Dispersion for Schrödinger equations

The Schrödinger equation forms the basic principles of quantum mechanics (like that of Newton’s second law in classical mechanics). It also plays an important role in describing waves at an appropriate regime in classical fluid dynamics (e.g., water waves) and plasma physics (e.g., Langmuir’s waves or oscillations in a plasma!). In this quick note, I shall present a few basic properties and classical results for the Schrödinger equations, focusing mainly on the defocusing cubic nonlinear equations

\displaystyle i\partial_t u + \Delta u = |u|^2 u \ \ \ \ \ (1)

on {\mathbb{R}_+ \times \mathbb{R}^d}, {d\ge 1} (also known as the Gross-Pitaevskii equation). These notes are rather introductory and classical (e.g., Tao’s lecture notes), which I’m using as part of my lectures at the summer school that P. T. Nam and I are running this week on “the Mathematics of interacting Bose gases” at VIASM, Hanoi, Vietnam (August 1-5, 2022)!

1. Free Schrödinger equation

Let us first consider the free Schrödinger equation

\displaystyle i\partial_t u + \Delta u =0\ \ \ \ \ (2)

with initial data {u(0,x) =u_0(x)}. Just like the heat equation (only except the “i” factor in front of the time derivative!), the Schrödinger equation is invariant under the parabolic scaling {(t,x) \mapsto (t/\lambda^2, x/\lambda)}, and it’s thus direct to construct the fundamental solution of (2)

\displaystyle G(t,x) = t^{-d/2} e^{i|x|^2/4t}

and therefore the solution to (2) is of the form

\displaystyle u(t,x) = e^{it\Delta} u_0(x) = G(t,x) \star_x u_0(x).

It follows that

\displaystyle \|e^{it\Delta} u_0\|_{L^2_x} = \| u_0\|_{L^2_x} , \qquad \|e^{it\Delta} u_0\|_{L^\infty_x} \le t^{-d/2} \|u_0\|_{L^1_x}.\ \ \ \ \ (3)

The first identity is the conservation of mass, while the second inequality gives dispersive decay of the solution. Roughly speaking, for compactly supported data in the unit ball, the solution is of order {t^{-d/2}} on the ball {\{|x|\ \le t\}} and rapidly decaying in time in the region {\{|x|\ge t\}}. Note that the initial data {u_0(x)} need not to be bounded, but the solution is immediately bounded for any positive time. By interpolation (e.g., the Riesz-Thorin interpolation theorem), we also have

\displaystyle \|e^{it\Delta} u_0\|_{L^p_x} \lesssim t^{-d (1/2-1/p)} \| u_0 \|_{L^{p'}_x}, \ \ \ \ \ (4)

for any {p\in [2,\infty]}. Here and in what follows, {p'} denotes the Sobolev dual index of {p}, that is {1/p+1/p' =1}.

Next, one has the following Strichartz estimates for the free Schrödinger equation, which play an important role in studying the nonlinear Schrödinger (or dispersive) equations.

Theorem 1 (Strichartz, 70′) For {d\ge 1} and any pair {(q,r)\in (2,\infty]\times [2,\infty]} satisfying

\displaystyle \frac 2q + \frac dr = \frac d2 ,\ \ \ \ \ (5)

there holds the estimate

\displaystyle \|e^{it\Delta} u_0\|_{L^q_tL^r_x} \lesssim \| u_0 \|_{L^2_x}, \ \ \ \ \ (6)

and its duality

\displaystyle \| \int_{\mathbb{R}} e^{-it\Delta} F(t) \; dt \|_{L^2_x} \lesssim \| F\|_{L^{q'}_t L^{r'}_x}. \ \ \ \ \ (7)

In addition, there holds

\displaystyle \begin{aligned} \| \int_{\mathbb{R}} e^{i(t-s)\Delta} F(s)\; ds \|_{L^{q_1}_t L^{r_1}_x} \lesssim \|F\|_{L^{q'}_tL^{r'}_x} \end{aligned} \ \ \ \ \ (8)

for any two admissible pairs {(q,r)} and {(q_1,r_1)}.

Observe that the admissibility (5) of indexes {(q,r)} for (6) to hold is consistent with the scaling invariance {(t,x) \mapsto (t/\lambda^2, x/\lambda)} of the Schrödinger equation: namely, {u_\lambda(t,x) = u(t/\lambda^2, x/\lambda)} is a solution iff {u(t,x)} is also a solution. Indeed, {\|u_\lambda(0,\cdot)\|_{L^2_x} = \lambda^{d/2}\|u_0(\cdot)\|_{L^2}}, while {\|u_\lambda\|_{L^q_tL^r_x} = \lambda^{2/q+d/r} \|u\|_{L^q_tL^r_x} } for arbitrary {\lambda}, leading to the constraint (5).

Proof: [Proof of Theorem 1] The proof follows from the standard {TT^*} argument. Indeed, first denote by {\langle \cdot \rangle_x} and {\langle \cdot \rangle_{t,x}} the standard inner product in {L^2(\mathbb{R}^d)} and in {L^2(\mathbb{R}^{1+d})}. By duality, the desired estimate (6) is equivalent to the bound

\displaystyle \langle e^{it \Delta} u_0, F \rangle_{t,x} \lesssim \|u_0\|_{L^2_x} \| F\|_{L^{q'}_t L^{r'}_x}

for any {F \in L^{q'}_t L^{r'}_x}. In addition, we note that

\displaystyle \langle e^{it \Delta} u_0, F \rangle_{t,x} = \langle u_0, e^{-it \Delta} F \rangle_{t,x} = \langle u_0, \int_{\mathbb{R}} e^{-it \Delta} F(t) \; dt \rangle_{x}

in which the last identity was due to the fact that {u_0(x)} is independent of {t}. Therefore, the estimate (6) follows from (7). Next, to prove (7), we write

\displaystyle \begin{aligned} \| \int_{\mathbb{R}} e^{-it\Delta} F(t) \; dt \|_{L^2_x}^2 &= \langle F(t,x) , \int_{\mathbb{R}} e^{i(t-s)\Delta} F(s,x)\; ds \rangle_{t,x} \\ &\lesssim \| F\|_{L^{q'}_t L^{r'}_x} \| \int_{\mathbb{R}} e^{i(t-s)\Delta} F(s)\; ds \|_{L^{q}_t L^{r}_x} \end{aligned}

which gives (7), upon using (8). Finally, we prove (8) for {(q_1,r_1) = (q,r)}. Indeed, using the dispersive estimates (4), we get

\displaystyle \begin{aligned} \| \int_{\mathbb{R}} e^{i(t-s)\Delta} F(s)\; ds \|_{L^{q}_t L^{r}_x} & \lesssim \| \int_{\mathbb{R}} |t-s|^{-d(1/2-1/r)} \|F(s)\|_{L^{r'}_x}\; ds \|_{L^{q}_t} \\ & \lesssim \| | t|^{-d(1/2-1/r)} \star_t \|F(t)\|_{L^{r'}_x} \|_{L^{q}_t} \\ & \lesssim \|F\|_{L^{q'}_tL^{r'}_x} \end{aligned}

in which the last inequality used the Hardy-Littlewood-Sobolev inequality in {\mathbb{R}} with {\gamma = d(1/2-1/r) <1} and { 1 - \gamma = \frac{1}{q'} - \frac1q} which is valid due to (5). The estimate (8) for {(q_1,r_1) \not= (q,r)} now follows from (6) and (7). \Box

2. Well-posedness theory

We now turn to the cubic nonlinear Schrödinger equations (1). Let’s start with the following simple theorem.

Theorem 2 Let {d\ge 1} and {s>d/2}. For any initial data {u_0 \in H^s}, there is a time {T>0} so that the Cauchy problem (1) has a unique solution {u(t,x)} in {C^0_t (0,T;H^s)}.

Proof: The proof is direct. Indeed, from the Duhamel representation,

\displaystyle u(t) = e^{it \Delta } u_0 -i \int_0^t e^{i(t-s)\Delta} N(u(s))\; ds \ \ \ \ \ (9)

where {N(u) = |u|^2 u}, we denote by {\mathcal{T}u} the right hand side, and prove that the map {\mathcal{T}u} maps the ball {\mathcal{B} = \{ \| u\|_{L^\infty_t H^s} \le 2\| u_0\|_{H^s}\}} into itself and it’s contractive, provided that {T} is sufficiently small. Since {s>d/2}, {H^s} is an algebra, and so

\displaystyle \| N(u)\|_{H^s} \le C_0 \| u\|_{H^s}^3.

Hence, noting {[\nabla, e^{it\Delta}]=0} and recalling the first identity in (3), we bound for any {u\in \mathcal{B}} and {t \in (0,T)},

\displaystyle \begin{aligned} \| \mathcal{T} u(t)\|_{H^s} &\le \|u_0\|_{H^s} + C_0\int_0^t \| u(s)\|_{H^s}^3 \; ds \\ &\le \|u_0\|_{H^s} + 8C_0 T\| u_0\|_{H^s}^3 \\ &\le \frac32\|u_0\|_{H^s} \end{aligned}

provided that {T} is sufficiently small so that {8 C_0 T \| u_0\|_{H^s}^2 \le 1/2}. That is, {\mathcal{T}} is a well-defined map from {\mathcal{B}} into itself. The contraction follows similarly. \Box

A natural question is whether the local wellposedness theory holds for less regular initial data. First, one observes that the cubic nonlinear Schrödinger equation (1) is invariant under the scaling

\displaystyle u_\lambda (t,x) = \lambda^{-1} u(t/\lambda^2, x/\lambda).

In particular, the homogenous {\dot H^s} Sobolev norm satisfies {\| u_\lambda\|_{\dot H^s} = \lambda^{-1-s + d/2}\| u\|_{\dot H^s}}. In particular, the {\dot H^s} norm is invariant at {s = s_c}, the critical exponent {s_c = \frac{d}{2} - 1}. One expects that the local wellposedness theory holds in {H^s} for {s>s_c}, while the problem is illposed for {s<s_c}. To complete this section, we give another quick theorem on the local wellposedness theory for data in {H^1} in {\mathbb{R}^3}. Note that {H^1} is no longer an algebra, and the Strichartz estimates developed in the previous section shall be used.

Theorem 3 For any initial data {u_0 \in H^1(\mathbb{R}^3)}, there is a time {T>0} so that the Cauchy problem (1) has a unique solution {u(t,x)} in {C^0_t (0,T;H^1\mathbb{R}^3)}.

Proof: Introduce the Strichartz norms

\displaystyle \begin{aligned} \| u\|_{S^0}: &= \sup_{(q,r) ~\mbox{admissible}} \| u\|_{L^q_t L^r_x} \\ \| u\|_{S^1}: &= \| u\|_{S^0}+ \| \nabla u\|_{S^0} , \end{aligned}\ \ \ \ \ (10)

where {(q,r)} satisfies (5). As in the proof of Theorem 2, it suffices to prove that the map {\mathcal{T}} is well-defined and contractive from the ball

\displaystyle \mathcal{B} = \{ \| u\|_{S^1} \le 2 \| u_0 \|_{H^1} \}

into itself. It’s direct from (6) that {\| e^{it \Delta } u_0 \|_{S^1} \le \| u_0\|_{H^1}}. Let’s check the inhomogeneous integral term. Using (7)(8), for {(q,r)=(4,3)}, we get

\displaystyle \| \int_0^t e^{i(t-s)\Delta} N(u(s))\; ds\|_{S^1} \lesssim \| N(u)\|_{L^{4/3}_t L^{3/2}_x} + \| \nabla N(u)\|_{L^{4/3}_t L^{3/2}_x} .

Note that {|\nabla N(u)| \lesssim |u|^2 |\nabla u|}, and so

\displaystyle \begin{aligned} \| \nabla N(u)\|_{L^{4/3}_t L^{3/2}_x} &\lesssim \| |u|^2 |\nabla u| \|_{L^{4/3}_t L^{3/2}_x} \\ &\lesssim T^{1/4} \| u\|^2_{L^{4}_t L^{12}_x} \| \nabla_x u \|_{L^\infty_t L^2_x} \lesssim T^{1/4} \| u\|_{S^1}^3, \end{aligned}

noting { \| u\|_{L^{4}_t L^{12}_x} \lesssim \| u\|_{L^{4}_t W^{1,3}_x} \le \| u\|_{S^1}}. The theorem thus follows, upon taking {T} sufficiently small as done in the proof of the previous theorem. \Box

Note that the above argument in Theorem 2 also shows that the local smooth solutions can be continued globally in time as long as {\| u\|_{L^1_t H^s}} remains finite. In fact, using { \| |u|^2 u\|_{H^s} \lesssim \| u\|_{L^\infty}^2\| u\|_{H^s}}, one obtains the following refined continuation criterium of solutions in {H^s}:

\displaystyle \int_0^T \| u (t)\|_{L^\infty}^2 \; dt < \infty .\ \ \ \ \ (11)

On the other hand, we have the following quick theorem which gives the global unique solution for {H^1} data in {\mathbb{R}^3}. The same result in fact also holds in {\mathbb{R}^2}. The result makes use of conservation of mass and energy for smooth solutions to (1):

\displaystyle \int_{\mathbb{R}^3}|u(t,x)|^2 \; dx = M_0, \qquad \int_{\mathbb{R}^3} (|\nabla u(t,x)|^2 + \frac14 |u(t,x)|^4)\; dx = E_0\ \ \ \ \ (12)

for all {t \in \mathbb{R}} and for some initial mass and energy constants {M_0,E_0}.

Theorem 4 There is a global in time unique solution in {C_t^0(\mathbb{R}; H^1(\mathbb{R}^3)} from any initial data in {H^1(\mathbb{R}^3)}.

Proof: The proof is again direct. Indeed, from the proof of Theorem 3, we may bound

\displaystyle \begin{aligned} \| \nabla N(u)\|_{L^{4/3}_t L^{3/2}_x} &\lesssim \| u\|^2_{L^{4}_t L^{6}_x} \| \nabla_x u \|_{L^4_t L^3_x} \lesssim \| u\|^2_{L^{4}_t L^{6}_x} \|u\|_{S^1} \end{aligned}

That is, the {H^1} solution can be continued globally in time, provided that { \| u\|_{L^{4}_t L^{6}_x}} remains finite for every finite time, which is in fact valid, since {\| u\|_{L^6_x} \lesssim \| u\|_{H^1_x} \lesssim 1}, using the conservation laws (12). This completes the proof of the theorem. \Box

We stress that Theorem 4 deals with the defocusing NLS and allows arbitrarily large initial data in {H^1}. In the focusing case, the energy may be negative and the global well-posedness theory is only known for small initial data.

3. Scattering theory in {\mathbb{R}^3}

The global unique {H^1} solution constructed in Theorem 4 in fact scatters (i.e. it behaves like a free Schrödinger evolution at the large time). Precisely,

Theorem 5 For initial data {u_0 \in H^1(\mathbb{R}^3)}, there are end states {u_{\pm,\infty} \in H^1(\mathbb{R}^3)} so that the global solution {u(t,x)} to (1) scatters to the free Schrödinger evolution {e^{it \Delta } u_{\pm,\infty}} as {t \rightarrow \pm \infty}. Precisely,

\displaystyle \| u(t) - e^{it \Delta} u_{\pm,\infty} \|_{H^1_x} \rightarrow 0\ \ \ \ \ (13)

as {t\rightarrow \pm \infty}.

Proof: Following Tao’s lecture notes, we work with the Strichartz norms as in (10). By Sobolev embedding, we note that {\| u\|_{L^q_{t,x}} \lesssim \| u\|_{S^1}} for {10/3\le q\le 10}. Now, in view of (9), we set

\displaystyle u_\pm(t) = u_0 -i \int_0^t e^{-is \Delta} N(u(s))\; ds .\ \ \ \ \ (14)

It thus suffices for (13) to prove that the integral term has a strong limit in {H^1}. For this, first using (7), we bound

\displaystyle \begin{aligned} \| \int_0^t e^{-is \Delta} N(u(s))\; ds\|_{H^1_x} &\lesssim \| N(u)\|_{L^{10/7}_t W^{1,10/7}_x} \\ & \lesssim \| u\|^2_{L^5_{t,x}} \| u\|_{L^{10/3}_t W^{1,10/3}_x} \lesssim \| u\|^3_{S^1}, \end{aligned}

uniformly in {t \in \mathbb{R}}. Therefore, the uniform boundedness of {\| u\|_{S^1}} would imply the weak convergence of {u_\pm(t)} in {H^1}. A similar calculation also proves that the sequence is Cauchy in {H^1}, since from the boundedness of {\| u\|_{S^1}} we have {\| u\|_{S^1(t,\infty)\times \mathbb{R}^3} \rightarrow 0} as {t\rightarrow \infty} and the strong convergence thus follows.

It therefore remains to prove {\| u\|_{S^1}\lesssim 1}. We have the following.

Proposition 6 Suppose that {\| u\|_{L^4_{t,x}}\lesssim 1}. Then, {\| u\|_{S^1}\lesssim 1}.

Proof: Take any small {\epsilon>0}. By assumption {\| u\|_{L^4_{t,x}}\lesssim 1}, we can decompose the whole line {\mathbb{R}} into finitely many subintervals {I } so that

\displaystyle \| u\|_{L^4_{t,x} (I\times \mathbb{R}^3)}\le \epsilon.

Recall that {\| u\|_{L^\infty_t L^6_x} \lesssim 1} from the conservation of energy. By interpolation, this yields

\displaystyle \| u\|_{L^{q_\theta}_t L^{r_\theta}_x} \lesssim \| u\|_{L^4_{t,x}}^\theta \| u\|_{L^\infty_tL^6_x}^{1-\theta} \lesssim \epsilon^\theta,\ \ \ \ \ (15)

for any {\theta \in(0,1)}, where {q_\theta = 4/\theta} and {r_\theta = 12/(\theta+2)}. Note that {q_\theta, r_\theta} may not be an admissible Strichartz pair.

Now on each interval {I = [t_0,t_1]}, from the Duhamel (9) and the Strichartz estimates (6)(8), we bound

\displaystyle \begin{aligned} \| u\|_{S^1(I\times \mathbb{R}^3)} &\lesssim \| u(t_0)\|_{H^1_x} + \||u|^2 u\|_{L^{q'}_t L^{r'}_x(I\times \mathbb{R}^3)}+ \||u|^2 \nabla u\|_{L^{q'_1}_t L^{r'_1}_x(I\times \mathbb{R}^3)} \\& \lesssim \| u_0\|_{H^1_x} + \| u\|^2_{L^{q_\theta}_t L^{r_\theta}_x(I\times \mathbb{R}^3)} \| u\|_{L^{q}_t W^{1,r}_x(I\times \mathbb{R}^3)} \end{aligned}

in which {(q,r)} and {(q_1,r_1)} are two admissible Strichartz pairs. The last estimate is only valid when

\displaystyle \frac{2}{q_\theta} + \frac{1}{q} = 1 - \frac{1}{q_1} , \qquad \frac{2}{r_\theta} + \frac{1}{r} = 1 - \frac{1}{r_1} .

Since {(q,r)} and {(q_1,r_1)} are admissible Strichartz pairs and satisfy (5), we must have

\displaystyle \frac{2}{q_\theta} + \frac{3}{r_\theta} = 1

which, together with {q_\theta = 4/\theta} and {r_\theta = 12/(\theta+2)}, implies that {\theta = 2/3} and {(q_\theta, r_\theta) = (6,9/2)}. Finally, we may take {(q,r) = (4,3)} and {(q_1,r_1) = (12/5, 9/2) }. Therefore, together with (15) with {\theta = 2/3}, we have proven

\displaystyle \begin{aligned} \| u\|_{S^1(I\times \mathbb{R}^3)} & \lesssim \| u_0\|_{H^1_x} + \| u\|^2_{L^{6}_t L^{9/2}_x(I\times \mathbb{R}^3)} \| u\|_{L^{4}_t W^{1,3}_x(I\times \mathbb{R}^3)} \\ & \lesssim \| u_0\|_{H^1_x} + \epsilon^{4/3} \| u\|_{S^1(I\times \mathbb{R}^3)} . \end{aligned}

Taking {\epsilon} sufficiently small yields the boundedness of {\| u\|_{S^1(I\times \mathbb{R}^3)} }. Summing over finitely many subintervals gives the proposition. \Box

Finally, we recall the following classical interaction Morawetz estimates:

\displaystyle 4\pi \iint |u(t,x)|^4 \; dxdt + \iiint \frac{|u(t,x)|^4|u(t,y)|^2}{|x-y|} \; dxdydt \le 2 M_0^{3/2} E_0^{1/2}

which in particular gives {\| u\|_{L^4_{t,x}}\lesssim 1}. That is, the assumption of the above proposition is verified, and the theorem thus follows. \Box

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