On the spectra of periodic waves for infinite-dimensional Hamiltonian systems

Transcription

1 On the spectra of periodic waves for infinite-dimensional Hamiltonian systems Mariana Hǎrǎguş Laboratoire de Mathématiques Université de Franche-Comté 16 route de Gray Besançon cedex, France Todd Kapitula Department of Mathematics and Statistics Calvin College Grand Rapids, MI 49546, USA March 19, 2008 Abstract. We consider the problem of determining the spectrum for the linearization of an infinitedimensional Hamiltonian system about a spatially periodic traveling wave. By using a Bloch-wave decomposition, we recast the problem as determining the point spectra for a family of operators J γ L γ, where J γ is skew-symmetric with bounded inverse and L γ is symmetric with compact inverse. Our main result relates the number of unstable eigenvalues of the operator J γ L γ to the number of negative eigenvalues of the symmetric operator L γ. The compactness of the resolvent operators allows us to greatly simplify the proofs, as compared to those where similar results are obtained for linearizations about localized waves. The theoretical results are general, and apply to a larger class of problems than those considered herein. The theory is applied to a study of the spectra associated with periodic and quasi-periodic solutions to the nonlinear Schr ĺodinger equation, as well as periodic solutions to the generalized Korteweg-de Vries equation with power nonlinearity. Contents 1. Introduction 1 2. General theory Symmetries Canonical form Refinements Eigenvalue problems of NLS-type NLS equation with periodic potential Example: Bose-Einstein condensates Quasi-periodic solutions to the defocusing NLS equation Eigenvalue problems of KdV-type Periodic waves to the gkdv equation Appendix: Proof of the SCS Basis Lemma 30 References 34 mharagus@univ-fcomte.fr tmk5@calvin.edu

2 M. Hǎrǎguş and T. Kapitula 1 1. Introduction We are interested in certain spectral-stability aspects of spatially periodic waves in Hamiltonian nonlinear partial differential equations. Upon linearizing about a periodic wave one is left with studying an eigenvalue problem of the form JLu = λu, 1.1) where L is a symmetric operator with spatially periodic coefficients, and J is a skew-symmetric operator. The main purpose of this paper is to obtain a general result relating the spectra of the operators L and JL, in the spirit of the results in [6, 20, 21, 27] for localized waves, in which the total number of negative point eigenvalues of L is related to the total number of unstable eigenvalues of JL. As is well-known, the spectra associated with periodic waves strongly depends upon the choice of the function space. While for spaces of periodic functions, e.g., H Lper[ L, 2 L]; C n ), the spectrum consists of isolated eigenvalues of finite multiplicity, for spaces of functions defined on the whole real line, e.g., H L 2 R; C n ), the spectrum is continuous, and in particular there are no isolated eigenvalues of finite multiplicity. Here, we are interested in the latter situation, which corresponds to studying the spectral stability of periodic waves with respect to localized perturbations. In this case, the fact that the spectrum of the operator is continuous is a serious impediment to implementing the results of [20, 21], which concern isolated eigenvalues of L and JL only. However, this difficulty can be overcome by considering a Blochwave decomposition see Section 3). Upon performing such a decomposition the eigenvalue problem 1.1) is replaced by a family of eigenvalue problems J γ L γ u = λu, π < γ π, 1.2) where for each γ the operator L γ is symmetric and has now a compact resolvent, while the operator J γ is skew-symmetric. As a consequence, J γ L γ has point spectrum only, so that one expects that the results and ideas presented in [20, 21] will be immediately applicable. However, one runs into another problem which must be overcome here: ImJ γ L γ ) 0, in contrast to the situation in [20, 21] where ImJL) = 0. There is one crucial difference between the cases ImJ γ L γ ) = 0 and ImJ γ L γ ) 0. In the former case the spectrum of J γ L γ is symmetric with respect to both the real and imaginary axes of the complex plane, i.e., one has the eigenvalue quartets {±λ, ±λ}, whereas in the latter case the symmetry is generically with respect to the imaginary axis only, i.e., one has the eigenvalue pairs {λ, λ}. Consequently, this changes the form of the general statement in Theorem 2.13 below as compared to that of [21, Theorem 1]. Nevertheless, under certain symmetry assumptions one recovers the symmetry with respect to the real axis, and hence the expected form of the result see Section 2.1 and Section 2.2). The results presented in [20, 21] rely heavily upon the previous work of [14, 15], in which it is assumed that the operator L has both point and continuous spectra. As it will be seen in Section 2, in our case the absence of continuous spectrum due to the compactness of the resolvent) allows us to recover these results via more elementary means. The proofs are in the spirit of those presented in [6, 27] for localized waves in that they involve a detailed analysis of the indefinite quadratic form, L γ ; however, the proofs presented herein allow us to remove the assumption that the point eigenvalues of J γ L γ are semi-simple see Remark 2.14b)). The paper is organized as follows. In Section 2 we consider general eigenvalue problems given in equation 1.2). Under the general Assumption 2.1 we obtain a relationship between the number of unstable eigenvalues of the operator J γ L γ and the number of negative eigenvalues of the symmetric operator L γ see Theorem 2.13). In particular, if ImJ γ L γ ) = 0, we recover the results presented in [20, 21], although the proof is different. Some particular cases are discussed in Section 2.1 and Section 2.2. The proofs presented in this section are self-contained. A crucial ingredient of the proof, and a result which is of independent interest, is the result of the SCS Basis Lemma Lemma 2.3). Therein it is shown that under general assumptions that the eigenvectors and generalized eigenvectors of the non-self-adjoint operator J γ L γ form a basis. In Section 3 we apply these general results to a class of eigenvalue problems of NLS-type in which J = J γ is a bounded operator. In particular, we study the spectra for periodic and quasi-periodic waves to nonlinear Schr ĺodinger equations. In Section 4 we apply the results to eigenvalue problems of KdV-type in which now J γ is an unbounded operator with compact resolvent. As an example, we study the

3 Spectra of periodic waves 2 spectra of periodic waves to the generalized Korteweg-de Vries equation gkdv). Finally, in the Appendix we give the proof of the SCS Basis Lemma. Acknowledgments. ACI JC 1039 M.H.). This work was partially supported by the Ministère de la Recherche through grant 2. General theory We consider the general eigenvalue problem JLu = λu 2.1) for a symmetric operator L and a skew-symmetric, invertible operator J. The problem will be considered on a complex Hilbert space H with inner product,. We make the following assumptions: Assumption 2.1. The operators J and L in equation 2.1) have the following properties: a) J is skew-adjoint J = J) and invertible with bounded inverse; b) L is self-adjoint L = L) and invertible with compact inverse; c) σl) R, where σl) denotes the spectrum of L, is a finite set; d) there exists a skew-adjoint operator A 0 with compact resolvent such that: i) the operator B JL A 0 is A 0 -compact and satisfies for some positive constants a, b, and r [0, 1); Bu a u + b A 0 r u, u DJL), 2.2) ii) the sequence of purely imaginary eigenvalues {iω n } n Z of A 0, where each eigenvalue is repeated as many times as the value of its multiplicity and ω n ω n+1, satisfies for some s 1; + n= ω n s <, iii) there exists a subsequence of eigenvalues {iω nk } k Z, and positive constants c > 0 and r > r, such that ω nk > 0 and ω nk +1 ω nk cω r n k +1 for k 0, ω nk < 0 and ω nk ω nk 1 c ω nk 1 r for k < ) Remark 2.2. a) Let {e k } k Z be an orthonormal basis of H consisting of eigenvectors of A 0. Then, the operator A 0 r can be defined through A 0 r u = ω k A 0 ) r u k e k, u = u k e k. k Z b) An immediate consequence of Assumption 2.1b) is that the spectrum of L is a countable set, with no accumulation point in C, consisting of real eigenvalues with finite multiplicity. Furthermore, the set of its eigenvectors form an orthonormal basis for H. c) Similarly, since A 0 is skew-adjoint with compact resolvent, by Assumption 2.1d), its spectrum is a countable set, with no accumulation point in C, consisting of purely imaginary eigenvalues with finite multiplicity, and the set of its eigenvectors forms an orthonormal basis for H. As we shall see in the next lemma, similar properties also hold for JL. k Z

4 M. Hǎrǎguş and T. Kapitula 3 d) If s = 1 in Assumption 2.1d)ii), then the operator A 0 is of trace class. e) Assumption 2.1d) is easily verified in the examples considered in Section 3 and Section 4, in which A 0 is an operator with constant coefficients. The eigenvalues of A 0 can then be explicitly computed via Fourier analysis; in particular, for NLS we have that actually B is bounded so that equation 2.2) holds with r = 0, and in equation 2.3) we find r = 1/2. For gkdv, we obtain r = 1/3 and r = 2/3. f) Assumption 2.1d)ii) is required to show that the set of eigenfunctions and generalized eigenfunctions of JL is complete in H. The assumptions equation 2.2) and equation 2.3) are needed to show that the set actually forms a basis of H. This section will be devoted to comparing the spectra of the operators L and JL. We start by collecting a number of properties of the operator JL which follow from the Assumption 2.1. We give the proof of the following lemma in the Appendix the acronym follows from Skew-symmetric Composed with Self-adjoint). Lemma 2.3 SCS Basis Lemma). Assume that Assumption 2.1 holds. following properties: a) JL is invertible and has a compact inverse L 1 J 1 ; Then the operator JL has the b) the spectrum σjl) of JL is a countable set with no accumulation point in C consisting of eigenvalues with finite algebraic multiplicity; c) the set of eigenfunctions and generalized eigenfunctions of JL forms a basis of H; d) given ϸ > 0, the eigenvalues of JL) 1, with a possible exception of a finite number, belong to the sector π/2 {ρe iθ : ρ R +, θ π/2 < ϸ or θ + π/2 < ϸ}. Remark 2.4. An important consequence of Lemma 2.3d) is that σjl) R is a finite set. In addition to the properties in the previous proposition, a key ingredient in the next arguments is the following symmetry property of the spectrum of JL. Proposition 2.5. If λ σjl), then λ σjl). multiplicity, m a λ) = m a λ). Furthermore, λ and λ have the same algebraic Proof: Suppose that λ σjl) is an eigenvalue of JL with associated eigenvector v, so that JLv = λv. Since J has a bounded inverse we have Lv = λj 1 v, and then JL) J 1 v) = LJJ 1 v) = Lv = λj 1 v. Consequently λ σjl) ), so that λ σjl). The equality of the algebraic multiplicities follows from the Fredholm alternative. As a consequence of Proposition 2.5 the eigenvalues of JL appear in pairs {λ, λ}, that is the spectrum of JL is symmetric with respect to the imaginary axis. In addition, if JL is a real operator, ImJL) = 0, then its spectrum is also symmetric with respect to the real axis, so that we have quartets of eigenvalues {±λ, ±λ}. However, this property is typically not true in our applications in which ImJL) 0. Notation 2.6. For an eigenvalue λ σjl), denote by E λ the corresponding spectral subspace, and by I λ the invariant subspace corresponding to the eigenvalue pair {λ, λ}, I λ = E λ E λ. The usual approach for comparing the spectra of JL and L relies upon the study of the quadratic form L, associated to L in the basis of H given by the eigenfunctions and generalized eigenfunctions of JL. The following properties of the quadratic form L, are similar to the ones in [23, Lemmas 2-4]. The main difference is that here they concern pairs of eigenvalues {λ, λ}, while in [23] they were obtained for quartets of eigenvalues {±λ, ±λ}. Lemma 2.7. If v E λ and w E σ with λ + σ 0, then Lv, w = 0.

5 Spectra of periodic waves 4 Proof: First suppose that v and w are not generalized eigenfunctions, so that JLv = λv and JLw = σw. Then Lv, w = λ J 1 v, w, v, Lw = σ v, J 1 w. Since L is symmetric and J 1 is skew-symmetric this implies that λ + σ) J 1 v, w = 0, from which one gets the desired result. Now suppose that either one or both of v and w are generalized eigenfunctions. In this case the proof will follow by induction. One first has the Jordan chains Arguing as above, we find JLv k+1 = λv k+1 + v k v 0 = 0), k = 0,..., k JLw l+1 = σw l+1 + w l w 0 = 0) l = 0,..., l. Lv k+1, w l+1 = λ J 1 v k+1, w l+1 + J 1 v k, w l+1 = σ v k+1, J 1 w l+1 + v k+1, J 1 w l, 2.4) so that We now show that λ + σ) J 1 v k+1, w l+1 + J 1 v k, w l+1 + J 1 v k+1, w l = ) λ + σ) J 1 v k+1, w l+1 = 0, k = 0,..., k, l = 0,..., l. 2.6) The result of equation 2.6) is known to be true for k, l) = 0, 0). First suppose that for some 0 n k 1 the equation 2.6) holds true for l = 0 and k = 0,..., n. Upon using equation 2.5) and the fact that w 0 = 0, one immediately gets that 2.6) holds for k = n + 1. Thus, the result holds for l = 0 and k = 0,..., k. Similarly, equation 2.6) holds for k = 0 and l = 0,..., l. Now consider equation 2.5) with k, l) = 1, 1). Since equation 2.6) holds for k, l) = 1, 0) and k, l) = 0, 1), we conclude that equation 2.6) holds for k, l) = 1, 1). Another induction argument gives the result for l = 1 and k = 0,..., k, as well as for k = 1 and l = 0,..., l. Continuing in this fashion leads to the desired result. Finally, since λ + σ 0 equation 2.6) gives J 1 v k+1, w l+1 = 0, k = 0,..., k, l = 0,..., l. Substituting the above equalities into equation 2.4) completes the proof. An immediate consequence of this lemma is the following property. Proposition 2.8. Suppose that w = v j, where v j I λj for distinct λ j. Then Lw, w = Lv j, v j. Finally, following the argument presented for [23, Lemma 4] we obtain: Proposition 2.9. Assume that λ σjl). Then the restriction L Iλ, of the quadratic form L, to I λ is non-singular. Proof: Recall that λ 0, because JL has a bounded inverse. Then if L Iλ, is singular, there exists a nonzero v I λ such that Lw, v = 0 for all w I λ. Together with Lemma 2.7 this implies that Lw, v = 0 for any eigenvector or generalized eigenvector w of JL. By Lemma 2.3c) this yields that v = 0, which is a contradiction. In other words, Proposition 2.8 and Proposition 2.9 state that the quadratic form associated to L when expanded over the eigenfunctions and generalized eigenfunctions of JL is the sum over the individual invariant subspaces I λ, and that each term in the sum is a nonsingular quadratic form. Thus, in order to study L,, on H, it is enough to consider the restrictions L Iλ, for each λ σjl).

6 M. Hǎrǎguş and T. Kapitula 5 Lemma Assume that λ σjl). Let Lλ) represent the Hermitian matrix associated with the quadratic form L Iλ, on I λ, and denote by nlλ)) and plλ)) the number of its negative and positive eigenvalues counted with algebraic multiplicities, respectively. If Re λ 0, then Lλ) C 2m aλ) 2m a λ), and If Re λ = 0, then Lλ) C m aλ) m a λ) and nlλ)) = plλ)) = m a λ). nlλ)) + plλ)) = m a λ). Proof: The properties on the dimension of the matrix Lλ) follow from the fact that dimi λ ) = 2m a λ), if Re λ 0, whereas dimi λ ) = m a λ), if Re λ = 0. Set k m a λ). First assume that Re λ 0. Let {u 1,..., u k } be a basis for E λ, and let {v 1,..., v k } be a basis for E λ. One has that ) L1 L Lλ) = c L, 2.7) c L 2 where L 1 ) ij = Lu i, u j, L 2 ) ij = Lv i, v j, L c ) ij = Lu i, v j. As a consequence of Lemma 2.7 one has that L 1 = L 2 = 0. Since µ 2 σl cl c ) if and only if ±µ σlλ)), and since L cl c is symmetric, positive semi-definite, and nonsingular, the first result now follows. The result for Re λ = 0 follows immediately as a consequence of Proposition 2.9. Definition Consider the spectral subsets σ r {λ σjl) : Re λ > 0, Im λ = 0}, σ c {λ σjl) : Re λ > 0, Im λ 0}, σ i {λ σjl) : Re λ = 0}. For an eigenvalue λ σjl), with Re λ 0, we define the following quantities: k r λ) m a λ), if λ σ r, k c λ) m a λ), if λ σ c, ki λ) nlλ)), if λ σ i, and set Remark k r k r λ), λ σ r k c k c λ), ki ki λ). 2.8) λ σ c λ σ i a) The union σ r σ c σ i is the set of eigenvalues of JL with Re λ 0. Since the spectrum of JL is symmetric with respect to the imaginary axis we have σjl) = {λ, λ : λ σ r σ c σ i }. b) The numbers k r, k c, and ki have been used in [20, 21] with a slightly different meaning. While in [20, 21] they were associated to quartets of eigenvalues {±λ, ±λ}, here they are associated to the pairs {λ, λ}. Of course this is due to the absence of the symmetry of the spectrum with respect to the real axis. However, if in addition the spectrum of JL is also symmetric with respect to the real axis for example if ImJL) = 0), so that we also have quartets of eigenvalues {±λ, ±λ}, then k r has the same significance, whereas k c, and ki are even numbers, and are precisely twice the numbers considered in [20, 21]. The following general result connecting the number of unstable eigenvalues of JL and the number of negative eigenvalues of L can now be proven.

7 Spectra of periodic waves 6 Theorem Consider the eigenvalue problem of equation 2.1) under Assumption 2.1. One has that k r + k c + k i = nl), where k r, k c, ki are the numbers introduced in Definition 2.11, and nl) denotes the number of negative eigenvalues including multiplicities) of L. Furthermore, if ImJL) = 0, then the spectrum of JL is symmetric with respect to the real axis and the numbers k c and ki are even. Proof: Consider the quadratic form L, and define the cone CL) by CL) {u H 1 : Lu, u < 0} {0}, where H 1 is the domain of L,, DL) H 1 H. Denote by dim[cl)] the dimension of the maximal linear subspace E CL). Since L is a symmetric operator with point spectrum only, upon using an orthogonal basis for H consisting of eigenvectors of L we find that dim[cl)] = nl). Indeed, the linear subspace spanned by the eigenvectors of L associated to the negative eigenvalues of L belongs to CL), so that dim[cl)] nl). Next, assuming that CL) contains a linear subspace of dimension nl) + 1, at least, this subspace necessarily contains a vector v = α j v j, with v j eigenvectors associated to the positive eigenvalues of L. Then Lv, v = α j 2 Lv j, v j > 0, which is in contradiction with v CL). This shows that dim[cl)] = nl). On the other hand, by Lemma 2.10, for any eigenvalue λ of JL with Re λ 0 there is a linear subspace H λ I λ CL) with dimension k r λ), Re λ > 0, Im λ = 0 dim[h λ ] = k c λ), Re λ > 0, Im λ 0 ki λ), Re λ = 0. By Lemma 2.7, the direct sum of these subspaces is contained in CL), so that dim[cl)] k r + k c + ki. Finally, by arguing as above, using the fact that the set of eigenvectors and generalized eigenvectors of JL are a basis, and Lemma 2.7, again, we conclude that dim[cl)] = k r + k c + ki. Together with the equality dim[cl)] = nl), this completes the proof. Remark a) It would be an interesting question to prove Theorem 2.13 by combining the results of [18, Appendix A] on Pontrjagin spaces and the SCS Basis Lemma. In particular, upon noticing that the operator ijl is symmetric on H equipped with the indefinite, sesquilinear, Hermitian scalar product {u, v} u, Lv, one is within the framework presented therein. A major hurdle to overcome in order to apply these ideas seems to be that completeness of the eigenfunctions of JL in H does not necessarily imply completeness in the space supplied with the inner product given above. b) The result of Theorem 2.13 is in the same vein as that presented in [20, 21] for localized waves. The primary difference is that here L is assumed to be invertible with compact inverse. If desired, the assumption that L is nonsingular can be removed, and a result which takes into account that L has a nontrivial null-space can be formulated. However, since in our applications L is typically invertible, we restrict to this case, only. The fact that L has compact inverse allows us to give a self-contained proof which no longer relies upon the results of [14, 15]. c) Recently [6, 27, 30] have initiated a program in which they prove, for localized waves, a result similar to that given in Theorem 2.13 without using any of the results in [14, 15]. In doing so, however, they typically assume that all of the eigenvalues of JL are semi-simple [27], and that the operators J and L have the special form J = 0 id id 0 ), L = ) L ) 0 L In [14, 15] equation 2.9) was also assumed; however, in [21] it was shown how this assumption could be circumvented.

8 M. Hǎrǎguş and T. Kapitula Symmetries The next goal is to refine the result of Theorem 2.13 under the assumption of some underlying symmetries. We will consider two different scenarios. In both cases we find the usual quadruplets {±λ, ±λ}. The second symmetry will be considered in more detail the next subsection. Assumption a) Assume that Im J = 0 and that there exists a bounded unitary operator U 1 such that U 1 J = JU 1 and U 1 L = LU 1. b) Assume that there exists a bounded unitary operator U 2 such that U 2 J = JU 2 and U 2 L = LU 2. Proposition Under Assumption 2.15a), if u H is a generalized) eigenfunction of JL associated with the eigenvalue λ, then U 1 u is a generalized) eigenfunction of JL associated with the eigenvalue λ. Under Assumption 2.15b), if u H is a generalized) eigenfunction of JL associated with the eigenvalue λ, then U 2 u is a generalized) eigenfunction of JL associated with the eigenvalue λ. Proof: Consider the implication of Assumption 2.15a). The proof of Assumption 2.15b) is similar to that of a), and is left for the interested reader.) Suppose that u is an eigenfunction with associated eigenvalue λ, so that JLu = λu. Upon taking the complex-conjugate of both sides one has that JLu = λu. Applying U 1 to both sides then yields the desired result. The case when u is a generalized eigenfunction is proved in the same way. As a consequence of Proposition 2.16 one has that the spectrum of JL is also symmetric with respect to the real axis. First assume the symmetry of Assumption 2.15a). For each invariant subspace I λ with Im λ 0 there is an associated invariant subspace I λ such that dimi λ ) = dimi λ ). Furthermore, the subspaces have the same Jordan block structure. Now, if Re λ 0, then the result of Lemma 2.10 immediately shows that k c λ) = k c λ). Now suppose that Re λ = 0, and let {u 1,..., u k } be a basis for I λ. Following the proof of Lemma 2.10 one has that Lλ)) ij = Lu i, u j. 2.10) As a consequence of Proposition 2.16 one has that {U 1 u 1,..., U 1 u k } is a basis for I λ. Upon using Assumption 2.15a) one has that Lλ)) ij = LU 1 u i, U 1 u j = U 1 Lu i, U 1 u j = Lu i, u j = Lu i, u j = Lu j, u i. Upon comparing with equation 2.10) one then sees that Lλ) = Lλ), which in particular implies that ki λ) = k i λ). A similar argument holds when considering Assumption 2.15b). The primary difference is that for λ ir, Lλ) = L λ). The result of Theorem 2.13 can now be modified in the following way. Theorem Consider the eigenvalue problem 2.1) under Assumption 2.1 and either Assumption 2.15a) or Assumption 2.15b). Then the spectrum of JL is symmetric with respect to both the imaginary and the real axis, and k r + k c + ki = nl), with k c and k i even numbers Canonical form We discuss in this section the particular case of the canonical form given in equation 2.9), when H = H H, and ) ) 0 id L+ 0 J =, L =, 2.11) id 0 0 L

9 Spectra of periodic waves 8 with L ± symmetric operators on H such that J and L satisfy Assumption 2.1. One recovers the result of Theorem 2.17, since JL satisfies Assumption 2.15b) with ) id 0 U 2 =. 2.12) 0 id The goal is to refine this result and state it in terms of the individual operators L ± see also [14, 15, 20, 27]). In order to do so, we review the basis result in Lemma 2.3, the orthogonality result in Lemma 2.7, and the structure of the matrix Lλ) in Lemma 2.10 in this case. Notation For an operator A we set σ + A) {λ σa) : Im λ > 0} {λ σa) : Im λ = 0 and Re λ > 0}. Proposition Consider the eigenvalue problem of equation 2.1) under the Assumption 2.1 with J and L given by equation 2.11). Suppose that λ is an eigenvalue of JL with λ σ + JL). Let {u λ 1,..., uλ k λ } be a basis for E λ consisting of generalized eigenvectors of JL, with u λ j = u λ j, v λ j )T. Then the following properties hold: a) The set {U 2 u λ 1,..., U 2u λ k λ } is a basis for E λ. Furthermore, U 2 u λ j = u λ j, v λ j )T. b) The set {u λ j, U 2u λ j : j = 1,..., k λ, λ σ + JL)} is a basis for H = H H. c) The sets {u λ j : j = 1,..., k λ, λ σ + JL)} and {v λ j : j = 1,..., k λ, λ σ + JL)} are basis for H. Proof: The property a) is an immediate consequence of the Assumption 2.15b) in Section 2.1 and of equation 2.12). Part b) follows from a) and the basis result in Lemma 2.3c), and c) is a consequence of a) and b). The following result is a refinement of Lemma 2.7. Lemma Let u λ u λ, v λ ) T E λ and u σ u σ, v σ ) T E σ. If λ ± σ 0, then L + u λ, u σ = L v λ, v σ = 0. Proof: Suppose that λ + σ 0, and further suppose that the eigenvectors u λ, u σ eigenvectors. From equation 2.11) one has that are not generalized L + u λ = λv λ, L v λ = λu λ ; L + u σ = σv σ, L v σ = σu σ, which upon using the fact that the operators L ± are symmetric yields L + u λ, u σ = λ v λ, u σ = σ u λ, v σ L v λ, v σ = λ u λ, v σ = σ v λ, u σ. By Lemma 2.7 one has that so that L + u λ, u σ + L v λ, v σ = 0, λ σ) u λ, v σ = λ σ) v λ, u σ = 0. If λ σ 0, then one gets the desired result. The proof for u λ, u σ being generalized eigenvectors follows that presented in Lemma 2.7 and is left for the interested reader. Finally, we come back to the structure of the matrix Lλ) in Lemma 2.10 in this case. We distinguish between Im λ = 0 and Im λ 0. Lemma Suppose that λ is an eigenvalue of JL with Im λ = 0. Let {u 1,..., u k } be a basis for E λ consisting of generalized eigenvectors of JL, with u j = u j, v j ) T. Then the following properties hold:

10 M. Hǎrǎguş and T. Kapitula 9 a) The Hermitian matrix Lλ) associated with the quadratic form L Iλ, on I λ = E λ E λ is of the form Lλ) = L 1 +λ) + L 1 λ), with where id id L 1 +λ) = L + λ) id id ), L 1 λ) = L λ) id id ) id, id L + λ)) ij L + u i, u j, L λ)) ij L v i, v j. 2.13) b) We have that L + λ) = L λ). Proof: The property a) is the same as Proposition 2.19a). Next, using Proposition 2.19a) and equation 2.11) a straightforward calculation gives the equalities in part b). Finally, part c) follows from the proof of Lemma 2.7 which shows that the terms on the diagonal of Lλ) vanish, so that L + λ)+l λ) = 0. Lemma Suppose that λ is an eigenvalue of JL with Im λ 0. Let {u 1,..., u k } be a basis for E λ consisting of generalized eigenvectors of JL, with u j = u j, v j ) T. Then the Hermitian matrix Lλ) associated with the quadratic form L Iλ, on I λ has the following properties: a) If Re λ = 0, then I λ = E λ and with Lλ) = L + λ) + L λ), L + λ) = L λ), L + λ)) ij L + u i, u j, L λ)) ij L v i, v j. b) If Re λ 0, let {x 1,..., x k } be a basis for E λ, with x j = x j, y j ) T. Then Lλ) = L + λ) + L λ), with L + λ) = L λ) = 0 Lc L c 0 ), L c ) ij L + u i, x j = L v i, y j. Proof: In both cases the equality Lλ) = L + λ) + L λ) is an immediate consequence of equation 2.11). a) Suppose that Re λ = 0, so that λ = λ and I λ = E λ. Let {u j,k, v j,k ) T : j = 1,..., m, k = 1,..., a j }, m a j = m a λ), j=1 be a basis that represents a Jordan block decomposition of E λ consisting of eigenvectors and generalized eigenvectors of JL such that L + u j,k = λv j,k v j,k 1, v j,0 = 0 L v j,k = λu j,k + u j,k 1, u j,0 = ) First note that upon using equation 2.14) and the fact that L + is symmetric, L + u j,k, u j,k = λ v j,k, u j,k v j,k 1, u j,k = λ u j,k, v j,k u j,k, v j,k ) As a consequence of the last equality in equation 2.15) one has that Similarly one sees that λ[ u j,k, v j,k + v j,k, u j,k ] = u j,k, v j,k 1 v j,k 1, u j,k. 2.16) L v j,k, v j,k = λ u j,k, v j,k + u j,k 1, v j,k = λ v j,k, u j,k + v j,k, u j,k 1, 2.17)

11 Spectra of periodic waves 10 which implies that λ[ u j,k, v j,k + v j,k, u j,k ] = v j,k, u j,k 1 u j,k 1, v j,k. 2.18) Combining equation 2.16) and equation 2.18) yields the identity 2λ[ u j,k, v j,k + v j,k, u j,k ] = u j,k, v j,k 1 + v j,k, u j,k 1 [ u j,k 1, v j,k + v j,k 1, u j,k ]. 2.19) Our aim is to prove that L + u j,k, u j,k = L v j,k, v j,k, which in view of equation 2.15) and equation 2.17) is equivalent to proving that λ[ u j,k, v j,k + v j,k, u j,k ] = u j,k, v j,k 1 + v j,k, u j,k 1. We claim that u j,k, v j,k + v j,k, u j,k = 0, 2.20) for any j, j, k, and k, which then gives the required equality. Indeed, the equality clearly holds when either k = 0 or k = 0, i.e., u j,0, v j,k + v j,0, u j,k = 0, k = 0,..., a j, and u j,k, v j,0 + v j,k, u j,0 = 0, k = 0,..., a j. Then using the equality 2.19), an induction argument for k = 0,..., a j and k = 0,..., a j, gives equation 2.20). b) Now suppose that Re λ 0; in particular, λ ± λ 0. In addition to the basis already given for E λ, following the discussion leading to equation 2.14), let be a basis for E λ such that {x j,k, y j,k ) T : j = 1,..., m, k = 1,..., a j } L + x j,k = λy j,k y j,k 1, y j,0 = 0 L y j,k = λx j,k + x j,k 1, x j,0 = ) First consider the restriction L ± λ) Eλ. The analogue to equation 2.15) is L + u j,k, u j,k = λ v j,k, u j,k v j,k 1, u j,k = λ u j,k, v j,k u j,k, v j,k 1, 2.22) and the analogue to equation 2.17) is L v j,k, v j,k = λ u j,k, v j,k + u j,k 1, v j,k = λ v j,k, u j,k + v j,k, u j,k ) As a consequence of Lemma 2.7 one has that L + u j,k, u j,k + L v j,k, v j,k = 0, which by equation 2.22) and equation 2.23) implies that where λ λ 0. Now λ λ) u j,k, v j,k = u j,k, v j,k 1 u j,k 1, v j,k, 2.24) u j,0, v j,k = u j,k, v j,0 = 0, k = 0,..., a j, k = 0,..., a j,

12 M. Hǎrǎguş and T. Kapitula 11 so that by an induction argument from equation 2.24) we conclude that for all j, j, k, and k, u j,k, v j,k = 0. As a consequence of equation 2.22) and equation 2.23) this yields L ± λ) Eλ L ± λ) E λ = 0. Finally, consider L + u j,k, x j,k = λ v j,k, x j,k v j,k 1, x j,k = λ u j,k, y j,k u j,k, y j,k 1. As a consequence of equation 2.25) one necessarily has that = 0. Similarly, one has that 2.25) Similarly one sees that which implies that λ[ u j,k, y j,k + v j,k, x j,k ] = u j,k, y j,k 1 v j,k 1, x j,k. 2.26) L v j,k, y j,k = λ u j,k, y j,k + u j,k 1, y j,k = λ v j,k, x j,k + v j,k, x j,k 1, 2.27) λ[ u j,k, y j,k + v j,k, x j,k ] = v j,k, x j,k 1 u j,k 1, y j,k. 2.28) Note the similarity of equation 2.26) with equation 2.16), as well as equation 2.28) with equation 2.18). We can then follow the induction argument leading to equation 2.20) to conclude that u j,k, y j,k + v j,k, x j,k = 0, for all j, j, k, and k. Together with equation 2.25) and equation 2.27) this gives L + u j,k, x j,k = L v j,k, y j,k, which completes the proof. Remark In [30] the unfolding of a purely imaginary eigenvalue with m g λ) = 1 and m a λ) = k was considered. Therein they showed that 2nL + λ)) {k 1, k + 1} if k is odd, and 2nL + λ)) = k if k is even. Notation For a Hermitian matrix S C define the cone CS) by For self-adjoint operators S define the cone CS) by CS) {c C : Sc, c < 0} {0}. CS) {u H : Su, u < 0} {0}. We can now refine the result in Theorem 2.17 in the following way. The result concerning k r strengthens that presented in [15, Theorem 1.3] also see [19] and [14, Theorem 2.6]). The result concerning k c + k i appears to be new. Theorem Consider the eigenvalue problem of equation 2.1) under the Assumption 2.1 with J and L given by equation 2.11). Then the spectrum of JL is symmetric with respect to both the imaginary and the real axis and k r + k c + k i = nl + ) + nl ), with k c and ki even numbers. Furthermore, let {u j, U 2 u j : j Z} be a basis consisting of generalized eigenvectors for H = H H, as in Proposition 2.19b), with u j = u j, v j ) T, and consider the Hermitian matrices L ± C defined as L + ) ij L + u i, u j, L ) ij L v i, v j. Then and k c + k i = 2 dim[cl + ) CL )], k r = nl + ) nl ) + 2 min{nl + ), nl )} dim[cl + ) CL )]) nl + ) nl ).

13 Spectra of periodic waves 12 Proof: The first part follows immediately from equation 2.11) and Theorem Now consider the second part, and first notice that by Lemma 2.20 L ± = L ± λ), λ σ + JL) where L ± λ) are the matrices defined in Lemma 2.21 and Lemma If Im λ > 0, then one has as a consequence of Lemma 2.22 that In addition, if Re λ 0 one has that and similarly, if Re λ = 0 CL + λ)) = CL λ)). k c λ) = nl + λ)) = dim[cl + λ))] = dim[cl + λ)) CL λ))], k i λ) = dim[cl +λ)) CL λ))]. Next, if Im λ = 0, then by Lemma 2.21c) one can conclude that CL λ)) CL + λ)) =. Thus, upon summing over the eigenvalues λ σ + JL) and following the argument at the end of the proof of Theorem 2.13 one can conclude that dim[cl + ) CL )] = k c λ) + ki λ). λ σ c σ + JL) λ σ i σ + JL) Consequently, upon using the equalities k c λ) = k c λ) and ki λ) = k i λ), one has that k c + k i = 2 dim[cl + ) CL )]. For the last property, assume without loss of generality that nl + ) nl ). Upon using the above result one has that k r = nl + ) + nl ) k c + k i ) = nl + ) nl ) + 2 nl ) dim[cl + ) CL )]). Furthermore, due to the basis result in Proposition 2.19c) one can conclude that dim[cl )] = nl ), and since dim[cl + ) CL )] dim[cl )] one gets the desired inequality Refinements The previous results can be refined so that they are not dependent upon the basis formed by the generalized eigenvectors. As a consequence of equation 2.14) one can write ) v j,k = λl 1 u j,k + 1 λ u j,k 1, u j,0 = 0; 2.29) hence, L v j,k, v j,k = λ 2 L 1 u j,k + 1 ) λ u j,k 1, u j,k + 1 ) λ u j,k ) In particular, this implies that if m a λ) = m g λ) for some λ σjl), then L λ) = λ 2 L 1 λ), where the latter matrix is formed in the obvious manner. The above results then are valid with L being replaced by L 1. Furthermore, since a cone is independent of the basis, one can replace CL + ) CL ) with CL + ) CL 1 ). This observation leads to the following result. The condition that λ σjl) R satisfy m a λ) = m g λ) appears to be necessary. This issue will be further discussed after the proof.

14 M. Hǎrǎguş and T. Kapitula 13 Corollary Consider the eigenvalue problem of equation 2.1) under the Assumption 2.1 with J and L given by equation 2.11). If all λ σjl) R + satisfy m a λ) = m g λ), then the result of Theorem 2.25 can be reformulated as k c + ki = 2 dim[cl + ) CL 1 )], and k r = nl + ) nl ) + 2 min{nl+ ), nl )} dim[cl + ) CL 1 )] ) nl + ) nl ). Proof: Suppose that Im λ > 0. Suppose that each Jordan block has length a j for j = 1,..., m g λ). Recall equation 2.29) and equation 2.30) for j = 1,..., m g λ) and k = 1,..., a j. Now order the eigenfunctions as {u 1,1,..., u 1,a1, u 2,1,..., u 2,a2,..., u mg λ),1,..., u mg λ),a mgλ) }, and define the nilpotent matrices N l C l l via 1, i = 1,..., l 1, j = i + 1 N l ) ij = 0, otherwise. If one sets Sλ) = diag S 1 λ),..., S mg λ)λ) ), S j λ) = id + 1 λ N a j, then upon substituting the expression of equation 2.29) into the quadratic form L λ)c, c yields that if Re λ = 0, then L λ) = λ 2 S λ)l 1 λ)sλ), 2.31) where the notation consistent with Lemma 2.22a) yields L 1 λ)) ij = L 1 u i, u j. Note that Sλ) = id +N, where N is a nilpotent matrix; hence, S is nonsingular. Re λ > 0, then the matrix L c λ) given in Lemma 2.22 is given by On the other hand, if L c λ) = λ 2 S λ)l 1 λ)sλ) where the notation consistent with Lemma 2.22b) yields L 1 λ)) ij = L 1 u i, x j. First suppose that λ ir with nl + λ)) = 0. The codimension of such eigenvalues is nl + ) + nl ). As a consequence of the above discussion and Lemma 2.22 one has that CL + λ)) = CL λ)) = {0}; consequently, by equation 2.31) one has that CL 1 λ)) = {0}. Now consider an eigenvalue for which nl + λ)) 1, and initially assume that all such eigenvalues are semisimple, so that Sλ) = id for each such eigenvalue. If Im λ > 0, then as a consequence of Lemma 2.22 one has In addition, if Re λ 0 one has that and similarly, if Re λ = 0 CL + λ)) = CL 1 λ)). k c λ) = nl + λ)) = dim[cl + λ))] = dim[cl + λ)) CL 1 λ))], k i λ) = dim[cl +λ)) CL 1 λ))]. Next, if Im λ = 0, then by Lemma 2.21c) one can conclude that CL + λ)) CL 1 λ)) = {0}.

15 Spectra of periodic waves 14 Thus, upon summing over the eigenvalues λ σ + JL) and following the argument at the end of the proof of Theorem 2.13 one can conclude that dim[cl + ) CL 1 )] = k c λ) + ki λ). λ σ c σ + JL) λ σ i σ + JL) Consequently, upon using the equalities k c λ) = k c λ) and ki λ) = k i λ), one has that k c + k i = 2 dim[cl + ) CL 1 )]. Recall that the eigenvectors and generalized eigenvectors form a basis. Since cones are independent of a given basis, and since CL + ) corresponds to taking a cone with a particular basis, one finally has that k c + k i = 2 dim[cl + ) CL 1 )]. All that is left to do is to remove the assumption that the finite number of eigenvalues with Im λ R + having nl + λ)) 1 are semisimple. It is important to keep in mind that there are only a finite number of such eigenvalues. Eigenvalues having a nontrivial Jordan block are nongeneric; hence, a generic perturbation will cause these eigenvalues to unfold to semisimple eigenvalues. Let L ϸ ± correspond to a relatively compact symmetric perturbation of the operators L ±. If the eigenvalue is originally purely imaginary, then it will unfold into eigenvalues which are either purely imaginary or have nonzero real part e.g., see [30]). If the eigenvalue is originally complex with nonzero real part, then it will unfold into eigenvalues having the same property. In either case, for the operators L ϸ ± one recovers the desired result since the eigenvalues are now all semisimple. Since cones are open sets which depend continuously upon ϸ, one one can conclude that the result holds in the limit. Remark If the assumption that all λ σjl) R + satisfy m a λ) = m g λ) in Corollary 2.26 is dropped, then all that can be said is that k c + ki 2 dim[cl + ) CL 1 )]. The above perturbation argument fails in the case of initially real-valued eigenvalues, for in this case the perturbed eigenvalues will either be purely real no cone intersection) or complex with nonzero imaginary part cones coincide). Hence, the limiting behavior is unclear. It is natural to wonder if a different argument will allow one to remove the condition that all λ σjl) R satisfy m a λ) = m g λ). Consider the following example, which suggests that it is indeed necessary. Let J R 4 4 with ) ) 2a a + b 1 0 L + =, L a + b 2b =. 0 1 It is not difficult to check that σjl) = {± b a}, and that each eigenvalue λ satisfies m g λ) = 1 and m a λ) = 2. Upon noting that L 1 = L one has that Note that CL + ) = {x, y) : ax 2 + a + b)xy + by 2 < 0}, CL + ) = {x, y) R 2 : y = x and y = a b x}, CL 1 ) = {x, y) : x 2 y 2 < 0}. CL 1 ) = {x, y) R 2 : y = ±x}. Assuming that b > a, so that σjl) R, it is not difficult to check that if a, b R +, then dim[cl + ) CL 1 )] = 0; otherwise, dim[cl + ) CL 1 )] = 1. Thus, it is not possible to use the dimension of the intersection of the cones in order to say anything definitive in this case. On the other hand, if b < a, so that σjl) ir, then one has dim[cl + ) CL 1 )] = 1. In practice it is difficult to precisely calculate dim[cl + ) CL 1 )]. However, one can derive a computable lower bound for this quantity. Let N + N ) CL + ) CL 1 )) be maximal subspaces, and suppose that an orthonormal basis for each is given by {φ ± 1,..., φ± n ± }, where n ± = nl ± ). It is clear that dimn + N ) dim[cl + ) CL 1 )].

16 M. Hǎrǎguş and T. Kapitula 15 If u N + N, then there exists vectors c ± C n ± such that n + n u = c + j φ + j = c j φ j. j=1 Upon using the orthonormality of the basis functions one sees that the above is equivalent to The above is equivalent to the system Thus, which yields j=1 c + = Nc, c = N c + ; N ij φ i, φ+ j. 2.32) c + = NN c +, c = N Nc. dimn + N ) = dim[kernn id)] = dim[kern N id)], dim[kernn id)] dim[cl + ) CL 1 )]. 2.33) As a consequence of equation 2.33) one now has the following result. Corollary Under the nondegeneracy assumption given in Corollary 2.26 one has that k c + k i 2 dim[kernn id)], where N is defined in equation 2.32). For a final theoretical result, we are now in position to refine the result of Theorem As is seen in [21], the eigenvalue problem of equation 2.1) is equivalent to that of equation 2.11) upon setting L + L, L JLJ. 2.34) The primary effect of looking at the original system in this fashion is that for equation 2.34) there is now an artificial symmetry with respect to the real axis. In particular, if λ; u λ ) is an eigenvalue/eigenfunction pair for equation 2.1), then for equation 2.34) one has the pairs λ; u λ, J 1 u λ ) T ) and λ; u λ, J 1 u λ ) T ). The major implication of introducing this artificial symmetry is that each of the quantities k r, k c, ki in Theorem 2.13 is now doubled for equation 2.34). The Jordan block structure of the eigenvalues are not effected, however. Taking into account the doubling of k c and ki, under the appropriate nondegeneracy condition one can now use Corollary 2.26 to state that for equation 2.1), k c + k i = dim[cl) C J 1 L 1 J 1 )]. Let us now derive a lower bound similar to that given in Corollary Let N CL) be a maximal subspace, and suppose that an orthonormal basis for N is given by {φ 1,..., φ n }, where n = nl). It is a straightforward calculation to show that an orthonormal basis for the maximal subspace of C J 1 L 1 J 1 ) is given by {Jφ 1,..., Jφ n }. Consequently, the matrix N derived in equation 2.32) satisfies N ij = ψ i, Jψ j. Since N is skew-symmetric, one has that NN = N 2, so that dim[kernn id)] = dim[kerin id)] + dim[kerin + id)]. Upon applying the above discussion to Corollary 2.28, we have now proved the following. Corollary Suppose that for the eigenvalue problem of equation 2.1) that all λ σjl) R + satisfy m a λ) = m g λ). Then one can refine Theorem 2.13 by further stating that k c + k i = dim[cl) C J 1 L 1 J 1 )] dim[kerin id)] + dim[kerin + id)], where N C nl) nl) is skew-symmetric and is given by N ij = ψ i, Jψ j.

17 Spectra of periodic waves Eigenvalue problems of NLS-type In this section we apply the general results of Section 2 to the eigenvalue problem where J = J R n n is a skew-symmetric, invertible matrix, and JLu = λu, 3.1) L D 2 2 x + D 1 x + Qx). 3.2) Here D 2 R n n is a positive definite, symmetric matrix, D 1 R n n a skew-symmetric matrix, and Q : R R n n a smooth map with Qx) symmetric for all x R. The form of the operator is motivated by the intended application, i.e., the study of spectra of periodic waves in equations of NLS type e.g., see [2 4, 11] and the references therein). Therefore, unlike in [20], we will assume here that the map Q is 2L-periodic for some L > 0, i.e., Qx + 2L) = Qx). The operators L and JL are considered in L 2 R; R n ). Consequently, all of the spectra are continuous, and there are no isolated point eigenvalues with finite multiplicity. However, upon using a Bloch-wave decomposition, i.e., Floquet theory, one has that solving equation 3.1) is equivalent to solving for each γ [ π, π) the eigenvalue problem JL γ u = λ γ u, u L) = ul), 3.3) where the operator L γ is formed by replacing x x + i γ 2L, so that More precisely, one has that L γ D 2 x + i γ ) 2 + D1 x + i γ ) + Qx). 3.4) 2L 2L σjl) = γ [ π,π) σjl γ ), 3.5) where the operator JL γ is taken in the Hilbert space H Lper[ L, 2 L]; C n ) equipped with the standard inner product e.g., see [28]). In contrast to JL the operator JL γ has compact resolvent, since its domain Hper[ L, 2 L]; C n ) is compactly embedded in H, and it satisfies Assumption 2.1 provided 0 σl γ ). Indeed, Assumption 2.1a) is clearly satisfied, and Assumption 2.1b) holds since L γ is self-adjoint, with compact resolvent, and invertible. ) 2, Moreover, L γ is a relatively compact perturbation of the positive operator D 2 x + i γ 2L which implies that its spectrum is bounded from below, σl γ ) [ M, ), for some M > 0. Together with the fact that σl γ ) is a discrete set, this shows that L γ has only a finite number of negative eigenvalues, which implies Assumption 2.1c). Next, JL γ is a relatively compact perturbation of the operator with constant coefficients A 0 γ JD 2 x + i γ ) 2 + JD1 x + i γ ), 2L 2L and the difference B JL γ A 0 γ = Qx), is bounded. Consequently, equation 2.2) holds with r = 0. A standard Fourier analysis shows that the spectrum of A 0 γ consists of eigenvalues λ k,j = iω k,j, k Z, j = 1,..., n, with λ k,j ) j=1,...,n eigenvalues of the skew-symmetric matrix JD 2 k + γ ) 2 + ijd1 k + γ ), for k Z. 2L 2L Analyzing the location of these eigenvalues on the imaginary axis, one finds that λ k,j = Ok 2 ), as k, and conclude that Assumption 2.1d)ii)-iii) holds for s = 1 and r = 1/2. This shows that J and L γ satisfy Assumption 2.1.

18 M. Hǎrǎguş and T. Kapitula 17 We point out that as a consequence of equation 3.4), one has that L γ = L γ, 3.6) so that if λ σjl γ ) with associated eigenfunction v, then λ σjl γ ) with associated eigenfunction v. Thus, one can conclude that for equation 3.1) one has the standard Hamiltonian quartet {±λ, ±λ} σjl). However, this quartet is recovered only by considering σjl γ ) σjl γ ) for 0 γ π. We can now apply the results in Section 2, more precisely Theorem 2.13, which together with equation 3.6) give the following. Theorem 3.1. Consider the eigenvalue problem 3.3). Let π γ < π be such that dim[kerl γ )] = 0. One has that k r γ) + k c γ) + k i γ) = nl γ). Furthermore, σjl γ ) = σjl γ ), and k r γ) = k r γ), k c γ) = k c γ), k i γ) = k i γ). Remark 3.2. a) As a consequence of the last part of Theorem 3.1, we may restrict from now on to γ [0, π]. b) In stability problems, JL represents the linearization about the periodic wave and if the problem possesses a translation invariance in space which is often the case) then the derivative of the periodic wave belongs to the kernel of JL 0, so that the condition dim[kerl γ )] = 0 does not hold for γ = 0. However, for γ 0 this zero eigenvalue of JL 0 typically moves off the origin, so that this condition is satisfied. One expects in such problems that this assumption will only be violated for a finite number of values of γ. As we shall see, in our examples here the condition dim[kerl γ )] = 0 is only violated for γ = 0. Notice, that since the full spectrum of JL is continuous, this restriction does not qualitatively effect the final result. We consider now the particular case D 1 = 0 and Q x) = Qx). Set U 1 [ux)] u x), 3.7) so that U 1 is a unitary operator on H. Then it is easy to verify that U 1 L = LU 1 and U 1 J = JU 1. For the Bloch operator JL γ it is not difficult to check that U 1 L γ = L γ U 1, and then U 1 L γ = L γ U 1. Consequently, U 1, J, and L γ satisfy Assumption 2.15a). Also note that in this case σjl γ ) = σjl γ ). By applying the results of Theorem 2.17 we obtain the following. Theorem 3.3. Under the assumptions in Theorem 3.1, further suppose that D 1 = 0 and Q x) = Qx). Then the spectrum of JL γ is symmetric with respect to both the imaginary and the real axis and the result in Theorem 3.1 holds with k c γ) and ki γ) even numbers. Finally, consider the canonical form J = which is achieved by taking 0 id id 0 ), L = diagl +, L ), 3.8) D 2 = diagd + 2, D 2), D 1 = 0, Qx) = diagq + x), Q x)). Then in 3.4) one finds L γ = diagl + γ), L γ)). As in the previous case, one has σjl γ ) = σjl γ ), and the result of Theorem 2.25 gives: Theorem 3.4. Assume that J and L are as in equation 3.8. Let π γ < π be such that dim[kerl ± γ))] = 0. Then the result in Theorem 2.25 holds for J and L γ.

19 Spectra of periodic waves NLS equation with periodic potential Consider the nonlinear Schr ĺodinger equation iq t + q xx + ωq + f x, q 2 )q = 0, 3.9) where f is a smooth, real-valued map, and f x + 2L, ) = f x, ). Here qx, t) C and x, t) R R +. A physically relevant example is f x, q 2 ) = g q 2 + V 0 sn 2 x, k), 3.10) where g = ±1 and snx, k) is the Jacobian elliptic sine function with 0 k < 1 e.g., see [3, 4]). In this case, if g = +1 then equation 3.9) is a model equation for an attractive Bose-Einstein condensate BEC), whereas if g = 1 equation 3.9) is a model equation for a repulsive BEC. The form of the potential allows one to explicitly solve the steady-state problem associated with equation 3.9) via the use of elliptic functions. We discuss this particular case in the next section. Suppose that Qx) is a real-valued, periodic steady solution to equation 3.9) which satisfies Qx + 2L) = ±Qx). Note that this implies that Q 2 x) is 2L-periodic. Upon linearizing about this solution one finds the eigenvalue problem given by equation 3.1) with J and L as in 3.8), where with L ± xx + Q ± x), Q ± x + 2L) = Qx), 3.11) Q + x) f x, Q 2 x)) 2D 2 f x, Q 2 x))q 2 x) ω, Q x) f x, Q 2 x)) ω. Note that L ± are operators of Hill type see [24]). Their spectra in L 2 R) consist of unions of bands with nontrivial interior [µ 0, µ 1 ] [µ 2, µ 1] [µ 2, µ 3 ], where the solution of the eigenvalue problem corresponding to the band edge µ j is periodic with period 2L the periodic eigenvalue), and the solution corresponding to the band edge µ j is periodic with period 4L the antiperiodic eigenvalue). In terms of Floquet theory the eigenvalue µ j has a Floquet exponent of γ = 0, while the eigenvalue µ j has a Floquet exponent of γ = π. In terms of the associated Bloch operators L ± γ), the band edge µ j corresponds to an eigenvalue for γ = 0, while the band edge µ j corresponds to an eigenvalue for γ = π. If µ = 2j 1 µ 2j or µ 2j 1 = µ 2j, then this corresponds to a closed gap, and such a point is referred to as a double point. For each γ 0, π) there is precisely one value of µγ) in the interior of each band such that γ is the Floquet exponent for that value of µγ); furthermore, µγ) depends continuously upon γ and µγ) = µ γ). At the band edge, in addition to the periodic or antiperiodic solution there is a solution which grows linearly, unless the band edge is a double point, in which case there are two linearly independent periodic or antiperiodic solutions. We are now in position to present another proof of the instability result in [2, Theorem 3.3], as well as its natural generalization. Theorem 3.5. The periodic wave Qx) is spectrally unstable in each of the following two cases: a) if µ = 0 is in the interior of a band of σl + ); b) if L + and L have a different number of bands contained in R {0}. In both cases, k r γ) 1, for some γ 0, π). Proof: We use the results of Theorem 3.4. Since L Q = 0, and Qx) satisfies Qx + 2L) = ±Qx), one has that µ = 0 is a band edge for L. The set σl ) R {0}) will then contain a finite number of bands, so that for each γ 0, π) one will have nl γ)) = k for some fixed k N 0 which is independent of γ. Now consider the operator L + under the assumption that µ = 0 is in the interior of a band, say without loss of generality) [µ j, µ j+1 ], for some j N 0. Let γ c 0, π) be such that µγ c ) = 0. One then has that j + 1, 0 < γ < γ c nl + γ)) = j, γ c < γ < π.