# Energy-shell factors and outgoing equations *Written by GPT-6.1 Sol (OpenAI) and GPT-6 Astra (OpenAI). Self-checked by the writing AI. Original exposition: CC0.* **Working question: Why does factorizing an energy graph leave an outgoing choice?** A first-order factor has homogeneous solutions as well as a particular solution. The outgoing condition supplies the direction of integration, just as in the transport model, but here the solution may initially have only endpoint ball bounds. The proof has to recover genuine Hilbert-space slices rather than assuming them at the start. Near a regular energy surface, one frequency is a smooth function of the others. A small long-range perturbation moves that graph. Dividing by the graph factor suggests a first-order equation in one spatial coordinate. Such an equation still has many homogeneous solutions. An outgoing condition selects one of them, even when the initial information permits squared mass to grow proportionally to the radius of a ball. We first prove the outgoing evolution estimate directly from its kernel. We then construct the perturbed energy graph and control every derivative of its polynomial factor and reciprocal. Read [Endpoint spaces and flat energy shells](endpoint-spaces-and-flat-energy-shells.md) for the spatial norms, and [Admissible differential perturbations](admissible-differential-perturbations.md) for the differential coefficient class. The [measure and Euclidean product proofs](../providers/analysis/finite-derivative-l2.md#measure-foundations) give the scalar integral, convergence and slice facts; the same reading proves [Fourier inversion and Plancherel](../providers/analysis/finite-derivative-l2.md#fourier-normalization). The [coordinate inverse and bump proofs](../providers/analysis/coordinate-inverses-and-integration.md#coordinate-inverse) supply the regular free graph and smooth cutoffs. Section 3 constructs the vector integrals used here. Freely accessible background references are Teschl's evolution and initial-value texts [T, ODE] and Yafaev's *Lectures on scattering theory* [Y]. The evolution, rough-slice recovery, root and quotient proofs are given below. ## 1. What an outgoing condition must control Write \(x=(s,z)\in\mathbb R\times\mathbb R^d\), where \(d=n-1\), and put \(\mathcal H=L^2(\mathbb R^d)\). For \(d=0\), use \(\mathcal H=\mathbb C\). Our convention is \(D_s=-i\partial_s\). Set \[ \begin{gathered} r_j=2^j,\qquad S_0=\{|x|<1\},\\ S_j=\{2^{j-1}\le |x|<2^j\}\quad(j\ge1),\\ \|f\|_B=\sum_{j\ge0}r_j^{1/2}\|f\|_{L^2(S_j)},\\ \|v\|_{B^*}=\sup_{j\ge0}r_j^{-1/2}\|v\|_{L^2(S_j)}. \end{gathered} \tag{1} \] Shells and balls give the equivalent bound \[ \int_{|x|d\), bounds \(A(s)\) uniformly. Reality of \(b\) identifies the kernel of \(A(s)-A(s)^*\) as the inverse Fourier integral of \(b(s,z,\eta)-b(s,y,\eta)\). The segment formula and (5), including its differentiated versions, bound that amplitude's \(\eta\) derivatives by \(C_\alpha w(s)|z-y|\). Repeating the integration by parts gives \[ \begin{aligned} &|K_s(z,y)-\overline{K_s(y,z)}|\\ &\le C_N w(s)|z-y|(1+|z-y|)^{-N},\\ &\|A(s)-A(s)^*\|_{\mathcal H\to\mathcal H}\\ &\le Cw(s). \end{aligned} \tag{9} \] Take \(N>d+1\) for the second line. A real left symbol need not give a self-adjoint operator. What we have proved is an integrable bound on its failure of self-adjointness. ## 3. Constructing the outgoing evolution **Theorem 3.1.** Under the hypotheses of Section 2, let \(g\in L^1(\mathbb R;\mathcal H)\). There is a solution of \[ (D_s-A(s))v=g \tag{10} \] with a continuous, locally absolutely continuous \(\mathcal H\)-valued representative, and \[ \|v(s)\|_{\mathcal H} \le C\int_{-\infty}^s\|g(t)\|_{\mathcal H}\,dt. \tag{11} \] Every distributional solution \(v\in B^*\) of (10) satisfying (3) has this representative and bound. In particular it is unique among the \(B^*\) solutions satisfying (3). The constant depends on the symbol bounds and \(\int w\), and is independent of \(g\). Here are the integration facts needed for the full \(L^1\) forcing class. The space \(\mathcal H=L^2(\mathbb R^d)\) is separable: the Euclidean rectangle-approximation proof makes step functions on finitely many rational boxes with rational real and imaginary values dense. In dimension zero use \(\mathbb C\). A strongly measurable \(\mathcal H\)-valued function with integrable norm is approximable in integral norm by finite-valued simple functions. Explicitly, first restrict to a bounded time interval and bounded values, losing arbitrarily little norm integral. Cover its values by balls of radius \(1/j\) centered in a countable dense subset, assign each value to the first such ball, and then keep finitely many of the resulting measurable sets. The discarded tail has vanishing measure; the values on the retained interval are bounded. This gives finite simple approximations with integral error tending to zero. For a simple function \(F=\sum_k h_k1_{E_k}\) on disjoint finite-measure sets define \(\int F=\sum_k |E_k|h_k\). The triangle inequality gives \(\|\int F\|\le\int\|F\|\), so completion defines the integral for the preceding \(L^1\) class and makes it independent of the approximation. Pairing with a fixed vector, or applying a bounded operator, commutes with the integral by the same norm estimate. In particular the primitive \(G(s)=\int_a^s F(t)\,dt\) is locally absolutely continuous: sums of its increments over disjoint intervals are bounded by the norm integral over their union, which tends to zero with that union's measure by truncating the integrable scalar function \(\|F\|\). Testing against a compact smooth scalar function and using scalar Fubini proves \(G'=F\) as a vector distribution. These facts also construct improper integrals whenever the norm has an integrable tail. For continuous operator-valued functions on compact intervals use Riemann sums in operator norm, as in the [continuous-vector integral proof](../providers/analysis/euclidean-approximation-and-convolution.md#oscillatory-integrals-and-averages). The same uniform-continuity estimate applies in the complete normed space of bounded operators: an operator-norm Cauchy sequence has a pointwise vector limit which is linear and bounded, and convergence to it is in operator norm. Thus no operator measurability assumption beyond the stated norm continuity is required. We begin with existence. Since \(\sup_s\|A(s)\|\le M\), iteration of \[ U(s,t)=I+i\int_t^s A(r)U(r,t)\,dr \tag{12} \] converges in operator norm on every bounded time interval. Its term of order \(k\) has norm at most \(M^k|s-t|^k/k!\). The same construction works for either orientation of the integral. Iterating the inequality for the difference of two solutions with zero initial value proves uniqueness of (12). Consequently \[ \begin{gathered} U(s,t)U(t,r)=U(s,r),\\ \qquad U(t,s)=U(s,t)^{-1}. \end{gathered} \tag{13} \] For \(u(s)=U(s,t)h\), its equation is \(\partial_su=iA(s)u\). The inner product is linear in the first variable. Equation (9) gives \[ \left|\frac{d}{ds}\|u(s)\|^2\right| \le Cw(s)\|u(s)\|^2. \tag{14} \] For \(\varepsilon>0\), divide (14) by \(\|u(s)\|^2+\varepsilon\). The absolute derivative of \(\log(\|u(s)\|^2+\varepsilon)\) is at most \(Cw(s)\). Integrate between the two times in either order, exponentiate and let \(\varepsilon\) decrease to zero. This proves \[ \begin{gathered} \|U(s,t)\| \\ \le \exp\!\left(\frac C2\int_{\mathbb R}w(r)\,dr\right) \\ =:C_0 \\ \quad(s,t\in\mathbb R). \end{gathered} \tag{15} \] Define \[ \begin{aligned} v(s)&=i\int_{-\infty}^s U(s,t)g(t)\,dt,\\ q(s)&=\int_{-\infty}^s\|g(t)\|\,dt. \end{aligned} \tag{16} \] The integral converges in \(\mathcal H\) by (15), and \(\|v(s)\|\le C_0q(s)\). Its integrand is strongly measurable: approximate \(g\) by simple functions; on every bounded time interval the operator factor is norm continuous, and each resulting vector factor is strongly measurable. Splitting at a finite time \(a\) and using (13) gives \[ v(s)=U(s,a)\left(v(a)+i\int_a^s U(a,t)g(t)\,dt\right). \] The bracket is locally absolutely continuous by the primitive construction, and \(U(s,a)\) is continuously differentiable in operator norm. The product is therefore locally absolutely continuous. Testing its product rule, or first using simple approximations to \(g\), gives \(\partial_sv=iAv+ig\) in distributions, including the sign in (10). The same formula gives the inhomogeneous integral equation on each finite interval. The bound and (4) give \(v\in B^*\). To prove (3), fix \(S<\min(T,0)\). For \(R\) large enough, \[ \begin{gathered} \frac1R\int_{\substack{|x|\tfrac12. \end{gathered} \tag{18} \] Indeed, the tail sum is bounded by a constant times \(\sum_j2^{j(1-2p)}\). Fubini therefore gives polynomially weighted \(L^2\) slices for almost every \(s\). Their strong measurability also follows from the rectangle proof: approximate the weighted function by finite product step functions in the space-time \(L^2\) norm and take a subsequence whose slice errors tend to zero almost everywhere. On bounded time intervals the weight is equivalent to a weight in \(z\) alone. The rapidly decreasing kernel (7) acts on these slices as a distribution and as a locally square-integrable function. Indeed, for bounded output \(z\), Cauchy–Schwarz bounds its value by the weighted slice norm times the \(L^2_y\) norm of \(K_s(z,y)(1+|y|)^p\), uniformly finite on bounded output sets by (7). Integration in bounded \(s,z\) proves local square integrability. Choose a real smooth \(\chi\), between zero and one, equal to one on the unit ball and zero outside the ball of radius two. Put \(\chi_R(x)=\chi(x/R)\) and \(v_R=\chi_Rv\). The distributional product rule gives \[ \begin{aligned} (D_s-A(s))v_R&=\chi_Rg+e_R,\\ e_R&=-iR^{-1}(\partial_s\chi)(x/R)v \\ &\quad-[A(s),\chi_R]v. \end{aligned} \tag{19} \] Define the finite quantity \[ M(r,T)=\|1_{\{|x|0\). For example, \(|y|\ge(r_j^2-4R^2)^{1/2}\ge\sqrt3\,r_j/2\) and \(|z|\le2R\le r_j/2\). For each fixed \(s\), both integrals in (8), restricted to these input and output sets, are at most \(C_Nr_j^{d-N}\). For the input integral, its volume is at most \(Cr_j^d\); for the output integral, it is at most \(CR^d\le Cr_j^d\). Equation (24) bounds the kernel by \(C_Nr_j^{-N}\). The \(j\)-th far term is consequently bounded in the slice \(L^1\) norm by \[ \begin{aligned} &C R^{1/2}r_j^{d-N} \|v1_{\{r_j<|x|\le2r_j\}}\|_2 \\ &\le C R^{1/2}r_j^{d+1/2-N}\|v\|_{B^*}. \end{aligned} \tag{25} \] Here (2) bounds the annular \(L^2\) norm by \(C r_j^{1/2}\|v\|_{B^*}\). Choose \(N=d+2\) and sum the geometric series. Its total is at most \(CR^{-1}\|v\|_{B^*}\). This also proves convergence of the kernel action on the far input. In dimension zero the commutator is identically zero. We have proved, in every dimension, \[ \|e_R\|_{L^1((-\infty,T);\mathcal H)}\longrightarrow0. \tag{26} \] Now \(v_R\) is globally \(L^2\) and has compact support. Equation (19) implies \(\partial_sv_R=iAv_R+i(\chi_Rg+e_R)\) in local \(L^1(\mathbb R;\mathcal H)\). It therefore has a locally absolutely continuous representative, as follows directly. Subtract the vector primitive of the right side constructed in Section 3, leaving a locally integrable vector distribution \(F\) with \(F'=0\). Choose \(\rho\in C_c^\infty(\mathbb R)\), \(\int\rho=1\), and set \(h=\int\rho(s)F(s)\,ds\in\mathcal H\). Every compact smooth test \(\varphi\) has \(\psi=\varphi-(\int\varphi)\rho\) of integral zero, so \(\Psi(s)=\int_{-\infty}^s\psi(t)\,dt\) is compact and smooth. The identity \(F'=0\) gives \(\int F\psi=0\), hence \(\int F\varphi=h\int\varphi\). Thus \(F=h\) as a vector distribution and almost everywhere. For the last conclusion pair with a countable dense subset of \(\mathcal H\); scalar convolution with compact smooth approximate identities and their local \(L^1\) convergence, proved in [Approximation and convolution](../providers/analysis/euclidean-approximation-and-convolution.md#mollification), identifies each scalar function almost everywhere. The common null set and density identify the vector. Adding the primitive back gives the claimed representative. This representative is zero for \(s<-2R\), since it is continuous and agrees there almost everywhere with zero. Variation of constants is valid for this rough forcing as well: multiplying the integral equation by \(U(a,s)\), or approximating its \(L^1\) right side by simple functions, gives the formula in Section 3. Uniqueness follows by iterating the homogeneous norm inequality, with bound \(M^k|s-a|^k/k!\) at its \(k\)-th iteration. Thus (15) implies, for \(s\le T\), \[ \|v_R(s)\|\le C_0q(s)+ C_0\|e_R\|_{L^1((-\infty,T);\mathcal H)}. \tag{27} \] Take \(R\) and the upper bound \(T\) through the positive integers. Outside one common null set, \(v_R(s,z)=\chi_R(s,z)v(s,z)\). Fatou's lemma in \(z\), followed by (26), proves \(\|v(s)\|_{\mathcal H}\le C_0q(s)\) for almost every \(s\). The slices now belong to \(\mathcal H\) and are uniformly bounded. Equation (10) gives a locally absolutely continuous representative, and continuity extends the inequality to every \(s\). For \(g=0\) this inequality forces \(v=0\). The difference of two \(B^*\) solutions satisfying (3) again satisfies (3), by the squared triangle inequality. Thus uniqueness follows. Solutions obeying (11) satisfy (3) by (17), so the existence statement is unique in its asserted class as well. This completes Theorem 3.1. \(\square\) ## 5. Moving a regular energy graph Let \(P(\xi)\) be a real elliptic polynomial of degree \(m\). Fix an energy \(\lambda\) and a regular graph patch \[ \begin{gathered} \xi=(\xi_1,\eta),\\ \qquad P(E(\eta),\eta)=\lambda,\\ \qquad \partial_{\xi_1}P>0. \end{gathered} \tag{28} \] Work on a compact set of \(\eta\) contained in an open graph patch. Shrink a collar around the graph so that the positive derivative has a fixed lower bound there. Let \(L(x,\xi)\) be a real polynomial in \(\xi\), of total degree at most \(m\), with smooth coefficients \(\ell_\gamma(x)\). Fix \(0<\delta<1/3\), put \(X=(1+|x|^2)^{1/2}\), and define the continuous concave function \[ \mu(t)= \begin{cases} \delta+t,&0\le t\le2,\\ 1+(1+\delta)t/2,&t\ge2. \end{cases} \tag{29} \] Assume the full bounds \[ |\partial_x^\beta\ell_\gamma(x)| \le C_{\gamma,\beta}X^{-\mu(|\beta|)} \tag{30} \] for every \(\beta\). Suppose also that \(L=0\) on \(|x|0\), increasing \(R_0\) makes \(F\) map a fixed small closed \(h\) interval into itself and makes \(|F_h|\le1/2\), uniformly in \(x,\eta\). Iteration is a contraction and produces the unique small root. Here is smooth dependence before differentiating that root. Put \(p=(x,\eta)\). On every compact parameter neighborhood, subtracting two fixed-point equations gives \(|h(p+q)-h(p)|\le2C|q|\). Taylor's segment formula then yields \[ \begin{gathered} \Delta h=h(p+q)-h(p),\\ j_p=1-F_h(p,h(p)),\\ j_p\Delta h=D_pF(p,h(p))q+o(|q|). \end{gathered} \] The coefficient has absolute value at least \(1/2\), so this proves differentiability and the continuous derivative formula \(D_ph=(1-F_h)^{-1}D_pF\). Inductively differentiating that formula proves smoothness at every finite order. Uniqueness of the small root makes these local constructions agree. For the derivative estimates, concavity gives, whenever \(b_0,\ldots,b_q\ge0\), \[ \sum_{j=0}^q\mu(b_j) \ge \mu\!\left(\sum_{j=0}^q b_j\right)+q\delta. \tag{34} \] To see this, fix their sum \(B>0\). A point in the simplex is the convex combination, with weights \(b_j/B\), of its vertices. Concavity of the sum bounds its value below by the common vertex value \(\mu(B)+q\mu(0)\). For \(B=0\) the inequality is equality. At order zero, (33) gives (31). Induct in total mixed derivative order. Differentiating the identity produces the term \(F_h\partial_\eta^\alpha\partial_x^\beta h\), which we move to the left. Every remaining finite chain term is either a derivative of \(F\) alone, or a derivative of \(F\) multiplied by \(q\ge1\) lower-order derivatives of \(h\). Their physical orders \(b_0,\ldots,b_q\) sum to \(|\beta|\). The induction hypothesis and (34) bound the term by \(C X^{-\mu(|\beta|)}\), with additional decay when \(q>0\). Division by \(1-F_h\), whose absolute value is at least \(1/2\), proves (31) at the next order. This includes all frequency derivatives. \(\square\) ## 6. Dividing the polynomial without losing decay Ellipticity implies that the homogeneous degree-\(m\) part of \(P\), evaluated at the first coordinate vector, is nonzero. Thus \(P\) has degree \(m\) in \(\xi_1\), with a nonzero constant leading coefficient. For \(R_0\) large the corresponding coefficient of \(L\) is uniformly small. Polynomial division therefore gives \[ \begin{aligned} P(\xi)-\lambda&=(\xi_1-E(\eta))Q_0(\xi),\\ P(\xi)+L(x,\xi)-\lambda&= (\xi_1-a(x,\eta))Q(x,\xi). \end{aligned} \tag{35} \] Both quotients have degree \(m-1\) in \(\xi_1\). On the chosen collar \(Q_0=J(\xi_1-E(\eta),\eta)\), so it is bounded away from zero. **Proposition 6.1.** On every fixed compact collar as above, every mixed derivative satisfies \[ \begin{aligned} |\partial_\xi^\alpha\partial_x^\beta(Q-Q_0)| &\le C_{\alpha,\beta}X^{-\mu(|\beta|)},\\ |\partial_\xi^\alpha\partial_x^\beta (Q^{-1}-Q_0^{-1})| &\le C'_{\alpha,\beta}X^{-\mu(|\beta|)}. \end{aligned} \tag{36} \] The second assertion holds after increasing \(R_0\) if necessary. **Proof.** Subtract the first factorization in (35), using the second root: \[ \begin{aligned} L+Q_0(a-E)&=\sum_{j=0}^m r_j(x,\eta)\xi_1^j,\\ Q-Q_0&=\sum_{j=0}^{m-1}q_j(x,\eta)\xi_1^j. \end{aligned} \tag{37} \] Every derivative of each \(r_j\), with physical order \(|\beta|\), has the first bound in (36), by (30), (31) and the product rule. Coefficient comparison gives \[ \begin{gathered} q_{m-1}=r_m,\\ \qquad q_{j-1}=r_j+a q_j\quad(1\le j1\). For fixed \(T\), only a bounded \(s\) interval contributes to (3), with a bounded slice norm; its quotient tends to zero. The strip \(|s|\le1\) likewise contributes \(O(1/R)\) to the total quotient. The part \(s>1\) gives \[ \begin{aligned} &|c|^2\int_{|z|1/2\), so both quotient denominators have a fixed positive lower bound. Theorem 5.1 and Proposition 6.1 apply directly. Alternatively, differentiate the first formula in (45): the square-root arguments stay in a fixed positive interval, and every physical chain term contains derivatives of \(\ell\). Equation (34) bounds their products by the required \(X^{-\mu(|\beta|)}\). Frequency derivatives have fixed bounded factors on the patch. The remaining formulas give the same estimates by the product and reciprocal rules. **Exercise 4 — Advanced: why reality is not operator symmetry.** In one transverse dimension take \(b(s,z,\eta)=q(s)a_0(z)\psi(\eta)\), with real smooth compactly supported \(a_0,\psi\), and \(q(s)=(1+s^2)^{-(1+\delta)/2}\). Show that the adjoint defect has integrable norm. Give choices for which it is nonzero. Derive the \(C/R\) bound on its commutator with \(\chi_R\). **Solution 4.** If \(k(w)=(2\pi)^{-1}\int e^{iw\eta}\psi(\eta)\,d\eta\), the kernel is \(q(s)a_0(z)k(z-y)\). Reality of \(\psi\) gives \(\overline{k(-w)}=k(w)\), so the adjoint-difference kernel is \[ q(s)\bigl(a_0(z)-a_0(y)\bigr)k(z-y). \tag{46} \] Since \(a_0\) is Lipschitz and \(k\) is rapidly decreasing, (8) bounds the norm by \(C|q(s)|\). Its integral is finite because \(q(s)=O(|s|^{-1-\delta})\). For a concrete nonzero defect, choose a nonnegative, even, nonzero \(\psi\); then \(k(0)>0\) and \(k\) is nonzero near zero. Choose \(a_0\) with \(a_0'(z_0)\ne0\). Nearby distinct \(z,y\) give a nonzero continuous kernel in (46), so the operator is nonzero: tests supported in sufficiently small neighborhoods of these points give a nonzero pairing. Finally the commutator kernel is \(q(s)a_0(z)k(z-y)(\chi_R(s,y)-\chi_R(s,z))\). The difference is bounded by \(C|z-y|/R\). Both absolute kernel integrals are bounded by \(C/R\), uniformly in \(s\), and (8) gives that operator norm. **Exercise 5 — Advanced: the larger forcing class is necessary.** Choose \(h\in C_c^\infty(\mathbb R^d)\), supported in \(|z|<1\), with \(\|h\|_2=1\), and set \[ \begin{aligned} \rho(s)&=\sum_{j\ge1}\frac{2^{-j}}j 1_{[3\cdot4^j,\ 3\cdot4^j+1]}(s),\\ g(s,z)&=\rho(s)h(z). \end{aligned} \tag{47} \] Show that \(g\in L^1(\mathbb R;L^2_z)\) but \(g\notin B\). For \(A=0\), verify the outgoing solution's existence, bounded slice norm and condition (3). In dimension zero use \(h=1\). **Solution 5.** The slice integral is \(\sum_{j\ge1}2^{-j}/j<\infty\). Each pulse is wholly inside the shell with outer radius \(2^{2j+2}=4\cdot4^j\) and inner radius \(2\cdot4^j\); the fixed transverse support and interval of length one preserve both strict inequalities. Distinct pulses occupy distinct shells. That shell's \(L^2\) norm is \(2^{-j}/j\), so its contribution to the \(B\) norm is \(2^{j+1}(2^{-j}/j)=2/j\). The sum diverges. Nevertheless \(v(s,z)=i h(z)\int_{-\infty}^s\rho(t)\,dt\) is continuous, locally absolutely continuous, and solves \(D_sv=g\). Its slice norm is at most \(\sum_j2^{-j}/j\), so (4) puts it in \(B^*\). For each fixed \(T\), its nonzero slices with \(s