Hamiltonian Floer theory for nonlinear Schr\"odinger equations and the small divisor problem

We prove the existence of infinitely many time-periodic solutions of nonlinear Schr\"odinger equations using pseudo-holomorphic curve methods from Hamiltonian Floer theory. For the generalization of the Gromov-Floer compactness theorem to infinite dimensions, we show how to solve the arising small divisor problem by combining elliptic methods with results from the theory of diophantine approximations.


Hamiltonian partial differential equations
Nonlinear Schrödinger equations play a very important role in mathematical physics and have applications in, e.g., solid state physics, condensed matter physics, quantum chemistry, nonlinear optics, wave propagation, protein folding and the semiconductor industry. In contrast to the well-known linear Schrödinger equation describing the time evolution of the quantum wave function of a single particle, nonlinear Schrödinger equations are classical field equations describing multiparticle systems, where the nonlinearity models the interaction between different particles. An example of a nonlinear Schrödinger equation is the so-called Gross-Pitaevskii equation i∂ t u = −∆u + c|u| 2 u + V (t, x)u, O. Fabert, VU Amsterdam, The Netherlands. Email: o.fabert@vu.nl. 1 which plays an important role in the theory of Bose-Einstein condensates. Here u = u(t, x) ∈ C is a complex-valued function depending on time and space, ∂ t is the derivative with respect to the time t ∈ R, ∆ denotes the Laplace operator with respect to the space coordinate x, V (t, x) is a time-dependent exterior potential and c ∈ R is a scalar whose sign depends on whether the particles are attracting or repelling each other. Here and in what follows we restrict ourself to the case of one spatial dimension for notational simplicity; we claim that everything, including our main theorem, can be generalized to the higher-dimensional case.
Nonlinear Schrödinger equations are important examples of Hamiltonian partial differential equations, where we refer to [9] for definitions, statements and further references. This means that they can be written in the form ∂ t u = X H t (u), where the Hamiltonian vector field X H t is determined by the choice of a (time-dependent) Hamiltonian function H = H t and a linear symplectic form ω. Here a bilinear form ω : H × H → R on a real Hilbert space H is called symplectic if it is anti-symmetric and nondegenerate in the sense that the induced linear mapping i ω : H → H * is an isomorphism. As in the finite-dimensional case it can be shown that for any symplectic form ω there exists a complex structure J 0 on H such that ω, J 0 and the real inner product ·, · R on H are related via ·, · R = ω(·, J 0 ·).
In the case of nonlinear Schrödinger equations on the circle S 1 = R /2π Z one chooses the complex Hilbert space H = L 2 (S 1 , C) of square-integrable complex-valued functions on the circle which naturally can be viewed as a real Hilbert space by identifying C with R 2 . The standard complex inner product ·, · C is related to the standard real inner product ·, · R and the standard symplectic form ω by ·, · C = ·, · R + iω and the symplectic form is related to the real inner product via ω = J 0 ·, · R with J 0 = i denoting the standard complex structure on H. In order to stress the relation with the finite-dimensional case of R 2n = C n , note that, using the Fourier series expansion u(x) = (2π) −1/2 ∞ n=−∞û (n) · exp(inx) and writinĝ u(n) = q n +ip n for all n ∈ Z, it follows that the symplectic Hilbert space L 2 (S 1 , C) can be identified with the space ℓ 2 (C) of square-summable complex-valued seriesû : Z → C equipped with the symplectic form ω = +∞ n=−∞ dp n ∧ dq n . The corresponding Hamiltonian function is of the form where f is a smooth, real-valued function on R + ×S 1 × R.

Nonlinear Schrödinger equations of convolution type
While the symplectic form ω is nondegenerate on L 2 (S 1 , C), the Hamiltonian H t is only well-defined and smooth on its dense subspace H 1,2 (S 1 , C). While this is apparent for the first summand as it involves the first derivative, observe that even the Hamiltonian F t modelling the nonlinearity is not defined on all of H when the resulting Schrödinger equation is truely nonlinear. This in turn leads to true problems with the existence of the corresponding Hamiltonian flow φ t = φ H t , describing the time-evolution of solutions of the nonlinear Schrödinger equation.
We start with the case of the free nonlinear Schrödinger equation, that is, when the nonlinearity f is equal to zero. Note that in this case the Hamiltonian H t simplifies to While the resulting Hamiltonian vector field X 0 (u) = i∆u is only defined on H 2,2 (S 1 , C), we can prove the following result about the corresponding flow φ 0 t . Proposition 2.1. The flow of the free Schrödinger equation is given by For fixed time t it preserves the L 2 -norm and hence defines a linear symplectomorphism on the full symplectic Hilbert space H = L 2 (S 1 , C), which restricts to a finite-dimensional linear symplectomorphism on every C 2k+1 := {u ∈ H :û(n) = 0 for all |n| > k}.
Proof. In order to see this, observe that, after applying the Fourier transform, the symplectic vector field X 0 (û)(n) = in 2 ·û(n) has a linear flow given by φ 0 t (û)(n) = exp(itn 2 ) ·û(n), n ∈ N . Since in every frequency it multiplies the Fourier coefficient by a complex number of norm one, the claims follow.
On the other hand, if the Hamiltonian F t describing the nonlinearity is only densly defined, it is typically a very hard problem to establish the existence of a corresponding Hamiltonian flow φ F t on the full phase space H. The problem is that the flow on H is no longer the unique solution of an ordinary differential equation given by the Hamiltonian vector field X F t . In order to circumvent problems arising from missing regularity in the nonlinear term, in this paper we hence work with a modification of the classical nonlinear Schrödinger equation, see [9]. Definition 2.2. A nonlinear Schrödinger equation of convolution type is a Hamiltonian PDE with Hamiltonian H t = H 0 + F t on the symplectic Hilbert space H = L 2 (S 1 , C), where the Hamiltonian defining the nonlinearity is now defined as is some fixed smoothing kernel, and f still denotes a smooth, real-valued function on Instead of considering density functions of the form f (|u(x)| 2 , x, t), from now on we hence consider density functions of the form f (|(u * ψ)(x)| 2 , x, t). Nonlinear Schrödinger equations of convolution type describe multi-particle systems with nonlocal interaction. The comparison with the above example and a short computation show that the resulting nonlinear Schrödinger equations are given by where ∂ 1 f means derivative with respect to the first coordinate and convolution is understood with respect to the space coordinate.
We collect important observations about these class of equations in the following Proposition 2.3. For every nonlinear Schrödinger equation of convolution type the resulting flow is given by the composition The time-dependent Hamiltonian functions F : R × H → R as well as G : R × H → R are smooth; in particular, for fixed time t the map φ t : H → H is a smooth symplectomorphism defined on the full Hilbert space H = L 2 (S 1 , C). Furthermore, everything descends to the projective Hilbert space P(H) equipped with the Fubini-Study form.
Proof. Since the convolution of a function u ∈ L 2 (S 1 , C) with the smooth function ψ ensures that the resulting function is smooth, i.e., u * ψ ∈ C ∞ (S 1 , C), it immediately follows that the Hamiltonian F is well-defined and smooth on all of R × H. Furthermore, since and φ 0 t is infinitely often differentiable with respect to t on C ∞ (S 1 , C) ⊂ H, it follows that the same continues to hold for G.
On the other hand, while it immediately follows from the fact that φ 0 t is a unitary linear map that φ 0 t descends to the projectivization P(H) of H, in order to prove the same for the Hamiltonian flows of F (and hence of G), we first introduce the new Hamiltonian function K : H → R given by (half the square of) the Hilbert space norm, K(u) = |u| 2 /2. With this it remains to prove that the Hamiltonian flow of F preserves the Hamiltonian K and vice versa. Since H is perpendicular to u with respect to the real inner product on H = L 2 (S 1 , R 2 ). First observe that the statement is immediately clear if u is truely real (that is, u(x) ∈ R ⊂ C for all x ∈ S 1 ) or truely imaginary, as in this case X F (u) is truely imaginary, or truely real, respectively. For the general case write u = u R + iu I and X F (u) = X F R (u) + iX F I (u) with real-valued functions u R , u I , X F R , X F I . In this case we can use the compatibility of the real L 2 -inner product with the product and convolution of functions to show that u R , In what follows we view the projective Hilbert space as quotient of the unit sphere S(H) in H by the action of U(1) = S 1 , P(H) = S(H)/S 1 . Note that studying the Schrödinger equation on P(H) in place of H is also natural from the view point of quantum physics.

Statement of the main theorem
From now on let us assume that the nonlinear term in the Schrödinger equation is one-periodic in time, that is, Then it follows that every nonlinear Schrödinger equation defines a flow φ t = φ H t on the projective Hilbert space P(H) where the underlying Hamiltonian is one-periodic in time, In the case of time-one-periodic smooth Hamiltonians on finitedimensional projective spaces CP n = P(C n+1 ) we have the following famous Theorem 3.1. ( [7], [14]) The time-one map of a Hamiltonian flow on CP n always has at least n+1 fixed points, that is, the degenerate version of the famous Arnold conjecture holds.
Viewing the Hamiltonian flow φ t on P(H) defined by the nonlinear Schrödinger equation of convolution type as an infinite-dimensional generalization, it is natural to ask whether an analogue of the Arnold conjecture also holds in this infinite dimensional context, establishing the existence of infinitely many fixed points of the time-one map.
But first, in order to show that the generalization to infinite dimensions is not trivial and we can only expect it to hold after imposing restrictions, we first give the following counterexample. Proof. The function L defined by decends to a function on the symplectic quotient P(H) = S(H)/S 1 , since its flow map is given by (φ L t (u))(x) = exp(itV (x)) · u(x) and hence preserves the L 2 -norm. In the same way it can be seen that u ∈ S(H) is a fixed point of φ L 1 on P(H) if and only if there exists some a ∈ R such that for all x ∈ S 1 we have exp(iV (x))·u(x) = exp(ia)·u(x) and hence either (V (x) − a)/2π ∈ Z or u(x) = 0. For a generic choice of the function V : S 1 → R it follows that u(x) = 0 almost everywhere and hence u = 0 ∈ S(H), resulting in the fact that its time-one map on P(H) has no fixed points at all.
Like for the linear Schrödinger equation, in our proof the appearance of the Laplace term in the nonlinear Schrödinger equations turns out to be essential to find infinitely many fixed points. Before we turn to the general case, we first have a look at the free Schrödinger equation where F t = 0. The proof of the following proposition is an easy exercise. Proposition 3.3. After passing to the projectivization, the time-one flow map φ 0 = φ 0 1 of the free Schrödinger equation on P(H) has infinitely many different fixed points u 0 n given by the complex oscillations, t denote the corresponding functions and flows on the projective Hilbert space P(H) = S(H)/S 1 . Furthermore recall that the Hofer norm of the timeperiodic Hamiltonian F t is defined as By generalizing Floer theory to the case of infinite-dimensional symplectic manifolds, in this paper we prove the following infinitedimensional version of the Arnold conjecture. Theorem 3.4. Assume that, after descending to the projectivization, the Hofer norm |||F ||| of the Hamiltonian defining the nonlinearity is smaller than π/4. Then for every fixed point u 0 n , n ≥ 4 of the time-one map φ 0 1 of the free Schrödinger equation there exists a fixed point u 1 n of the time-one map φ 1 of the given nonlinear Schrödinger equation of convolution type, and a Floer curveũ n : R × R → P(L 2 (S 1 , C)) which connects u 0 n and u 1 n . Furthermore all fixed points u 1 n , n ≥ 4 of φ 1 : P(L 2 (S 1 , C)) → P(L 2 (S 1 , C)) are different.
Note that every fixed point u 1 n ∈ P(L 2 (S 1 , C)) corresponds to a weak solution of the nonlinear Schrödinger equation which, after taking the modulus, is periodic with respect to space and time. Since it follows from our proof that we actually have u 1 n ∈ P(C ∞ (S 1 , C)) ⊂ P(L 2 (S 1 , C)), we indeed get the following stronger which are periodic in space and time with respect to the amplitude, and are normalized in the sense that We first explain the statement about the existence of a Floer curve connecting the fixed point u 0 n of the free Schrödinger equation with a fixed point u 1 n of the given Schrödinger equation of convolution type, which we view as a path u 1 n : R → P(H) with u 1 n (t + 1) = φ 0 (u 1 n (t)) and ∂ t u 1 n = X G t (u 1 n ). With this we mean a smooth mapũ =ũ n : R × R → P(H) withũ(·, t + 1) = φ 0 (ũ(·, t)) satisfying the Floer equation 0 =∂ũ − ϕ(s) · ∇G t (ũ), where∂ = ∂ s +i∂ t denotes the standard Cauchy-Riemann operator and ϕ is a smooth cut-off function with ϕ(s) = 0 for s ≤ −1 and ϕ(s) = 1 for s ≥ 0 and slope 0 ≤ ϕ ′ (s) ≤ 2. It connects u 0 n and u 1 n in the sense that there exist two sequences (s ± γ ) of real numbers, s ± γ → ±∞ withũ n (s − γ , ·) → u 0 n ,ũ n (s + γ , ·) → u 1 n as γ → ∞; the latter weaker asymptotic condition is a consequence of the fact that we do not want to assume that the nonlinearity is generic in the sense that all orbits are isolated. Since our proof indeed shows thatũ =ũ n has image in P(C ∞ (S 1 , C)) ⊂ P(L 2 (S 1 , C)), every Floer curve provides us with a strong solutionũ : R × R × R → C of a partial differential equation of perturbed Cauchy-Riemann-Schrödinger type, ∂ sũ + i∂ tũ = −∆ũ + ∂ 1 f (|ũ * ψ| 2 , x, t)(ũ * ψ) * ψ, satisfying periodicity and asymptotic conditions.
We will show below that the Hofer norm is indeed always finite for nonlinearities of convolution type, so that the condition can always be fulfilled after rescaling the function f or, equivalently, rescaling the time or space variables. Furthermore, we do not claim that the bound on the Hofer norm is sharp in any sense; indeed we only want to stress the fact that this is not a perturbative result in the spirit of KAM theory.
On the other hand, in contrast to the case of Floer theory in finite dimensions, we are for the first time faced with the famous small divisor problem that plays an important role in KAM theory, see [5] for the case of the nonlinear Schrödinger equation. Following the proof of proposition 2.1, the complex eigenvalues of the (linear) time-one flow map φ 0 1 = exp(i∆) are given by the sequence exp(in 2 ) ∈ C, n ∈ N. After restricting to a finite-dimensional subspace C 2k+1 ⊂ H and passing to the projectivization, it follows that all fixed points of φ 0 1 are nondegenerate in the sense that one is not an eigenvalue of the linearized return map. On the other hand, after passing to the infinite-dimensional case, the latter is no longer true, as a subsequence of eigenvalues converges to 1. In order to deal with the resulting lack of nondegeneracy of the timeone flow map of the free Schrödinger equation, we will use a deep result from the theory of diophantine approximations, proved using methods from analytic number theory. In essence, we have to use that the space period 2π cannot be approximated too well by rational multiples of the time period 1.
Remark 3.6. Before we turn to the proof of the main theorem, the followings remarks in order: i) While KAM theory for Hamiltonian PDE's has already been studied in a number of papers, the only global (i.e., nonperturbative) results that are known to the author are generalizations of the seminal work of P. Rabinowitz in [11] on the existence of time-periodic solutions of the nonlinear wave equation, see [2] for an overview. In order to avoid the problem with small divisors, Rabinowitz only considers the case where the time period is a natural multiple of the space period. ii) While the result of Rabinowitz hence only holds for very special space periods, our theorem continues to hold when the space period 2π is replaced by any other space period L and the time period 1 is replaced by any other time period T as long as the quotient θ := L 2 /(2π·T ) is still a diophantine real number in the sense that there exist r > 0 and c > 0 such that inf p∈Z |θ−p/q| ≥ c/q r for all q ∈ N. Since the set of diophantine numbers has full Lebesgue measure, it follows that our theorem actually holds true for generic choices for the space and for the time period.
Forgetting for the moment that we are working in the setting of infinite-dimensional symplectic manifolds, following Gromov's existence proof of symplectic fixed points in [8] the idea would be to study moduli spaces of Floer curves (ũ, T ), where T > 0 0 is some non-negative real number andũ =ũ n,T now denotes a smooth map u : R × R → P(H) withũ(·, t + 1) = φ 0 (ũ(·, t)), which now satisfies the asymptotic conditionũ(s, t) → u 0 n as s → ±∞ as well as the Tdependent perturbed Cauchy-Riemann equation Assuming that, as in the case of finite-dimensional projective spaces, one could compactify the above moduli space by just adding broken holomorphic curves corresponding to the case that T converges to +∞, see [4] and the references therein, one would be able to show that for every n ∈ N there exists a Floer curve connecting the fixed point u 0 n of the free Schrödinger equation with a fixed point u 1 n of the given Schrödinger equation of convolution type. In order to see that there are indeed infinitely many different fixed points u 1 n , we will use bounds for their symplectic action.
On the other hand, it is quite apparent that the underlying theory of pseudo-holomorphic curves does not instantly carry over from finite to infinite dimensions. In particular, the non-compactness of the target manifold leads to the fact that Gromov's compactness theorem does not naturally generalize from finite-dimensional projective spaces to P(H). For our proof we will make use of the fact that the Hamiltonian function F on P(H) defining the nonlinearity can be approximated uniformly by finite-dimensional Hamiltonian functions F k on CP 2k ⊂ P(H). Since the existence of Floer curves the idea is to show that a subsequence converges in the C ∞ loc -sense to a Floer curvẽ u =ũ n : R × R → P(H) as in the main theorem. The fact that Gromov-Floer compactness still holds now relies on the following two key observations: First, the bubbling-off argument needed to uniformly bound the derivatives of the sequenceũ k still works. And secondly, although the target manifold is not compact and the time-one map is degenerate in the sense that a sequence of eigenvalues converges to 1, C 0 -convergence of the Floer curves can still be established as long as the infinite-dimensional nonlinearity is better approximated by finite-dimensional ones than that the space period 2π is approximated by rational multiples of the time period 1. For the proof of the latter we use the aforementioned result from number theory. This paper is organized as follows: While in section 4 we show how finite-dimensional symplectic Floer theory can be used to prove our main theorem in the special case of so-called finite-dimensional nonlinearities, in section 5 we discuss finite-dimensional approximations for general infinite-dimensional nonlinearities. In section 6 we show that a corresponding sequence of finite-dimensional Floer curves has bounded derivatives, using a bubbling-off argument in projective Hilbert space. Together with a result from the theory of diophantine approximations, we will show that this will be sufficient to control C 0 -convergence in infinite dimensions in section 7 and finish the proof of our main theorem in section 8.

Floer curves in complex projective spaces
Before we turn to the general case, we first restrict ourselves to the case of finite-dimensional nonlinearities. Based on Floer's proof of the Arnold conjecture in finite dimensions we show Proposition 4.1. Assume the nonlinearity is finite-dimensional in the sense that the support of the Fourier transformψ : Z → C of the smoothing kernel ψ is finite, that is, supp(ψ) ⊂ {−k, . . . , +k} for some natural number k. Then the statement of the main theorem holds.
Identifying the symplectic Hilbert space H with ℓ 2 (C) using the Fourier transform, it follows that G just depends on its value after applying the projection π k : H → C 2k+1 onto the finite-dimensional symplectic subspace C 2k+1 = {û ∈ H :û(n) = 0 for all |n| > k}. In other words we have G = G k := G • π k , so that at every point the gradients ∇G t and X G t are vectors in C 2k+1 ⊂ H. Note that, together with proposition 2.1, this implies that the flow φ t on P(H) restricts to a symplectic flow φ k t on CP 2k .
With this the proof essentially relies on the following existence result of Floer curves in finite-dimensional complex projective spaces. From now on let us assume that the Hofer norm |||F ||| of F t on P(H) is strictly smaller than π/4. Furthermore, let ϕ T : R → R, T > 0 ∪ {0} denote a smooth family of smooth cut-off functions as in section 3, i.e. with ϕ 0 = 0 and ϕ T (s) = 0 for s ≤ −1 and s ≥ 2T + 1, ϕ T (s) = 1 for 0 ≤ s ≤ 2T and slope 0 ≤ ϕ ′ T (s) ≤ 2 for s < 0 and −2 ≤ ϕ ′ T (s) ≤ 0 for s > 0 as long as T ≥ 1. Furthermore the natural Riemannian norm on CP 2k is denoted by | · |. Floer curve, satisfying the periodicity conditionũ(·, t + 1) = φ 0 1 (ũ(·, t)), the asymptotic conditionũ n (s, ·) → u 0 n as s → ±∞, and the perturbed Cauchy-Riemann equation . Furthermore, for the resulting families of mapsũ =ũ n we have that the energy E(ũ n ) defined by Proof. For the proof one observes that, for every k ∈ N, CP 2k is a closed symplectic manifold where the energy of a holomorphic sphere is bounded from below by π. Since, for every n ≤ k, u 0 n ∈ P(H) given by u 0 n (x) = exp(inx) is a fixed point of φ 0 1 in CP 2k , it follows that the existence of the mapũ =ũ k n,T for every T > 0 ∪ {0} can be deduced using the properties of the moduli space of Floer curves (ũ, T ) established in [4] and the references therein; an alternative proof is provided by [3] by translating everything into a statement about holomorphic curves in almost complex manifolds with cylindrical ends.
First we emphasize that an easy computation shows that the standard complex structure satisfies the periodicity condition (φ 0 1 ) * i = i required in [4]. Assuming for the moment that transversality for the Cauchy-Riemann operator∂ G given by∂ G (ũ, T ) =∂ T Gũ holds, it follows that the moduli space of tuples (ũ, T ) is a one-dimensional manifold. Since for T = 0 the constant curveũ k n,0 (s, t) = u 0 n staying over the fixed point u 0 n ∈ CP 2k of φ is the unique solution, the existence of a Floer curveũ k n,T for all T > 0 follows from the Gromov-Floer compactness result. Here we emphasize that bubbling-off of holomorphic spheres is excluded due to fact that the energy E(ũ) is bounded from above by twice the Hofer norm of the Hamiltonian G k , and the Hofer norm of G k is smaller than 1/2 the minimal energy of a holomorphic sphere in CP 2k ; for an analogous statement see proposition 9.1.4 in [10]. Finally, since the Cauchy-Riemann operator∂ G cannot be expected to be transversal, one first has to approximate i by a family of t-dependent al- Then the Gromov-Floer compactness result in [4] can be used again to deduce the existence of Floer curves for ν = 0 from the existence of Floer curves for ν = 0 for all T > 0. In particular, we emphasize that one does not need more elaborate technology like Kuranishi structures or polyfolds to establish the desired properties of the moduli space. Concerning the additional statement, observe that the bound on E(ũ) has already been used to exclude bubbling-off for compactness and it can be found in [10] (for the case of φ being the identity).
Note that, in the case of the free Schrödinger equation, the fixed points u 0 n of the time-one map φ 0 are distinguished by their symplectic action, defined in [4]. For this choose u 0 0 = 1 as reference fixed point and choose for each u 0 n a holomorphic curveũ 0n : R × R → P(H) with u 0n (·, t + 1) = φ 0 (ũ 0n (·, t)),ũ 0n (s, ·) → u 0 n as s → +∞ andũ 0n (s, ·) → u 0 0 as s → −∞ given by a gradient flow line u 0n : R → P(H) of H 0 byũ 0n (s, t) = φ 0 t (u 0n (s)). Then it can be shown that the symplectic action A(u 0 n ) defined in [4] is given by the Hamiltonian H 0 itself, n ) = n 2 /2, in particular, it grows quadratically with n ∈ N. Note that here we use the translation between Hamiltonian Floer theory and Floer theory for general symplectomorphisms. Furthermore we are building on the fact that H 0 restricts to a smooth function on a finite-dimensional projective space CP 2k ⊂ P(ℓ 2 (C)) and every gradient flow line connecting u 0 0 and u 0 n indeed stays inside this CP 2k as long as n ≤ k.
In order to prove that we obtain sufficiently many different fixed points of the time-one map φ 1 of the given nonlinear Schrödinger equation of convolution type in the end, we want to make use of the fact that, just as in the case of the free Schrödinger equation, they also can be distinguished by their symplectic action. This is the content of the following Proposition 4.3. For every s ∈ [0, 2T ] the symplectic action A(ũ n (s)) := A n (ũ n (s)) of the pathũ n (s) =ũ n (s, ·) : R → CP 2k , defined as is well-defined in the sense that it is independent of n ∈ N. Furthermore, for m > n ≥ 4 it holds that |A(ũ m (s)) − A(ũ n (s ′ ))| > 1.
Proof. For the definition of the symplectic action following [4] we use that, after concatenation with the curveũ 0n : R × R → CP 2k used in the definition of the symplectic action A(u 0 n ), the Floer curveũ =ũ n can be used to connectũ n (s) with u 0 0 . For the well-definedness let us assume thatũ m (s) =ũ n (s ′ ). Then one has to show that the concatenated curvesũ 0m #ũ m | (−∞,s] andũ 0n #ũ n | (−∞,s ′ ] are homotopic, that is, the homotopical difference betweenũ m | (−∞,s] and the concatention of a holomorphic curveũ nm : R × R → CP 2k connecting u 0 n with u 0 m given by a gradient flow line of H 0 withũ n | (−∞,s ′ ] represents the zero class in π 2 (CP 2k ). But since every gradient flowline is contractible, it suffices to observe that the energy bound in proposition 4.2 implies that so that the homotopical difference still has to be represented by the zero class in π 2 (CP 2k ). For the last statement note that the same proof as used to establish the energy bound proves the inequality A n (ũ n (s)) − n 2 /2 ≤ 2 · |||G k ||| < π/2. Together with the fact that With this we can show that the main theorem holds for true in the case of finite-dimensional nonlinearities.
Proof. (of proposition 4.1) As T tends to +∞, it follows from the uniform energy bound together with elliptic bootstrapping that the sequenceũ T =ũ k n,T of Floer curves converges in the C ∞ loc | -sense to a smooth mapũ : R × R → CP 2k withũ(·, t + 1) = φ 0 1 (ũ(·, t)). The map u satisfies the Floer equation 0 =∂ũ − ϕ(s) · ∇G t (ũ), where ϕ is now a smooth cut-off function with ϕ(s) = 0 for s ≤ −1 and ϕ(s) = 1 for s ≥ 0. It connects the fixed point u 0 n of the free Schrödinger equation with a fixed point u 1 n of φ 1 in the sense that there exist sequences (s ± γ ) of real numbers, s ± γ → ±∞ withũ(s − γ , ·) → u 0 n andũ(s + γ , ·) → u 1 n as γ → ∞. In order to see the latter, note that by the bound for the energy E(ũ) from proposition 4.2, it follows that for every γ ∈ N there exists γ ≤ |s γ | ≤ 2γ such that By compactness of CP 2k we know, possibly after passing to a subsequence, that the sequenceũ(s ± γ , 0) converges to a fixed point of φ 0 1 or φ 1 , respectively. In order to see thatũ(s − γ , ·) indeed converges to the fixed point u 0 m of φ 0 1 with m = n, it suffices to observe that for m > n ≥ 4 the action difference |A(u 0 m ) − A(u 0 n )| = (m 2 − n 2 )/2 is greater than the uniform energy bound π/2. Further recall that the weaker asymptotic condition is a consequence of the fact that we do not want to assume that the nonlinearity is generic in the sense that all orbits are isolated. In order to see that u 1 m = u 1 n for all m > n ≥ 4, observe that the inequality |A(ũ m,T (s)) − A(ũ n,T (s ′ ))| > 1 for all T > 0 and s, s ′ ∈ [0, 2T ] implies that |A(u 1 m ) − A(u 1 n )| > 1, where the action of u 1 n is defined as andũ n now denotes the Floer curve connecting u 0 n and u 1 n ; furthermore this definition of action is independent of n ∈ N by the same arguments as used in the proof of proposition 4.3.
On the other hand, for m > n it follows that the fixed points u 0 n with n > k of the free Schrödinger equation are also fixed points of φ 1 , where the corresponding Floer curveũ =ũ n is just the constant curvẽ u n,0 (s, t) = u 0 n = u 1 n , (s, t) ∈ R × R. In the case of finite-dimensional nonlinearities we see that the main theorem can be proven by studying Floer curves in finite-dimensional complex projective spaces. In preparation for the general statement, we first show that everything is independent of the chosen ambient finite-dimensional projective space. Proof. Letũ ℓ = π ℓ •ũ : R × R → CP 2ℓ denote the composition of u : R × R → CP 2k with the projection from CP 2k to CP 2ℓ . We claim that, in the case when n ≤ ℓ, that is, u 0 n ∈ CP 2ℓ , we indeed have thatũ has image in CP 2ℓ ⊂ CP 2k and henceũ =ũ ℓ . Sinceũ = u 0 n ∈ CP 2n ⊂ CP 2ℓ for T = 0 andũ continues to be asymptotic to u 0 n for T > 0, we will assume without loss of generality that the Floer curvẽ u =ũ k n,T sits in the image of the coordinate map ϕ n : C 2k → CP 2k around ϕ n (0) = u 0 n ; note also that the symplectomorphism φ 0 1 maps the coordinate neighborhood to itself. With this we claim that we can writeũ as a pair of maps, ⊥ remembers the normal component. Note that else we may still assume that the Floer curveũ sits in a tubular neighborhood of CP 2ℓ in CP 2k , possibly after passing to 0 < T ′ < T ; in that case we can still writeũ as a pair of maps where u ℓ ⊥ is a section in the normal bundle to the image ofũ ℓ , which still remains trivial. Now the important observation is that, since G = G ℓ = G • π ℓ , the projection of ∇G t to C 2k−2ℓ vanishes. This however implies that the perpendicular componentũ ℓ ⊥ is truely holomorphic, that is, solves the unperturbed Cauchy-Riemann equation∂ũ ℓ ⊥ = 0. Sinceũ ℓ ⊥ (s, t) → 0 for s → ±∞ as u 0 n ∈ CP 2ℓ , we can employ Liouville's theorem to show that we haveũ ℓ ⊥ = 0, that is,ũ =ũ ℓ . Note that, instead of referring to Liouville's theorem, the result can be viewed as a consequence of the minimal surface property of pseudo-holomorphic curves. Indeed, in the proof of proposition 7.1 below, we see that the result also follows from the fact the L 2 -norm of ∂ sũ ℓ ⊥ is zero. Remark 4.5. The following observations are immediate: i) By the same arguments it follows that, even if we first allowed the Floer curveũ to live in the infinite-dimensional manifold P(H), the finite-dimensionality of the nonlinearity ensures that it actually lives in the finite-dimensional submanifold CP 2ℓ . ii) Along the same lines using Liouville's theorem or the minimal surface property, it is immediate to see that, in the case of n > ℓ, the Floer curve is constant and the fixed point u 0 n ∈ CP 2n of the free Schrödinger equation thus agrees with the corresponding fixed point u 1 n of the nonlinear Schrödinger equation with convolution term.

From finite to infinite dimensions
For the general case everything furthermore relies on a finitedimensional approximation result of the flow of G t (F t ), where we now crucially make use of the special form of the nonlinearity. To this end, define for the given convolution kernel ψ with Fourier series expansion ψ(x) = (2π) −1/2 +∞ n=−∞ψ (n) exp(inx) for each k ∈ N the approximating kernel and define the resulting sequence of Hamiltonians G k t (F k t ) by for all k ∈ N. Note that F k t then defines a finite-dimensional nonlinearity in the sense of proposition 4.1.
Lemma 5.1. For each k ∈ N the Hamiltonian flow φ G,k t of the Hamiltonian G k t restricts to a finite-dimensional Hamiltonian flow on CP 2k ⊂ P(H). Furthermore the sequence of time-dependent Hamiltonians G k t converges uniformly on P(H) with all derivatives to the original Hamiltonian G t as k → ∞. In particular, the same holds true for the symplectic gradients X G,k and X G , the symplectic time-one maps φ G,k 1 and φ G 1 , and the Hofer norm |||G||| of G t : P(H) → R is finite. Proof. Based on the fact that the flow φ 0 t restricts to finite-dimensional flows by proposition 2.1, for the first statement it suffices to observe that the symplectic gradient X F,k t of F k t : H → R given by has vanishing Fourier coefficients, X F,k t (u)(n) = 0, for |n| > k. Since the supremum norm of u * ψ − u * ψ k can be bounded by 2 → 0 as k → ∞ and u 2 = 1 for all u ∈ S(H) that u * ψ k → u * ψ uniformly as k → ∞. On the other hand, since f is assumed to be smooth, it immediately follows that F k t (u) → F t (u) and hence G k t (u) → G t (u) as k → ∞, uniformly with all derivatives. Since the smoothing kernel ψ is assumed to be smooth, we actually know that ψ k converges exponentially fast to ψ, that is, ψ−ψ k 2 ·k δ → 0 as k → ∞ for every δ ∈ N. But this immediately implies that the sequence of time-dependent Hamiltonians G k t converges not only uniformly but also exponentially fast with all derivatives to the original Hamiltonian G t as k → ∞. This proves Lemma 5.2. Indeed we have that |||G − G k ||| · k δ → 0 and |X G (u) − X G,k (u)| · k δ → 0 as k → ∞ for all δ ∈ N.
In this section we start with the proof of the main theorem.  C)) which connects u 0 n and u 1 n . Furthermore all fixed points u 1 n , n ≥ 4 of φ 1 : P(L 2 (S 1 , C)) → P(L 2 (S 1 , C)) are different.
Until further notice let us fix the natural number n ∈ N. Recall that a Floer curve connecting the fixed point u 0 n of the free Schrödinger equation with a fixed point u 1 n of the given Schrödinger equation of convolution type is a smooth mapũ =ũ n : R × R → P(H) withũ(·, t + 1) = φ 0 (ũ(·, t)) satisfying the Floer equation 0 =∂ũ − ϕ(s) · ∇G t (ũ), where∂ = ∂ s + i∂ t denotes the standard Cauchy-Riemann operator and ϕ is a smooth cut-off function with ϕ(s) = 0 for s ≤ −1 and ϕ(s) = 1 for s ≥ 0. It connects u 0 n and u 1 n in the sense that there exist two sequences (s ± γ ) of real numbers, s ± γ → ±∞ withũ n (s − γ , ·) → u 0 n , u n (s + γ , ·) → u 1 n as γ → ∞; the latter weaker asymptotic condition is a consequence of the fact that we do not want to assume that the nonlinearity is generic in the sense that all orbits are isolated.
As mentioned in section 3, as a starting point we make use of the fact that the infinite-dimensional nonlinearity F can be uniformly approximated by the finite-dimensional Hamiltonians F k , together with the fact that for every finite-dimensional Hamiltonians F k Floer curves are known to exist.
To this end, choose T k := k for all k ∈ N. By proposition 4.2, for every n ≤ k there exists a Floer curveũ k =ũ k n =ũ k n,T k : R × R → CP 2k for the finite-dimensional Hamiltonian F k satisfying the periodicity conditionũ k (·, t + 1) = φ 0 1 (ũ k (·, t)), the asymptotic conditionũ k n (s, ·) → u 0 n as s → ±∞, and the perturbed Cauchy-Riemann equation is bounded by 2|||G k |||, which is strictly less than π/2 for k sufficiently large.
In what follows we are going to show that, possibly after passing to a subsequence, the sequence of Floer curves u k =ũ k n : R × R → CP 2k ⊂ P(H) will converge in the C ∞ loc -sense to a Floer curveũ =ũ n : R × R → P(H) for the infinite-dimensional Hamiltonian F as in the main theorem. Note that with the latter we mean thatũ k converges in the C ∞ loc -sense toũ. Our proof will rely on bubbling-off analysis in infinite dimensions as well as a result from the theory of diophantine approximations in order to deal with the small divisor problem.

Bounded derivatives using bubbling-off analysis
As a first step we prove the following Proposition 6.1. The C α -norm of the mapsũ k =ũ k n,T k is uniformly bounded in k, that is, Since for given n ∈ N every Floer curveũ k is asymptotic to the fixed point u 0 n of φ 0 1 (and for T = 0 is even given by the constant map to u 0 n ), we will assume without loss of generality that every Floer curveũ k has image in the image of the corresponding coordinate chart ϕ n : C 2k → CP 2k with ϕ n (0) = u 0 n . Identifyingũ k with the map ϕ n •ũ k , the above statement is to be understood with respect to the standard norm on C 2k . We stress that the symplectomorphism φ 0 1 maps the coordinate neighborhood to itself and preserves the norm. For the proof we use an infinite-dimensional version of the classical bubbling-off argument from ( [10], chapter 4) together with elliptic regularity from ( [10], appendix B). Here the crucial observation is that the bubbling-off argument can indeed be adapted to the infinite-dimensional setting.
Lemma 6.2. The first derivatives ∂ sũ k (s, t), ∂ tũ k (s, t) can be uniformly bounded in k, that is, Proof. For the proof we use an infinite-dimensional version of the classical bubbling-off argument from ( [10], chapter 4). In order to show that the supremum norm of ∂ sũ k is bounded, we are now essentially going to use that the energy E(ũ k ) ofũ k is strictly smaller than the minimal energy of a holomorphic sphere in CP 2k ⊂ P(H), at least as long as k is sufficiently large. Under the assumption that the gradient explodes, in contrast to the classical proof from finite dimensions, we are not going to prove the existence of a holomorphic sphere in order to derive a contradiction. In order to circumvent the generalization of the underlying Gromov compactness statement to infinite dimensions, we instead show that, when the gradient is large enough, a bubble can be formed by adding in a small disk of diameter smaller than the injectivity radius. The latter will be sufficient to derive a contradiction. Note that, since∂ũ k − ϕ T (s)∇G k t (ũ k ) = 0 and ∇G k → ∇G as k → ∞ due to lemma 5.1, a bound for the supremum norm of ∂ sũ k implies that also the supremum norm of ∂ tũ k is bounded.
To the contrary, assume that, possibly after passing to a subsequence, we have that C k := max{|∂ sũ k (z)| : z ∈ R × R} → ∞ and choose for every k ∈ N a point z k 0 ∈ S 2 such that |∂ sũ k (z k 0 )| = C k . In the proof of this lemma the norm refers to the standard Riemannian metric on CP 2k ; note that establishing a bound for the Riemannian metric on CP 2k establishes a bound in terms of the Euclidean metric on C 2k after applying the coordinate chart ϕ n . Note that the maximum exists, due to the asymptotic condition and we assume without loss of generality that z k 0 = (0, 1/2). As in the classical bubbling-off proof we defineṽ k : B 2 √ C k (0) → CP 2k byṽ k (z) :=ũ k (z/C k + z k 0 ), such that |∂ sṽ k (0)| = 1 and |∂ sṽ k (z)| ≤ 1 for all z ∈ B 2 √ C k (0). For each r ∈ [0, √ C k ] ⊂ R + define γ k r : S 1 → CP 2k by γ k r (θ) :=ṽ k (re iθ ). Denote by ℓ the map which assigns to each loop γ : S 1 → CP 2k its length with respect to the canonical Riemannian metric and E ω (v) := v * ω the symplectic area of a disk map v : B 2 r (0) → CP 2k . With this we can formulate the following Claim: For every k there exists some ρ k ∈ [ √ C k /2, √ C k ] such that ℓ(γ k ρ k ) → 0. Furthermore, for sufficiently large k, we have for the symplectic area of the restricted mapṽ k ρ k =ṽ k : Settingw k (r, θ) =ṽ k (re iθ ) and using the finiteness of the C 1 -norm of G k , an easy computation shows that converges to 0 as C k → ∞. Together with E(ũ k ) ≤ 2|||G k ||| and Cauchy-Schwarz, this implies that for k sufficiently large. In particular, by setting ℓ k min := min{ℓ(γ k r ) : r ∈ [ C k /2, C k ]}, it follows that ℓ k min ≤ 2π 2 /( √ C k /2)) → 0 as k → ∞; in other words, for every ǫ > 0 there exists k 0 ≥ n such that ℓ k min < ǫ if k ≥ k 0 .
For the result on the symplectic area it suffices to observe that E ω (ṽ k ) < 2|||G k ||| < π for k sufficiently large. On the other hand, for the a priori estimate, we just need to observe that For this we make use of a quantative version of the maximum principle.
On the other hand, setting v := ∂ sṽ k , it follows from Stokes' theorem for ǫ > 0 sufficiently small that which together with implies the claim.
In order to finish the proof of the lemma we observe that, due to the fact that the length of γ k ρ k converges to zero, for k sufficiently large there exists a filling diskγ k ρ k : Indeed it is shown in remark 4.4.2 in [10] that, when the length of γ : S 1 → CP 2k is smaller than half of the injectivity radius, the map γ has a canonical local extension to a mapγ from the disk defined using the exponential map; further it is shown there that there exist ℓ max > 0 and c > 0 such that E ω (γ) ≤ cℓ(γ) 2 if ℓ(γ) ≤ ℓ max . Even more important, due to the symmetries of the canonical Riemannian metric on CP 2k (and the fact that the embedding of CP 2ℓ into CP 2k respects the metric for ℓ ≤ k), it follows that the constants ℓ max > 0 and c > 0 are actually independent of the complex dimension 2k.
Proof. (of the proposition) With the help of the above lemma, we can now give the proof of proposition 8.1. As discussed above, in order to keep the setup sufficiently simple, we will assume that u k =ũ k n,T k has image in the image of the corresponding coordinate chart ϕ n : C 2k → CP 2k with ϕ n (0) = u 0 n , and we will identifyũ k with the map ϕ n •ũ k with image in C 2k . For the proof we have to show that the C α -norm ofũ k is bounded uniformly in k ∈ N.
Since from the lemma we know that the maximum norms of ∂ sũ k and ∂ tũ k are bounded (uniformly in k), we already know that ũ k C 1 is bounded. In order to show that ũ k C α is bounded for all α ∈ N, we apply the classical elliptic regularity result, together with lemma 5.1. For this we fix some p > 2 and introduce for every α ≥ 1 the Sobolev H α,p -norm · α,p = · H α,p which assigns toũ k a non-negative real number ũ k α,p ; note that here we restrict the map to R ×[0, 1] ⊂ R × R. By the well-known Sobolev embedding theorem relating the Sobolev H α,p -norms with the C β -norms for different α, β ∈ N, note that for all β ≤ α − 2/p we have p) is independent of the dimension of the target space.
We now prove by induction that ũ k α,p is bounded for all α ≥ 1. For the induction start, note that the bound on ũ k C 1 implies that ũ k H 1,p is bounded. For the induction step, let us assume that ũ k α,p is uniformly bounded in k.
is bounded if and only if the H α,p -norm of ∇G k t (ũ k ) is bounded. Since by lemma 5.1 we have for all α ∈ N that ∇G k C α → ∇G C α as k → ∞, it follows that ∇G k C α is bounded. By the second inequality in ( [10], proposition B.1.7) we have with a constant c 1 > 0 which is independent of the dimension of the target space. Since ũ k α,p is bounded, it follows that the H α,p -norm of ∇G k t (ũ k ) and hence η k α,p is bounded. In order to complete the induction step, we apply the local regularity for the∂-operator in ( [10], theorem B.3.4) in order to obtain ũ k α+1,p ≤ c 2 ∂ũ k α,p + ũ k p , where the constant c 2 > 0 in ([10], theorem B.3.4) is again independent of the dimension of the target space. Together with the boundedness of ∂ũ k α,p = η k α,p and ũ k p ≤ ũ k α,p this proves that ũ k α+1,p is still bounded.

Normal component and the small divisor problem
For some k ≥ ℓ ≥ n letũ k =ũ k n,T k : R × R → CP 2k be the Floer curve from proposition 4.2 and consider CP 2ℓ ⊂ CP 2k . As in the discussion about the case of finite-dimensional nonlinearities, we will assume without loss of generality that the Floer curveũ k sits in the image of the coordinate map ϕ n : C 2k → CP 2k around ϕ n (0) = u 0 n . With this we claim that we can again writeũ k as a pair of maps, ⊥ remembers the normal component. We stress that the symplectomorphism φ 0 = φ 0 1 respects this splitting. Furthermore, a computation in the local coordinates ϕ n shows that its sequence of eigenvalues is given by e i(m 2 −n 2 ) , m ∈ N.
In order to be able to show that the Floer curvesũ k converge in the C ∞ loc -sense to a Floer curveũ : R × R → P(H) as in the main theorem, the key challenge is to be able to control the normal componentũ k,ℓ ⊥ . We emphasize that the proof of the following proposition uses not only the bounds for the derivatives ofũ k obtained using bubbling-off analysis in infinite dimensions, but also crucially relies on a deep result from the theory of diophantine approximations obtained using methods from analytic number theory.
The proof of proposition 7.1 relies on the following lemmata. In what follows we assume, without loss of generality, that the normal component is small, so that the Riemannian metric is sufficiently well approximated by the product metric.
Lemma 7.2. The L 2 -norm of ∂ sũ k,ℓ ⊥ can be bounded from above in terms of the Hofer norm, Proof. The proof of this lemma builds on lemma 8.1.6, remark 8.1.7 and the proof of theorem 9.1.1 in [10]; although they only treat the case where the symplectomorphism φ is the identity, we claim that everything generalizes immediately to the case of φ = φ 0 1 . Introducing the energies E(ũ k ), E(ũ k,ℓ ), E(ũ k,ℓ ⊥ ) to be the L 2 -norms of the corresponding partial derivatives ∂ sũ k , ∂ sũ k,ℓ and ∂ sũ k,ℓ ⊥ after restriction to R ×[0, 1] ⊂ R × R, we clearly have E(ũ k ) = E(ũ k,ℓ ) + E(ũ k,ℓ ⊥ ). On the other hand, following lemma 8.1.6 in [10], we know that t in the sense of ([10], 8.1). Note that the first summands in both (in)equalities are indeed zero due to homotopical reasons and in the second case we indeed just expect an inequality, as in generalũ k,ℓ itself does not satisfy the Floer equation. On the other hand, it is easy to see from the definition of the Hamiltonian curvature that Following remark 8.1.7 and the proof of 9.1.1 in [10], we know that the last expression can be bounded by the Hofer norm of the Hamiltonian curvature of G k,T k − G ℓ,T k , which itself agrees with 2 |||G k − G ℓ |||.
In the case of finite-dimensional nonlinearities with supp(ψ) ⊂ {−ℓ, . . . , +ℓ}, note that, instead of using Liouville's theorem as in section 4, we can alternatively employ the above lemma to prove that E(ũ k,ℓ ⊥ ) = 0 which in turn again immediately impliesũ k,ℓ ⊥ = 0. Lemma 7.3. The supremum norm of ∂ sũ k,ℓ ⊥ can be bounded in terms of its L 2 -norm and the C 1 -norm of the differential Tũ k of the Floer curveũ k , Proof. The proof of this lemma follows from elementary estimates for the integral of the non-negative function f = |∂ sũ ℓ ⊥ | 2 : R × R → R. Note that over the disk of radius f ∞ /(2 T f ∞ ) around any maximum, the integral of f can be bounded from below by f ∞ /2 · R 2 π, which yields that Computing all norms of f in terms of the corresponding norms of ∂ sũ ℓ ⊥ gives the above estimate.
Finally, we now employ a deep result from the theory of diophantine approximations, proven using methods from analytic number theory. In essence we have to use that the irrational number π cannot be approximated too well by rational numbers. Lemma 7.4. For every integer m with |m| > ℓ we have that the norm corresponding componentũ k m (s, t) ofũ k,ℓ ⊥ (s, t) can be bounded by |ũ k m (s, t)| ≤ c · (m 2 − n 2 ) 7 · |φ 0 1 (ũ k,ℓ ⊥ (s, t)) −ũ k,ℓ ⊥ (s, t)| with some fixed constant c > 0.
Proof. First we observe that for every m as above we have where e i(m 2 −n 2 ) is just the corresponding eigenvalue of φ 0 1 . In order to get the above bound for |ũ k m (s, t)|, it hence suffices to bound the first factor away from zero. On the other hand, it is well-known that the sequence of eigenvalues e i(m 2 −n 2 ) , m ∈ N of the time-one map of the free Schrödinger equation φ 0 1 in the coordinates around u 0 n has a subsequence converging to 1, which is known as the small divisor problem. Assuming without loss of generality that e i(m 2 −n 2 ) is close to 1, note that |e i(m 2 −n 2 ) − 1| is approximated by with q = m 2 − n 2 , some constant c > 0 and r > 0 denoting the irrationality measure of (2π) −1 which equals the one of π and is known to be smaller than 8 following [13].
Note that here we crucially use that the infinite-dimensional nonlinearity is better approximated by finite-dimensional ones than that the space period 2π is approximated by rational multiples of the time period 1. On the other hand, in the case when the image ofũ k is not contained in the coordinate chart around u n , then the normal componentũ k,ℓ ⊥ is a section in an appropriate normal bundle to the image ofũ k,ℓ which remains trivial even after the image ofũ k,ℓ is no longer contained in the coordinate chart.

Completing the proof
We show now that the sequence of Floer curvesũ k =ũ k n,T k : R × R → CP 2k ⊂ P(H), k ≥ n converges in the C ∞ loc -sense to a Floer curvẽ u : R × R → P(H) as in the main theorem, possibly after passing to a suitable subsequence. As in the finite-dimensional case, the idea is to make use of elliptic bootstrapping to find a limit in the C ∞ locsense. While we have already proven in proposition 6.1, using bubblingoff analysis in P(H), that all the derivatives ofũ k are bounded as k converges to infinity, note that this is not sufficient to establish the existence of a C ∞ loc -limit due to the non-compactness of P(H). On the other hand, we show below that the result in proposition 7.1 about the normal component, proven using the diophantine approximation result, provides us with the missing piece.
By using a diagonal sequence argument, we know that, after passing to a subsequence, we may assume that the sequence of maps u k,ℓ : R × R → CP 2ℓ converges in the C ∞ loc -sense to a smooth map u ℓ : R × R → CP 2ℓ as k → ∞ for all ℓ ≥ n simultaneously. Note that here we crucially make use from the fact that, for fixed ℓ, the mapsũ k,ℓ have image in the same finite-dimensional manifold CP 2ℓ as k → ∞. Now fix α ∈ N and restrict all maps to a given open subset. For given ǫ > 0, we find ℓ ≥ n such that sup k≥ℓ ũ k,ℓ ⊥ C α < ǫ/3. Sinceũ k,ℓ →ũ ℓ for this given ℓ, we find k 0 ≥ ℓ such that ũ k,ℓ −ũ k ′ ,ℓ C α < ǫ/3 for all k, k ′ ≥ k 0 . But together this gives ũ k −ũ k ′ C α ≤ ũ k,ℓ ⊥ C α + ũ k,ℓ −ũ k ′ ,ℓ C α + ũ k ′ ,ℓ ⊥ C α < ǫ. Note that here we again assume without loss of generality that all Floer curves have image in the coordinate neighborhood around u 0 n . With this we can now finish the proof of the main theorem.
as γ → ∞, the claim follows. Recall that the weaker asymptotic condition is a consequence of the fact that we do not want to assume that the nonlinearity is generic in the sense that all orbits are isolated.
In order to prove that we obtain sufficiently many different fixed points of the time-one map φ 1 of the given nonlinear Schrödinger equation of convolution type in the end, we want to make use of the fact that, just as in the finite-dimensional case, they also can be distinguished by their symplectic action. A literal translation of proposition 4.3 again shows that the symplectic action A(u 1 n ) of the path u 1 n : R → P(H), defined as ũ * 0n ω + ũ * n ω + 1 0 G t (u 1 n (t)) dt, is well-defined in the sense that it is independent of n ∈ N. Furthermore, for m > n ≥ 4 it holds that |A(u 1 m ) − A(u 1 n )| > 1.
Finally, we show how our findings imply the existence of infinitely many different strong solutions of the original Hamiltonian PDE.
Proof. (of the corollary) In order to see that the path u 1 n : R → P(L 2 (S 1 , C)) provides us with a strong solution of the nonlinear Schrödinger equation of convolution type as stated in the corollary, we first choose a lift of u 1 n to a path in S(L 2 (S 1 , C)) ⊂ L 2 (S 1 , C) which is a weak solution of the nonlinear Schrödinger equation. This provides us with a map u : R 2 → C which automatically satisfies the periodicity condition |u(t + 1, x)| = |u(t, x)| = |u(t, x + 2π)| for all (t, x) ∈ R 2 . In order to prove that u is a strong solution of the nonlinear Schrödinger equation, it suffices to prove that u is smooth.
Recall from the proof of proposition 7.1 that for the sequence of Floer curvesũ k : R × R → CP 2k converging to the Floer curveũ : R × R → P(H) we know that sup k≥ℓ ũ k,ℓ ⊥ C 0 · ℓ δ → 0 as ℓ → ∞ for all δ ∈ N, which in turn implies that ũ ℓ ⊥ C 0 · ℓ δ → 0. But with this it follows that for the Fourier coefficients of u(t, ·) : R → C, t ∈ R we have u(t, ·)(m) · m δ → 0 as m → ∞ for all δ ∈ N, which is equivalent to the fact that every point (t, x) ∈ R 2 all partial derivatives ∂ δ x u(t, x), δ ∈ N exist. In order to show that u is also smooth in the t-direction, we observe that by the proof of lemma 8.1 we actually have sup k≥ℓ ũ k,ℓ ⊥ C α · ℓ δ → 0 and hence ũ ℓ ⊥ C α · ℓ δ → 0 as ℓ → ∞ for all (α, δ) ∈ N 2 . But this now implies that for the Fourier coefficients of ∂ α t u(t, ·) : R → C, t ∈ R we have ∂ α t u(t, ·)(m) · m δ → 0 as m → ∞ for all (α, δ) ∈ N, which is equivalent to the fact that every point (t, x) ∈ R 2 all partial derivatives ∂ δ x ∂ α t u(t, x), (α, δ) ∈ N 2 exist.