RESONANT HAMILTONIAN SYSTEMS ASSOCIATED TO THE ONE-DIMENSIONAL NONLINEAR SCHRÖDINGER EQUATION WITH HARMONIC TRAPPING

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1 ESONANT HAMILTONIAN SYSTEMS ASSOCIATED TO THE ONE-DIMENSIONAL NONLINEA SCHÖDINGE EQUATION WITH HAMONIC TAPPING JAMES FENNELL Abstract. We study two resonant Hamiltonian systems on the phase space L ( C: the quintic one-dimensional continuous resonant equation, and a cubic resonant system that appears as the modified scattering limit of a specific NLS equation. We prove that these systems approximate the dynamics of the quintic and cubic one-dimensional NLS with harmonic trapping in the small data regime on long times scales. We then pursue a thorough study of the dynamics of the resonant systems themselves. Our central finding is that these resonant equations fit into a larger class of Hamiltonian systems that have many striking dynamical features: non-trivial symmetries such as invariance under the Fourier transform and the flow of the linear Schrödinger equation, a robust wellposedness theory, including global wellposedness in L, and an infinite family of orthogonal, explicit stationary wave solutions in the form of the Hermite functions.. Introduction In recent years resonant systems have emerged as extremely useful tools for studying nonlinear Schrödinger equations (NLS. esonant equations have been used to construct solutions of the cubic NLS on T that exhibit large growth of Sobolev norms []. They have appeared as modified scattering limits for a number of equations, including the cubic NLS on T d [], the cubic NLS on d with d 5 and harmonic trapping in all but one direction [], and a coupled cubic NLS system on T [5]. The continuous resonant equation (C was originally shown to approximate the dynamics of small solutions of the two-dimensional cubic NLS on a large torus T L over long times scales (longer than L /ɛ, where ɛ is the size of the initial data []. ecent work has extended this by showing that a whole family of C equations approximate the dynamics of NLS on T d L for arbitrary dimension and arbitrary analytic nonlinearity [8]. The original two-dimensional cubic C equation is the same resonant system that appears in the modified scattering limit in [] for d = 3; it has also been shown to be a small data approximation for the cubic NLS with harmonic trapping set on [7]. One of the principal reasons that resonant systems are useful is that they generally exhibit a large amount of structure. They are often Hamiltonian and usually possess many symmetries, a good wellposedness theory, and an infinite number of orthogonal, explicit solutions. Extensive work has been done on studying such purely dynamical properties of the C equations: starting in the paper that introduced the original two-dimensional cubic equation [], in subsequent works again on this cubic case [6, 7], and a more recent paper on the general case [7]. This research fits into a larger program of studying the dynamics of nonlocal Hamiltonian PDEs; we mention, for example, work on the Szegö equation [5] and the lowest Landau level equation [4]. The two-dimensional cubic C equation has, in particular, been found to have many remarkable dynamical properties. The PDE is symmetric under many non-trivial actions such as the Fourier transform and the linear flow of the Schrödinger equation (with or without harmonic trapping; it is Hamiltonian, and through these symmetries admits a number of conserved quantities. The equation is globally wellposed in L and all higher Sobolev spaces. It has many explicit stationary wave solutions, including all of the Hermite functions and the function / x. All stationary waves that are in L are automatically analytic and exponentially decaying in physical space and Fourier space. The present work was initiated by the question of whether these striking properties also hold for the only other continuous resonant equation that scales like L : the one-dimensional quintic continuous Mathematics Subject Classification. 35Q55; 37K5.

2 ESONANT HAMILTONIAN SYSTEMS resonant equation. Our investigation subsequently broadened to include another one-dimensional resonant equation that is somewhat more physically relevent, and turns out to be the modified scattering limit in [] for d =. Our overall finding is that these Hamiltonian systems do display much of the remarkable dynamical structure of the two-dimensional cubic C. In fact, we are able to show that both systems belong to a large class of Hamiltonian systems on the phase space L ( C, and that each system is this class bears many of the features of L critical C: they have a strong symmetry structure, global wellposedness in L and other Sobolev spaces, and many explicit stationary wave solutions in the form of the Hermite functions. Typical members of the class lack much of the structure of both cubic two-dimensional and quintic one-dimensional C for example, it is not the case that all L stationary waves are analytic but our findings do suggest that a number of the properties of the L critical C equations are generic... Presentation of the equations. The two systems we study in this article are resonant systems corresponding to the nonlinear Schrödinger equation with harmonic trapping, (. iu t u + x u = iu t + Hu = u k u, where the spatial variable is x and k is an integer, so that the nonlinearity is analytic. The cubic k = equation is physically relevant: in this case, (. is the Gross Pitaevskii equation and is a model in the physical theory of Bose Einstein condensates [9]. Let us first see how the resonant equations arise. Looking at the profile v(t = e ith u(t (where e ith is the propagator of the linear equation iu t + Hu =, we find it satisfies iv t = e ( ith e ith v k e ith v. Expressing v(t in the basis of eigenfunctions of the operator H (namely the Hermite functions, the equation on v can be written as, [ k ] ( (. iv t (t = e itl Π nk+ (Π nm v(t(π nk++m v(t Π nk+ v(t, n,...,n k+ Z + m= where Π n v is the projection onto the eigenspace of H corresponding to eigenvalue n +. The phase L in (. is given by L = n n k+ (n k n k+. The resonant terms in the sum in (. are the terms that are not oscillating in time; that is, those satisfying L =. The resonant system corresponding to (. is obtained by considering only the resonant terms; namely, (.3 iw t (t = n,...,n k+ Z + L= Π nk+ [ k m= ( (Π nm w(t(π nk++m w(t Π nk+ v(t We will show in Section that this resonant PDE may be written more compactly in terms of a certain time average of the nonlinearity, (.4 iw t (t = π π/4 π/4 e ish ( e ish w(t k e ish w(t ds. From this expression we are able to infer that the resonant system is the Hamiltonian flow on the phase space L ( C corresponding to the Hamiltonian, (.5 H k+ (f = π/4 (e ith f(x k+ dxdt. π π/4 The overall resonant program is to gain information on the dynamics of solutions to (. by studying the associated resonant system (.4. This program has two, distinct components. The first is to establish approximation results that rigorously demonstrate that solutions of the resonant system well approximate solutions of the full system in certain function spaces and over certain timescales. The second component of the program is to understand the dynamics of the resonant equation itself. One then projects these dynamics back to the original equation through the approximation results. We start the article by proving an approximation result that is valid for all positive integers k. We then analyze the resonant system (.4 in depth for the cubic case, when k =, and the quintic case, when k =. These two cases are particularly significant for separate reasons. ].

3 ESONANT HAMILTONIAN SYSTEMS 3 The cubic case k = is physically relevant, as previously mentioned. In addition, the resonant equation here is exactly the resonant equation obtained in [] as the modified scattering limit of the NLS equation, (.6 iu t u xx u yy + y u = u u, where the space variable is (x, y. Precisely, consider small initial data u (x, y. Suppose that u(x, y, t solves (.6 with initial data (x, y u (x, y. For each fixed x, let w(x, y, t be the solution of the resonant equation (.3 with initial data y u (x, y. Then, lim u(x, y, t e it( xx yy+ y w(x, y, ln(t =, HN ( t where H N is the usual Sobolev space. (This holds for any N 8 so long as the initial data is sufficiently small. In the quintic case, k =, we will prove that the resonant system (.4 is precisely the onedimensional quintic continuous resonant equation. It is the only C equation, other than the original two-dimensional cubic C equation, that scales like L. One of the central motivations of this work is to understand the dynamics of the C system in this important special case... Obtained results.... An approximation theorem. We begin, in Section, by proving the following theorem, which shows that solutions of the resonant equation (.4 well-approximate solutions of the full equation (. on a long time scale. This theorem is essentially a lower dimensional version of Theorem 3. in [7], and our proof follows theirs closely. Theorem (Theorem.3, page. Define the space H s by the norm f H s = H s/ f L ; this is equivalent to the norm x s f L + ξ s f L. Fix s > / and initial data u H s. Let u be a solution of the nonlinear Schrödinger equation with harmonic trapping (. and w a solution of the resonant equation (.4, both corresponding to the initial data u. Suppose that the bounds u(t H s, w(t H s ɛ hold for all t [, T ]. Then for all t [, T ], u(t e ith w(t H s ( t(k + ɛ 4k+ + ɛ k+ exp ( (k + tɛ k. In particular if t ɛ k then u(t e ith w(t H s ɛ k+.... epresentation formulas for the Hamiltonians. Following the approximation result, we focus on studying the resonant system (.4 in the cases k = and k =. ight away we note that the Hamiltonians (.5 for these systems arise from multilinear functionals, in the following way. In the quintic case (k =, so that the nonlinearity in (.4 is order 5 the Hamiltonian is, (.7 H 6 (f = π eith f 6 L = π/4 (e ith f(x 6 dxdt, 6 t L6 x π which arises from the multilinear functional, (.8 E 6 (f, f, f 3, f 4, f 5, f 6 = π/4 (e ith f (e ith f (e ith f 3 (e π ith f 4 (e ith f 5 (e ith f 6 dxdt, π/4 π/4 through H 6 (f = E 6 (f, f, f, f, f, f. The cubic Hamiltonian, corresponding to k =, is, (.9 H 4 (f = π eith f 4 L = π/4 (e ith f(x 4 dxdt, 4 t L4 x π π/4 which is associated to the multilinear functional, (. E 4 (f, f, f 3, f 4 = π/4 (e ith f (e ith f (e π ith f 3 (e ith f 4 dxdt, π/4 through H 4 (f = E 4 (f, f, f, f. The fact that the Hamiltonians can be expressed in terms of multilinear functionals is a nontrivial structural property that guides much of the analysis. The symmetries of the Hamiltonian, its wellposedness theory and the existence of certain stationary wave solutions can

4 ESONANT HAMILTONIAN SYSTEMS 4 all be determined from studying the associated multilinear functional (see Theorems 3.8 and 3. for examples of this in practice. We will prove that Hamilton s equations corresponding to H 6 and H 4 are given by, iu t = T 6 (u, u, u, u, u and iu t = T 4 (u, u, u, respectively, where the multilinear operators T 6 and T 4 are defined implicitly by the formulas, T 6 (f, f, f 3, f 4, f 5, g L = 6E 6 (f, f, f 3, f 4, f 5, g and T 4 (f, f, f 3, g L = 4E 4 (f, f, f 3, g. Hamilton s equations are precisely the resonant equations (.4 in the cases k = and k =. In the study of resonant equations, it has turned out to be fundamental to determine alternative representations for the Hamiltonian H, the associated multilinear functional E and associated multilinear operator T. These alternative representations often reveal structure that is concealed by specific representaitons such as (.7 and (.9. In Sections 4. and 5. we derive numerous representations for E 6 and E 4 respectively. First, for E 6, we find the two formulas, E 6 (f, f, f 3, f 4, f 5, f 6 = (. (e it f (e it f (e it f 3 (e π it f 4 (e it f 5 (e it f 6 dxdt = π 6 f (y f (y f 3 (y 3 f 4 (y 4 f 5 (y 5 f 6 (y 6 δ y+y +y 3=y 4+y 5+y 6 δ y +y +y 3 =y 4 +y 5 +y 6 dy, where, in the first equation, e it denotes the propogator of the linear Schrödinger equation. These representations both show that the quintic Hamiltonian system is the one-dimensional quintic continuous resonant equation [8]. To describe our next representations, we require some notation. For an isometry A : 3 3, let E A be the multilinear functional, (. E A (f, f, f 3, f 4, f 5, f 6 = f ((Ax f ((Ax f 3 ((Ax 3 f 4 (x f 5 (x f 6 (x 3 dx dx dx 3, 3 where (Ax k = Ax, e k. The functional E A is a special case of the type of functional that appears in Brascamp Lieb inequalities [6]. We then have the following representations: for the quintic equation, we prove that, (.3 E 6 (f, f, f 3, f 4, f 5, f 6 = 3π π E (θ (f, f, f 3, f 4, f 5, f 6 dθ, where (θ is the rotation of 3 by θ radians about the axis (,, ; while for the cubic equation, we prove that, (.4 E 4 (f, f, f 3, f 4 = π π E S(θ (G, f, f, G, f 3, f 4 dθ, where G(x = e x /, and S(θ is the rotation of 3 by θ radians about the axis (,, ; The two representations (.3 and (.4 are extremely beneficial for studying H 6 and H 4. They also place the two Hamiltonians in a larger class of Hamiltonians that, we will find, share much of the same structure. This is not obvious: a priori we might expect the Hamiltonians H 6 and H 4 to be quite unalike. The differences in (.3 and (.4 are also of note. The presence of the Gaussians G in (.4 ultimately causes the symmetry group of the cubic equation to be smaller than that of the quintic equation; it is also prevents the cubic equation from having a scaling law, which has consequences for the possible stationary waves we can construct...3. Properties of a class of multilinear functionals. Formulas (.3 and (.4 suggest that one can learn much about the dynamics of the Hamiltonian systems H 6 and H 4 by studying the class of functionals E A. This is precisely what we do in Section 3. Our overall finding is that many of the remarkable properties of the two-dimensional cubic C equation may be found at the level of the functionals E A. By (.3 and (.4, these properties are inherited directly by E 6 and E 4. We will show that the functionals E A have a large group of symmetries, that the associated PDE problem is locally wellposed in every Sobolev space and globally wellposed in L, and that all of the Hermite

5 ESONANT HAMILTONIAN SYSTEMS 5 functions are stationary wave solutions of the associated PDE problem. We emphasize that A is always assumed to be an isometry of 3, as it is in (.3 and (.4. Theorem (Theorems 3. and 3.3, page 4. The functional E A is invariant under the following actions (for any λ: (i Fourier Transform: f k f k. (iv Quadratic modulation: f k e iλx f k. (ii Modulation: f k e iλ f k. (v Schrödinger group: f k e iλ f k. (iii L scaling: f k (x λ / f k (λx. (vi Schrödinger with harmonic trapping group: f k e iλh f k. If, in addition, A satisfies A(,, = (,, (as is the case for (θ in (.3, then E A is invariant under the following actions (for any λ: (vii Linear modulation: f k e iλx f k. (viii Translation: f k f k ( + λ. All of these actions give symmetries for H 6, by (.3. For H 4, the symmetries are not inherited automatically because of the Gaussian terms in (.4, however we find that five of the eight symmetries do hold. Corollary (Theorems 4.8, page 8, and 5.5, page 4. The functional E 6 is invariant under all the actions (i through (viii. The functional E 4 is invariant under the actions (i, (ii, (vi, (vii and (viii. These symmetries, seen purely at the level of E A, are used to generate conserved quantities for the resonant equations, using Noether s Theorem. Corollary (Tables 4., page 8, and 5., page 43. The following are conserved quantities of the resonant equation (.4 in the cases k = and k =, f(x dx, x f(x dx, if (xf(xdx, xf(x + f (x dx. In the quintic case k =, we have the additional conserved quantities, [ixf (x + f(x] f(xdx, xf(x dx, We next examine the L boundedness of E A. f (x dx. Theorem (Theorem 3.6, page 8. There holds the bound E A (f, f, f 3, f 4, f 5, f 6 6 f k L. In particular, H A (f := E A (f, f, f, f, f, f f 6 L. If A is not a signed permutation, there is equality in the Hamiltonian bound if and only if f is a Gaussian. This bound is actually an example of a geometric Brasscamp-Lieb inequality, and the classification of the maximizers is already known []. We prove the inequality and classify the maximizers in our case in a way that appears to be new. The L bound on E A directly gives L bounds for H 6 and H 4 ; indeed, H 6 (f 3π π E (θ (f, f, f, f, f, f dθ 3π f 6 L, and similarly, H 4 (f (/ π f 4 L. There is equality in these only if f is a Gaussian. We find that for H 6 any Gaussian gives equality, whereas for H 4 not every Gaussian does, essentially because of the lack of a scaling law; see Proposition 5.7. Using the representation H 6 (f = (/π e it f L 6 t L 6 x, from (., the L bound on H 6 reads, e it f 6 L 6 t L6 x 3 f 6 L, which is the Strichartz inequality in dimension one. Our work shows that the constant here is the best possible, and that there is equality if and only if f is a Gaussian. These facts were previously determined in [3].

6 ESONANT HAMILTONIAN SYSTEMS 6 We then turn to the PDE problem associated to E A. Given E A, a multilinear operator T A is defined implicitly by, T A (f,..., f 5, g = E A (f, f, f 3, g, f 4, f 5 + E A (f, f, f 3, f 4, g, f 5 + E A (f, f, f 3, f 4, f 5, g. From the representations (.3 and (.4, we find representations for the resonant equations, (.5 iu t = T 6 (u, u, u, u, u = 3π and (.6 iu t = T 4 (u, u, u = π π π T (θ (u, u, u, u, udθ, T S(θ (G, u, u, G, udθ, where (θ and S(θ are the same matrices as in (.3 and (.4. Theorem (Theorem 3.7, page 8. The mutilinear operator T A is bounded from X 5 to X for (i X = L, (ii X = L,σ for any σ, and (iii X = H σ for any σ. This theorem automatically implies analogous bounds for T 6 and T 4, and this leads directly to local wellposedness for the resonant equations in all the spaces in the theorem. By pairing this local wellposedness result with the conservation of the L norm, we get global wellposedness in L. Corollary (Theorems 4., page 3, and 5.9, page 44. Hamilton s equations corresponding to H 6 and H 4 are locally wellposed in X for (i X = L, (ii X = L,σ for any σ, and (iii X = H σ for any σ. They are globally wellposed in L. Finally, we find that the functional E A interacts well with the Hermite functions. Theorem (Theorem 3.9, page. Let {φ n } n= be the Hermite functions. If n +n +n 3 n 4 +n 5 +n 6, then E A (φ n, φ n, φ n3, φ n4, φ n5, φ n6 =. It follows that, for some C and n 6 = n + n + n 3 n 4 n 5. T A (φ n, φ n, φ n3, φ n4, φ n5 = Cφ n6, By using the representations of T 6 and T 4 in (.5 and (.6, and the fact that G = Cφ, we immediately discover that, T 6 (φ n, φ n, φ n, φ n, φ n = C n φ n and T 4 (φ n, φ n, φ n = D n φ n, for some constants C n and D n. This immediately implies that e icnt φ n (x and e idnt φ n (x are explicit solutions of the resonant equations (.5 and (.6 respectively. A solution of the form e iωt ψ(x is a stationary wave solution. Corollary. For every n, φ n (x is a stationary wave of the Hamiltonian systems H 6 and H 4. By letting the symmetries of each of the equations act on φ n we can construct more stationary waves; see (4.37 and ( The quintic Hamiltonian, H 6. The previous subsection outlined results on the quintic Hamiltonian H 6 that all arise from the representation (.7 along with relevant properties of the functional E A. Such results also apply to any composite Hamiltonian of the form, (.7 H(f = φ(ωe A(ω (f,..., fdω, Ω where A(ω is always an isometry and φ is integrable. One of the aspirations of the present work is that other Hamiltonian systems may be cast into the framework of (.7, and that our results on the functional E A may be applied therein. The Hamiltonian H 6, however, has more structure than a generic Hamiltonian of type (.7. In Section 4 we present a number of results that are based on this additional structure and that do not follow simply from analogous properties of E A. We first prove that if a stationary wave is in L, then it is automatically analytic and exponentially decaying in physical space and Fourier space.

7 ESONANT HAMILTONIAN SYSTEMS 7 Theorem (Corollary 4.6, page 33. Suppose that φ L is a stationary wave solution of the quintic resonant equation (.3. Then there is α, β > such that φe αx L and ˆφe βx L. In particular, φ can be extended to an analytic function on the complex plane. We then investigate further boundedness properties of E 6, which lead directly to local wellposedness of Hamilton s equation in the relevant spaces. Our first result is that T 6 is smoothing: it maps Sobolev data to a higher Sobolev space. The second result concerns boundedness in weighted L spaces. Theorem. (i (Theorem 4.7, page 34 For any σ >, there is a δ > and a constant C such that, 5 5 T 6 (f, f, f 3, f 4, f 5 L,σ+δ C f k L,σ and T 6 (f, f, f 3, f 4, f 5 H σ+δ C f k H σ. (ii (Theorem 4.8, page 36 For any s > / there is a constant C such that, 5 T 6 (f, f, f 3, f 4, f 5 L,s C f k L,s. It is expected that item (ii here can be sharpened to show that T 6 is bounded from ( L,/ 5 to L,/ (note these are homogeneous weighted L spaces. As discussed after the proof of Theorem 4.8, this is equivalent to / x being a stationary wave of the quintic Hamiltonian system, which we conjecture...5. The cubic Hamiltonian, H 4. As in the quintic case, we present a number of results on the cubic resonant equation that rely on further structure of H 4 beyond that given by the representation (.6. Again, we prove that stationary waves are analytic and exponentially decaying in physical space and Fourier space once they are in L, and we examine boundedness in weighted L spaces. Our stationary waves theorem, and the broad plan of the proof, are the same as those of the quintic equation, but the technical details are quite different. Theorem (Theorem 5.3, page 48. Suppose that φ L is a stationary wave solution of the cubic resonant equation (.3. Then there is α, β > such that φe αx L and ˆφe βx L. In particular, φ can be extended to an analytic function on the complex plane. Theorem. For any s > / there is a constant C such that, T 4 (f, f, f 3 L,s C 3 f k L,s..3. Plan of the article. In Section we prove the main approximation result. Section 3 is devoted to studying the functionals E A defined in (.. We study the functionals in somewhat more generality than indicated above: we assume the matrix A is an isometry from n to n and, then, that E A takes n inputs. In Section 4 we present results concerning the quintic Hamiltonian system defined by H 6, including the representation formulas and the details of how our findings on E A translate to E 6. The cubic Hamiltonian system is treated in a similar fashion in Section 5. There is one appendix that deals with the technical classification of the maximizers of the L bound on E A..4. Notations and conventions. For x, the Japanese bracket is x = + x. f, g L = f(xg(xdx. The Sobolev space H σ is defined by the norm f H σ = x σ f L. The weighted space L,σ is defined by the norm f L,σ = x σ f L. H = + x is the operator corresponding to the quantum harmonic oscillator. The Fourier transform of f is F(f(ξ = ˆf(ξ = (π / e ixξ f(xdx. With this convention, the map f ˆf is an isometry of L (, and the identity F(F(f(x = f( x holds. We will frequently use the Fourier inversion formula, (.8 (π n e ia w,x φ(wdwdx = ˆφ(awdw = n (π n n/ a φ(. n We set G(x = e x /. For all a >, F (e ax (ξ = a / e ξ a and e ax dx = π/a.

8 ESONANT HAMILTONIAN SYSTEMS 8 A B means there is an absolute constant C such that A CB. A B means A B and B A.. An approximation theorem We begin the article by treating more precisely the derivation of the resonant equation (.4 and then proving the approximation theorem described in the introduction. Before studying the nonlinear problem, we recall some basic properties of the linear problem corresponding to (.. These facts will be used extensively throughout the article. The linear equation corresponding to (. is simply the equation for the quantum harmonic oscillator, (. iu t + Hu = iu t u + x u =, where H = + x. For any initial data u L there is a unique solution to (., which we denote e ith u. An explicit representation of this solution is given by the Mehler formula, (. e ith u (x = e i[(x /+y / cos(t xy]/ sin(t u (ydy. π sin(t (This and other properties of the linear flow may be found in [9]. From this expression we see that the solution is time-periodic with period π. An alternative representation of the solution of (. may be found by examining the Hermite functions {φ n } n=. The Hermite functions are eigenfunctions of H they satisfy Hφ n = (n + φ n and they form an orthonormal basis of L. Each of these functions is a polynomial multiplied by the Gaussian e x / ; for example, φ (x = c e x /, φ (x = c xe x /, φ (x = c ( x e x /, where the constants c n are normalizing constants that ensure φ n L =. Using the eigenfunction property one finds that e ith φ n = e it(n+ φ n. Let Π n u = u, φ n φ n be the orthogonal projection onto the eigenspace spanned by φ n. Given any u L we may expand u (x = n= (Π nu (x, and then find, e ith u (x = e it(n+ (Π n u (x, n= so the flow has a simple description in the Hermite function coordinates. We finally note that the Hermite functions satisfy φ n ( x = ( n φ n (x, as may be infered from the formula φ n (x = c n e x / (d n /dx n e x from [9]. We now turn to the nonlinear problem (.. The linear part of the equation may be absorbed into the nonlinearity by changing variables to the profile v(x, t = e ith u(x, t. The function v satisfies the equation, (.3 iv t = e ith ( e ith v k e ith v := N t (v,..., v, where N t is the (k + multilinear functional, [( k ] (.4 N t (f,..., f k+ = e ith (e ith f m (e ith f k++m (e ith f k+. We expand each of the functions f m in the basis of Hermite functions, ( e ith f m = e ith Π nm f m = e it(nm+ Π nm f m, n,...,n k+ n m= m= and then substitute into (.4. This yields, [( k ] (.5 N t (f,..., f k+ = e ilt Π nk+ (Π nm f m (Π nk++m f k++m Π nk+ f k+, where L = k+ m= n m n k+m+. m= n m=

9 ESONANT HAMILTONIAN SYSTEMS 9 In (.5, when L the associated term in the sum is oscillating, while when L = the associated term is not. The resonant equation arises simply from neglecting the oscillatory terms. Define the multilinear functional T by [( k ] (.6 T (f,..., f k+ = Π nk+ (Π nm f m (Π nk++m f k++m Π nk+ f k+. n,...,n k+ L= The resonant PDE is then given by, m= (.7 iw t = T (w,..., w. Lemma.. The resonant functional T is the time average of the functionals N r over the interval [ π/4, π/4]; that is, (.8 T (f,..., f k+ = π π/4 π/4 N r (f,..., f k+ dr. Proof. We integrate the sum in (.5 over [ π/4, π/4] term by term. If L = nothing changes and we get the associated term in (.6. If L is even then π/4 π/4 eilr dr =, and the term in (.5 is. Finally if L is odd, then either n k+ is even and L n k+ is odd, or n k+ is odd and L n k+ is even. In the first case we have, using the Hermite function property (Π n f( x = ( n (Π n f(x, that, ( k ((Π nm f m ( x((π nk++m f k++m ( x (Π nk+ f k+ ( x m= ( k = ( L n k+ ((Π nm f m (x((π nk++m f k++m (x (Π nk+ f k+ (x, m= and hence the function here is odd. Projecting onto the eigenspace spanned by the even function φ nk+ gives the vector. The associated term in the sum (.5 is thus. In the case when n k+ is even and L n k+ is odd a similar analysis shows that the term in the sum is again. In conclusion, all of terms corresponding to L vanish, while those corresponding to L = are unchanged. By virtue of the lemma the resonant equation can be written as, (.9 iw t = T (w,..., w = π which is precisely (.4. Hamiltonian, π/4 π/4 N r (w(t,..., w(tdr = π π/4 π/4 e irh ( e irh w(t k e irh w(t dr, One can show that the resonant equation is the flow corresponding the H k+ (f = π π/4 π/4 e irh f(x k+ dxdr. The details of this Hamiltonian correspondence a presented in Theorem 4. below. We now prove the approximation theorem. The theorem is essentially a lower dimensional analog of Theorem 3. in [7], and our proof follows theirs closely. The function space in our theorem is, H s = {u L : H s/ u L }, with the norm u H s = H s/ u L. From [6], we have the norm equivalence u H s x s/ u L + ξ s/ û L. This space H s is useful for two reasons: first, if s > /, then the space is an algebra (as a direct consequence of the norm equivalence; and, second, the space interacts well with the linear propagator e ith, as seen in the following Lemma. Lemma.. Fix s. For all u H s and t we have e ith u H s u H s. A general L p version of this lemma appears in [4]; for L, there is the following shorter proof.

10 ESONANT HAMILTONIAN SYSTEMS Proof. First let s be an even non-negative integer. Write u L in the basis of Hermite functions as u = n= a nφ n. Becuase s is a non-negative even integer, for every n we have H s/ φ n = (n+ s/ φ n. This then gives, e ith u H = H s/ a s n e it(n+ φ n a n e it(n+ (n + s/ φ n = = n= n= L = n= a n e it(n+ (n + s/ φ n L (by orthogonality a n (n + s/ φ n L = u H s. n= L The result for general s follows from interpolation. Theorem.3. Fix s > / and initial data u H s. Let u be a solution of the nonlinear Schrödinger equation with harmonic trapping (. and w a solution of the resonant equation (.7, both corresponding to the initial data u. Suppose that the bounds u(t H s, w(t H s ɛ hold for all t [, T ]. Then for all t [, T ], u(t e ith w(t H s ( t(k + ɛ 4k+ + ɛ k+ exp ( (k + tɛ k. In particular if t ɛ k then u(t e ith w(t H s ɛ k+. Proof. Let v(x, t = e ith u(x, t, so that v satisfies the PDE (.3. We note that v(x, = u(x, = u (x. Using the lemma, we find that, (. u(t e ith w(t H s = e ith v(t e ith w(t H s v(t w(t H s. To prove the theorem it therefore suffices to show that v and w are close in H s. Therefore let v and w be solutions of the equations (.3 and (.7 respectively with the same initial data u, (. (. (.3 iv t (t = N t (v(t,..., v(t = e ith ( e ith v(t k e ith v(t, iw t (t = T (w(t,..., w(t = π u (x = v(x, = u(x,. π/4 π/4 e irh ( e irh w(t k e irh w(t dr, Set, (.4 D t (f,..., f k+ = N t (f,..., f k+ T (f,..., f k+ [( k ] (.5 = e itl Π nk+ (Π nm f m (Π nk++m f k++m Π nk+ f k+. n,...,n k+ L From the expressions of the multilinear operators N t and T in (. and (. (or their multilinear versions (.4 and (.8, from Lemma., and from the fact that H s is an algebra, it follows that N t and T are uniformly bounded from (H s k+ to H s. The same holds for D t from (.4. Set φ(t = v(t w(t. Because φ( =, the Duhamel form of the equation on φ is, (.6 iφ(t = t m= [T (v(r,..., v(r T (w(r,..., w(r + D r (v(r,..., v(r] dr

11 ESONANT HAMILTONIAN SYSTEMS We will determine a priori bounds on φ. For the first term in the integrand here, we can expand by multilinearity to find, (.7 T (v(r,..., v(r T (w(r,..., w(r H s (.8 (.9 k m= k m= T (v(r,..., v(r, v(r w(r, w(r,..., w(r }{{}}{{} m times k m times v(r m H s v(r w(r H s w(r k m H s (k + ɛ k v(r w(r H s. For the second term in the integrand in (.6 we need to look more closely at the operator D t. We first observe the identity, e irl = d dr r π r π eiθl dθ, where x is the smallest integer less that x. (ecall from the proof of the first lemma that only even values of L contribute to the sum in (.5. The interval of integration here has length less than. We can then handle the second term in (.6 as follows, t = = [D r (v(r,..., v(r] ds [( t k ] e irl Π nk+ (Π nm v(r(π nk++m v(r Π nk+ v(r dr n,...,n k+ L n,...,n k+ L t Using integration by parts, we have (left hand side = t = n,...,n k+ L + k r t m= ( [( d r k dr π r π eiθl dθ Π nk+ ( r n,...,n k+ L π r π eiθl dθ m= (Π nm v(r(π nk++m v(r Π nk+ v(r ] H s [( k ] d dr Π n k+ (Π nm v(r(π nk++m v(r Π nk+ v(r dr m= ( [( t k ] π t π eiθl dθ Π nk+ (Π nm v(t(π nk++m v(t Π nk+ v(t m= π m= r π D θ(v(r,..., v(r, v r (r, v(r,..., v(r dθdr }{{}}{{} m times k m times ( t + π t π D θ(v(t,..., v(tdθ. Because the interval of integration [ π t ] π, t has length less than, we get, t [D r (v(r,..., v(r] ds t(k + sup (. H s r [,t] t(k + ɛ 4k+ + ɛ k+, ( v(r k H s v r (r H s + v(t k+ H s where in the last line we have used v t H s v k+ H s ɛk+, coming from (.3. dr.

12 ESONANT HAMILTONIAN SYSTEMS Combining the estimates (.9 and (. we get t φ(t H s (k + ɛ k φ(s H sds + t(k + ɛ 4k+ + ɛ k+. Gronwell s inequality then implies that, v(t w(t H s = φ(t H s ( t(k + ɛ 4k+ + ɛ k+ exp ( (k + tɛ k, which with (. gives the theorem. 3. Analysis of a class of multilinear functionals In the introduction we presented two formulas (.7 and (.9 that represent the Hamiltonians H 6 and H 4 in terms of simpler functionals of the form E A, π H 6 (f = (3. E 3π (θ (f, f, f, f, f, fdθ, H 4 (f = E π S(θ (G, f, f, G, f, fdθ. Here, for an isometry A : n n, the n multilinear functional E A is defined by, (3. E A (f,..., f n = f k ((Ax k f n+k (x k dx, n where x n, x k = x, e k and (Ax k = Ax, e k. (In the introduction, and in formulas (3., n is set to 3, but the work in this section is for arbitrary n. As stated in the introduction, it turns out that one can gain significant insight into the dynamics of the systems associated to the Hamiltonians H 6 and H 4 by understanding properties of the functionals E A. This section, therefore, is a general study of this family of functionals. We will examine the symmetries of E A, its boundedness in L and higher Sobolev spaces, and its relationship to the Hermite functions. Our motivation throughout is to relate these findings back to the dynamics of the Hamiltonian systems defined by H 6 and H 4. For this reason we will also present a number of results that relate properties of a generic multilinear functional E(f,..., f n, to the flow induced by the Hamiltonian H(f = E(f,..., f associated to it. Our approach here is abstract, but we consider the abstraction justified for three reasons. First, it is efficient. Once we have proved, for example, L local wellposedness for the partial differential equation induced by (3., it will immediately imply L local wellposedness for the two distinct systems defined by (3.. Second, our approach clarifies which structure in the Hamiltonians (3. is responsible for certain dynamics. As a byproduct, it suggests that many of the remarkable properties of these Hamiltonian systems (large number of symmetries, wellposedness in many spaces, Hermite functions as stationary waves are generic. Third, one might expect that there are other Hamiltonian systems of mathematical or physical interest that can be cast into the framework suggested by the representations in (3.. Our results here would immediately give significant insight into the dynamics of such systems. 3.. The multilinear functional framework. In what follows, plain Latin letters such as E A, T A and H A will denote the specific multilinear functional defined by (3. and objects associated to it. Curly letters E, T, and H will denote a generic multilinear functional, multilinear operator and Hamiltonian respectively. All multilinear functionals take n arguments, are linear in the first n arguments and conjugate linear in the last n arguments, as in (3.. The functional properties of E A depend strongly on the matrix properties of A. In the representations in (3., the matrices involved are all isometries, and we will find that this structural property plays a key role in the analysis. We will therefore assume throughout that the matrix A is an isometry. Definition 3.. (i To each matrix A we associate a multilinear operator T A defined implicitly by the formula, (3.3 T A (f,..., f n, g L = E A (f,..., f n+k, g, f n+k,..., f n. π (ii To each matrix A we associate a function H A defined by H A (f = E A (f,..., f. These definitions are motivated by the following theorem.

13 ESONANT HAMILTONIAN SYSTEMS 3 Theorem 3.. Suppose that a multilinear functional E(f,..., f n has the permutation symmetry, (3.4 E(f,..., f n, f n+,..., f n = E(f n+,..., f n, f,..., f n. Then H(f = E(f,..., f is a real valued function and hence a Hamiltonian on the phase space L ( C. Hamilton s equation of motion is given by iu t (t = T (u(t,..., u(t where the multilinear operator T is defined implicitely by (3.5 T (f,..., f n, g L = E(f,..., f n+k, g, f n+k,..., f n. Proof. First, if (3.4 is satisfied, then H(f = E(f,..., f = E(f,..., f = H(f, and therefore H(f. In order to find Hamilton s equation of motion corresponding to H, we first recall the Hamiltonian phase space structure of L ( C. A symplectic form on L is given by ω(f, g = Im f, g L. Given a Hamiltonian H : L, the symplectic gradient ω H is defined as the unique solution of the equation (3.6 ω( ω H(f, g = d dɛ H(f + ɛg. ɛ= Hamilton s equation is then u t = ω H(u. In the case when H(f = E(f,..., f, we have, by multilinearity, d dɛ H(f + ɛg = d ɛ= dɛ E(f + ɛg,..., f + ɛg ɛ= (3.7 = E(f,..., f, g, f,..., f = e E( f,..., f, g, f,..., f, }{{}}{{}}{{}}{{} k times n k times n+k times n k times where in the last step we used the permutation symmetry (3.4. On the other hand, setting i w H(f = T (f,..., f, we find, (3.8 ω( ω H(f, g = Im it (f,..., f, g = e T (f,..., f, g. By the definition of the symplectic gradient in (3.6, the right hand sides of (3.7 and (3.8 must match for all f and g. By replacing g by ig and using conjugate linearity, we see that this equality condition holding for all g actually implies that, T (f,..., f, g = E( f,..., f, g, f,..., f, }{{}}{{} n+k times n k times which, in polarized from, is precisely (3.5. Finally, Hamilton s equation is iu t = i w H(u = T (u,..., u. For a generic isometry A, the functional E A does not satisfy the permutation symmetry condition (3.4. However, if we define, for example, Ẽ A (f,..., f n = [ ] E A (f,..., f n + E A (f n+,..., f n, f,..., f n, then ẼA does satisfy (3.4, and all the properties of E A we prove below carry over to ẼA. We will not be concerned with this point, because while the functionals E A do not have the permutation symmetry (3.4, the functionals E 6 and E 4 defined in (.8 and (. do. Before presenting general results on E A, we give two concrete examples. These two examples illustrate how different isometries A can give rise to very different partial differential equations iu t = T A (u,..., u.

14 ESONANT HAMILTONIAN SYSTEMS 4 Example 3.. Take n = and let A : be the identity matrix. Then, E A (f, f, f 3, f 4 = f (x f (x f 3 (x f 4 (x dx = f, f 3 L f, f 4 L, and hence H A (f = f 4 L. We calculate, [ [ ] [ ] ] T A (f, f, f 3 (y = f (x f (x f 3 (x δ y=x dx + f (x f 3 (x f (x δ y=x dx = f (y f, f 3 L + f (y f, f 3 L. Hamilton s equation is then iu t = T A (u, u, u = 4u u L, which has a unique solution for initial data u L given by u(x, t = e 4i u L t u (x. ( Example 3.. Take n = again, and let A = be the rotation of by π/4 radians. Then, ( ( x x x + x E A (f, f, f 3, f 4 = f f 3 (x f 4 (x dx. f and, [ ( ( ( ( ] y s y + s s y s + y (3.9 T A (f, f, f 3 (y = f f f 3 (sds + f f f 3 (sds. In this case it is not clear that the general solution of the equation iu t = T A (u, u, u can be written explicitly. However it is still possible to determine many properties of the flow. For example, one may verify by substitution that, for any α >, the functions u(x, t = e i 8π/αt e αx and u(x, t = xe αx, are explicit solutions of iu t = T A (u, u, u (the second solution does not depend on time. These solutions were both produced using Corollary 3. below. 3.. Symmetries of the functional and associated conservation laws. In this subsection we uncover some of the rich symmetry structure of the functional E A. We recall that A is assumed to be an isometry throughout. Theorem 3.. The functional E A is invariant under the Fourier transform, that is, (3. E A ( f,..., f n = E A (f,..., f n. It follows that T A ( f,..., f n (ξ = T A (f,..., f n (ξ. Proof. Because A is an isometry, we have ξ, Ax = A ξ, x for all ξ, x n. Now calculating, E A ( f,..., f n = f k ((Ax k f n+k (x k dx n = (π n = (π n n n ( ( e iξ k(ax k f k (ξ k dξ k e iν kx k f n+k (ν k dν k dx e i A ξ ν,x n f k (ξ k f n+k (ν k dξdνdx, where in the last line we have used ξ, Ax + ν, x = A ξ ν, x. We first change variables y(ξ = A ξ ν, or ξ(y = Ay+Aν. The determinant of this change of variables is because A is an isometry. Performing the change of variables then gives the required identity, E A ( f,..., f n = (π n = n n n n e i y,x f k (Aν k + Ayf n+k (ν k dydνdx f k ((Aν k f n+k (ν k dν = E A (f,..., f n, where in the second equality we used the Fourier inversion identity (.8 with a =.

15 ESONANT HAMILTONIAN SYSTEMS 5 For the operator statement, let g be an arbitrary element of L. Using the definition of T A in (3.3 we have, (3. T A (f,..., f n, ĝ = T A (f,..., f n, g = = The operator identity follows. E A (f,..., f n+k, g, f n+k,..., f n E A ( f,..., f n+k, ĝ, f n+k,..., f n = T A ( f,..., f n, ĝ. Theorem 3.3. The functional E A is invariant under the following actions (for any λ: (i Modulation: f k e iλ f k. (ii L scaling: f k (x λ / f k (λx. (iii Quadratic modulation: f k e iλ x f k. (iv Schrödinger group: f k e iλ f k. (v Schrödinger with harmonic trapping group: f k e iλh f k, where H = + x. If, in addition, A satisfies Ae = e, where e = (,..., n, then E A is invariant under the following actions (for any λ: (vi Linear modulation: f k e iλx f k. (vii Translation: f k f k ( + λ. Proof. (i We have, E A (e iλ f,..., e iλ f n = e iλ f ((Ax k e iλ f n (x k dx = E A (f,..., f n. n (ii Let fk λ(x = λ/ f k (λx. We write out E A and perform the change of variables y = λx (with dy = λ n dx to find, E A (f λ,..., fn λ = λ n f k (λ(ax k f n+k (λx k dx = f((ay k f(x k dx = E A (f,..., f n. n n (iii Because A is an isometry, Ax = x for all x n. Using this, we have, E A (e iλ x f,..., e iλ x f n = e iλ (Ax k f k ((Ax k e iλ x k f n+k (x k dx = n n e iλ Ax e n iλ x f k ((Ax k f n+k (x k dx = E A (f,..., f n. (iv Using the previous part and the invariance of the Hamiltonian under the Fourier transform, we find, E A (e iλ f,..., e iλ f n = E A (e iλ x f,..., e iλ x fn = E A ( f,..., f n = E A (f,..., f n. (v In this part we use t instead of λ, and show invariance of the functional under e ith. First, we note that if n is an integer then e i(π/+nπh f = f (from, for instance, the Mehler formula (.. The t = π/ + nπ case thus follows from Theorem 3.. If t π/ + nπ then we may represent e ith f using the lens transform [4]. This transform relates solutions of the free linear Schrödinger to the linear Schrödinger equation with harmonic trapping. Precisely, there holds, (3. (e ith f k (x = cos(t (e i(tan(t/ f k ( x e ix tan(t/. cos(t

16 ESONANT HAMILTONIAN SYSTEMS 6 We substitute this expression into the functional. Using in turn the symmetries (iii (with λ = tan(t/, (ii (with λ = / cos(t, and (iv (with λ = tan(t/, we determine that, E A (e ith f,..., e ith f n ( = E A (e i(tan(t/ f cos(t ( x,..., cos(t ( x (e i(tan(t/ f n cos(t = E A ((e i(tan(t/ f (x,..., (e i(tan(t/ f n (x = E A (f,..., f n. cos(t (vi In these last two parts we assume that, in addition to being an isometry, the matrix A also satisfies Ae = e for e = (,..., n. We then have, E A (e iλx f,..., e iλx f n = e iλ(ax k f k ((Ax k e iλx k f n+k (x k dx n = e iλ Ax,e e λ x,e n f k ((Ax k f n+k (x k dx = E A (f,..., f n. where in the last step we used Ax, e = x, A e = x, e. (vii This follows immediately from the previous part and the invariance of the functional under the Fourier transform, as in item (iv, noting that the Fourier transform takes x e iλx f(x to ξ f(ξ + λ. The symmetries of the functional E A lead directly to commutator identities for the operator T A. Corollary 3.4. We have the following commuter identities, (3.3 (3.4 e iλq T A (f,..., f n = T A (e iλq f,..., e iλq f n QT A (f,..., f n = T A (f,..., f k, Qf k, f k+,... f n n k=n+ T A (f,..., f k, Qf k, f k+,... f n where Q is any of the following operators. ( For a generic isometry A, Q =, Q = x, Q = and Q = H. ( If in addition Ae = e, where e = (,...,, Q = x, Q = id/dx. Proof. For each of the operators Q, the flow map e iλq is an isometry of L for all λ, and, from Theorem 3.3. For each g L, we thus have, E A (e iλq f,..., e iλq f n = E A (f,..., f n, e iλq T A (f,..., f n, g L = T A (f,..., f n, e iλq g L ( = E A f,..., f n+k, e iλq g, f n+k,..., f n, = ( E A e iλq f,..., e iλq f n+k, g, e iλq f n+k,..., e iλq f n, = T A (e iλq f,..., e iλq f n, g L, which gives (3.3. To get (3.4, differentiate (3.3 with respect to λ and set λ =. In Hamiltonian mechanics, the primary purpose of finding symmetries is to determine conservation laws. These two concepts are linked through Noether s theorem. We have seen, in Theorem 3., that in the present context if a functional E satisfies the permutation symmetry (3.4, then the functional

17 ESONANT HAMILTONIAN SYSTEMS 7 gives rise to a Hamiltonian H and Hamilton s equation of motion is iu t = T (u,..., u. With this Hamiltonian structure, a version of Noether s Theorem applies. Theorem 3.5 (Noether s Theorem. Let E be a multilinear functional that satisfies the permutation symmetry (3.4. Suppose that Q is a self-adjoint operator on L such that (3.5 E(e iλq f,..., e iλq f n = E(f,..., f n. Then the quantity Qf, f L is conserved by the Hamiltonian flow of H(f = E(f,..., f. Proof. We first show that T (f,..., f, Qf. Differentiating equation (3.5 with respect to λ and setting λ = gives, (3.6 E(f,..., f, Qf, f,..., f = E( f,..., f, Qf, f,..., f ; }{{}}{{}}{{}}{{} k times n k times n+k times n k times the sign being determined by the linearity or conjugate linearity of each component. Now using the permutation symmetry (3.4 followed by (3.6 gives, T (f,..., f, Qf = E( f,..., f, Qf, f,..., f = E(f,..., f, Qf, f,..., f }{{}}{{}}{{}}{{} n+k times n k times k times n k times = E( f,..., f, Qf, f,..., f = T (f,..., f, Qf, }{{}}{{} n+k times n k times which shows that T (f,..., f, Qf is real. Then, if if t = T (f,..., f, we have d Qf, f = i [ QT (f,..., f, f Qf, T (f,..., f ] dt = i [ T (f,..., f, Qf Qf, T (f,..., f ] = i Im T (f,..., f, Qf =, so Qf, f is constant. The following table summarizes the relationships between the symmetries of E A described in Theorem 3.3, the associated commuting operators in Corrolary 3.4, and the conserved quantities given by Noether s Theorem. As discussed previously, the functional E A does not automatically satisfy the permutation symmetry (3.4 so Noether s Theorem does not apply directly; however, the Hamiltonian systems defined by H 6 and H 4 do satisfy (3.4 and so will have a number of these conserved quantities as a consequence of symmetries induced by Theorem 3.3. Symmetry e iλq of E Operator Q commuting with T Conserved quantity Qf, f f e iλ f f(x dx f f λ [ixf (x + f(x] f(xdx f e iλ x f x xf(x dx f e iλ f f (x dx f e iλh f H xf(x + f (x dx f e iλx f x x f(x dx f f( + λ id/dx if (xf(xdx 3.3. Boundedness and wellposedness. In this subsection we establish an L bound on E A, and bounds of the form, (3.7 T A (f,..., f n X C f X f n X, for X = L, X = L,σ and X = H σ. We will then show how these bounds imply local existence of solutions to Hamilton s equation iu t = T A (u,..., u in the space X. By employing some of the

18 ESONANT HAMILTONIAN SYSTEMS 8 conservation laws derived in the last section, it is possible to establish global existence in L and other spaces for certain functionals. Theorem 3.6. There holds the bound, (3.8 E A (f,..., f n f k L. In particular we have H A (f f n L. Both of these bounds are sharp. Proof. The proof involves one use of the Cauchy Schwarz inequality. We have, E A (f,..., f n = f k ((Ax k f n+k (x k dx n ( / ( / f k ((Ax k dx f n+k (x k dx. n n In the first integral we perform the change of variables y = Ax. Because A is an isometry, the determinant of this change of variables is, and so by Fubini s Theorem, ( / ( / E A (f,..., f n f k (y k dy f n+k (x k dx = f k L, n n which is the bound for E A. Setting f k = f for all k gives the bound H A (f f L. From the Cauchy Schwarz inequality, we have the equality condition, (3.9 f k ((Ax k = f n+k (x k. By assumption, A is an isometry, so that n (Ax k = Ax = x = n x k. This immediately gives that the Gaussian G(x = e αx satisfies the equality condition (3.9, and hence that H(G = G n L. The inequality (3.8 is thus sharp. In a later subsection we discuss the classification of all functions {f k } n that saturate the multilinear functional inequality (3.8. The two examples on page 4 show that, in general, such a classification will depend on the matrix properties of A. In Example 3., H A (f = f 4 L, and so the bound H A (f f 4 L in Theorem 3.6 is always equality. This is not the case in Example 3.. In Theorem 3., we will find that if A is not a signed permutation matrix namely that there is at least one basis element e k such that Ae k is a linear sum of at least two other e j basis elements then the equality H A (f = f k L holds only if f is a Gaussian. In the meantime, we use the inequality (3.8 to establish bounds for the multilinear operator T A. Corollary 3.7. There holds the bound, n T A (f,..., f n X C n for the following spaces. (i X = L with C n = n; (ii X = L,σ, for any σ with C n = n +σ. (iii X = H σ, for any σ with C n = n +σ. f k n X It is important that the boundedness constant C n is independent of A. This implies that if we have a composite Hamiltonian of the form, φ(λh A(λ (fdλ, Ω then the associated operator will be bounded once φ is integrable. This is precisely how Corollary 3.7 will be applied to the Hamiltonian systems H 6 and H 4.

19 ESONANT HAMILTONIAN SYSTEMS 9 Proof. (i We argue by duality using the implicit representation (3.3. Theorem 3.6 we have T A (f,..., f n, g L k=n+ n k=n+ E A (f,..., f n+m, g, f n+m,..., f n f k L g L = ( n n By the bound on E A from f k L g L This gives the result for X = L. (ii Fix x n. Because A is an isometry we have, for every m, x m x = Ax = n (Ax k. Therefore, for fixed m, there is an integer l such that x m n (Ax l. We then have x m n (Ax l and so, x m n,k m (Ax k x k (Ax m, because in all cases t. In terms of the functional E A, this gives, (3. E A ( f,..., f k, t σ f k, f k,..., f n Now applying this to T A, we have, T A (f,..., f n, g L,σ = T A (f,..., f n, t σ g L = n n +σ ( n σ E A ( t σ f,..., t σ f k, f k, t σ f k,..., t σ f n. k=n+ E A (f,..., f k, t σ g, f k,..., f n k=n+ n +σ n E A ( t σ f,..., t σ f k, t σ g, t σ f k,..., t σ f n f k L,σ g L,σ, which gives the result for X = L,σ. (iii This follows from (ii using the invariance of the operator T A under the Fourier transform. The next proposition shows that, in general, bounds of the form determined in the previous proposition imply local wellposedness. Proposition 3.8. Suppose that T : X m X is a multilinear operator that is bounded on a complete normed vector space X; that is, T (f,..., f m X C T m f k X for some C T. Set T (f = T (f,..., f. For every f X there is a T > and unique local solution to the Cauchy problem for Hamiltonian s equation, (3. if t = T (f, f(t = = f, in the space L ([, T ], X. The time of existence T depends only on f X. Proof. The Duhamel formulation of the Cauchy problem (3. is f(t := (f(t = f + t T (f(sds.