∀ ζ ∈ R , A ,γ = A + γ, A j , γ = A j e γξ j , ∀ j = 1 . This transformation Σ γ arises from a change of variables V ( ξ ) = e γξ U ( ξ ) in (1.19) with constant coefficient A = (cid:18) K , (cid:16) A j (cid:17) j ∈J (cid:19) . One can then easily check that∆ Σ γ ·A ( ν ) = ∆ A ( ν − γ ) , ν ∈ C , so that Σ γ shifts all eigenvalues to the right by an amount of γ . Proposition 4.7.
Suppose that ν = iℓ , with ℓ ∈ R , is a simple root of the characteristic equation (2.3) associated to A , and suppose that there are no other roots with ℜ λ = 0 . Then for γ ∈ R , < | γ | < η sufficiently small, we have that ι (Σ − γ · A , Σ γ · A ) = − sign( γ ) . (4.7)19 roof. With A = (cid:18) K , (cid:16) A j (cid:17) j ∈J (cid:19) , we make the change of variable V ( ξ ) = W γ ( ξ ) U ( ξ ), with W γ ( ξ ) = e γ √ ξ +1 , in equation (1.19). This leads to a nonautonomous equation of the form V ′ ( ξ ) = N [ A γ ( ξ )] · V ( ξ )which is asymptotically hyperbolic with limiting operators A ± γ = Σ ± γ · A . It is easy to check that theexponential localization of K in L η ( R , M n ( C )) and (cid:16) A j,γ (cid:17) j ∈J in ℓ η ( R , M n ( C )) ensures that T A γ satisfiesHypotheses 1.1 and 1.2, and thus is Fredholm for 0 < γ < η . Similarly, we make the change of variable V ( ξ ) = W − γ ( ξ ) U ( ξ ) in the adjoint equation U ′ ( ξ ) = N [ A ] ∗ U ( ξ )which results in the nonautonomous equation V ′ ( ξ ) = N [ A γ ( ξ )] ∗ · V ( ξ ) . Without loss of generality, we suppose that γ > ν = iℓ is the only root ofdet(∆ A ) = 0 in the strip |ℜ ( ν ) | ≤ γ < η . Suppose that V is a nonzero element of the kernel of T A γ , then V is bounded and U = W − γ V is also a bounded solution of U ′ ( ξ ) = N [ A ] · U ( ξ ), hence U ( ξ ) = e iℓξ p forsome nonzero vector p ∈ C n . Indeed, as ν = iℓ is a simple root of det(∆ A ) = 0, there exists p ∈ C n , suchthat p belongs to the kernel of ∆ A and thus e iℓξ p is in the kernel of T A using Fourier transform. But, V ( ξ ) = W γ ( ξ ) e iℓξ p is now unbounded which is a contradiction, and so ker T A γ = { } . Applying a similarargument to the adjoint equation, one sees that V ( ξ ) = W − γ ( ξ ) e − iℓξq , q ∈ C n , is the one-dimensionalspan of ker T ∗A γ . Thus, applying Theorem 2, we have ι (Σ − γ · A , Σ γ · A ) = ind T A γ = dim ker T A γ − dim ker T ∗A γ = − . The following proposition shows that without loss of generality we may assume that roots of the char-acteristic equation cross the imaginary axis by means of a rigid shift of the spectrum with the operatorΣ γ . Proposition 4.8.
Let
A ∈ P be such that N H ( A ) is a finite set and has only simple crossings. Thenthere exists e A ∈ P such that:(i) A ± = e A ± ;(ii) N H ( A ) = N H ( e A ) ;(iii) for each ρ j ∈ N H ( A ) , we have ℜ ( ˙ ν j ( ρ j )) = ℜ ( ˙ e ν j ( ρ j )) , with e ν j corresponding to e A ;(iv) e A has only simple crossings.In addition, the family e A has the form e A ρ = Σ γ j ( ρ − ρ j ) · A ρ j , γ j := ℜ ( ˙ ν j ( ρ j )) , (4.8) for ρ in a neighborhood of each ρ j .
20e omit the proof of this result, as it is identical to that in [12].
Proof. [of Theorem 4] Let
A ∈ C (cid:0) R , L η ( R , M n ( C )) × ℓ η ( M n ( C )) (cid:1) be a one-parameter family as in thestatement of Theorem 4. Without loss, by Lemma 4.1, we may assume that A has only simple crossings.Let e A ∈ C (cid:0) R , L η ( R , M n ( C )) × ℓ η ( M n ( C )) (cid:1) as in statement of Proposition 4.8. Then for any sufficientlysmall ǫ >
0, using the Corollary 3.4, we have that ι ( A − , A + ) = ι ( A − , e A ρ − ǫ ) + m − X j =1 ι ( e A ρ j + ǫ , e A ρ j +1 − ǫ ) + m X j =1 ι ( e A ρ j − ǫ , e A ρ j + ǫ ) + ι ( e A ρ m + ǫ , A + ) . For each ρ in the intervals: [ ρ j + ǫ, ρ j +1 − ǫ ], 1 ≤ j ≤ m −
1, ( −∞ , ρ − ǫ ] and [ ρ n + ǫ, + ∞ ), equation (1.19)is hyperbolic, and one concludes that ι ( A − , e A ρ − ǫ ) = m − X j =1 ι ( e A ρ j + ǫ , e A ρ j +1 − ǫ ) = ι ( e A ρ m + ǫ , A + ) = 0 . On each interval [ ρ j − ǫ, ρ j + ǫ ], 1 ≤ j ≤ m , we have a simple crossing and we can apply the result ofProposition 4.7: m X j =1 ι ( e A ρ j − ǫ , e A ρ j + ǫ ) = − m X j =1 sign ( ℜ ( ˙ ν j ( ρ j ))) . This implies that ι ( A − , e A ρ − ǫ ) = − cross( A ) which concludes the proof. We now give a first application of Theorem 3 to operators posed on exponentially weighted spaces. Assumethat A = (cid:18) K , (cid:16) A j (cid:17) j ∈J (cid:19) ∈ P is a constant coefficient operator and consider the associated operator T A = ddξ − N [ A ] on the space e L η ( R , C n ) with norm k U k e L η = k U ( · ) e γ · k L ( R , C n ) . Using the isomorphism e L γ ( R , C n ) −→ L ( R , C n ) , U ( ξ ) U ( ξ ) e γξ , the operator T A for U on e L γ ( R , C n ) is readily seen to be conjugate to T γ A = ddξ − N [Σ γ · A ] for V on L ( R , C n ). We conclude that T γ A is Fredholm for γ in open subsets of the real line. When A has onlyfinitely many simple crossings, we can consider the family of operators T γ A with γ close to zero. Moregenerally, we introduce a two-sided family of weights via k U k γ − ,γ + = k U χ + k e L γ + + k U χ − k e L γ − where χ ± ( ξ ) = ( ± ξ >
00 otherwise . The operator T A on L γ − ,γ + is conjugate to an operator T γ − ,γ + A on L whose coefficients are Σ γ + · A for ξ > γ − · A for ξ <
0. The following corollary is a direct consequence of the above discussion andTheorem 3. 21 orollary 4.9.
Suppose that ν = iℓ , with ℓ ∈ R , is a root of the characteristic equation associated to A of multiplicity N , and suppose that there are no other roots with ℜ λ = 0 . Then, the operator T γ − ,γ + A isFredholm for all γ ± close to zero with γ − γ + = 0 and for γ ∈ R , γ = 0 sufficiently small, we have that ι (Σ − γ · A , Σ γ · A ) = ind T − γ,γ A = − sign( γ ) N. (4.9) We give two applications of our main result. We first consider the effect of small inhomogeneities innonlocal conservation laws. We then show how our results can be used to study edge bifurcations fornonlocal eigenvalue problems, replacing Gap Lemma constructions with Lyapunov-Schmidt and far-fieldmatching constructions.
Consider the nonlocal conservation laws U t = ( K ∗ F ( U ) + G ( U )) x , U ∈ R n , x ∈ R , (5.1)with appropriate conditions on convolution kernel K , and fluxes F, G . Nonlocal conservation laws arisein a variety of applications and pose a number of analytic challenges; see [6] for a recent discussion andreferences.In the absence of the nonlocal, dispersive term
K ∗ F , the system of conservation laws is well known todevelop discontinuities in finite time which are referred to as shocks. Shocks can usually be classifiedaccording to ingoing and outgoing characteristics. In the presence of viscosity, shocks are smooth travelingwaves, and characteristic speeds can be characterized via the group velocities of neutral modes in thelinearization. In our case, the linearization at a constant state V t = ( K ∗ dF U (0) + dG U (0)) V x , V ∈ R n , x ∈ R , can be readily solved via Fourier transform, with dispersion relation d ( λ, iℓ ) = det (cid:16) iℓ b K ( iℓ ) dF U (0) + iℓdG U (0) − λ I n (cid:17) . We find an eigenvalue λ = 0 with multiplicity n . Assuming that b K ( iℓ ) dF U (0) + dG U (0) possesses real,distinct eigenvalues − c j , we obtain expansions λ j ( iℓ ) = − c j ℓ + O( ℓ ), so that the negative eigenvalues c j naturally denote speeds of transport in different components of the system. As with viscous approximationsto local conservation laws, instabilities can enter for finite wavenumber ℓ for non-scalar diffusion, so thatwe will need an extra condition on the nonlocal part that guarantees stability of the homogeneous solution.Rather than studying existence of large-amplitude shock profiles, we focus here on a perturbation result,exploiting the linear Fredholm theory developed in the previous sections. It will be clear from the techniquesemployed here and in the subsequent section that our results can be used to develop a spectral theory forlarge amplitude shock profiles in the spirit of [24]. Our results parallel the results in [21], where viscous22egularization of conservation laws were analyzed. Roughly speaking, our results show that at smallamplitude, nonlocal, dispersive terms act in a completely analogous fashion to viscous regularizing terms.Our analysis considers spatially localized source terms of the nonlocal conservation law (5.1), U t = ( K ∗ F ( U ) + G ( U )) x + ǫH ( x, U, U x ) , U ∈ R n (5.2)for a kernel K ∈ L η ( R , M n ( R )), with fixed η >
0, and a smooth hyperbolic flux g withdet( dG U (0)) = 0 (5.3a) σ (cid:16) dG U (0) + b K (0) dF U (0) (cid:17) = {− c > − c > · · · > − c n } (5.3b)det (cid:16) dG U (0) + b K ( iℓ ) dF U (0) (cid:17) = 0 , ∀ ℓ ∈ R , ℓ = 0 (5.3c) b K ( ν ) dF U (0) , dG U (0) ∈ S n ( R ) = (cid:8) M ∈ M n ( R ) | M = M t (cid:9) ∀ ν ∈ C (5.3d)and a smooth, spatially localized, source term H so that there exist constant C, δ > k H ( x, U, V ) k ≤ Ce − δ | x | (5.4)for all x ∈ R and all ( U, V ) near zero in R n × R n .Here, the first condition guarantees that steady-states are solutions to ODEs, hence smooth; the secondcondition enforces strict hyperbolicity of the nonlocal linear part, the third condition guarantees that zerois not in the essential spectrum of the linearization for any nonzero wavenumber. The last condition refersto the usual requirement of symmetric fluxes.We look for small bounded solutions of the nonlocal equation0 = ( K ∗ F ( U ) + G ( U )) x + ǫH ( x, U, U x ) . (5.5)Contrary to hyperbolic conservation laws where the viscous term is typically BU xx with a positive definite,symmetric viscosity matrix B , we cannot use spatial dynamics techniques for (5.5) because of the nonlocalterm K ∗ F ( U ). Instead, following [21], we will use an approach based only on functional analysis andLyapunov-Schmidt reduction, thus exploiting the Fredholm and spectral flow properties developed in theprevious sections. The key point of our approach is the linearization of equation (5.5) at the solution U = 0and ǫ = 0 L U = K x ∗ ( dF U (0) U ) + dG U (0) U x . (5.6)The adjoint L ∗ of (5.6) is given by L ∗ U = − dF U (0) t K t − ∗ U x − dG U (0) t U x (5.7)where K t − ( x ) = K t ( − x ). Assuming that dG U (0) is invertible, we can associate the operator e L U = U x + dG U (0) − K x ∗ ( dF U (0) U ) (5.8)which is of the form of a constant operator studied in Section 3 as K x ∈ L ( R , M n ( R )). Both L and e L canbe viewed as unbounded linear operators on L ( R , R n ) but also can be considered as unbounded operatorson L η ( R , R n ) for 0 < η < η as K ∈ L η ( R , M n ( R )) with norm k U k L η ( R , R n ) = k U ( x ) e η | x | k L ( R , R n ) . emma 5.1. Assume that c j = 0 for all j , then there is an η ∗ > with the following property. For eachfixed η with < η < η ∗ , the operator L defined on L η ( R , R n ) is Fredholm with index − n and has trivialnull space. Proof.
The characteristic equation associated to the linearized system (5.8) is0 = det( ν I n + νdG U (0) − b K ( ν ) dF U (0)) = ν n det( dG U (0) − ) det (cid:16) dG U (0) + b K ( ν ) dF U (0) (cid:17) , (5.9)so that ν = 0 is an root with multiplicity n , and all other roots have nonzero real part due to (5.3c). Wecan apply Corollary 4.9 and find that the Fredholm index of e L and thus of L on L η ( R , R n ) is equal to − n as claimed. Since (5.6) is translation invariant, we can use Fourier transform to analyze the kernel. Anyfunction U in the kernel of L satisfies0 = iℓ (cid:16) b K ( iℓ ) dF U (0) + dG U (0) (cid:17) b U ( ℓ ) . As U ∈ L η ( R , R n ), b U ( ℓ ) is a bounded analytic function in the strip S η , and thus b U ( ℓ ) = 0 for all ℓ ∈ R .This proves that the kernel of L in the exponentially weighted space is trivial.Lemma 5.1 implies that the kernel of the L -adjoint L ∗ of L considered on L − η ( R , R n ) is n -dimensionaland thus spanned by the constants e j for j = 1 , . . . , n where e j form an orthonormal basis of R n such that (cid:16) dG U (0) + b K (0) dF U (0) (cid:17) e j = − c j e j . To find shock-like transition layers, caused by the inhomogeneity h for small ǫ , we make the followingansatz U ( x ) = n X j =1 a j e j χ + ( x ) + n X j =1 b j e j χ − ( x ) + W ( x ) , (5.10)where a j , b j ∈ R and W ∈ L η ( R , R n ), and χ ± ( x ) = (1 + tanh( ± x )) /
2. Substituting the ansatz into (5.2),we obtain an equation of the form F ( a, b, W ; ǫ ) = 0 , F ( · ; ǫ ) : R n × R n × D ( L ) −→ L η ( R , R n ) (5.11)for a = ( a j ) , b = ( b j ). For small enough η , the map F is smooth and the its linearization at ( a, b, W ) = 0is given by F W (0; 0) = L , F a j (0 ,
0) = K x ∗ ( dF U (0) e j χ + )+ dG U (0) e j χ ′ , F b j (0 ,
0) = K x ∗ ( dF U (0) e j χ − ) − dG U (0) e j χ ′ where F a (0 ,
0) and F b (0 ,
0) lie in L η ( R , R n ). Lemma 5.2.
Under the hypotheses of Lemma 5.1, the operator F a,W (0; 0) : R n × L η ( R , R n ) −→ L η ( R , R n ) , ( a, W ) a (0; 0) a + F W (0; 0) W is invertible. roof. We first note that the n partial derivatives with respect to a j are linearly independent. To seethis, we integrate F a j (0 ,
0) over the real line to find Z R F a j (0 , dx = Z R K x ∗ ( dF U (0) e j χ + ) dx + dG U (0) e j = Z R K ∗ (cid:0) dF U (0) e j χ ′ + (cid:1) dx + dG U (0) e j = b K (0) dF U (0) e j c χ ′ + (0) + dG U (0) e j = (cid:16) b K (0) dF U (0) + dG U (0) (cid:17) e j = − c j e j , and we exploit the fact that all c j = 0, and that the e j form a basis of R n . Next, we evaluate the scalarproduct of F a j (0 ,
0) with the elements e k of the kernel of the adjoint L ∗ : Z R hF a j (0 , , e k i dx = h b K (0) dF U (0) + dG U (0) e j , e k i = − c j δ j,k which, for fixed j , is nonzero for j = k . Hence, the partial derivative F a j (0 ,
0) are not in the range of L .This proves the lemma.We can now solve (5.11) with the Implicit Function Theorem and obtain unique solutions ( a, W )( b ; ǫ ) andthus a solution U of the form (5.10) to (5.5). As outlined in [21], the physically interesting quantity is thejump U ( ∞ ) − U ( −∞ ) = a ( b ; ǫ ) − b . A straightforward expansion in ǫ gives U ( ∞ ) − U ( −∞ ) = a ( b ; ǫ ) − b = − ǫ Z R (cid:16) b K (0) dF U (0) + dG U (0) (cid:17) − H ( x, , dx + O ( ǫ )which is independent of b to leading order.The preceding analysis also allows us to study the case where precisely one characteristic speed c j vanishes.In this situation we may further assume that h b K ′ (0) dF U (0) e j , e j i 6 = 0, such that ν = 0 is a simple zeroof det( b K ( ν ) dF U (0) + dG U (0)) = 0 and ( b K (0) dF U (0) + dG U (0)) e j = 0. We directly see that the Fredholmindex of e L and thus L in L η ( R , R n ) is now − ( n + 1), since ν = 0 has multiplicity n + 1 as a solution of(5.9). The kernel of the adjoint operator L ∗ is spanned by the constant functions e j and the linear function xe j . Indeed, we have L ∗ ( xe j ) = − dF U (0) t K t − ∗ e j − dG U (0) e j = − (cid:16) dF U (0) t b K t − (0) + dG U (0) t (cid:17) e j = − (cid:16) b K (0) dF U (0) + dG U (0) (cid:17) e j = 0 . We can once again use the ansatz (5.10) and arrive at the function F given in (5.11). Lemma 5.3.
Assume that b K (0) dF U (0) + dG U (0) has distinct real eigenvalues with a simple eigenvalue at ν = 0 with eigenvector e j . We also suppose that h b K ′ (0) dF U (0) e j , e j i 6 = 0 . Then the linearization of F with respect to ( a, b j , W ) is invertible at (0; 0) . roof. One readily verifies that the partial derivatives with respect to ( a j ) j =1 ,...,n and b j are linearlyindependent and that for each fixed j = 1 , . . . , n , j = j , we have Z R hF a j (0 , , e k i dx = h b K (0) dF U (0) + dG U (0) e j , e k i = − c j δ j,k , which is non zero for j = k . Lastly, Z R hF a j (0 , , xe j i dx = − Z R h ( K ∗ ( dF U (0) χ + ) + dG U (0) χ + ) e j , e j i dx = −h b K ′ (0) dF U (0) e j , e j i 6 = 0and similarly Z R hF b j (0 , , xe j i dx = h b K ′ (0) dF U (0) e j , e j i 6 = 0so that F a j (0; 0) and F b j (0; 0) do not lie in the range of F W (0; 0). Thus F a,b j ,W (0; 0) is invertible.We can therefore solve (5.11) using the Implicit Function Theorem and obtain a unique solution ( a, b j , W )as functions of (( b j ) j =1 ,...,n, j = j ; ǫ ). In that case we have that the solution U selects both a j and b j via a j = M ǫ + O ( ǫ ) , b j = − M ǫ + O ( ǫ ) , M := Z R x h H ( x, , , e j ih b K ′ (0) dF U (0) e j , e j i dx. When M = 0, the difference between the number of positive characteristic speeds at ∞ and −∞ is two,and the viscous profile is a Lax shock or under compressive shock of index 2.Summarizing, we have shown that nonlocal conservation laws behave in a very similar fashion as localconservation laws when subject to local source terms. Sources that move with non-characteristic speedcause a jump across the inhomogeneity, while number of ingoing and outgoing characteristics are equal.Sources that move with characteristic speed are able to act as sources with respect to the characteristicspeed, so that the number of outgoing characteristics exceeds the number of incoming characteristics bytwo.In both cases, stationary profiles are smooth , similar to what one would expect from a viscous conservationlaw. Loosely speaking, smoothing here is provided by dispersal through the nonlocal term rather thansmoothing by viscosity. We show how our methods can be used to study eigenvalue problems near the edge of the essential spectrum.Motivated most recently by questions on stability of coherent structures, such as solitons in dispersiveequations and viscous shock profiles, there has been significant interest in studying spectra of operatorsnear the edge of the essential spectrum. In the original works [8, 11], a Wronskian-type function that trackseigenvalues and multiplicities via its roots was extended into the essential spectrum, exploiting the fact thatcoefficients of the linearized problem converge exponentially as | x | → ∞ . While Wronskians are usuallyfinite-dimensional, extensions are sometimes possible to infinite-dimensional systems, using exponentialdichotomies and Lyapunov-Schmidt reduction to obtain reduced Wronskians.26ap Lemma type arguments had been used routinely in the theory of Schr¨odinger operators, providingextensions of scattering coefficients into and across the continuous spectrum. One is often interested intracking how eigenvalues may emerge out of the essential spectrum when parameters are varied. It wasobserved early that small localized traps inserted into a free Schr¨odinger equation will create bound statesin dimensions n ≤
2; see [23]. The bound state corresponds to an eigenvalue emerging from the edge ofthe continuous spectrum.We show here how a result analogous to [23] can be proved for nonlocal eigenvalue problems. We thereforeconsider the system T ( λ, ǫ ) · U := U ξ + (cid:16) K + ǫ e K ξ (cid:17) ∗ U − λBU = 0 , U ∈ R n . (5.12)Here, K , e K ξ ∈ L η ( R , M n ( R )), B ∈ M n ( R ), and e K ξ −→ ξ →±∞ L η ( R , M n ( R )) such that there existconstants C > δ > (cid:13)(cid:13)(cid:13) e K ( ζ ; ξ ) (cid:13)(cid:13)(cid:13) n ≤ Ce − δ | ξ | , ∀ ζ ∈ R . We think of (5.12) as coming from a higher-order differential operator such as ∂ ξξ , including nonlocalterms, after rewriting the eigenvalue problem as a first-order system of (nonlocal) differential equations in ξ . Proposition 5.4.
We assume that the dispersion relation d ( ν, λ ) = det (cid:16) ν I n + b K ( ν ) − λB (cid:17) is diffusive near λ = 0 :1. d (0 ,
0) = d ν (0 ,
0) = 0 ;2. d νν (0 , · d λ (0 , < ; and3. d ( iℓ, = 0 for all ℓ ∈ R , ℓ = 0 .We also assume that the localized perturbation is generic: M := D e K ξ e , e ∗ E L ( R , R n ) D I n + ∂ ν b K (0)) e + ∂ νν b K (0) e , e ∗ E R n s − d νν (0 , d λ (0 , = 0 . Then there exists ǫ > , such that for all < M ǫ < ǫ there exist = U ǫ ∈ H ( R , R n ) and λ ∗ ( ǫ ) > sothat T ( λ ∗ ( ǫ ) , ǫ ) · U ǫ = 0 . We also have the asymptotic expansion: lim ǫ → + λ ∗ ( ǫ ) ǫ = M . (5.13)27e prepare the proof of this proposition by reformulating the eigenvalue problem as a nonlinear equationthat can be solved with the Implicit Function Theorem near a trivial solution. We first introduce λ = γ ,so that the dispersion relation has local analytic roots γ ν ± ( γ ) ∈ C . Expanding d ( ν, γ ) in γ , wearrive at the expansion d ( ν, γ ) = ν d νν (0 , γ d λ (0 ,
0) + O (cid:0) | ν | + | γ | (cid:1) , so that to leading order we have ν ± ( γ ) = ± s − d λ (0 , d νν (0 , γ + O ( γ ) . Associated with these roots can be analytic vectors in the kernel, γ e ± ( γ ) ∈ C n , with (cid:16) ν ± ( γ ) I n + b K ( ν ± ( γ )) − γ B (cid:17) e ± ( γ ) = 0 , (5.14)and e = e ± (0) = 0 solves b K (0) e = 0.Following the analysis of the previous section, there exists η ∗ > η with 0 < η < η ∗ ,the linear operator L L : U ddξ U + K ∗ U, defined on L η ( R , R n ), is Fredholm with index − d ( ν,
0) = det (cid:16) ν I n + b K ( ν ) (cid:17) = ν e d ( ν ) , e d (0) = 0 , with d ( iℓ, = 0 for all ℓ ∈ R , ℓ = 0. This implies that ν = 0 is a root with multiplicity 2 and all otherroots have nonzero real part. Thus the Fredholm index of L is − L in the exponentially weighted space L η ( R , R n ) is trivial. Thus the kernel of the L -adjoint L ∗ of L considered on L − η ( R , R n ) is two-dimensional. Here, the adjoint L ∗ is given via L ∗ : U ddξ U + K t − ∗ U, where K t − ( ξ ) = K t ( − ξ ) for all ξ ∈ R . Note thatdet (cid:16)c L ∗ ( ν ) (cid:17) = det (cid:16) − ν I n + b K t ( − ν ) (cid:17) = d ( − ν,
0) = ν e d ( − ν ) , so that there exists e ∗ ∈ R n with b K t (0) e ∗ = 0 and thus L ∗ ( e ∗ ) = 0. As d ν (0 ,
0) = 0, the following scalarproduct vanishes: D ( I n + ∂ ν b K (0)) e , e ∗ E R n = 0 , (5.15)which ensures the existence of e ∗ ∈ R n so that − (cid:16) I n + ∂ ν b K t (0) (cid:17) e ∗ + b K t (0) e ∗ = 0 . (5.16)28ndeed, the above equation can be solved if D(cid:16) I n + ∂ ν b K t (0) (cid:17) e ∗ , e E R n = 0, which holds true because of(5.15). We now claim that ξe ∗ + e ∗ belongs to the kernel of L ∗ : L ∗ ( ξe ∗ + e ∗ ) = (cid:2) − e ∗ + K t − ∗ ( ξe ∗ ) (cid:3) + b K t (0) e ∗ = h − e ∗ − ∂ ν b K t (0) e ∗ i + b K t (0) e ∗ = 0 . Summarizing, the kernel of L ∗ , considered on L − η ( R , R n ), is spanned by the functions e ∗ and ξe ∗ + e ∗ .In the same way, we also define e ∈ R n via (cid:16) I n + ∂ ν b K (0) (cid:17) e + b K (0) e = 0 . (5.17)Furthermore, differentiating (5.14) with respect to γ and evaluating at γ = 0 we obtain ± s − d λ (0 , d νν (0 , (cid:16) I n + ∂ ν b K (0) (cid:17) e + b K (0) e ′± (0) = 0 . We see from the above equation and (5.17) that e ′± (0) = ± q − d λ (0 , d νν (0 , e . Moreover, combining equations(5.16) and (5.17) we have the equality D ( I n + ∂ ν b K (0)) e , e ∗ E R n = − D ( I n + ∂ ν b K (0)) e , e ∗ E R n . (5.18)The fact that d νν (0 , = 0 ensures that the following quantity is not vanishing: D ( I n + ∂ ν b K (0)) e , e ∗ E R n + 12 D ∂ νν b K (0) e , e ∗ E R n = 0 . (5.19)To find solutions of the eigenvalue problem (5.12), for small ǫ , we make the following ansatz U ( ξ ) = a + e + ( γ ) χ + ( ξ ) e ν + ( γ ) ξ + a − e − ( γ ) χ − ( ξ ) e ν − ( γ ) ξ + w ( ξ ) , (5.20)where a + , a − ∈ R and w ∈ L η ( R , R n ). Here χ + ( ξ ) = ρ ( ξ )2 , where ρ ∈ C ∞ ( R ) is a smooth even functionsatisfying ρ ( ξ ) = − ξ ≤ − ρ ( ξ ) = 1 for all ξ ≥ χ − ( ξ ) = 1 − χ + ( ξ ). Substituting the ansatzinto (5.12), we obtain an equation of the form F ( a, γ, w ; ǫ ) = 0 , F ( · ; ǫ ) : R × R × R n × D ( L ) −→ L η ( R , R n ) (5.21)for a = ( a + , a − ). We have that F ((1 , , ,
0; 0) = 0. For small enough η , following the analysis conductedin [17] and exploiting the localization of e K ξ , we have that F is a smooth map. Its linearization at ( a , γ, w ) =( , ,
0) (here for convenience we have denoted = (1 , F w ( , ,
0; 0) = L , F a ± ( , ,
0; 0) = L ( χ ± e ) , F γ ( , ,
0; 0) = s − d λ (0 , d νν (0 ,
0) [ L ( χ + e ) + L ( ξχ + e )] − s − d λ (0 , d νν (0 ,
0) [ L ( χ − e ) + L ( ξχ − e )]where F a ( , ,
0; 0) and F γ ( , ,
0; 0) lie in L η ( R , R n ).29 emma 5.5. Under the above assumptions, the operator F a − ,γ,w ( , ,
0; 0) : R × R × L η ( R , R n ) −→ L η ( R , R n )( a − , γ, w ) a − ( , ,
0; 0) a − + F γ ( , ,
0; 0) γ + F w ( , ,
0; 0) w is invertible. Proof.
We first recall that the cokernel of F w (0; 0) is spanned by e ∗ and ξe ∗ + e ∗ . We next evaluate thefunctional L u = hL ( ue ) , e ∗ i L ( R , R n ) , with associated symbol c L ( ν ) = D(cid:16) ν I n + b K ( ν ) (cid:17) e , e ∗ E R n . We have that c L (0) = ∂ ν c L (0) = 0, so thatthere exists H ∈ L η ( R , M n ( R )) such that c L ( ν ) = ν D b H ( ν ) e , e ∗ E R n = (cid:28) [ d dξ H ( ν ) e , e ∗ (cid:29) R n with 2 b H (0) = ∂ νν b K (0). We can rewrite L u as L u = (cid:28) H ∗ (cid:18) d dξ ue (cid:19) , e ∗ (cid:29) L ( R , R n ) . It is now a straightforward computation to evaluate the following quantities: L χ − = (cid:28) d dξ H ∗ ( χ − e ) , e ∗ (cid:29) L ( R , R n ) = 0 , L ( ξχ ± ) = (cid:28) d dξ H ∗ ( ξχ ± e ) , e ∗ (cid:29) L ( R , R n ) = D b H (0) e , e ∗ E R n (cid:18) lim ξ → + ∞ (cid:20) ddξ ( ξχ ± ( ξ )) (cid:21) − lim ξ →−∞ (cid:20) ddξ ( ξχ ± ( ξ )) (cid:21)(cid:19) = ± D ∂ νν b K (0) e , e ∗ E R n . We can also define the functional L u = hL ( u e ) , e ∗ i L ( R , R n ) such that c L ( ν ) = D(cid:16) ν I n + b K ( ν ) (cid:17) e , e ∗ E R n and c L (0) = 0. Thus, we can find H ∈ L η ( R , M n ( R )) suchthat c L ( ν ) = ν D b H ( ν ) e , e ∗ E R n = D d ddξ H ( ν ) e , e ∗ E R n with b H (0) = I n + ∂ ν b K (0). Using (5.18) we find that L χ ± = (cid:28) ddξ H ∗ ( χ − e ) , e ∗ (cid:29) L ( R , R n ) = ± D ( I n + ∂ ν b K (0)) e , e ∗ E R n = ∓ D ( I n + ∂ ν b K (0)) e , e ∗ E R n . We have thus shown that hL ( χ ± e ) + L ( ξχ ± e ) , e ∗ i L ( R , R n ) = L χ ± + L ( ξχ ± )= ∓ D ( I n + ∂ ν b K (0)) e , e ∗ E R n ± D ∂ νν b K (0) e , e ∗ E R n = 0 . hL ( χ − e ) , e ∗ + ξe ∗ i L ( R , R n ) = − D ( I n + ∂ ν b K (0)) e , e ∗ E R n + 12 D ∂ νν b K (0) e , e ∗ E R n = 0 . Summarizing our results, we have proved that: (cid:10) F a − ( , ,
0; 0) , e ∗ (cid:11) L ( R , R n ) = 0 , (cid:10) F a − ( , ,
0; 0) , e ∗ + ξe ∗ (cid:11) L ( R , R n ) = 0 , hF γ ( , ,
0; 0) , e ∗ i L ( R , R n ) = 0 . Thus F a − ,γ (0; 0) span the cokernel of L , which implies that F a − ,γ,w ( , ,
0; 0) is invertible, as a Fredholmindex 0 operator that is onto.
Proof. [of Proposition 5.4] Using Lemma 5.5, we can solve using the Implicit Function Theorem andobtain a unique solution ( a − , γ, w ) as a function of ( a + , ǫ ). First, the asymptotic expansion (5.13) followsdirectly by noticing that, to leading order in ǫ , we have γ hF γ ( , ,
0; 0) , e ∗ i L ( R , R n ) + ǫ D e K ξ e , e ∗ E L ( R , R n ) + O ( ǫ ) = 0 . Here, we have used the fact that (cid:10) F a − ( , ,
0; 0) , e ∗ (cid:11) L ( R , R n ) = hL e , e ∗ i L ( R , R n ) = 0. Our above computa-tions lead to hF γ ( , ,
0; 0) , e ∗ i L ( R , R n ) = 2 s − d λ (0 , d νν (0 , (cid:28) ( I n + ∂ ν b K (0)) e + 12 ∂ νν b K (0) e , e ∗ (cid:29) R n = 0 . This gives the desired expansion (5.13) and implies that γ = − M ǫ + O ( ǫ ) is of negative sign for M ǫ > λ ∗ ( ǫ ) > U ǫ ( ξ ) given in the ansatz(5.20) belongs to L ( R , R n ). For small M ǫ >
0, we have that ν ± ( γ ) = ∓ q − d λ (0 , d νν (0 , M ǫ + O ( ǫ ), such that ∓ℜ ( ν ± ( γ )) > U ǫ is exponentially localized. Since for λ >
0, there are no roots ν ∈ i R , we know that T ( λ, ǫ ) is Fredholm with index zero. Together, this implies that T ( λ, ǫ ) possesses a kernel for λ = λ ∗ ( ǫ ).This completes the proof of Proposition 5.4. Remark 5.6.
Following [17, Prop. 5.11], one can show uniqueness and simplicity of the eigenvalue λ ∗ ( ǫ ) for M ǫ > . Also, the analysis here gives a natural extension of the eigenvalue concept into the essentialspectrum: for M ǫ < , we can track the eigenvalue λ ∗ ( ǫ ) in smooth fashion as a resonance pole, thatis, a function with particular prescribed exponential growth. In this sense, our method here provides analternative to the Gap Lemma [8, 11], were this possibility of tracking eigenvalues into the essential spectrumwas the main objective. Acknowledgments:
GF was partially supported by the National Science Foundation through grantNSF-DMS-1311414. AS was partially supported by the National Science Foundation through grant NSF-DMS-0806614. 31 eferences [1] R. Abraham and J. Robbin.
Transversal mappings and flows.
Benjamin, New-York, 1970.[2] P.W. Bates, P.C. Fife, X. Ren and X. Wang.
Traveling waves in a convolution model for phasetransitions.
Arch. Rational Mech. Anal., vol 138, pages 105–136, 1997.[3] X. Chen
Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolutionequations.
Advences in Differential Equations, 2, pages 125–160, 1997.[4] A.J.J. Chmaj.
Existence of traveling waves for the nonlocal Burgers equation.
Applied MathematicsLetters, vol 20, pages 439–444, 2007.[5] A. De Masi, T. Gobron and E. Presutti.
Traveling fronts in non-local evolution equations.
Arch. Rat.Mech. Anal, 132, pp 143–205, 1995.[6] Q. Du, J.R. Kamm, R.B. Lehoucq and M.L. Parks.
A new approach for nonlocal, nonlinear conser-vation laws.
SIAM J. Appl. Math., vol 72, no 1, pages 464–487, 2012.[7] G. B. Ermentrout and J. B. McLeod.
Existence and uniqueness of travelling waves for a neuralnetwork.
Proc. Roy. Soc. Edin., 123A, pp. 461–478, 1993.[8] R.A. Gardner and K. Zumbrun.
The gap lemma and geometric criteria for instability of viscous shockprofiles.
Comm. Pure Appl. Math. , vol 51, no. 7, 797–855, 1998.[9] J. Harterich, B. Sandstede and A. Scheel.
Exponential dichotomies for linear non-autonomous func-tional differential equations of mixed type.
Indiana Univ. Math. J., vol 51, No. 5 , pages 1081–1110,2002.[10] H.J. Hupkes and B. Sandstede.
Traveling pulse solutions for the discrete FitzHugh-Nagumo system.
SIAM J. Applied Dynamical Systems, vol 9, no 3, pages 827–882, 2010.[11] T. Kapitula and B. Sandstede.
Stability of bright solitary-wave solutions to perturbed nonlinearSchr¨odinger equations.
Phys. D 124, no. 1-3, 58–103, 1998.[12] J. Mallet-Paret.
The Fredholm alternative for functional differential equations of mixed type.
Journalof Dynamics and Differential Equations, vol 11, no 1, pages 1–47, 1999.[13] J. Mallet-Paret.
The global structure of traveling waves in spatially discrete dynamical systems.
Journalof Dynamics and Differential Equations, 11, 1, pages 49–127, 1999.[14] J. Mallet-Paret and S.M. Verduyn-Lunel.
Exponential dichotomies and Wiener-Hopf factorizationsfor mixed-type functional differential equations.
Journal of Differential Equations, to appear, 2001.[15] K.J. Palmer.
Exponential dichotomies and Fredholm operators.
Proc. Amer. Math. Soc. 104, pages149–156, 1988.[16] D.J. Pinto and G.B. Ermentrout.
Spatially structured activity in synaptically coupled neuronal net-works: 1. Traveling fronts and pulses.
SIAM J. of Appl. Math., vol. 62, pages 206–225, 2001.3217] A. Pogan and A. Scheel.
Instability of Spikes in the Presence of Conservation Laws.
Z. Angew. Math.Phys. 61, pages 979–998, 2010.[18] A. Pogan, A. Scheel
Layers in the Presence of Conservation Laws.
J. Dyn. Diff. Eqns., vol. 24 , pages249–287, 2012.[19] J. Robbin and D. Salamon.
The spectral flow and the Maslov index.
Bull. London Math. Soc., vol 27,pages 1–33, 1995.[20] B Sandstede.
Stability of traveling waves.
In: Handbook of Dynamical Systems II (Edited by BFiedler), Elsevier, pages 983–1055, 2002.[21] B. Sandstede and A. Scheel.
Relative Morse indices, Fredholm indices, and group velocities.
Discreteand Continuous Dynamical Systems, vol 20, no 1, pages 139–158, 2008.[22] M. Schwarz.
Morse Homology.
Progress in Mathematics, vol. 111, Birkhauser Verlag, Basel, 1993.[23] B. Simon.
The bound state of weakly coupled Schr¨odinger operators in one and two dimensions.
Ann.Phys. , vol 97, pages 279–288, 1976.[24] K. Zumbrun and P. Howard.