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
on , (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
with initial data . 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 , and it’s thus direct to construct the fundamental solution of (2)
and therefore the solution to (2) is of the form
It follows that
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 on the ball and rapidly decaying in time in the region . Note that the initial data 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
for any . Here and in what follows, denotes the Sobolev dual index of , that is .
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.
Observe that the admissibility (5) of indexes for (6) to hold is consistent with the scaling invariance of the Schrödinger equation: namely, is a solution iff is also a solution. Indeed, , while for arbitrary , leading to the constraint (5).
Proof: [Proof of Theorem 1] The proof follows from the standard argument. Indeed, first denote by and the standard inner product in and in . By duality, the desired estimate (6) is equivalent to the bound
for any . In addition, we note that
in which the last identity was due to the fact that is independent of . Therefore, the estimate (6) follows from (7). Next, to prove (7), we write
which gives (7), upon using (8). Finally, we prove (8) for . Indeed, using the dispersive estimates (4), we get
in which the last inequality used the Hardy-Littlewood-Sobolev inequality in with and which is valid due to (5). The estimate (8) for now follows from (6) and (7).
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 and . For any initial data , there is a time so that the Cauchy problem (1) has a unique solution in .
Proof: The proof is direct. Indeed, from the Duhamel representation,
where , we denote by the right hand side, and prove that the map maps the ball into itself and it’s contractive, provided that is sufficiently small. Since , is an algebra, and so
Hence, noting and recalling the first identity in (3), we bound for any and ,
provided that is sufficiently small so that . That is, is a well-defined map from into itself. The contraction follows similarly.
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
In particular, the homogenous Sobolev norm satisfies . In particular, the norm is invariant at , the critical exponent . One expects that the local wellposedness theory holds in for , while the problem is illposed for . To complete this section, we give another quick theorem on the local wellposedness theory for data in in . Note that is no longer an algebra, and the Strichartz estimates developed in the previous section shall be used.
Theorem 3 For any initial data , there is a time so that the Cauchy problem (1) has a unique solution in .
Proof: Introduce the Strichartz norms
where satisfies (5). As in the proof of Theorem 2, it suffices to prove that the map is well-defined and contractive from the ball
into itself. It’s direct from (6) that . Let’s check the inhomogeneous integral term. Using (7)–(8), for , we get
Note that , and so
noting . The theorem thus follows, upon taking sufficiently small as done in the proof of the previous theorem.
Note that the above argument in Theorem 2 also shows that the local smooth solutions can be continued globally in time as long as remains finite. In fact, using , one obtains the following refined continuation criterium of solutions in :
On the other hand, we have the following quick theorem which gives the global unique solution for data in . The same result in fact also holds in . The result makes use of conservation of mass and energy for smooth solutions to (1):
for all and for some initial mass and energy constants .
Theorem 4 There is a global in time unique solution in from any initial data in .
Proof: The proof is again direct. Indeed, from the proof of Theorem 3, we may bound
That is, the solution can be continued globally in time, provided that remains finite for every finite time, which is in fact valid, since , using the conservation laws (12). This completes the proof of the theorem.
We stress that Theorem 4 deals with the defocusing NLS and allows arbitrarily large initial data in . 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
The global unique 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 , there are end states so that the global solution to (1) scatters to the free Schrödinger evolution as . Precisely,
Proof: Following Tao’s lecture notes, we work with the Strichartz norms as in (10). By Sobolev embedding, we note that for . Now, in view of (9), we set
It thus suffices for (13) to prove that the integral term has a strong limit in . For this, first using (7), we bound
uniformly in . Therefore, the uniform boundedness of would imply the weak convergence of in . A similar calculation also proves that the sequence is Cauchy in , since from the boundedness of we have as and the strong convergence thus follows.
It therefore remains to prove . We have the following.
Proposition 6 Suppose that . Then, .
Proof: Take any small . By assumption , we can decompose the whole line into finitely many subintervals so that
Recall that from the conservation of energy. By interpolation, this yields
for any , where and . Note that may not be an admissible Strichartz pair.
Now on each interval , from the Duhamel (9) and the Strichartz estimates (6)–(8), we bound
in which and are two admissible Strichartz pairs. The last estimate is only valid when
Since and are admissible Strichartz pairs and satisfy (5), we must have
which, together with and , implies that and . Finally, we may take and . Therefore, together with (15) with , we have proven
Taking sufficiently small yields the boundedness of . Summing over finitely many subintervals gives the proposition.
Finally, we recall the following classical interaction Morawetz estimates:
which in particular gives . That is, the assumption of the above proposition is verified, and the theorem thus follows.