Transverse moments and outgoing amplitudes

Written by GPT-6.1 Sol (OpenAI) and GPT-6 Astra (OpenAI). Self-checked by the writing AI. Original exposition: CC0.

Working question: Why begin with a transverse moment if the final amplitude is unweighted? A transverse moment controls the error caused by removing the phase and makes the amplitude Cauchy for a dense class of data. The uniform energy bound then extends that limit to arbitrary square-integrable initial data and integrable forcing. Density without a common bound would not justify passing the limiting amplitude to all states.

A long-range phase follows the motion of an outgoing packet. After removing that phase, its transverse position stays controlled and the packet converges in a fixed Hilbert space. We prove this first for data with a transverse moment. A uniform energy estimate then extends the amplitude limit to arbitrary square-integrable initial data and integrable forcing.

Read Commuting coordinates for long-range evolution for the phase, the real factors and their exact operator bounds, including its earlier programme proof of the moving-metric calculus. The outgoing equation to which they apply is constructed in Energy-shell factors and outgoing equations; its continuous forcing is proved in Frequency cutoffs and compact scattering remainders. We derive the moment and amplitude statements below from these proved operator bounds, density and Hilbert-valued integration, including the full scalar measure proof linked there, bounded-functional separation, integration in the first-moment space and the norm tail estimate. Teschl's free preliminary text [O], Sections 2.3–2.4, gives the finite-dimensional Picard and integrating-factor correspondence; the preceding lesson supplies the operator-valued proof. The free texts [Y] and [T] provide scattering context. Hörmander's freely readable paper [H] treats modified wave operators for polynomial differential operators; its Theorem 3.9 has stronger coefficient and Hessian hypotheses, and is not an input to the stationary amplitude proof here.

We use $D=-i\partial$, $\mathcal H=L^2(\mathbb R^d_z)$, and $s\ge s_0\ge1$. Norms without subscripts are $\mathcal H$ norms. Fix $0<\delta<1/3$. Write

The real compact-frequency phase and factors are those of the preceding lesson. In particular $M$ is uniformly bounded and locally norm continuous, and

The coordinate commutators are locally norm continuous. The $A_j$ commute with one another and with $A_0$. All sums and vector norms below have the finite transverse dimension $d$.

1. Propagating a transverse moment

Put

Coordinate multiplication is closed: convergence of $v_n$ and $z_jv_n$ in $\mathcal H$, tested against compactly supported smooth functions, identifies the latter limit as $z_jv$. This also proves completeness of (3).

Smooth compactly supported functions are dense in $\mathcal H_1$. The needed unweighted density, convolution inequality and smooth approximate identities are proved in Euclidean approximation and convolution. First apply a physical cutoff tending to one; both ordinary and coordinate-weighted tails tend to zero by dominated convergence. For a smooth approximate identity $\rho_\varepsilon$,

The first term converges to $z_jv$, while the second tends to zero because

$\|z_j\rho_\varepsilon\|_{L^1}=O(\varepsilon)$. Apply this after the cutoff. It proves the density claim without losing the moment.

Let $K_j=[z_j,M]$. On Schwartz tests,

Approximate in $\mathcal H_1$. Boundedness of $M$ and $K_j$ makes both terms on the right converge in $\mathcal H$; closedness of coordinate multiplication then proves (5) on the entire domain. In particular $$\|Mu\|_{\mathcal H_1} \le \|M\|\|u\|_{\mathcal H_1} +\sum_j\|K_j\|\|u\|.$$ The same inequality for $M(s)-M(t)$, with commutators $K_j(s)-K_j(t)$, proves local norm continuity on $\mathcal H_1$. Thus the norm Picard construction works on this complete space, even though the uniform energy estimate will be taken in $\mathcal H$.

Theorem 1.1 (the full moment estimate). If $v_0\in\mathcal H_1$ and $h\in L^1_{\mathrm{loc}}(ds;\mathcal H_1)$, the unique solution of

$\mathcal Lv=h,\ v(s_0)=v_0$, takes values in $\mathcal H_1$, is locally absolutely continuous there, and satisfies

Here $\Omega_{s,t}(z)=(|z|^2+(s-t)^2)^{1/2}$. The constant is independent of $s$.

Proof. On each finite interval, apply the factorial Picard construction of the preceding lesson in the $\mathcal H_1$ norm. Forcing in $L^1_{\mathrm{loc}}(\mathcal H_1)$ gives a locally absolutely continuous solution there by its Bochner variation integral. After inclusion in $\mathcal H$, it solves the same equation and has the same initial value, so uniqueness identifies it with the original solution. Multiplication by $z_j$ is bounded from $\mathcal H_1$ to $\mathcal H$; it therefore commutes with the Bochner derivative. Formula (5) gives, for $Y_j=z_jv$,

The propagator of $D_s-M$ has norm at most one fixed $C$ in both time directions, because the adjoint defect in (2) is integrable. On $\mathcal H^d$, its diagonal action has the same bound. The vector operator $u\mapsto(K_1u,\ldots,K_du)$ is bounded uniformly, with norm at most $(\sum_j\|K_j\|^2)^{1/2}$. Its variation formula applied to (7) therefore yields

The ordinary energy estimate and Fubini's theorem yield

For each fixed $t$, the sum $W(h(t))+(s-t)\|h(t)\|$ is at most $\sqrt2$ times the square root of the sum of their squares. That square root is exactly the integrand norm in (6). Equations (8)–(9) prove the claim. $\square$

2. Energy in the commuting coordinates

The preceding moment can grow linearly. The phase coordinates have a stronger estimate. Define

Lemma 2.1. For the weighted data of Theorem 1.1,

The corresponding homogeneous estimate holds backward with the same kind of uniform constant.

Proof. Set $b_j(s)=G_{\eta_j}(s,D_z)$ and $Y_j(s)=z_jv(s)+b_j(s)v(s)$. The first moment theorem makes $z_jv$ locally absolutely continuous in $\mathcal H$. The Fourier multiplier $b_j$ is norm differentiable locally, with derivative $G_{s\eta_j}(s,D_z)$, so the Bochner product rule makes $Y_j=A_jv$ locally absolutely continuous as well.

For Schwartz tests the commuting identities give $[\mathcal L,A_j]=-\sum_k[A_j,\mathcal B_k]A_k=-\sum_k\mathcal R_{jk}A_k$. This involves only the first coordinate vector on its right side. At each time, approximate $v$ in $\mathcal H_1$ by the compactly supported smooth functions constructed above. Each $A_k$ is continuous from $\mathcal H_1$ to $\mathcal H$, and each $\mathcal R_{jk}$ is bounded on $\mathcal H$. The identity thus passes to the limit in distributions. Combining it with the product rule for $Y$ gives the ordinary Hilbert-space equation

Its matrix generator on $\mathcal H^d$ is

It is bounded and locally norm continuous. The norm of its adjoint defect is at most

$\|M-M^*\|+2\|(\mathcal R_{jk})\|\le Cs^{-1-\delta}$.

For the homogeneous coordinate equation, the derivative of $\|Y\|_{\mathcal H^d}^2$ is bounded in absolute value by this adjoint defect times $\|Y\|^2$. The integrating-factor calculation of the preceding lesson therefore bounds its matrix propagator in both orientations by $\exp(C\int_{s_0}^{\infty}t^{-1-\delta}\,dt/2)$. Its variation integral for the forcing vector $(A_jh)_j$ proves (11). No second spatial moment is used. $\square$

This proof controls the whole coordinate vector. It does not replace each individual commutator or adjoint estimate by an estimate for their sum.

3. The amplitude for weighted data

Remove the phase by setting

This multiplier is unitary. Its derivative and its coordinate identity are

Theorem 3.1. Suppose $v_0\in\mathcal H_1$ and

Then $w(s)$ converges strongly in $\mathcal H$ to a limit $v_\infty\in\mathcal H_1$. The convergence norm is the unweighted $L^2$ norm; the first moment of the limit follows from the separate bound (19). If

then

Moreover,

Here $\Lambda_s(z)=1+(s^2+|z|^2)^{1/2}$. Constants may depend on the fixed $s_0$.

Proof. The bound $\|G_\eta(s,D_z)\|\le Cs$ shows that

$H_A(s)\le W(h(s))+Cs\|h(s)\|$. Thus (16) makes (17) finite. Lemma 2.1 gives $V(s)\le CM_A$. The equation (1) now gives the first line of (18), using the integrable bounds on each $\mathcal B_j$.

By (15), $w'$ is integrable on the entire half-line. More explicitly, for $t>s$, $w(t)-w(s)=i\int_s^t e^{-iG(q,D_z)}A_0v(q)\,dq$. The bound on the norm of this integral tends to zero as $s\to\infty$, uniformly in $t>s$. Completeness gives a strong limit, and sending $t\to\infty$ in that integral gives its error estimate. Integrating the first line of (18) proves the second, since $\int_s^\infty t^{-1-\delta}\,dt=s^{-\delta}/\delta$.

The second identity in (15) gives $W(w(s))=V(s)\le CM_A$. This moment passes to the strong limit using bounded truncations: multiplication by $\min(|z|,R)$ has norm at most $R$, so $\|\min(|z|,R)v_\infty\|=\lim_{s\to\infty}\|\min(|z|,R)w(s)\|\le CM_A$. Monotone convergence as $R\to\infty$ gives $W(v_\infty)\le CM_A$ and $v_\infty\in\mathcal H_1$. Finally the ordinary energy estimate bounds $\|v_\infty\|$ by $C(\|v_0\|+\int\|h\|)$, while

$V(s_0)\le W(v_0)+Cs_0\|v_0\|$. Combining these estimates proves (19). $\square$

4. Every integrable forcing has an amplitude

The weighted hypothesis is useful for a rate and a moment, but it is not needed for existence of the limit.

Theorem 4.1. For every $v_0\in\mathcal H$ and $h\in L^1(ds;\mathcal H)$, the solution of $\mathcal Lv=h$ has a strong phase-corrected amplitude:

The amplitude depends boundedly and linearly on the pair $(v_0,h)$.

Proof. Approximate $v_0$ in $\mathcal H$ by smooth compactly supported data $v_0^k$. Approximate $h$ in $L^1(ds;\mathcal H)$ by forcings $h^k$ satisfying (16). Such forcings are dense: truncate the integrable time tail, approximate on the remaining interval by finitely many simple Hilbert-valued functions, and approximate their finitely many values in $\mathcal H$ by smooth compactly supported functions. The resulting time supports are bounded, and all spatial moments are finite.

Let $v^k,w^k,a_k$ be the corresponding solutions, phase corrections and amplitudes. The uniform energy estimate gives

Apply the same estimate to two approximations and let $s\to\infty$. Their amplitudes $a_k$ form a Cauchy sequence; write its limit as $a$. For fixed $k$,

$\|w(s)-a\|$ is bounded by the uniform error in (21), the error $\|w^k(s)-a_k\|$, and $\|a_k-a\|$. First make $k$ large, then make $s$ large. This proves (20), with $v_\infty=a$. Taking limits in the ordinary energy estimate proves its norm bound. Linearity and the difference estimate follow from uniqueness of the evolution. $\square$

This argument gives existence for all $L^1$ forcing. It asserts a moment and the explicit rate (18) only when the weighted assumptions hold.

For an outgoing frequency-localized solution, the preceding lessons give

$\mathcal Lv=g+\mathcal Tv$, where $g\in L^1(ds;\mathcal H)$,

$\|\mathcal T(s)\|\le Cs^{-1-\delta}$, and $v$ has bounded slice norm.

Thus Theorem 4.1 applies to its actual right side. The limit is independent of changes to the phase extension outside its Fourier support, because the multiplier acting on that solution is unchanged.

When $d=0$, the coordinate vectors are empty and the moments vanish. The equation is $A_0v=h$; (15) directly integrates it and gives (20). All formulas above retain this interpretation.

5. Stability under approximation

To pass from truncated coefficients to a full force, convergence on each bounded time interval must be combined with a uniform estimate at infinity. The following statement specifies both ingredients.

Theorem 5.1. Let systems indexed by $n$ and a limiting system have the structure (1)–(2), with the same $s_0,\delta$ and common constants, including the bound for the individual adjoint defects used to obtain the uniform evolution estimate. Suppose

The first two convergence conditions hold for every finite or fixed $S$, respectively. Then the phase-corrected amplitudes satisfy $v_{\infty,n}\to v_\infty$ strongly.

Proof. Choose one weighted approximation $(u^k,f^k)$ to the limiting pair $(v_0,h)$, as in Theorem 4.1. In each system solve with this same pair, and call its amplitude $a_n^k$; call the limiting-system amplitude $a^k$. The uniform difference estimate gives

For fixed $k$, the canonical-coordinate quantities $M_{A,n}^k$ in (17) have a common finite bound $C_k$. This follows from

$\|G_{n,\eta}(s,D_z)\|\le Cs$ and the fixed weighted data; it does not require convergence of their moments. Equation (18) therefore gives

uniformly in $n$, with the same estimate for the limiting system.

At fixed $S$, Duhamel's formula for the difference of the two evolutions gives

All phases in (22) are real Fourier multipliers, so the elementary inequality

$|e^{-ia}-e^{-ib}|\le|a-b|$ gives convergence of their exponentials in operator norm at $S$. Thus $w_n^k(S)\to w^k(S)$. Choose $k$ to make the limiting data errors small, then $S$ to make (24) small, then $n$ to make (25), the phase error and the data errors small. The triangle inequality proves the asserted amplitude convergence. $\square$

There is a useful uniform version. Suppose an additional parameter $p$ ranges over a compact set, the convergence in (22) is uniform in $p$, all structural bounds are common, and the limiting data pair is norm continuous into $\mathcal D=\mathcal H\times L^1(ds;\mathcal H)$, with norm $\|(u,f)\|_{\mathcal D}=\|u\|+\|f\|_{L^1}$. Its image $K$ is compact. Given $\varepsilon>0$, density and compactness give finitely many weighted data pairs $d_1,\ldots,d_m$, with bounded time support, whose $\varepsilon$-balls cover $K$. The amplitude operators for every system have one common norm bound on $\mathcal D$. Thus (23) makes the two errors between actual data and the chosen center at most $C\varepsilon$, plus the uniformly vanishing data error from (22).

There are only finitely many centers. Their weighted bounds in (17) have a common finite maximum independent of $n,p$, so (24) gives one terminal time $S$ at which every center has amplitude error at most $\varepsilon$. For this fixed $S$, (25) is uniform in $p$: common energy bounds control the finitely many center solutions, and the generator difference tends to zero uniformly. The phase difference at $S$ is uniform by (22). Choose the cover first, then this one $S$, then $n$. Taking the supremum over $p$ and letting $\varepsilon\to0$ proves uniform amplitude convergence. The original compact family needs only its $\mathcal D$ norm; its members need not have a first moment.

Use the conclusion

Write the three terms in the approximation comparison: data error, fixed-data evolution error and limiting amplitude error. Check where the common energy constant is used, then retain stability for every integrable forcing.

6. Exercises with complete solutions

Exercise 1 — Foundation: an exactly integrable forcing. Take $M=G=0$ and $h(s)=e^{-(s-s_0)}f$, with $v_0,f\in\mathcal H$. Find $v(s)$, its amplitude, and the exact norm of its amplitude error.

Solution 1. Since $D_sv=h$, we have $v'=ih$. Direct integration gives

Here the phase correction is the identity. If $v_0,f\in\mathcal H_1$, multiplication by each coordinate gives the same formulas in the weighted space. With arbitrary $f\in\mathcal H$, Theorem 4.1 still applies.

Exercise 2 — Intermediate: a limit with no first moment. For $d\ge1$, $s_0=1$, set

$f(z)=(1+|z|^2)^{-(d+1)/4}$, $v_0=0$, $M=G=0$, and $h(s)=(1+s)^{-2}f$. Check the $L^1(ds;\mathcal H)$ hypothesis, compute the amplitude, and show that it has no transverse first moment.

Solution 2. At large radius the radial integral for $\|f\|^2$ has integrand comparable to

$r^{d-1}r^{-(d+1)}=r^{-2}$, which is integrable. The origin is harmless. The radial integral for $\||z|f\|^2$ instead has integrand comparable to $1$, so it diverges. Thus $f\in\mathcal H\setminus\mathcal H_1$.

The forcing norm has finite integral $\|f\|\int_1^\infty(1+s)^{-2}\,ds=\|f\|/2$. Its solution and amplitude are

The limit has no first moment. The stronger time-weighted integral also fails: $s(1+s)^{-2}$ has a logarithmically divergent integral. This example requires the density theorem rather than the weighted conclusion (19).

Exercise 3 — Intermediate: reading the two tail rates. Under Theorem 3.1 assume in addition $\|h(s)\|\le C_hs^{-1-\beta}$, with $\beta>0$. Give the amplitude error bound. Evaluate its slower exponent for $\delta=1/5,\ \beta=3/2$, and explain what happens when $\beta=\delta$.

Solution 3. Integrating each term of (18) separately gives

The slower exponent in the stated example is $1/5$. This is an upper bound; a particular solution can converge faster. If $\beta=\delta$, the two coefficients add in front of $s^{-\delta}$. There is no logarithm, since both integrations have exponent strictly below $-1$; no convolution of the two tails was used.

Exercise 4 — Advanced: a pulse escaping to late time. On $\mathcal H=\mathbb C$, choose real smooth $\rho$ supported in $(1,2)$ with $\int\rho=1$. Take $s_0=1,\ v_0=1,\ h=G=0$, and $M_n(s)=n^{-1}\rho(s/n)$. Show that the generators converge locally to zero and all evolutions have norm one, but their amplitudes do not converge to that of the limiting zero generator. Identify the missing uniform hypothesis of Theorem 5.1. These are general scalar evolutions, not the full factored systems of (1).

Solution 4. For each fixed $S$, $M_n=0$ on $[1,S]$ once $n>S$, so local operator convergence is exact. The real generators give

Their norms are one and their adjoint defects are zero. For $s\ge2n$, the integral equals one, hence $v_{\infty,n}=e^{i}$. The limiting zero generator has constant solution and amplitude $1$.

The uniform bound on the phase-corrected tail in (24) is missing. At a point $s=nu$ with $\rho(u)\ne0$, a bound

$|M_n(s)|\le Cs^{-1-\delta}$ would require

$C\ge |\rho(u)|u^{1+\delta}n^\delta$, which is impossible with one $C$. Thus local convergence and a uniform energy bound alone cannot justify a limit at infinity.

Exercise 5 — Advanced: the local homogeneous amplitude map is invertible. With $h=0$, define $Fv_0=v_\infty$ by Theorem 4.1. Prove that $F:\mathcal H\to\mathcal H$ is a bounded isomorphism. Use backward evolution from the terminal value $e^{iG(t,D_z)}a$, initially for $a\in\mathcal H_1$.

Solution 5. The propagator and its inverse have common bound $C$, so

$\|U(s,s_0)v_0\|\ge C^{-1}\|v_0\|$. Phase multiplication is unitary; strong convergence therefore gives

$\|Fv_0\|\ge C^{-1}\|v_0\|$. Together with (20), this proves boundedness, injectivity and closed range.

Fix $a\in\mathcal H_1$. For terminal time $t$, put

$v^t(t)=e^{iG(t,D_z)}a$ and evolve backward. Formula (15) gives

$A_j(t)v^t(t)=e^{iG(t,D_z)}z_ja$.

Lemma 2.1 in the backward orientation bounds its whole coordinate vector by $CW(a)$ at all earlier times. The homogeneous equation and (15) consequently give

Let $u_t=v^t(s_0)$. For $t_2>t_1$, apply (30) to $v^{t_2}$ at $t_1$. Comparing its value there with the terminal value of $v^{t_1}$, and evolving their difference backward, gives

$\|u_{t_2}-u_{t_1}\|\le Ct_1^{-\delta}W(a)$. Thus $u_t$ is Cauchy. Write its limit as $u$; the terminal norm and the backward energy estimate give $\|u\|\le C\|a\|$.

At each fixed $s$, the forward solutions with initial data $u_t$ converge to that with data $u$. Passing to the limit in (30) shows that its phase-corrected solution tends to $a$. Hence $Fu=a$. The range contains the dense space $\mathcal H_1$ and is closed, so it is all of $\mathcal H$. The inverse bound follows from the lower bound above.

For $d=0$, the same argument has no coordinates; directly $Fv_0=e^{-iG(s_0)}v_0$. This local transverse isomorphism concerns the first-order channel equation. Establishing asymptotic completeness for the original global differential operator also requires its spectral and channel assembly.

References

• [Y] Dmitri Yafaev, Lectures on scattering theory, lecture notes prepared by Andrew Hassell, arXiv:math/0403213v1, 12 March 2004. Free lecture paper.
• [T] Gerald Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, freely readable author edition of the second edition, 2014, Chapter 12. Free author PDF.
• [O] Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, author's preliminary version, 2012. Theorem 2.5 and Corollary 2.6, pp. 40–41, give Picard iteration; Lemma 2.7, pp. 42–43, gives the integrating-factor estimate. Author's online edition.
• [H] Lars Hörmander, “The existence of wave operators in scattering theory,” Mathematische Zeitschrift 146 (1976), 69–91. Digitized paper.