Abstract. In these notes we prove the existence of a wave map operator W^{+} for the Schrödinger operator, and describe some of its properties.

SCATTERING FOR THE THREE-DIMENSIONAL

SCHRÖDINGER EQUATION WITH COMPACTLY SUPPORTED POTENTIAL

TERENCE TAO

1. The free Helmholtz equation

Let k > 0 be a real number. In this section we study solutions w to the free Helmholtz equation

(Δ + 4π^{2}k^{2})w = f

on R^{3}, where f is an L^{2} function with compact support.

This is best studied by first considering the perturbed equation

(Δ + 4π^{2}(k + iε)^{2})w_{ε} = f (1)

Note that this is of the form T + is, where T is a closed self-adjoint operator and is is a non-zero imaginary number. Such operators are always invertible on L^{2}. To see this, note that the identity

∥(T + is)f∥_{2}^{2} = ∥T f∥_{2}^{2} + s^{2}∥f∥_{2} ≥ s^{2}∥f∥_{2}

shows that T + is has no kernel, while the dual identity

∥(T + is)^{∗}f∥_{2}^{2} = ∥T f∥_{2}^{2} + s^{2}∥f∥_{2} ≥ s^{2}∥f∥_{2}

shows that (T + is)^{∗} has no kernel, i.e. T + is has no co-kernel.

The equation (1) has solution

w_{ε} = f ∗ G_{k}^{ε}

where the Green’s function G_{k}^{ε} is given by

Ĝ_{k}^{ε}(ξ) = \frac{1}{4π^{2}[(k + iε)^{2} − |ξ|^{2}]},

or in other words that

G_{k}^{ε}(x) = ∫ \frac{e^{2πix·ξ} dξ}{4π^{2}[(k + iε)^{2} − |ξ|^{2}]}.

This function can be rescaled via the easily verified identity

G_{k}^{ε}(x) = kG_{1}^{ε/k}(kx).

So we’d like to know what happens to G_{1}^{ε} as ε → 0.

Proposition 1.1. We have lim_{εto0} G_{1}^{ε}(x) = G_{1}(x) for all non-zero x, where

|G_{1}(x)| ≤ C|x|^{−1} for |x| ≤ C

G_{1}(x) = C \frac{e^{2πi|x|}}{|x|} + O(|x|^{−3/2}) for |x| ≥ C.

Furthermore, we have the radiation condition

∂_{r}G_{1}(x) − 2πiG_{1}(x) = o(|x|^{−1})

as |x| → ∞, where ∂_{r} is the radial derivative. also, we have the distributional identity

(Δ + 4π^{2}k^{2})G_{1} = δ

where δ is the Dirac delta function.

Proof Since G_{1} is clearly radial we may assume that x = x_{1}e_{1} for some x_{1} > 1. We shall assume that ε is extremely small, in particular ε ≲ (1 + |x|)^{−100}. We will obtain bounds for G_{1}^{ε} which are independent of ε; convergence as ε → 0^{+} is left as an exercise to the reader.

We partition unity as

1 = ϕ_{0}(r) + ψ_{0}(r) + ∑_{k>0} ψ_{k}(r),

where ϕ_{0}, ψ_{0}, ψ_{k} are C_{0}^{∞} functions supported on {|r| < 1/2}, {1/4 < |r| < 4}, {2^{k+1} < |r| < 2^{k+3}} respectively. We consider the corresponding contributions to G_{1}(x), namely

∫ \frac{e^{2πix·ξ}ϕ_{0}(|ξ|) dξ}{4π^{2}[(1 + iε)^{2} − |ξ|^{2}]}, (2)

∫ \frac{e^{2πix·ξ}ψ_{0}(|ξ|) dξ}{4π^{2}[(1 + iε)^{2} − |ξ|^{2}]} (3)

and

∫ \frac{e^{2πix·ξ}ψ_{k}(|ξ|) dξ}{4π^{2}[(1 + iε)^{2} − |ξ|^{2}]} (4)

The first term (2) is the Fourier transform of a Schwarz function and is therefore Schwarz, so that it has no impact on the estimates. To estimate (4) we rewrite it as

2^{−2k} ∫ e^{2πix·ξ}ψ̃_{k}(|ξ|) dξ

where ψ̃_{k} satisfies the estimates

|\frac{d^{N}}{dr^{N}} ψ̃_{k}(r)| ≲ C2^{−Nk}

for all non-negative integers N. If we estimate this integral by replacing everything by absolute values, we obtain a bound of O(2^{k}). On the other hand, if we repeatedly integrate by parts using the identity

e^{2πix·ξ} = \frac{1}{2πix_{1}} \frac{∂}{∂ξ_{1}} e^{2πix·ξ}

we can rewrite the integral as

2^{−2k} \frac{C}{|x|^{1}00} ∫ e^{2πix·ξ} \frac{∂^{100}}{∂x_{1}^{100}} ψ̃_{k}(|ξ|) dξ.

If we now estimate everything by absolute values we get a bound of

O(2^{k}(\frac{2^{k}}{|x|})^{100}).

Combining the two estimates we obtain

(4) ≲ 2^{k} min(1, \frac{2^{k}}{|x|})^{100}.

Summing this in k we obtain

∑_{k>0} (4) ≲ min(|x|^{−1}, |x|^{−100})

which is acceptable.

We now turn to the most difficult contribution, which is (3). We rewrite this in polar co-ordinates ξ = ρω as

∫ ∫_{S^{2}} \frac{e^{2πiρω_{1}x_{1}} ψ̃(ρ)}{1 + iε − ρ}dωdρ,

where we have absorbed various harmless factors into the ψ cutoff. We partition unity in the ω variable as

1 = Φ_{1}(ω) + Φ_{2}(ω) + Φ_{3}(ω)

where Φ_{1}, Φ_{2}, Φ_{3} are supported on the sets ω_{1} > 1/4, |ω_{1}| < 1/2, and ω_{1} < −1/4.

We first treat the contribution of Φ_{2}. We subdivide this further into the portion where |ρ − 1| > |x|^{−50} and |ρ − 1| ≤ |x|^{−50}.

To estimate the former contribution, we first observe that the ω integral is an oscillatory integral with a non-stationary phase, and it can be bounded by O(|x|^{−100}) with constants independent of everything. So the contribution of this part is at most

∫_{|ρ−1|>|x|^{−50}} \frac{|x|^{−100}}{|1 + iε − ρ|} dρ ≲ |x|^{−100} log(1 + |x|),

which is certainly acceptable. To handle the second contribution, we write

e^{2πiρω_{1}x_{1}} ψ̃(ρ) = e^{2πiω_{1}x_{1}} ψ̃(1) + O(|ρ − 1|)

to rewrite this integral as

∫_{|ρ−1|≤|x|^{−50}} ∫_{S^{2}} \frac{e^{2πiω_{1}x_{1}} ψ̃(1)}{1 + iε − ρ}dωdρ + O(∫_{|ρ−1|≤|x|^{−50}} ∫_{S^{2}} \frac{|ρ − 1|}{|1 − ρ + iε|} dωdρ).

The error term is O(|x|^{−50}), which is acceptable. For the main term, we compute the ρ integral to be πi + O(|x|^{−50}). The error term is clearly acceptable, so we’re left with

C ∫_{S^{2}} e^{2πiω_{1}x_{1}}Φ_{2}(ω)dω.

But this is O(|x|^{−100}) by the principle of non-stationary phase.

We now turn to the contribution of Φ_{1}(ω). We first deal with the ρ integral by observing that

∫ \frac{e^{2πiρω_{1}x_{1}} ψ̃(ρ)}{1 + iε − ρ} dρ = −2πiψ̃(1)e^{2πiω_{1}x_{1}} + O((1 + |x|)^{−100}).

To see this, we rewrite ψ̃(ρ) as ψ̃(1)χ(ρ) + (ρ − 1)ψ′(ρ) where ψ′, χ are bump functions and χ = 1 near 1. The contribution of ψ′ converges to an oscillatory integral with non-stationary phase as ε → 0, and so is O((1 + |x|)^{−100}).

To handle the χ integral, we extend χ smoothly to the complex plane so that χ = 1 near 1. The integrand is then analytic near 1 except for a pole at 1 + iε. We can then shift the contour upward, so that it stays a fixed distance away from 1. The integral is then an oscillatory integral with a bounded amplitude (in fact, it becomes exponentially small as the contour moves into the upper half-plane), and is O((1 + |x|)^{−100}). The main contribution then comes from the Cauchy residue theorem.

Discarding the error term, we are left with

C ∫_{S^{2}} Φ_{1}(ω)e^{2πiω_{1}x_{1}} dω.

When |x| < 1 this integral is under control, so let’s assume |x| > 1. The phase is stationary when ω = e_{1} and is non-degenerate, with Hessian comparable to |x|^{2} (admittedly the manifold of integration is curved, but one can fix that by a change of variables). The integral is therefore

Ce^{2πix_{1}}|x|^{−1} + O(|x|^{−3/2})

as desired.

Finally, we look at Φ_{3}. We treat this the same way as Φ_{1}, except that we shift the contour downwards instead of upwards. Thus we do not any contribution from the pole, and this term is just O((1 + |x|)^{−100}).

The distributional formula comes from inspecting the Fourier transform of G_{1}. To show the radiation condition, it suffices to show it for x = x_{1}e_{1}, i.e.

(\frac{∂}{∂x_{1}} − 2πi)G_{1}(x_{1}e_{1}) = o(|x|^{−1}).

But the left-hand side can be given as a Fourier transform just like G_{1}, except that there is an extra factor of 2πi(ξ_{1} − e_{1}) in the integral. This factor vanishes at the stationary point ξ = e_{1}, and so one gets the desired improvement o(|x|^{−1}).

In particular, we have the estimate

G_{k}(Rω − x) = CR^{−1}e^{2πikR}e^{−2πikx·ω} + O(R^{−3/2}) (5)

for large R, ω in S^{2}, and x in a bounded set.

As a corollary we see that if f is L^{2} with compact support, then the solution w = f ∗ G_{k} satisfies the Sommerfeld radiation condition

w = O(|x|^{−1}), w_{r} − 2πikw = o(|x|^{−1}) as |x| → ∞. (6)

To see this, we simply observe from (5) that the function w = G(· − x_{0}) obeys (6) for all x_{0} in a bounded set.

2. Elliptic estimate

Let R ≫ 1 be a large number. We will prove the elliptic estimate

∥∇w∥_{L^{2}(B(0,R))}^{2} ≤ C(∥w∥_{L^{2}(B(0,2R))}^{2} + ∥Δw∥_{L^{2}(B(0,2R))^{2}}) (7)

where the constant is independent of R.

To see this, we estimate the left-hand side by

Re ∫ ∇w∇\overline{w}χ

where χ is a cutoff function adapted to B(0, 2R) that equals one on B(0, R). We integrate this by parts as

−Re ∫ Δw\overline{w}χ − Re ∫ \overline{w}∇w · ∇χ.

By inequality |ab| ≤ \frac{1}{2}(|a|^{2} + |b|^{2}), the first term is bounded by

∫ \frac{1}{2}(|Δw|^{2} + |w|^{2})χ,

which is acceptable. The second term we rewrite as

− ∫ \frac{1}{2}∇(|w_{ε}|^{2}) · ∇χ,

which we integrate by parts as

∫ \frac{1}{2}|w_{ε}|^{2}Δχ

which is also acceptable. Note that every time we differentiate χ we actually gain a factor of 1/R.

As a corollary, we observe that if

(Δ + 4π^{2}(k + iε)^{2} + V)w_{ε} = f,

then

∥∇w_{ε}∥_{L^{2}(B(0,R))}^{2} ≤ C_{k}(∥w_{ε}∥_{L^{2}(B(0,2R))}^{2} + ∥f∥_{L^{2}(B(0,2R))^{2}}). (8)

Furthermore, applying (7) with w = ∇w_{ε}, we obtain

∥∇^{2}w_{ε}∥_{L^{2}(B(0,R))}^{2} ≤ C_{k}(∥w_{ε}∥_{L^{2}(B(0,4R))}^{2} + ∥f∥_{H^{1}(B(0,4R))^{2}}).

From Sobolev embedding we therefore have

∥w_{ε}∥_{L^{∞}(B(0,R))}^{2} ≤ C_{k}(∥w_{ε}∥_{L^{2}(B(0,4R))} + ∥f∥_{H^{1}(B(0,4R))}). (9)

One could continue in this fashion, and estimate just about any local norm of w_{ε} in terms of a suitable norm of f, but we shall not do so here.

3. Green’s theorem

On a ball B(0, R), Green’s theorem is

∫_{B(0,R)} uΔv − vΔu = ∫_{∂B(0,R)} uv_{r} − vu_{r}.

We rewrite this slightly as

∫_{B(0,R)} u(Δ + 4π^{2}k^{2} + V)v − v(Δ + 4π^{2}k^{2} + V)u = ∫_{∂B(0,R)} u(v_{r} − 2πikv) − v(u_{r} − 2πiku). (10)

4. Uniqueness for the Helmholtz equation

We now show that uniqueness for the Helmholtz condition given the radiation condition (6). In other words, we want to show that if w is a locally L^{2} function satisfying (6) and

(Δ + 4π^{2}k^{2} + V)w = 0

then w = 0.

Let w be as above. We first observe from (8) and higher iterates that w is everywhere smooth.

Next, we apply (10) with u = w and v = G_{k}(· − x_{0}) to obtain

∫_{B(0,R)} −w(x)V (x)G_{k}(x−x_{0}) dx−w(x_{0}) = ∫_{∂B(0,R)} w(∂_{r}−2πik)G_{k}(·−x_{0})−G_{k}(·−x_{0})(∂_{r}−2πik)w.

Since both w and G(· − x_{0}) satisfy (6), the right-hand side is o(1) as R → ∞. Taking limits we thus obtain

w(x_{0}) = − ∫ w(x)V (x)G_{k}(x − x_{0}) dx, (11)

or in other words that

w = (−V w) ∗ G_{k}.

Note that this is consistent with the defining equation for w, since G_{k} is a Green’s function for Δ + 4π^{2}k^{2}.

Next, we apply (10) with u = w and v = \overline{w} to obtain

0 = ∫_{∂B(0,R)} w(∂_{r} − 2πik)\overline{w} − \overline{w}(∂_{r} − 2πik)w.

The latter term is o(1) by (6). To treat the former term we observe from conjugating

(6) that

(∂_{r} − 2πik)\overline{w} = −4πik\overline{w} + o(1/R).

Inserting this into the above we see that

∫_{∂B(0,R)} |w|^{2} dS → 0 as R → ∞.

On the other hand, from (11) and (5) we see that

w(Rω) = CR^{−1}e^{2πikR} ∫ w(x)V (x)e^{−2πikx·ω} dx + O(R^{−3/2})

so that

|w(Rω)| = CR^{−1}|V̂ w(kω)| + o(R^{−1}).

Inserting this into the above and taking R → ∞, we see that

∫ |V̂ w(kω)|^{2} dω = 0,

or in other words that the Fourier transform of V w vanishes on the sphere of radius k.

Now since V w is compactly supported and smooth, we see that the Fourier transform of V w is uniformly analytic and rapidly decreasing. In particular, we have an estimate of the form

|∂^{α}V̂ w(ξ)| ≤ CA^{|α|}(1 + |ξ|)^{−10}

for all multi-indices α and some constants C, A > 0 which do not depend on α. Since V̂ w vanishes on the sphere, the function

ŵ(ξ) = \frac{V̂ w(ξ)}{4π^{2}(k^{2} − |ξ|^{2})}

is thus also uniformly analytic and rapidly decreasing, so that

|∂^{α}ŵ(ξ)| ≤ C(2A)^{|α|}(1 + |ξ|)^{−10}

It suffices to show this when ||ξ| − k| ≤ A/10, since the claim is clear otherwise. By Taylor’s theorem it suffices to prove the bound (with 2A replaced by A) when |ξ| = k. Without loss of generality we may take ξ = ke_{1}. By a locally analytic diffeomorphism we may perturb the sphere to a hyperplane ξ_{1} = k and the denominator to just k − ξ_{1}, without significantly worsening the bounds on the derivatives. But this follows from a straightforward application of Taylor’s theorem, and the observation that all the terms in the Taylor expansion have a factor of (k − ξ_{1}) in them.

From our bounds on ŵ we see that w is compactly supported. To see this, let x = x_{1}e_{1} with x_{1} > 4πA. From the Fourier inversion formula

w(x) = ∫ e^{2πix_{1}ξ_{1}} ŵ(ξ) dξ

and many integrations by parts using the identity e^{2πix_{1}ξ_{1}} = \frac{1}{2πix_{1}} ∂_{1}e^{2πix_{1}ξ_{1}}, we obtain

w(x) = (−2πix_{1})^{−N} ∫ e^{2πix_{1}ξ_{1}} \frac{∂^{N}}{∂x_{1}^{N}} ŵ(ξ) dξ.

From our bounds on ŵ we thus obtain

|w(x)| ≤ C(\frac{2A}{2πix_{1}} )^{N}

where C is independent of N. Taking N → ∞ we obtain the result.

To finally show that w = 0, we observe the Carleman estimate

∥e^{2πNx_{1}}w∥_{2} ≤ CN^{−2}∥e^{2πNx_{1}}Δw∥_{2}

valid for all compactly supported w and large N, where C is a constant independent of N. To show this, we observe that the Fourier transform of w, e^{Nx_{1}}w, and e^{Nx_{1}}Δw are entire, and are related by the identities

{ \atop e^{Nξ}w}(ξ) = ŵ(ξ − iN), { \atop e^{Nξ}Δw}(ξ) = −4π^{2}(x − iN)^{2}ŵ(ξ − iN).

The claim then follows from Plancherel’s theorem and the observation that

1 ≤ N^{2}|(ξ − iN)^{2}|.

From the Carleman estimate and the defining equation for w we see that

∥e^{2πNx_{1}}w∥_{2} ≤ CN^{−2}∥e^{2πNx_{1}}(V + 4π^{2}k^{2})w∥_{2}

Since V + 4π^{2}k^{2} is bounded, we thus obtain

∥e^{2πNx_{1}}w∥_{2} ≤ C_{k}N^{−2}∥e^{2πNx_{1}}w∥_{2}

If N is sufficiently large, we therefore obtain

∥e^{2πNx_{1}}w∥_{2} = 0,

so w = 0 as desired.

5. Existence for the Helmholtz equation

We now show that for any f which is in L^{2} with compact support, there exists a solution to

(Δ + 4π^{2}k^{2} + V)w = f (12)

which satisfies the radiation condition (6).

Firstly, we define w_{ε} by

(Δ + 4π^{2}(k + iε)^{2} + V)w_{ε} = f; (13)

the operator on the left is a self-adjoint operator plus a small imaginary multiple of the identity, so w_{ε} is well-defined and in L^{2}. The issue is to show that w_{ε} converges in L^{2} to some function w.

Let us first make the assumption that w_{ε} is bounded in L^{2} as ε → 0; we shall justify this shortly. Then from (7) we see that w_{ε} is thus bounded on H^{1}(B(0, R)). By the Rellich embedding theorem (see appendix), w_{ε} thus lives in a compact subset of L^{2}, and one can extract a subsequence which converges to some limit w in L^{2}.

From (13) we have

w_{ε} = (f − V w_{ε}) ∗ G_{k}^{ε}.

Taking limits in L^{2} for the subsequence above, we obtain

w = (f − V w) ∗ G_{k}

on B(0, R) at least. Since w is in L^{2}, we thus see that w obeys (6) and

(Δ + 4π^{2}k^{2} + V)w = f

on B(O, R). By the way, even though we only obtained convergence for a subsequence, we see from the uniqueness theory that w_{ε} has only one limit point, so we must have w_{ε} → w in L^{2} for all ε → 0^{+}. This also allows us to take R → ∞ and obtain the desired properties on all of R^{3}.

Now we explain why w_{ε} is bounded in L^{2}. Suppose for contradiction that there was a subsequence of ε for which

∥w_{ε}∥_{2} → ∞.

For this subsequence, define the normalized solutions w̃_{ε} = w_{ε}/∥w_{ε}∥_{2}, so that ∥w̃_{ε}∥ = 1. From (13) we have

(Δ + 4π^{2}(k + iε)^{2} + V)w̃_{ε} → 0 in L^{2}.

By (7) w̃_{ε} therefore lives in a compact subset of L^{2}(B(0, R)), and by extracting a further subsequence we can make w̃_{ε} converge in L^{2} to some w̃ ∈ L^{2}. By repeating the previous argument we may take limits in L^{2} and conclude that

(Δ + 4π^{2}k^{2} + V)w̃ = 0.

But by uniqueness this implies that w̃ = 0. But this contradicts the fact that w̃_{ε} → w̃ in L^{2}, since we have ∥w̃_{ε}∥_{2} = 1. This concludes the proof of existence.

Note that the last argument allows us to show the estimate

∥w∥_{L^{2}(B(0,R))} ≤ C_{R}∥f∥_{2}. (14)

By (7) and its iterated versions, we can also control w in just about any local norm (e.g. L^{∞}(B(0, R))).

We summarize our results in the following theorem.

Theorem 5.1. If f is in L^{2} with compact support, then there exists a unique solution w in L_{loc}^{2} which satisfies (12) and (6). Furthermore, this solution satisfies

w = (f − V w) ∗ G_{k} (15)

and is the limit in L_{loc}^{2} of the functions w_{ε} defined by (13). Finally, we have the bound (14) for all R.

6. Distorted plane waves, and Greens functions

Let ξ be an element of R^{3}. By the above existence and uniqueness theory we may find an L^{2} function w_{ξ} satisfying (6) and

(Δ + 4π^{2}|ξ|^{2} + V)w_{ξ}(x) = −V e^{2πix·ξ}. (16)

If we write Φ_{ξ}(x) = e^{2πix·ξ} + w_{ξ}(x), we therefore have

(Δ + 4π^{2}|ξ|^{2} + V)Φ_{ξ} = 0.

The function Φ_{ξ} is thus a generalized eigenfunction of Δ + V, and plays the same role with respect to that operator as the plane wave e^{2πix·ξ} does with Δ. The function Φ_{ξ} is usually called a distorted plane wave with frequency ξ.

In a similar fashion, let y be an element of R^{3} and k > 0, and let h(x, y, k + iε) be the unique L^{2} function such that

(Δ + 4π^{2}(k + iε)^{2} + V)h(·, y, k + iε) = −V G_{k}^{ε}(· − y);

note that our estimates for G_{k} allow us to conclude that the right-hand side is uniformly in L^{2} with compact support. Then if we write G_{H}(x, y, k + iε) = G_{k}^{ε}(x − y) + h(x, y, k + iε), we obtain

(Δ + 4π^{2}(k + iε)^{2} + V)G_{H}(·, y, k + iε) = δ_{y}.

Furthermore, as ε → 0 h(·, y, k +iε) converges to an L^{2} function h(·, y, k) satisfying (6) and

(Δ + 4π^{2}k^{2} + V)h(·, y, k) = −V G_{k}(· − y); (17)

in particular, we have

(Δ + 4π^{2}k^{2} + V)G_{H}(·, y, k) = δ_{y}.

We have thus found a Green’s function G_{H} for the operator Δ + 4π^{2}k^{2} + V. Note that G_{H} is symmetric in x and y since the operator is self-adjoint.

Suppose y = rω for some large r. Then from the asymptotics of G_{k} we have

G_{k}(x − y) = Cr^{−1}e^{2πikr}e^{2πix·kω} + O(r^{−3/2})

as r → ∞, if x remains bounded. From this, (16), (17), and improved versions of (14), we thus obtain the following important asymptotic relationship:

G_{H}(x, rω, k) = Cr^{−1}e^{2πikr}Φ_{kω}(x) + O(r^{−3/2}) (18)

if r → ∞ and x remains bounded. A similar argument gives

∂_{r}G_{H}(x, rω, k) = 2πikCr^{−1}e^{2πikr}Φ_{kω}(x) + O(r^{−3/2}). (19)

We can be far more precise about the relationship between G_{H} and Φ - at least for the imaginary part of G_{H}. More precisely:

Proposition 6.1.

ImG_{H}(x, y, k) = CkIm ∫_{S^{2}} Φ_{kω}(x)\overline{Φ_{kω}(y)} dω. (20)

Proof Let R be a large number, and fix x, y, k. From Green’s theorem applied to G_{H}(·, x, k) and \overline{G_{H}(·, y, k)} we have

∫_{B(0,R)} G_{H}(·, x, k)Δ\overline{G_{H}(·, y, k)}−Δ_{y}G_{H}(·, x, k)\overline{G_{H}(·, y, k)} = ∫_{∂B(0,R)} G_{H}(·, x, k)∂_{r}\overline{G_{H}(·, y, k)}−∂_{r}G_{H}(·, x, k)\overline{G_{H}(·, y, k)} dS.

The left-hand side is just 2iImG_{H}(x, y, k) since G_{H} is a Green’s function for Δ+V. By (18) and (19), the right-hand side is

R^{2} ∫_{S^{2}}(CR^{−1}e^{2πikR}Φ_{kω}(x))(CkR^{−1}e^{−2πikR}\overline{Φ_{kω}(y)})−(CkR^{−1}e^{2πikR}Φ_{kω}(x))(CR^{−1}e^{−2πikR}\overline{Φ_{kω}(y)}) dω+O(R^{−1}).

The result then follows by letting R → ∞.

7. Negative eigenvalues

We have studied the invertibility of the operator Δ + λ + V when λ is a positive real, or is within epsilon of being a positive real. When λ is not real the invertibility of this operator is straightforward, so it remains to consider the case of negative λ. (There is also the case λ = 0, but we will finesse this issue).

Let λ < 0. On the case V = 0 then Δ + λ is a Fourier multiplier with symbol −4π^{2}|ξ|^{2}+λ, which is uniformly bounded away from zero. Thus for the free problem invertibility is automatic. When one has a potential the situation is almost as good. More precisely:

Proposition 7.1. For any λ < 0, the spectral projection χ_{(−∞,λ)}(Δ+V) has finite rank.

In other words, the spectrum of Δ + V on the negative real axis is discrete and consists only of pure point spectrum (i.e. L^{2} eigenfunctions).

Proof Let X denote the range of χ_{(−∞,λ)}(Δ + V), so that

⟨(Δ + V)f, f⟩ ≤ λ∥f∥^{2}

for all f ∈ X. By integrating in parts as in the proof of (7) we obtain

∥∇f∥_{2} ≤ C_{λ,V} ∥f∥_{2}.

Thus the unit ball in X (with the L^{2} topology) is a bounded subset of H^{1} and is therefore compact by Rellich embedding. But a ball in a Hilbert space can only be compact if X is finite dimensional.

These eigenfunctions are very bad for scattering theory. For, suppose f was an eigenfunction of Δ+V with eigenvalue λ < 0. Then the solution to the Schrödinger equation with initial data f is easily seen to be e^{iλt}f, which does not tend toward a solution to the free equation as t → ∞. (For instance, it does not decay in time, whereas solutions to the free equation always do). So we are forced to exclude these eigenfunctions from our scattering theory.

Let H_{c} be the orthogonal complement in L^{2} of the space of eigenfunctions of Δ+V. Then Δ+V leaves H_{c} invariant, and the restriction of Δ+V to H_{c} has no spectrum on the negative real axis.

We want to understand functions ϕ(Δ + V) of the Hamiltonian, and more Let ϕ be an arbitrary function which is analytic on a neighbourhood of the real axis and rapidly decreasing near +∞. From contour integration we have the Plemelj formula

ϕ(x) = \frac{1}{2πi} lim_{ε→0} ∫_{0}^{∞} ϕ(λ)[(x + λ − iε)^{−1} − (x + λ + iε)^{−1}] dλ

for all positive x; we rewrite this as

ϕ(x) = \frac{1}{pi} lim_{ε→0} ∫_{0}^{∞} ϕ(λ)Im[(x + λ − iε)^{−1}] dλ.

By functional calculus we therefore have

ϕ(Δ + V) = \frac{1}{2πi} lim_{ε→0} ∫_{0}^{∞} ϕ(λ)Im[(Δ + V + λ − iε)^{−1}] dλ

on H_{c}; this can be proven rigorously first for functions in the range of the spectral projection χ_{(0,R)}(Δ + V), and then letting R → ∞. Note that the imaginary part of an operator T is simply (T − T^{∗})/2i.

Writing λ = 4π^{2}k^{2} and using our Green’s function representation, we thus obtain

ϕ(Δ + V)f(x) = C ∫_{0}^{∞} ∫_{R_{3}} ϕ(4π^{2}k^{2})ImG_{H}(x, y, k)f(y) dy kdk.

By (20) we thus have

ϕ(Δ + V)f(x) = C ∫_{0}^{∞} ∫_{R_{3}} ∫_{S^{2}} ϕ(4π^{2}k^{2})Φ_{kω}(x)\overline{Φ_{kω}(y)} dωdyk^{2}dk,

or after the substitution ξ = kω,

ϕ(Δ + V)f(x) = C ∫_{R_{3}} ∫_{R_{3}} ϕ(4π^{2}|ξ|^{2})Φ_{ξ}(x)\overline{Φ_{ξ}(y)} dydξ.

We rewrite this as

ϕ(Δ + V) = F_{H}^{∗}ϕ(4π^{2}|ξ|^{2})F_{H} on H_{c}

where F_{H} : H_{c} → L^{2}(R^{3}) is the perturbed Fourier transform

F_{H}f(ξ) = C ∫_{R_{3}} f(y)\overline{Φ_{ξ}(y)} dy.

In particular, setting ϕ = 1 (or more rigorously, setting ϕ = e^{−εx} and then letting ε → 1, using the monotone convergence theorem, we see that

I = F_{H}^{∗}F_{H},

i.e. that F_{H} is an isometry on Hilbert spaces.

It can also be proved that F_{H}^{∗} is injective, so that F_{H} is actually unitary. But we will not need that here.

Finally, setting ϕ = e^{−itx} (again putting in an e^{−εx} factor to be rigorous), we have a representation for the Schrödinger operator e^{−itH} on H_{c}:

e^{−itH} = F_{H}^{∗}e^{−it4π^{2}|ξ|^{2}}F_{H} (21)

where H = Δ + V. This should be compared with the Fourier representation of the solution to the free Schrödinger equation

e^{−itΔ} = F^{∗}e^{−it4π^{2}|ξ|^{2}}F (22)

where F is the Fourier transform.

8. The wave map operator

We now show that the Schrödinger operator with potential e^{−itH} behaves asymptotically like the free Schrödinger operator e^{−itΔ}. In other words, for every L^{2} function f there exists an L^{2} function g such that

e^{−itH}f − e^{−itΔ}g → 0 as t → +∞,

where we shall be vague about the nature of convergence. We write W_{+}f for g. (There is an analogous situation as t → −∞, but it gives a function W_{−}f which is generally distinct from W_{+}f. The composition W_{+}W_{−}^{−1} is known as the scattering operator and is important for applications).

Since e^{itΔ} is unitary, we can define W_{+} as

W_{+}f = lim_{t→∞} e^{itΔ}e^{−itH}f,

where the convergence is in (say) the weak topology of L^{2}. (We can actually improve this to strong L^{2}, but we shall not do so here).

We first show that the limit actually exists, i.e. that

lim_{t→∞}⟨e^{itΔ}e^{−itH}f, g⟩

converges for all f, g ∈ L^{2}. Since this expression is bilinear in f, g and is bounded by ∥f∥_{2}∥g∥_{2} by Cauchy-Schwarz and unitarity (at least if we temporarily replace the lim by a lim sup), it suffices to show this when f and g are test functions. By the fundamental theorem of calculus,

∫_{1}^{∞} |\frac{d}{dt}⟨(e^{itΔ}e^{−itH}f), g⟩| dt < ∞.

By the product rule for operators, we have

\frac{d}{dt}(e^{itΔ}e^{−itH}f) = e^{itΔ}(iΔ)e^{−itH}f + e^{itΔ}(−iH)e^{−itH}f = −ie^{itΔ}V e^{−itH}f

and so it suffices to show

∫_{1}^{∞} |⟨e^{itΔ}V e^{−itH}f, g⟩| dt < ∞.

We estimate this by unitarity and Cauchy-Schwarz as

∫_{1}^{∞} ∥e^{−itH}f∥_{2}∥V e^{−itΔ}g∥_{2} dt.

But since g is a test function, we see from the integral representation

e^{−itΔ}g(x) = \frac{C}{t^{3/2}} ∫ e^{−i|x−y|^{2}/4t}g(y) dy

that e^{−itΔ}g decays like O(t^{−3/2}). Inserting this into the above and using the L^{2} boundedness of e^{−itH}f we see the integrand is O(t^{−3/2}), which is absolutely integrable. This proves existence for W_{+}.

We now write W_{+} as a Cesaro mean:

W_{+}f = lim_{ε→0} ε ∫_{0}^{∞} e^{−εt}e^{itΔ}e^{−itH}f dt,

where the limit is in the weak topology. We wish to show the formula

W_{+} = F^{−1}F_{H},

or in other words that

lim_{ε→0}⟨ε ∫_{0}^{∞} e^{−εt}e^{itΔ}e^{−itH}f dt, g⟩ = ⟨F_{H}f, Fg⟩.

To show this, it suffices as before to prove it for test functions f, g. By integration by parts, the left-hand side is

⟨f, g⟩ + lim_{ε→0} ∫_{0}^{∞}⟨e^{−εt} \frac{d}{dt}(e^{itΔ}e^{−itH}f), g⟩ dt,

which by the same calculation as before becomes

⟨f, g⟩ − lim_{ε→0} ∫_{0}^{∞} e^{−εt}⟨ie^{itΔ}V e^{−itH}f, g⟩ dt.

16 TERENCE TAO

Since ⟨f, g⟩ = ⟨F_{H}f, F_{H}g⟩, it suffices to show that

lim_{ε→0} ∫_{0}^{∞} e^{−εt}⟨ie^{itΔ}V e^{−itH}f, g⟩ dt = ⟨F_{H}f, F_{H}g − Fg⟩. (23)

We expand out the left-hand side as

lim_{ε→0} ∫_{0}^{∞} ∫ ie^{−εt}e^{−itH}f(x)V (x)e^{itΔ}\overline{g}(x) dxdt.

By (21) this becomes

lim_{ε→0} ∫_{0}^{∞} ∫ ∫ ie^{−εt}Φ_{ξ}(x)e^{4π^{2}it|ξ|^{2}}F_{H}f(ξ)V (x)e^{itΔ}\overline{g}(x) dξdxdt.

We rewrite this as

lim_{ε→0} ∫_{0}^{∞} ∫ ∫ iΦ_{ξ}(x)F_{H}f(ξ)V (x)e^{it[Δ+4π^{2}|ξ|^{2}+iε]}\overline{g}(x) dtdξdx.

Computing the t integral, we obtain

lim_{ε→0} ∫ ∫ Φ_{ξ}(x)F_{H}f(ξ)V (x)(Δ + 4π^{2}|ξ|^{2} + iε)^{−1}\overline{g}(x) dξdx.

Taking ε → 0, we thus obtain

− ∫ ∫ Φ_{ξ}(x)F_{H}f(ξ)V (x)\overline{g} ∗ G_{|ξ|}(x) dxdξ.

We rewrite this as

∫ F_{H}f(ξ)⟨(−V Φ_{ξ} ∗ G_{|ξ|}, g⟩ dξ.

But Φ_{ξ}(x) = e^{2πix·ξ} + w_{ξ}(x) where

w_{ξ} = (−V e^{2πix·ξ} − V Φ_{ξ}) ∗ G_{|ξ|}

by (15). We therefore have

(−V Φ_{ξ} ∗ G_{|ξ|}) = Φ_{ξ} − e^{2πix·ξ}.

The above integral therefore simplifies to

∫ F_{H}f(ξ)(⟨Φ_{ξ}, g⟩ − ⟨e^{2πix·ξ}, g⟩) dξ.

But this equals

∫ F_{H}f(ξ)\overline{F_{H}g(ξ) − Fg(ξ)} dξ

as desired.

9. Appendix: The Rellich embedding theorem

In this appendix we show that the unit ball in H^{1}(B(0, R)) is relatively compact in L^{2}.

It suffices to show that H^{1}(B(0, R)) is in the ε-neighbourhood of a compact subset of L^{2} for every ε > 0; this will imply that every sequence in H^{1}(B(0, R)) has a convergent subsequence in L^{2} by the Arzeli-Ascoli diagonalization argument.

A function in H^{1}(B(0, R)) can be written as a Fourier series

∑_{k∈Z^{3}} c_{k}e^{2πik·x/4R}

where by Plancherel’s theorem

∑_{k∈Z^{3}} |k|^{2}|c_{k}|^{2} ≲ 1.

We subdivide this for some large N into the low-frequency part

∑_{|k|≤N} c_{k}e^{2πik·x/4R}

and the high-frequency part

∑_{|k|≥N} c_{k}e^{2πik·x/4R}.

The low-frequency part lives in the unit ball of a finite dimensional vector space, so it is compact. By Plancherel, the high-frequency part has an L^{2} norm of

∑_{|k|≥N} |c_{k}|^{2} ≤ N^{−2} ∑_{k≥N} |k|^{2}|c_{k}|^{2} ≲ N^{−2}

which is arbitrarily small. Thus the unit ball in H^{1}(B(0, R)) is within an εneighbourhood of a compact set for any ε, as desired.

References

1. Michael Taylor, “Partial differential equations. II. Qualitative studies of linear equations.”, Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996.

Department of Mathematics, UCLA, Los Angeles CA 90095-1555

E-mail address: tao@math.ucla.edu