On the Benjamin-Ono equation on $\mathbb{T}$ and its periodic and quasiperiodic solutions

In this paper, we survey our recent results on the Benjamin-Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions, which are not $C^\infty$-smooth.


Introduction
In this paper we consider the Benjamin-Ono (BO) equation on T, (1) where u ≡ u(t, x) is real valued and H denotes the Hilbert transform, defined as the Fourier multiplier We refer to the recent survey [25] for a discussion of the origin of this equation as a model for long, unidirectional internal gravity waves in a two layer fluid and for a comprehensive bibliography.
Our study of the equation (1) focuses on the following topics: wellposedness in Sobolev spaces, traveling waves and their orbital stability, long time behaviour of solutions (i.e., on properties of their orbits such as boundedness, orbital stability, recurrence), aspects of integrability, and the construction of periodic and quasiperiodic solutions.
Sharp wellposedness in Sobolev spaces.The wellposedness of (1) in Sobolev spaces H s r ≡ H s (T, R) has been extensively studied in the last 40 years -see e.g.references in [20].To state our results we need to recall the following result of [1] (cf. also [24]).It says that for any initial data u in H s r with s > 3/2, there exists a unique solution v in C(R, H s r ) of (1) with v| t=0 = u such that the solution map S(t)u := v(t), S : R × H s r → H s r , (t, u) → S(t, u) ≡ S(t)u, is continuous.Theorem 1. (i) For any s > −1/2, S extends as a continuous map, S : R × H s r → H s r ( [11]).(ii) No such extension exists for s = −1/2.In more detail, there exists a sequence (u (k) ) k≥1 in C ∞ (T, R) with the property that u (k) → 0 in H −1/2 r , but the function t → S(t)u (k) | e ix does not converge to 0 on any time interval I ⊆ R with |I| > 0 ( [11]).
Note that 2π 0 udx is a prime integral of (1).In particular, for any a ∈ R and s > −1/2, the solution map S(t) leaves the subspace invariant.We denote by S a (t) the restriction of S(t) to H s r,a .Addendum to Theorem 1.In [13], [14] we prove that for any t ∈ R, a ∈ R, and −1/2 < s < 0, S a (t) : H s r,a → H s r,a is nowhere locally uniformly continuous and that for any s ≥ 0, S a (t) : H s r,a → H s r,a is real analytic.
Remark 1. Theorem 1(i) improves on a result by Molinet [18] (cf.also [20]), saying that for any s ≥ 0, S : R × H s r → H s r extends as a continuous map and Theorem 1(ii) improves on a result by Angulo Pava & Hakkaev [4], saying that no such extension exists for s < −1/2.The Addendum to Theorem 1 improves on a result by Molinet [18]- [19], saying that for any a ∈ R and s ≥ 0, S a (t) : H s r,a → H s r,a is analytic near the zero solution.
We refer to [20] for a comprehensive bibliography on the wellposedness of (1).We note that our method of proof is different from the methods used in the papers cited above (cf.Theorem 4 below).
Traveling waves.Recall that a solution t → S(t)U of (1) with U ∈ H s r , s > −1/2, is said to be a traveling wave with profile U and velocity c ∈ R if S(t)U = U(• − ct).Amick & Toland [2] listed all C ∞ -smooth profiles of traveling waves of (1), 1−2r cos x+r 2 , 0 < r < 1, are the traveling wave profiles with corresponding velocity c r := 1+r 2 1−r 2 , found by Benjamin [5].Theorem 2. For any s > −1/2 the following holds: (i) Any traveling wave in H s r has a profile of the form U r,N,α,a and hence in particular is C ∞ -smooth ( [11]).(ii) Any traveling wave is orbitally stable in H s r ( [11], Remark 5 below).Remark 2. Theorem 2(ii) improves on a result by Angulo Pava & Natali [3], saying that the traveling waves with profile of the form U r,N,α,a are orbitally stable in Long time behaviour of solutions.For solutions of the Benjamin-Ono equation on the line, the question of main interest concerning their long time behaviour is to know whether they admit an asymptotic description as t → ∞.Since T is compact, such a description typically does not exist for solutions of (1).In such a case, one is interested to know properties of the orbits of solutions such as boundedness, orbital stability, or recurrence.Theorem 3.For any u ∈ H s r with s > −1/2 the following holds: 11]).(iii) sup t∈R S(t)u s ≤ M where M > 0 can be chosen uniformly on bounded subsets of H s r ( [11]).Addendum to Theorem 3. The solutions of Theorem 1 of (1) are orbitally stable in the sense explained in Remark 5 below.Remark 3. We point out that a solution R → H s r , t → S(t)u of (1) being almost periodic implies that it is Poincaré recurrent.In particular, Theorem 3(ii) improves on results by Deng & Tzvetkov & Visciglia [8] and Deng [7].In these papers (cf.also references in [8], [7]) invariant measures are constructed on Sobolev spaces of various order of regularity, which then are used to show that for a.e.initial data, the corresponding solutions are Poincaré recurrent.Theorem 3(iii) improves on results of similar type, which can be derived from the BO hierarchy, obtained in [6], [22].The BO hierarchy consists of a sequence H j (u), j ≥ 0, of prime integrals of (1).The boundedness of (H j ) 0≤j≤n can be shown to be equivalent to the boundedness of the H n/2 -norm.Furthermore, Talbut [26] proved such estimates for the H s -norms, −1/2 < s < 0, for smooth solutions of (1).
Nonlinear Fourier transform.Our proofs of Theorem 1 -Theorem 3 rely on the integrability of the BO equation.In fact, we show that this equation is integrable in the strongest possible sense.To state this result, we first need to introduce some more notation.As already mentioned above, 2π 0 udx is a prime integral of (1).Furthermore, for any solution u(t, x) of (1) and any a ∈ R, u a (t, x) = a + u(t, x − 2at) is again a solution.We therefore restrict ourselves to consider equation (1) on the Sobolev spaces H s r,0 .By h σ + ≡ h σ (N, C), σ ∈ R, we denote the weighted ℓ 2 -sequence spaces defined by h σ Theorem 4. ( [10], [11]) There exists a map so that the following properties hold for any s > − It follows that for any n ≥ 1, |ζ n (S(t)u)| 2 is independent of t.
(NF3) The map Φ does not continuously extend to a map H −1/2 → h 0 + .Addendum to Theorem 4. In [13], [14], we prove that for any s > −1/2, Φ : . Furthermore, the linearization of (1) at the zero solution is given by ∂ t v = H∂ 2 x v.The solution of the latter equation in H s r,0 are given by n =0 e i sign(n)n 2 t v(n)e inx .For this reason, we refer to Φ as a nonlinear Fourier transform.(ii) It is well known that (1) is Hamiltonian, where ∂ x is the Poisson structure, which corresponds to the Poisson bracket, defined for functionals F, G on H s r,0 with sufficiently regular L 2 -gradients, We prove that for any n, m ≥ 1, , is an invariant torus for (1).Any such torus is Lyapunov stable in the sense that for any initial data u ∈ H s r,0 near Iso(ξ), the solution S(t)u stays close to Iso(ξ) for all t ∈ R.
In the remaining part of this introduction, we briefly comment on applications of Theorem 4 and on elements of its proof.We keep our exposition as short as possible and refer to our papers for more details.
The Benjamin-Ono equation admits finite dimensional integrable subsystems.To define them, we need to introduce some more notation.We say that u ∈ s>−1/2 H s r,0 is a finite gap potential if there exists N ∈ N so that ζ n (u) = 0 for any n > N. We denote by U N the set of all such potentials in s>−1/2 H s r,0 with ζ N = 0. Furthermore, we say that u ∈ s>−1/2 H s r,0 is a one gap potential if there exists N ≥ 1 so that ζ n (u) = 0 if and only if n = N.In particular such a potential is in U N .Theorem 4 implies that for any where 0 < |q j | < 1 for any 1 ≤ j ≤ N ( [10]).The time evolution of potentials in U N can be explicitly described, using the frequencies, defined in (NF2) of Theorem 4. These solutions coincide with the ones constructed by Satsuma & Ishimori [23] and further studied by Dobrokhotov & Krichever [9].We refer to these solutions as finite gap solutions or (periodic in x) multi-solitons.They are quasiperiodic in time.The one gap solutions coincide with the traveling waves of Theorem 2 and are periodic in time.
In Section 3 we address the questions whether there are periodic and quasiperiodic solutions in time of (1) which are not (multi-)solitons.Both questions are answered affirmatively -see Theorem 5, Proposition 3, and Theorem 6.The proof of these results is based on the action to frequency map, studied in Section 2. To the best of our knowledge, results of this type are not known for integrable PDEs such as the Korteweg-de Vries (KdV) equation or the nonlinear Schrödinger (NLS) equation.We expect, but have not verified, that such results also hold for many of these PDEs although the action to frequency map might be significantly more complicated and hence the results more difficult to prove.In this connection, we only mention that the Hessian of the KdV and the NLS Hamiltonian are known to be strictly convex in a neighborhood of the zero solution (cf.[16], [21] for details).
A key ingredient of the proof of Theorem 4 is the Lax pair formulation of (1), ) , and T u denotes the Toeplitz operator, defined for potentials u in H s r,0 , s > −1/2.Here, the pseudo-differential operators L u and B u act on the Hardy space [11]).The Lax pair formulation implies that the spectrum of L u is preserved by (1).For any u ∈ H s r,0 , the latter is discrete, bounded from below and consists of a sequence of simple real eigenvalues, which we list in increasing order, [10] (s = 0) and [11] (−1/2 < s < 0)).They satisfy (4) where γ n , referred to as the nth gap of the spectrum of L u (cf.[10, Appendix C]), turns out to be the action action variable |ζ n | 2 , mentioned in Remark 4. The spectrum of L u is encoded by the generating function, (5) This function is at the heart of the construction of the map Φ.It admits an expansion at λ = ∞, whose coefficients constitute the BOhierarchy, mentioned in Remark 3. In Appendix A we show that H λ (u) can be viewed as the relative determinant of L u + λ + 1 with respect to L u + λ.
Acknowledgements.We are grateful to E. Fouvry for drawing our attention to the reference [27].
For the sequel, it is convenient to set for any y = (y n ) n≥1 in c, Denote by c ↑ the subset of c of sequences y = (y n ) n≥1 satisfying By ( 7)-( 8), for any γ ∈ ℓ 1,1 ≥0 , the sequence (ω n (γ)) n≥1 is in c ↑ .The normalized action to frequency map is defined as To the best of our knowledge, comparable results for integrable PDEs such as the KdV equation or the NLS equation are not known.For partial results in this direction for the KdV equation, we refer to [16] and references therein.
Proof.By (8), Ω is one-to-one.To see that Ω is onto, consider y = (y n ) n≥1 in c ↑ .Let (9) γ Then by the definition of c ↑ , γ n ≥ 0 for any n ≥ 1.By telescoping, one has for any n ≥ 1, and for any n ≥ 0, Finally, note that Ω is the restriction of a linear map ℓ 1,1 → c, whose norm is bounded by 2. Indeed, since for any (x n ) n≥1 ∈ ℓ 1,1 and any n ≥ 1, This implies that Ω : ℓ 1,1 ≥0 → c ↑ is continuous.Going through the proof of the ontoness of Ω : ≥0 is continuous as well.Indeed, assume that (y (k) ) k≥1 is a sequence in c ↑ , converging to y = (y n ) n≥1 ∈ c ↑ .By (9) it follows that for any n ≥ 1, the nth component γ ) converges to the nth component γ n of γ := Ω−1 (y) and that n≥1 nγ one then concludes that γ (k) converges to γ in ℓ 1,1 .
Remark 7. A result similar to the one of Proposition 1 can be derived for the restriction ΩJ of Ω to the subset ≥0,J and c ↑,J are defined in these two cases as follows.(i) Case J infinite.In this case, the subset J is of the form J := and c ↑,J the set of strictly increasing sequences y J := (y np ) p≥1 of positive numbers, satisfying where we set n 0 = 0 and y 0 = 0. Note that for any where we recall that ω0 = 0.For any p ≥ 1, one then has (ii) Case J finite.In this case, the subset J is of the form and c ↑,J the set of strictly increasing finite sequences y J := (y np ) 1≤p≤N of real numbers, satisfying where we set n 0 = 0, y 0 = 0 and n N +1 = n N + 1, y n N+1 = y n N .Note that for any In this case, ωn = ωn N for any n ≥ n N and for any 1 ≤ p ≤ N, It is convenient to extend Ω to a linear map Ω : ℓ 1,1 → c.This extension, is given by Then Ω is a bounded by (10).Denote by Q the quadratic form, induced by Ω.For any x ∈ ℓ 1,1 , Q(x) is given by As a quadratic form, Q extends to ℓ 1,1/2 ≡ ℓ 1,1/2 (N, R) and (We mention that the quadrant ℓ ≥0 , filled out by the actions, corresponds to the phase space of potentials H −1/4 r,0 , for which the map Φ is well defined by Theorem 4.) By ( 11) one has Q(x) = n≥1 n x 2 n + 2x n k>n x k .Hence by completing squares one obtains ) where s n := k≥n x k .It implies In particular, one sees that Q is a positive semidefinite quadratic form on ℓ 1,1/2 and that −Q, when restricted to ℓ 1,1 ≥0 , coincides with the quadratic part the Hamiltonian H of the BO equation, when expressed in the action variables γ = (γ n ) n≥1 (cf.[10, Proposition 8.1]), ( 12) Since −2Q is the Hessian of H, the latter can thus be viewed as a concave function.In summary, Q has the following properties.

Applications
As an illustration of our analysis of the action to frequency map we apply our results to construct families of periodic solutions of the BO equation, which are not traveling waves, and families of quasiperiodic solutions, which are not finite gap solutions.
Periodic solutions.Our first result addresses the question, whether there are periodic in time solutions of (1), which are not finite gap solutions.
Theorem 5. (i) For T > 0 with T /π rational, any T -periodic solution in L 2 r,0 of (1) is a finite gap solution.
(ii) For any positive irrational number b, there exists a strictly increasing sequence (n p ) p≥1 in N and a sequence of actions γ = (γ n ) n≥1 in ℓ 1,1 ≥0,J , J := {n p : p ∈ N}, satisfying where ω n (γ) is given by (6).As a consequence, any potential u 0 in the torus (cf.[10, Section 3]) ( 14) r,0 and the solution u(t) of the BO equation with u(0) = u 0 (cf.Theorem 1) is periodic in time with period T = 2π/b.Therefore, Iso γ is entirely filled up with T -periodic solutions.
Proof.(i) Let u(t) be the solution of (1) with initial condition u(0 Hence u is T -periodic if and only if, for every n ≥ 1 with ζ n (0) = 0, By assumption, ω is rational.Choose p, q ∈ Z with q ≥ 1 so that ω = p/q.Since u 0 is in L 2 r,0 , γ = (γ n ) n≥1 ∈ ℓ 1,1 ≥0 and formula (6) for the frequencies hold.It then follows that for any n ≥ 1 with ζ n (0) = 0, Assume that there are infinitely many integers n with ζ n (0) = 0. Since Consequently, for infinitely many integers n, one has 2 k>n (j − n)|ζ k (0)| 2 ∈ 1 q Z, which contradicts that k>n (k − n)γ k converges to 0. Hence there are only finitely many integers n with ζ n (0) = 0, which implies that u(t) is a finite gap solution.(ii) Our task is to find a strictly increasing sequence (n p ) p≥1 of N and a sequence γ = (γ n ) n≥1 in ℓ 1,1 ≥0,J , J := {n p | p ∈ N}, (cf.Remark 7(i)), so that there exists a sequence (m p ) p≥1 in Z with the property that ( 15) To find such sequences, we use that by a result due to Weyl [27], the set Given an arbitrary positive real number y ∞ , choose a sequence (ε p ) p≥1 of the form ( 17) where ε 0 > 0 is chosen so that (18) ε 0 < 4y ∞ .
For any p ≥ 1, we then choose integers n p ≥ 0 and k p so that ( 19) . By the definition of the sequence (ε p ) p≥1 , (ρ p ) p≥1 is a strictly decreasing sequence of positive numbers, converging to 0 as n → ∞, By induction on p, it is possible to choose n p ≥ 1 for any p ≥ 1, so that n 1 ≥ 1 and n p+1 ≥ 2n p .(Indeed, for every integer N, the set so what is left over after removing it from D b is still dense in R.) Thinking of ρ p as 2y ∞ − 2ω np and hence of 2y ∞ − ρ p as 2ω np , we define for any p ≥ 2 (cf.Remark 7(i)), (20) γ Using that n p ≥ n p − n p−1 and n p+1 − n p ≥ n p , one sees that ( 21) Since by (19), ε p ≤ ρ p ≤ 2ε p and by (17), ε p−1 − 4ε p = 0 , one gets from ( 20) and ( 21), Consequently, γ np > 0 and, for some c > 0, On the other hand by (20), hence by telescoping and by the bound (18) of ε 0 , Now define γ n 1 > 0 so that the second identity in ( 16) holds, It remains to check the identities in (15).Using the definition (20) of γ np , p ≥ 2, one verifies that for any p ≥ 1, ρ p = 2 q>p (n q − n p )γ nq and thus, by the definition of This completes the proof of item (ii).
Remark 8. (i) It is possible to choose the sequence (ε p ) p≥1 , constructed in the proof of Theorem 5(ii), so that u / ∈ H s r,0 for some 0 < s < 1. (ii) The sequence (n p ) p≥1 , constructed in the proof of Theorem 5(ii), needs to be sparse in the following sense: if (n p ) p≥1 is a strictly increasing sequence in N and γ = (γ k ) k≥1 a sequence of actions in ℓ ≥0,J , J := {n p | p ≥ 1}, with the property that there exists an infinite set of integers n in J so that n − 1 and n + 1 are also contained in J, then the frequencies cannot satisfy (15).Indeed, the frequencies satisfy on ℓ 1,1 ≥0 the identities (with ω 0 = 0) Hence if ω np ∈ bZ for any p ≥ 1, it then would follow that 2 + 2γ n (u) ∈ bZ for infinitely many n in J, implying that 2 ∈ bZ.This however contradicts the assumption of b being irrational.
The following result says that that there are many finite gap solutions of the BO equation which are periodic in time, but not traveling waves.Proposition 3.For any rational number of the form 1/a, a ∈ N, any N ∈ N , and any strictly increasing sequences (where we set n 0 := 0, n N +1 := n N + 1, k 0 := 0, and k N +1 := k N ), the following holds: the sequence of actions, γ = (γ n ) n≥1 ∈ ℓ 1,1 ≥0,J , defined by and As a consequence, any potential u 0 in the torus Iso γ (cf.(14)) is a finite gap potential, the solution u(t) of (1) with u(0) = u 0 periodic in time with period T = 2πa, and hence Iso γ entirely filled with T -periodic solutions.
Remark 9. Since by Theorem 2, the traveling waves of the BO equation coincide with the one gap solutions, it follows from Proposition 3 that there is a plentitude of periodic in time solutions of (1) which are finite gap solutions, but not traveling waves.
Proof.The claimed results follow from Remark 7(ii).
Quasiperiodic solutions.The aim of this paragraph to construct quasiperiodic solutions of (1), which are not finite gap solutions.We begin with describing the ω-quasiperiodic in time solutions of (1) in terms of the map Φ of Theorem 4 where ω is a frequency vector in R d , d ≥ 2, with Q-linearly independent components.
Here, by notational convenience, the vector tω denotes also the class of vectors tω + (2πZ) d .The function U is referred to as the profile of u. where and (k (n) ) n≥1 is a sequence in Z d with the property that for any n ≥ 1, ( 23) Remark 10.For any ω-quasiperiodic solution of (1) with action vari- , the invariant torus (1).The corresponding profiles are given by (22) with (ζ n ) n≥1 being an arbitrary element in the set Φ(Iso γ ).
Proof.Let U be the profile of an ω-quasiperiodic solution u(t) in H s r,0 (T) of (1).It is to show that ( 22)-( 23) hold.Let (ξ n (ϕ) , ϕ → (ξ n (ϕ)) n≥1 is continuous and by (NF2) in Theorem 1, for any n ≥ 1, (24) ζ n e itωn = ξ n (tω) , . Since by assumption, the components of ω are linearly independent in Q, the Fourier coefficients ξ n (k), k ∈ Z d , of ξ n can be computed as Furthermore, by the formula (24) for ξ n (tω), one has lim Note that the right hand side of the latter identity vanishes if ω n = k•ω, and equals ζ n if ω n = k • ω.Consequently, for any given n ≥ 1, the following dichotomy holds: in the case where there is no k ∈ Z d , satisfying ω n = k • ω, it follows that ξ n (k) = 0 for any k ∈ Z d .Hence the continuous function ξ n vanishes, implying that ζ n = ξ n (0) = 0. Otherwise, since the components of ω are linearly independent over Q, there exists exactly one k (n) ∈ Z d such that ω n = k (n) • ω and ξ n (ϕ) equals ζ n e ik (n) •ϕ .We thus have proved that ( 22)-( 23) hold.
The following result illustrates how Proposition 4 can be used to construct ω-quasiperiodic solutions of (1), which are not C ∞ -smooth, hence in particular not finite gap solutions.Theorem 6.Let b be an irrational real number and ω := (1, b) ∈ R 2 .For any s > −1/2, there exists a ω-quasiperiodic solution of the BO equation in H s r,0 \ σ>s H σ r,0 .Proof.Let s > −1/2 be given.In view of Proposition 4, it suffices to find a sequence The latter identities imply that for any n ≥ 1, It is convenient to reformulate our problem.Let 1) , n ≥ 1, our problem can be described equivalently as follows: find a sequence (γ n ) n≥1 in R >0 , belonging to ℓ 1,1+2s + \ ∪ σ>s ℓ 1,1+2σ + , a sequence (ℓ (n) ) n≥1 in Z 2 , and k (1) ∈ Z 2 so that k (0) = 0 and ( 25) where for any n ≥ 1, γ n is related to Using the density of the additive subgroup ω • Z 2 = Z + bZ in R, it is straightforward to construct sequences (γ n ) n≥1 and (ℓ (n) ) n≥1 , which satisfy the first set of identities in (25).But it is more involved to construct such sequences satisfying at the same time the second identity in (25), which can be rephrased as Accordingly, we proceed in two steps: Step 1.Let (ε n ) n≥1 be a sequence in R >0 that belongs to the set ℓ 1,1+2s ≥0 \ ∪ σ>s ℓ 1,1+2σ ≥0 and satisfies (27) 4 By the density of ω For any n ≥ 1, let γ n be the number in [ε n /2, ε n ], defined by By ( 27) it then follows that Step 2. We correct m (n) and γ n so that ( 26) is satisfied with k (1) = 0.
Proposition 5.For any u ∈ H s r,0 with s > −1/2 the following holds: (i) For any λ > −λ 0 , (L u + 1 + λ) −1 − (L u + λ) −1 is of trace class.(ii) L u + 1 + λ admits a determinant relative to L u + λ and Proof.We use arguments developed in the proof of [10,Lemma 3.2].(i) By functional calculus, ( One then concludes from (29) and the decay properties of the γ n that (L u + 1 + λ) −1 − (L u + λ) −1 is of trace class for any λ > −λ 0 and that its trace is given by − (ii) Denote by S the shift operator on H + and by S * its adjoint, where w λ := (L u + λ) −1 1. Combining the identities (30)-(31), one gets where Note that II λ is an operator of rank 1 and hence in particular of trace class.Its trace can be computed as follows.Denote by (f n ) n≥0 the orthonormal basis of eigenfunctions of L u , introduced in [10] (s = 0) and [11]  Since Since when expanding H λ (u) = (L u + λId) −1 1| 1 with respect to the orthonormal basis (f n ) n≥0 , one obtains and since lim λ→∞ H λ (u) = 0 we proved that H λ (u) is the determinant of L u + 1 + λ relative to L u + λ.
In the remaining part of this appendix we study in more detail, how for any given u ∈ H s r,0 , s > −1/2, H λ (u) is related to the spectrum of L u .First note that it follows from (35) that implying that (cf.[10, Proposition 3.1], [11]) Therefore, H λ (u) is determined by the periodic spectrum of L u .Since the latter is invariant by the flow of ( 1), H λ (u) is a one parameter family of prime integrals of this equation.The question arises if conversely, H λ (u) determines the spectrum of L u .To this end we take a closer look at the product representation (36) of H λ (u).Setting ν n := λ n−1 + 1 for any n ≥ 1, one has Furthermore, for any n ≥ 1, ν n = λ n if and only if γ n = 0. Hence Note that ν n , n ∈ J u , are the zeros of H λ (u) and λ n , n ∈ J u ∪ {0}, its poles.All the poles and zeros of H λ (u) are simple.The question raised above can now be rephrased as follows: does H λ (u) besides ν n (u), n ∈ J u , and λ n (u), n ∈ J u ∪ {0}, also determine the eigenvalues λ n (u) with γ n (u) = 0? The following result says that this is indeed the case.Proposition 6.For any u ∈ H s r,0 , s > −1/2, the generating function H λ (u) determines the entire spectrum of L u .
Proof.Since λ 0 is a pole of H λ (u), it is determined by the generating function.If ν 1 ≡ λ 0 + 1 is a zero of H λ (u), then ν 1 = λ 0 + 1 < λ 1 and hence λ 1 is a pole of H λ (u).If λ 0 + 1 is not a zero of H λ (u), then λ 0 + 1 is the periodic eigenvalue λ 1 of L u and hence also determined by H λ (u).Arguing inductively, the claimed result follows.