The nonlinear Schr\"odinger equations with harmonic potential in modulation spaces

We study nonlinear Schr\"odinger $i\partial_tu-Hu=F(u)$ (NLSH) equation associated to harmonic oscillator $H=-\Delta +|x|^2$ in modulation spaces $M^{p,q}.$ When $F(u)= (|x|^{-\gamma}\ast |u|^2)u, $ we prove global well-posedness for (NLSH) in modulation spaces $M^{p,p}(\mathbb R^d)$ $ (1\leq p<2d/(d+\gamma), 0<\gamma<\min \{ 2, d/2\}).$ When $F(u)= (K\ast |u|^{2k})u$ with $K\in \mathcal{F}L^q $ (Fourier-Lebesgue spaces) or $M^{\infty,1}$ (Sj\"ostrand's class) or $M^{1, \infty},$ some local and global well-posedness for (NLSH) are obtained in some modulation spaces. When $F$ is real entire and $F(0)=0$, we prove local well-posedness for (NLSH) in $M^{1,1}.$ As a consequence, we can get local and global well-posedness for (NLSH) in a function spaces$-$which are larger than usual $L^p_s-$Sobolev spaces.


Introduction
We study Cauchy problem for the nonlinear Schrödinger equation with the harmonic oscillator H = −∆ + |x| 2 : x i , and F : C → C is a nonlinearity. Mainly we consider nonlinearity of the Hartree and power type. Specifically, we study (1.1) with the Hartree type nonlinearity F (u) = (K * |u| 2k )u, (1. 2) where * denotes the convolution in R d , k ∈ N, and K is of the following type: (1. 6) where F L q is a Fourier-Lebesgue space, and M ∞,1 (R d ) ⊃ F L 1 (R d ) and L 1 (R d ) ⊂ M 1,∞ (R d ) are modulation spaces (see Definition 2.1 below). The homogeneous kernel of the form (1. 3) is known as Hartree potential. The kernel of the form (1.5) is sometimes called Sjöstrand class (particular modulation space). We also study (1.1) when F is real entire and F (0) = 0 (see Definition 4.5 below). In this case power-type nonliearity is covered, and in particular, when F (u) = −|u| 2 u, equation (1.1) is the well-known Gross-Pitaevskii equation.
The harmonic oscillator (also known as Hermite operator) H is a fundamental operator in quantum physics and in analysis [25]. Equation (1.1) models Bose-Einstein condensates with attractive interparticle interactions under a magnetic trap [8,23,31,16]. The isotropic harmonic potential |x| 2 describes a magnetic field whose role is to confine the movement of particles [8,23,27]. A class of NLS with a "nonlocal" nonlinearity that we call "Hartree type" occurs in the modeling of quantum semiconductor devices (see [11] and the references therein, cf. [13]).
It is known that the free Schrödinger propagator e it∆ is bounded on Lebesgue spaces L p (R d ) if and only if p = 2. Hence, we cannot expect to solve Schrödinger equation on L p −spaces. This has inspired to study in other function spaces (e.g., modulation spaces M p,q ) arising in harmonic analysis. In fact, in contrast to L p −spaces, in [29,2] it is proved that the Schrödinger propagator e it(−∆) α 2 (0 ≤ α ≤ 2) is bounded on M p,q (R d ) for all 1 ≤ p, q ≤ ∞. Local well-posedness results of the corresponding nonlinear equations, with nonlinearity of power-type, or more generic real entire, were obtained in [3,6]. And global well-posedness results, with nonlinearity of power or Hartree type, were obtained in [28,4,29]. We refer to excellent survey [22] and the reference therein for details.
Coming back to harmonic oscillator H, we note that well-posedness for (1.1) with Hartree and power type nonlinearity were obtained in the energy space in [10,9,11,10,12,13,31,23]. Recently Bhimani-Balhara-Thangavelu [7] have proved that Schrödinger propagator associated with the harmonic potential (see (2.2) and Theorems 2.6 below) is bounded on M p,p (R d ) (1 ≤ p < ∞) (cf. [19,15]). However, there is no work so far for nonlinear Schrödinger equation with harmonic potential in modulation spaces. We also note that Cauchy data in modulation spaces are rougher (see Proposition 2.5 below) than L p −Sobolev spaces (see definition in Subsection 2.1 below). Taking these considerations into account, we are inspired study (1.1) in modulation spaces. Specifically, we have following theorem.
Theorem 1.1. Let F (u) and K be given by (1.2) and (1.3) respectively with k = 1, and Then there exists a unique global solution of (1.1) such that We note that up to now cannot know (1.1) is globally well-posed in for s > d (see Proposition 2.5 below). Theorem 1.1 reveals that we can get the global well-posedness for (1.1) with Cauchy data rougher than L p −Sobolev spaces. We do not know whether the range of p in Theorem 1.1 is sharp or not for the Hartree potential. However, if we take potential from Sjöstrand's class, the range of p can be improved. Since modulation spaces enlarges as their exponents are increasing (see Lemma 2.3 (1) below), we can solve (1.1) with Cauchy data in relatively more low regularity spaces. Specifically, we have following theorem.
Theorem 1.2. Let F (u) be given by (1.2) with k = 1. Assume that K is given by (1.5) and . Then there exists a unique global solution of (1.1) such that Now we note that formally the solution of (1.1) satisfies (see for e.g., [11,31,13]) the conservation of mass: Theorem 1.4. Let F (u) be given by (1.2).
Now we briefly mention the mathematical literature. Carles-Mauser-Stimming [12] and Cao-Carles [13] have studied well-posedness for (1.1) with Hartree type nonlinearity. Zhang [31] and Carles [10,9,11] have studied well-posedness for (1.1) with power type nonlinearity. We would like to mention that so far all previous authors have studied (1.1) in the energy space Finally, we note that the existence of solution for (1.1) is shown under very low regularity assumption for the initial data. Specifically, to see how typical Cauchy data Theorems 1.1, 1.2, and 1.3 can handle, see Proposition 2.5 and Lemma 2.3 (2) below. Theorems 1.1, 1.2 1.3 and 1.4 highlights that modulation spaces are a good alternative as compared to Sobolev and Besov spaces to study equation (1.1). And we hope that our results will be useful for the further study (e.g., stability and scattering theory) of equation (1.1). This paper is organized as follows. In Section 2, we introduce notations and preliminaries which will be used in the sequel. In particular, in Subsections 2.1 and 2.2, we introduce L p s −Sobolev spaces and modulation spaces(and their properties) respectively. In Subsection 2.3, we review boundedness of Schrödinger propagator associated with harmonic potential on modulation spaces. As an application of Strichartz's estimates and conservation of mass, we obtain global well-posedness for (1.1) in L 2 (R d ) in Section 3. We shall see this will turn out to be one of the main tools to obtain global well-posedness in modulation spaces. In Subsections 4.1, 4.2, 4.3 and 4.4, we prove Theorems 1.1, 1.2, 1.3 and 1.4 respectively.

Preliminaries
2.1. Notations. The notation A B means A ≤ cB for some constant c > 0. The symbol A 1 ֒→ A 2 denotes the continuous embedding of the topological linear space Then the norm of the space-time Lebesgue space L p (I, L q (R d )) is defined by If there is no confusion, we simply write t,x . The Schwartz class is denoted by S(R d ) (with its usual topology), and the space of tempered distributions is denoted by S ′ (R d ). For x = (x 1 , · · · , x d ), y = (y 1 , · · · , y d ) ∈ R d , we put Then F is a bijection and the inverse Fourier transform is given by The Fourier transform can be uniquely extended to F : The F L p (R d )−norm is denoted by The standard Sobolev spaces W s,p (R d ) (1 < p < ∞, s ≥ 0) have a different character according to whether s is integer or not. Namely, for s integer, they consist of L p −functions with derivatives in L p up to order s, hence coincide with the L p s −Sobolev spaces (also known as Bessel potential spaces), defined for s ∈ R by

Modulation Spaces.
In 1983, Feichtinger [17] introduced a class of Banach spaces, the so called modulation spaces, which allow a measurement of space variable and Fourier transform variable of a function or distribution on R d simultaneously using the short-time Fourier transform(STFT). The STFT of a function f with respect to a window function g ∈ S(R d ) is defined by whenever the integral exists. For x, y ∈ R d the translation operator T x and the modulation operator M y are defined by T In terms of these operators the STFT may be expressed as where f, g denotes the inner product for L 2 functions, or the action of the tempered distribution f on the Schwartz class function g.
Definition 2.1 (modulation spaces). Let 1 ≤ p, q ≤ ∞, and 0 = g ∈ S(R d ). The modulation space M p,q (R d ) is defined to be the space of all tempered distributions f for which the following norm is finite: Next we justify Remark 2.2(2): we shall see how the Fourier-Wigner transform and the STFT are related. We consider the Heisenberg group Let π be the Schrödinger representation of the Heisenberg group which is realized on L 2 (R d ) and explicitly given by and the representation ρ(x, y, e it ) acting on L 2 (R d ) is given by We now write the Fourier-Wigner transform in terms of the STFT: Specifically, we put ρ(x, y)φ(ξ) = e ix·ξ φ(ξ + y), z = (x, y), and we have This useful identity (2.1) reveals that the definition of modulation spaces we have introduced in Remark 2.2(2) and Definition 2.1 is essentially the same.
The following basic properties of modulation spaces are well-known and for the proof we refer to [18,17,30].
Proof. For the proof of statements (1), (2), (3) and (4) which is easy to obtain. The proof of the statement (7) is trivial, indeed, we have f M p,q = f M p,q .
We remark that there is also an equivalent definition of modulation spaces using frequencyuniform decomposition techniques (which is quite similar in the spirit of Besov spaces B p,q (R d )), independently studied by Wang et al. in [29,28], which has turned out to be very fruitful in PDE. Since Besov and Sobolev spaces are widely used PDE, we recall some inclusion relations between Besov, Sobolev and modulation spaces. Besides, this gives us a flavor how far low regularity initial Cauchy data in Theorems 1.1, 1.2, 1.3, and 1.4 one can take. .
The Hermite functions Φ α are eigenfunctions of H with eigenvalues (2|α| + d) where |α| = α 1 +...+α d . Moreover, they form an orthonormal basis for L 2 (R d ). The spectral decomposition of H is then written as where ·, · is the inner product in L 2 (R d ). Given a function m defined and bounded on the set of all natural numbers we can use the spectral theorem to define m(H). The action of m(H) on a function f is given by This operator m(H) is bounded on L 2 (R d ). This follows immediately from the Plancherel theorem for the Hermite expansions as m is bounded. On the other hand, the mere boundedness of m is not sufficient to imply the L p boundedness of m(H) for p = 2 (see [25]).
In the sequel, we make use of some properties of special Hermite functions Φ α,β which are defined as follows. We recall (2.1) and define Then it is well known that these so called special Hermite functions form an orthonormal basis for L 2 (C d ). In particular, we have ([25, Theorem 1.3.5]) We define Schrödinger propagator associated to harmonic oscillator m(H) = e itH , denoted by U(t), by equation (2.2) with m(n) = e itn (n ∈ N, t ∈ R). Next proposition says that U(t) is uniformly bounded on M p,p (R d ). Specifically, we have Proof. Let f ∈ S(R d ). Then we have the Hermite expansion of f as follows: Now using (2.5) and (2.3), we obtain Since {Φ α } forms an orthonormal basis for L 2 (R d ), (2.6) gives Therefore, for m(H) = e itH , we have . In view of (2.1) and (2.7), we have By using polar coordinates z j = r j e iθ j , r j := |z j | ∈ [0, ∞), z j ∈ C and θ j ∈ [0, 2π), we get z α = r α e iα·θ and dz = r 1 r 2 · · · r d dθdr (2.9) where r = (r 1 , · · · , r d ), θ = (θ 1 , · · · , θ d ), dr = dr 1 · · · dr d , dθ = dθ 1 · · · dθ d , |r| = d j=1 r 2 j . By writing the integral over C d = R 2d in polar coordinates in each time-frequency pair and using (2.9), we have By a simple change of variable (θ j − 2t) → θ j , we obtain Combining (2.8), (2.10), (2.11), and Lemma 2.4(4), we have e itH f M p,p = f M p,p for f ∈ M p,p (R d ).
Remark 2.7. In view of (2.6) and (2.7), in Theorem 2.6, we cannot expect to replace M p,p norm by M p,q norm. See also [19].

Global wellposedness in
In this section we prove global well-posedness for (1.1) with Cauchy data in L 2 (R d ). To this end, we need Strichartz estimates, and hence we recall it. (1) There exists C r such that There exists C = C(|I|, q 1 ) (constant) such that for all intervals I ∋ 0, where p ′ i and q ′ i are Hölder conjugates of p i and q i respectively.
We also need to work with the convolution with the Hartree potential |x| −γ , so for the convenience of reader we recall: Proposition 3.3 (Hardy-Littlewood-Sobolev inequality). Assume that 0 < γ < d and 1 < p < q < ∞ with Then the map f → |x| −γ * f is bounded from L p (R d ) to L q (R d ) : Proposition 3.4. Let F (u) and K be given by (1.2) and (1.3) respectively with k = 1, and ). In addition, its L 2 −norm is conserved, and for all admissible pairs (p, q), u ∈ L p loc (R, L q (R d )).
Proof. By Duhamel's formula, we write (1.1) as We introduce the space Then (Y, d) is a complete metric space. Now we show that Φ takes Y (T ) to Y (T ) for some T > 0. We put Note that (q, r) is admissible and Let (q,r) ∈ {(q, r), (∞, 2)}. By Proposition 3.2 and Hölder inequality, we have Since 0 < γ < min{2, d}, by Proposition 3.3, we have t,x . (In the last inequality we have used inclusion relation for the L p spaces on finite measure spaces: t,x . This shows that Φ maps Y (T ) to Y (T ). Next, we show Φ is a contraction. For this, as calculations performed before, first we note that Put δ = 8 4−γ and notice that 1 Now using Hölder inequality, we obtain In view of the identity We start with decomposing the Fourier transform of Hartree potential into Lebesgue spaces: indeed, in view of Proposition 4.1, we have
Proof of Theorem 1.1. By Duhamel's formula, we note that (1.1) can be written in the equivalent form We first prove the local existence on [0, T ) for some T > 0. We consider now the mapping By Minkowski's inequality for integrals, Theorem 2.6, and Lemma 4.8, we obtain M p,p . By Theorem 2.6, and using above inequality, we have We choose a T such that cT M 2 ≤ 1/2, that is, T ≤T ( u 0 M p,q , d, γ) and as a consequence we have that is, J u ∈ B T,M . By Lemma 4.3, and the arguments as before, we obtain Therefore, using the Banach's contraction mapping principle, we conclude that J has a fixed point in B T,M which is a solution of (4.10). Taking Proposition 3.4 into account, to prove Theorem 1.1, it suffices to prove that the modulation space norm u M p,p cannot become unbounded in finite time. In view of (4.1) and to use the Hausdorff-Young inequality we let 1 < d d−γ < q ≤ 2, and we obtain For a given T > 0, h satisfies an estimate of the form provided that 0 ≤ t ≤ T, and where we have used the fact that β ′ is finite. Using the Hölder's inequality we infer that Raising the above estimate to the power β ′ , we find that In view of Gronwall inequality, one may conclude that h ∈ L ∞ ([0, T ]). Since T > 0 is arbitrary, h ∈ L ∞ loc (R), and the proof of Theorem 1.1 follows.

4.2.
Global well-posedness in M p,p for the potential in M ∞,1 (R d ). In this section we prove Theorem 1.2.
Proof. We note that L 1 ⊂ M 1,∞ (see Lemma 2.3 (2)) and using Theorem 2.4, we have This proves the first inequality. In view of (4.9), Theorem 2.4, and exploiting ideas from the first inequality gives the second inequality.
Proof of Theorem 1.2. We note that (1.1) can be written in the equivalent form (4.11) u(·, t) = U(t)u 0 − iA(K * |u| 2 )u where U(t) = e itH and (Av)(t, We first prove the local existence on [0, T ) for some T > 0. We consider now the mapping By Minkowski's inequality for integrals, Theorem 2.6, and Lemma 4.4, we obtain for some universal constant c. By Theorem 2.6 and the above inequality, we have which is the closed ball of radius M, and centered at the origin in C([0, T ], M p,q (R d )). Next, we show that the mapping J takes B T,M into itself for suitable choice of M and small T > 0. Indeed, if we let, M = 2 u 0 M p,p and u ∈ B T,M , it follows that We choose a T such that cT M 2 ≤ 1/2, that is, T ≤T ( u 0 M p,p ) and as a consequence we have that is, J u ∈ B T,M . By Lemma 4.4, and the arguments as before, we obtain Therefore, using Banach's contraction mapping principle, we conclude that J has a fixed point in B T,M which is a solution of (1.1). Indeed, the solution constructed before is global in time: in view of the conservation of L 2 norm, Theorem 2.4, and Lemma 2.3, we have a mn s m t n (4.12) that converges absolutely for every (s, t) ∈ R 2 .
(2) If F is real entire function given by (4.12), then we denote byF the function given by the power series expansioñ Note thatF is real entire if F is real entire. Moreover, as a function on [0, ∞)×[0, ∞), it is monotonically increasing with respect to each of the variables s and t. (1) F (u) M p,1 F ( u M p,1 , u M p,1 ).