Schr\"odinger equations with rough Hamiltonians

We consider a class of linear Schr\"odinger equations in R^d with rough Hamiltonian, namely with certain derivatives in the Sj\"ostrand class $M^{\infty,1}$. We prove that the corresponding propagator is bounded on modulation spaces. The present results improve several contributions recently appeared in the literature and can be regarded as the evolution counterpart of the fundamental result of Sj\"ostrand about the boundedness of pseudodifferential operators with symbols in that class. Finally we consider nonlinear perturbations of real-analytic type and we prove local wellposedness of the corresponding initial value problem in certain modulation spaces.


Introduction
It is well-known the the free Schrödinger propagator e it∆ in R d is not bounded on the Lebesgue spaces L p , except for p = 2. This has motivated the study in other function spaces arising in Harmonic Analysis. Among them recently much attention has been given to the so-called modulation spaces. They can be defined similarly to the Besov spaces, but dyadic annuli in the frequency domain are replaced by isometric boxes. Actually, for our purposes it will be more useful an equivalent definition, in terms of the short-time Fourier tranform, or Bargmann transform, of temperate distributions (see [17,18] or Section 2 below, where the weighted variant is considered). Namely, for x, ω ∈ R d , consider the time-frequency shifts The short-time Fourier tranform (STFT) of a temperate distribution f ∈ S ′ (R d ) with respect to a Schwartz function g ∈ S(R d ) is defined as (with the pairing ·, · skew-linear in the second factor). Then, for 1 ≤ p, q ≤ ∞ and 0 ≡ g ∈ S(R d ), we define the modulation spaces x )) < ∞}, and M p (R d ) = M p,p (R d ) (changing window yields equivalent norms). In particular we have M 2 (R d ) = L 2 (R d ), whereas L 2 -based Sobolev spaces can be regarded as weighted modulation spaces as well. In short, the modulation space norm measures the position-momentum (or time-frequency) concentration in phase space of a function. Now, it was proved in [1,44] that the propagator e it∆ is in fact bounded on M p,q (R d ), 1 ≤ p, q ≤ ∞; see also [23,24,25,26]. As shown in [30], in the case of the fractional Laplacian (−∆) κ/2 , with κ > 2, a loss of derivatives instead occurs.
Local well-posedness of the corresponding nonlinear equations, with nonlinearity of power-type, or even entire real-analytic, were also considered in [2,9]. Remarkably, modulation spaces revealed to provide a good framework for the global wellposedness as well, as shown recently in [40,44,41,42] for several dispersive equations. The main results in this connection are now available in the recent book [43] or in the survey [33].
A strictly related issue is the sparsity of the Gabor matrix representation, in phase space, for the corresponding propagator; this property can be in fact considered as a microlocal form of boundedness and implies the boundedness on modulation spaces in the usual sense. However it contains much more refined information which is essential, e.g., for the problem of propagation of singularities; we refer to [6,7,11,12,13,19,20,31,32,38] and the references therein for more detail.
A case of special interest is given by the Schrödinger equation with potential, i.e. D t − ∆ + V (t, x), with V real-valued. The literature about wellposedness in L 2based Sobolev spaces is enormous and we refer e.g. to [15] and the references therein. Concerning the wellposedness in modulation spaces, it was proved in [26] that if V is smooth and has quadratic growth, i.e. ∂ α , for |α| ≥ 1, one has boundedness on every M p,q (R d ), 1 ≤ p, q ≤ ∞. The main motivation of the present paper is an extension of these results in the case of nonsmooth potential, say with minimal regularity. Indeed, our result will apply to more general equations.
Looking for optimal results, one is led to consider potentials with derivatives in the so-called Sjöstrand class, which is noting but the modulation space M ∞,1 (R d ). In fact, in the theory of pseudodifferential operator that function space was first introduced in [36,37] as a natural symbol class with minimal regularity which still gives rise to bounded operators on L 2 (R d ), and also on M p,q (R d ) [18] (similar results hold for certain Fourier integral operators as well [3,4,5,12]). Several deep results in Micolocal Analysis, such as the Fefferman-Phong inequality, keep valid for symbols (with a certain number of derivatives) in M ∞,1 (R 2d ) [21,28,29]. Hence that class looks the natural choice when dealing with boundedness of linear operators in modulation spaces.
Let us now state our results in a simplified form for a model equation; we refer to Section 2 below for general statements, which involve weighted modulation spaces as well.
Denote by M p,q (R d ) and M p (R d ) the closure of the Schwartz space in the corresponding modulation spaces (hence Assume moreover a mild continuity dependence with respect to t (narrow convergence; see Section 2 below) and let 1 ≤ p ≤ ∞.
Then for every (1). Moreover the corresponding propagator is bounded on M p (R d ).
In the case of first order potentials we have a better conclusion. Theorem 1.2. Suppose in addition V 2 ≡ 0 and let 1 ≤ p, q ≤ ∞. Then for every u 0 ∈ M p,q (R d ) there exists a unique solution u ∈ C([0, T ], M p,q (R d )) to (1). Moreover the corresponding propagator is bounded on M p,q (R d ).
It is shown in Remark 5.3 that similar results do not hold if one replaces M ∞ (R d ) by the larger space M ∞ (R d ).
For comparison, observe that , but functions in M ∞,1 generally do not possess any derivative. Hence, the above results represent a significant improvement of those in [26].
Notice that in the special case of 0th order potentials (V 2 = V 1 ≡ 0) more refined results were obtained in [8], where the propagator was shown to be a generalized metaplectic operator; see also [14] for the corresponding problem of propagation of singularities.
Actually, as anticipated, we will consider much more general equations, namely of the form D t u+a w (t, x, D x )u = 0, where a(t, x, ξ) is a second order pseudodifferential operator, whose symbol has some derivatives in M ∞,1 (R 2d ). In this connection our results can be regarded as the evolution counterpart of the boundedness results for pseudodifferential operators with symbols in M ∞,1 (R 2d ) proved in [36,18].
The proof of wellposedness relies on the construction of a parametrix for the forward Cauchy problem, in the spirit of [27,34,35,38,39]. Namely, one decomposes the initial datum in coherent states T x M ω g, i.e. Gabor atoms, where g is a fixed window; then a parametrix is constructed as a generalized localization operator in phase space (a type of operators introduced in their basic form in [16]) that moves the Gabor atoms in phase space according to the Hamiltonian flow, together with a phase shift. Notice that at this low level of regularity the more classical approach via WKB expansions and Fourier integral operators turns out inapplicable.
Finally, we will consider the corresponding nonlinear equations, with a nonlinearity F (u) which is entire real-analytic in C (e.g. a polynomial in u, u) and we will prove that all the above results extend in that setting if the initial datum is in M 1 (R d ) or even in M p,1 (R d ) (in the case of first order potentials), at least for small time.
Briefly, the paper is organized as follows. In Section 2 we recall the main definition and properties of modulation spaces and we state the results in full generality. Section 3 is devoted to some preliminary estimates, whereas in Section 4 we introduce a class of generalized localization operators which will appear subsequently. Section 5 is devoted to the proof of the main results (linear case). Finally Section 6 deals with the extension to nonlinear equations, at least for small time. [22]. The Fourier transform is normalized as

Weyl quantization
and the Weyl quantization of a symbol a(x, ξ) is correspondingly defined as We recall the following easy properties, which can be checked directly: [17,18]. We have already defined in the Introduction the time-frequency shifts and the STFT of a temperate distribution, as well as the unweighted modulation spaces M p,q (R d ). Here we extend the definition in the presence of a weight. We consider a positive submultiplicative even continuous function v in R 2d (v(z + w) v(z)v(w), v(−z) = v(z)); moreover we suppose in the sequel that v satisfies

Modulation spaces
as well as , satisfying in addition the following condition: for every constant C 1 > 0 there exists C 2 > 0 such that, for z, w ∈ R 2d , This implies that m • χ ≍ m is χ : R 2d → R 2d is any invertible transformation, Lipschitz together with its inverse. As prototype one can consider the standard weights v r (z) = z r = (1 + |z| 2 ) r/2 , z ∈ R 2d , for which we have v s ∈ M vr if and only if |s| ≤ r. Now, for 1 ≤ p, q ≤ ∞, m ∈ M v and 0 ≡ g ∈ S(R d ), we define the spaces . We now recall the definition of the narrow convergence. In this connection there are several related definitions in the literature; the present one has first appeared in [36] and is different e.g. from that in [5].
Here 0 ≡ g ∈ S(R 2d ) is a fixed window and the definition is independent of the choice of g, as a consequence of Lemma 3.1 below. It is also clear from the very definition that the set of symbols {a ζ : ζ ∈ Ω} is then bounded in M ∞,1 1⊗v (R 2d ).

Statement of the results.
Let T > 0 be fixed. Consider the Cauchy problem x, ξ ∈ R d , and a specified below. This is our main result.
, where a 2 , a 1 are real-valued and suppose that, for j = 0, 1, 2 and j ≤ |α| ≤ 2j, the map (8). Moreover the corresponding propagator is bounded on M p m (R d ). In the case of symbols that can be written as a sum of symbols depending only on x or ξ less regularity may be assumed. (8). Moreover the corresponding propagator is bounded on M p m (R d ). Theorem 2.4. Under the same assumption as in Theorem 2.3, suppose moreover that (8). Moreover the corresponding propagator is bounded on M p,q m (R d ). Theorem 2.3 applies of course to problems of the form under the hypotheses given there on the potentials V j , j = 0, 1, 2. If V 2 ≡ 0 then Theorem 2.4 applies as well. We can also consider the case of the fractional Laplacian, i.e. (12), the corresponding propagator being bounded on (12), the corresponding propagator being bounded on M p,q m (R d ).

Preliminary estimates
In the sequel we will use the following covariance property of the STFT, which can be verified by direct inspection: We also recall the following pointwise inequality of the short-time Fourier transform [18,Lemma 11.3.3]. It is useful when one needs to change window functions.
The following lemma is proved in [19,Lemma 3.1].
Proof. Clearly it suffices to consider the un-weighted case The second term in the right-hand side can be estimated as Consider now the first term in the right-hand side of (14). It turns out This last expression tends to zero as t → t 0 by the dominated convergence theorem, because a t → a t 0 in S ′ (R 2d ) implies that V g (a t − a t 0 ) → 0 pointwise, whereas the assumption about narrow convergence yields This concludes the proof.
The following result is essentially known. We shall give the proof for the benefit of the reader.
It suffices to prove the first part of the statement. Consider the case of x j (similar arguments apply to D j ). Let g ∈ S(R d ). We have On the other hand, with G(x) = x j g(x) , for every M > 0, where we used (13) and the fact that the STFT of Schwartz functions is Schwartz.
Hence we get which gives the desired boundedness of (6)). Similarly, from (16) we have which gives the boundedness of The following result shows the usefulness of the notion of narrow convergence.
It is well known, see e.g. [19], that the set {a w ζ (x, D)} is bounded in the space of linear continuous operators on M p,q m . Hence it is sufficient to prove the strong continuity on M 1 m , since M 1 m ⊂ M p,q m with inclusion continuous and dense. Now, we can suppose that a ζ → 0, say for ζ → ζ 0 , for the narrow convergence.
Now, we have a w ζ (x, D)π(z)g, π(w)g → 0 pointwise if ζ → ζ 0 . On the other hand it follows from Lemma 3.2 that for a new window Φ ∈ S(R 2d ), with j(z 1 , z 2 ) = (z 2 , −z 1 ), for some H ∈ L 1 v (R 2d ), where for the last inequality we used the hypothesis of narrow continuity of ζ → a ζ . Since we also have mV g f ∈ L 1 by assumption, we get a w ζ (x, D)f M 1 m → 0 from (17) and the dominated convergence theorem. The next results will be used often in the subsequent sections.
1⊗v (R 2d ) be continuous for the narrow convergence, and γ(τ ) be a continuous function of τ ∈ [0, 1]. Then the map is still continuous in M ∞,1 1⊗v (R 2d ) for the narrow convergence.
The same holds true for the map Proof. We prove only the first part of the statement, because the last part follows similarly from an easier argument. The continuity of (t, ζ) → b t,ζ in S ′ (R 2d ) is clear. Let us estimate the STFT of b t,ζ . Let g be a Gaussian function, with g L 2 = 1. We have Using the change-of-window formula in Lemma 3.1 we get We now take the supremum with respect to z and we get, by Young inequality, where we used the estimate (see e.g. [10,Formula (19)]) , by assumption. Hence we obtain (19) sup It suffices to prove that this last expression is in L 1 v . This follows at once from the Fubini-Tonelli theorem, using the following two estimates: it turns out v(τ ·) H * |V g g τ | (v(τ ·)H) * |v(τ ·)V g g τ | (vH) * |v(τ ·)V g g τ |, which follows because v is submultiplicative and satisfies (5), and we also have v(τ ·)V g g τ L 1 (R 2d ) ≤ C, 0 < τ ≤ 1.
Proposition 3.7. Let a(t, ·) satisfy the assumption in Theorem 2.2, and m ∈ M v , 1 ≤ p, q ≤ ∞. Then the operator a w (t, and the map t → a w (t, ·) is strongly continuous on the these spaces.

A class of generalized localization operators
We consider the Hamiltonian flow (x t , ξ t ), as a function of t ∈ [0, T ], x, ξ ∈ R d , given by the solution of Under the assumptions of Theorems 2.2-2.5 we see that the functions ∂ α x,ξ a 2 (t, x, ξ), |α| = 2, are continuous by Proposition 3.3. Moreover they are bounded, because . Hence the solution of the above initial value problem exists globally in time for every initial datum, and the flow is a map of class C 1 in all variables t, x, ξ. The map χ(t, s) : (x s , ξ s ) → (x t , ξ t ) is moreover symplectic.
Further consider the real-valued phase ψ(t, x, ξ) defined by (23) ψ We now introduce the class of operators used in the next Section for the construction of the parametrix (cf. [16,38]).
We first apply the change of variable (y, η) = χ(t, 0)(x, ξ), which has Jacobian = 1, and we call again (x, ξ) for the new variables; we obtains We can write is of course bounded, and the operators U t,s , R t,s are defined as follows. The operator R t,s F = e iψ(t,χ −1 (t,0)·)−iψ(s,χ −1 (t,0)·) F • χ −1 (t, s), F ∈ L p m (R 2d ), is strongly continuous on L p m (R 2d ), 1 ≤ p < ∞, and on the closure of the Schwartz space in L ∞ m (R 2d ); this is straightforward to check. The operator is bounded on L p m (R 2d ), because by the assumption on G and (13) we can dominate its integral kernel by a convolution kernel in To prove the strong continuity of U t,s we can assume F ∈ L 1 v , and the conclusion then follows by the dominated convergence theorem, as in the proof of Proposition 3.5.

Proofs of the main results (Theorems 2.2-2.5)
We first construct a parametrix for the Cauchy problem (8), in the form of generalized localization operator as in the previous section.
Hence we obtain Using repeatedly (3) and (4), the assumption (9) and Proposition 3.6 we can write where a α,β (t, x, ξ, y, η) are a family of symbols in M ∞,1 1⊗v (R 2d ), continuous with respect to the narrow convergence as a function of t, x, ξ.
Hence we get To see that this operator enjoys the properties in (1), by Proposition 4.1 we have to verify that G(t, x, ξ, ·) := b w (t, x, ξ, y, D y )g is continuous as a function of t, x, ξ in S ′ (R d ), which is clear, and that for some function H ∈ L 1 v (R 2d ) independent of t, x, ξ. To this end observe that, using (26), we are left to prove that if c(t, x, ξ, ·, ·) is a family of symbols in M ∞,1 1⊗v (R 2d ), continuous for the narrow convergence with respect to t, x, ξ, it turns out, for g, γ ∈ S(R d ), | c w (t, x, ξ, y, D y )γ, π(w)g | ≤ H(w) First we observe that for K(t, s) as in Theorem 5.1, for every This is a consequence of the contraction mapping theorem applied in the space If v is such a solution, we then set whereS(t, s) is the parametrix in Theorem 5.1. It is then straightforward to check that (28) holds true, as well as the claimed boundedness of the propagator.

Proof of Theorem 2.3.
A carefully inspection of the proof of Theorem 2.2 and the needed preliminary results shows that the only point were we used the condition on ∂ α a 2 (t, ·) for |α| = 3, 4, and ∂ α a 1 (t, ·) for |α| = 2, is to pass to the algebraic expression in (20) to the corresponding quantization in the form (21)where, using (2)-(4), additional derivatives fall on the symbol-and similarly in the proof of Theorem 5.1 (from (25) to (26)). However, when the symbol a(t, x, ξ) has the special form in Theorem 2.3 the factorization at the level of symbol for σ j (t, ξ), V j (t, x) gives a corresponding exact factorization at the level of operators, and therefore the conditions ∂ α a 2 (t, ·) ∈ M ∞,1 1⊗v (R d ), |α| = 2, and ∂ α a 1 (t, ·) ∈ M ∞,1 1⊗v (R d ), |α| = 1, suffice in that case.

Nonlinear Schrödinger equations
In this section we briefly discuss the extension of the above results in presence of a non-linearity of the type F (u), where the function F : C → C is entire realanalytic, with F (0) = 0 (F (z) has a Taylor expansion in z, z, valid in the whole complex plane). In particular we can take a polynomial in z, z.
To avoid repetitions we summarize the results in a unique statement. It is understood that v = v r , 0 ≤ r < κ, in the case of Theorem 2.5. where S(t, s), 0 ≤ s ≤ t ≤ T , is the linear propagator corresponding to initial data at time s. The classical iteration scheme works in X if the following properties are verified: a) S(t, s) is strongly continuous on M 1 s (R d ) for 0 ≤ s ≤ t ≤ T (which also implies a uniform bound for the operator norm with respect to s, t, by the uniform boundedness principle); b) We have F (u) − F (v) X ≤ C u − v X , for u, v ∈ X in every fixed ball. The estimate in b) was proved in [9,Formula (28)] in the case m(x, ξ) = ξ s , s ≥ 0, but the same proof extends to any weight m ∈ M v satisfying m 1, since one just uses the fact that M 1 m is a Banach algebra for pointwise multiplication (the same holds for M p,1 m ).
As far as a) is concerned, it follows from the above linear results that S(t, s) is bounded on M 1 m uniformly with respect to s, t and it is strongly continuous in M 1 m as a function of t, for fixed s. The same holds for s, t exchanged, because the equation is time-reversible. To prove its strong continuity jointly in (t, s) we observe that if s ′ ≤ s ≤ t, u 0 ∈ M 1 m (R d ), S(t, s)u 0 − S(t, s ′ )u 0 M 1 m ≤ C 1 u 0 − S(s, s ′ )u 0 M 1 m . Hence, the map s → S(t, s)u 0 is in fact continuous in M 1 m (R d ) uniformly with respect to t. This yields the strong continuity of S(t, s) on M 1 m , as a function of s, t and concludes the proof.