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The model's stationary scattering wave, its outgoing amplitude and the differential and total cross-section readings

Physical scope

The declared short-range scattering model with a real bounded compactly supported potential, in the pair's stationary elastic beam reading at fixed wave number and incident direction: existence and uniqueness of the outgoing stationary solution on the interaction region, its local regularity, its uniform far-field expansion with a smooth amplitude and O(r^-2) remainder, the scattered radial current, and the resulting differential and total cross-section readings in m^2 per steradian and m^2. Conditional on the declared model and the cited mathematics; the stationary wave is not a state-space element and the item asserts no preparation, no detector acceptance, no count rate and no measured cross section.

Statement

Adopt the declared short-range scattering model The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units with mass $m > 0$ , spin zero and action constant $ℏ > 0$ , with a compact-support-class potential $V$ supported in the ball $B_{R}$ , and adopt the declared stationary elastic beam model The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections at wave number $k > 0$ , direction $d \in S^{2}$ and the declared $n_{i n c} > 0$ in $m^{- 3}$ , with incident energy $E = ℏ^{2} k^{2} / (2 m)$ . The construction below retains the dimensionless unit-incident shape $Ψ$ ; its physical beam field is $Φ = \sqrt{n_{i n c}} Ψ$ . Put $W : = \frac{2 m}{ℏ^{2}} V, G_{k} (z) : = \frac{e^{i k ∣ z ∣}}{4 π ∣ z ∣}, (K_{k} f) (x) : = \int_{B_{R}} G_{k} (x - y) W (y) f (y) d y$ for $f \in L^{2} (B_{R}; C)$ and $x \in B_{R}$ . Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Then:

(a) $I + K_{k}$ is boundedly invertible on $L^{2} (B_{R}; C)$ ; with $u : = (I + K_{k})^{- 1} (e^{i k d \cdot x} ∣_{B_{R}})$ and $Ψ (x) : = e^{i k d \cdot x} - \int_{B_{R}} G_{k} (x - y) W (y) u (y) d y,$ one has $Ψ \in H_{l o c}^{2} (R^{3})$ , $(- Δ + W - k^{2}) Ψ = 0$ distributionally on $R^{3}$ , and $Ψ ∣_{B_{R}} = u$ ;

(b) with $r = ∣ x ∣ > 2 R$ and $Ω = x / r$ , the scattering amplitude $f_{k} (Ω) : = - \frac{1}{4 π} \int_{B_{R}} e^{- i k Ω \cdot y} W (y) u (y) d y$ is smooth on $S^{2}$ and $Ψ (x) = e^{i k d \cdot x} + \frac{e^{i k r} f_{k} (Ω)}{r} + E (x),$ where $∣ E (x) ∣ + ∣ \partial_{r} E (x) ∣ \leq C (k, R, ∥ W u ∥_{1}) r^{- 2}$ uniformly in $Ω$ , and for every fixed product $D_{Ω}$ of smooth tangential differential operators on $S^{2}$ also $∣ D_{Ω} E ∣ + ∣ D_{Ω} \partial_{r} E ∣ \leq C_{D} r^{- 2}$ ;

(c) the physical scattered radial current of the beam model, with $Φ_{s c} = \sqrt{n_{i n c}} (Ψ - e^{i k d \cdot x})$ , is $j_{r}^{s c} (r, Ω) = n_{i n c} (\frac{ℏ k}{m} \frac{∣ f_{k} (Ω) ∣^{2}}{r^{2}} + O (r^{- 3}))$ uniformly in $Ω$ , with physical incident flux density $J_{i n c} = n_{i n c} ℏ k / m$ in $m^{- 2} s^{- 1}$ ; the common density factor cancels, and the cross-section readings are $\frac{d σ}{d Ω} (Ω) = ∣ f_{k} (Ω) ∣^{2} in m^{2} s r^{- 1}, σ = \int_{S^{2}} ∣ f_{k} (Ω) ∣^{2} d Ω < \infty in m^{2};$

(d) the solution $Ψ$ of (a) is a generalized stationary solution of the beam model at energy $E$ and direction $d$ in the sense of clause (c) of The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections, and no other function satisfying the same equation on the complement of $supp V$ and the same outgoing reading is produced by the construction.

Every assertion is conditional on the declared model and the cited mathematics. The stationary wave is not an element of the state space; the item asserts no preparation, no beam apparatus, no detector acceptance and no measured count rate.

Facts & Assumptions

Given: the declared model and beam model with $m, ℏ > 0$ , compact-support-class $V$ supported in $B_{R}$ , wave number $k > 0$ , direction $d$ , density scale $n_{i n c} > 0$ , reduced potential $W = (2 m / ℏ^{2}) V$ , kernel $G_{k}$ , operator $K_{k}$ and choice hypotheses.

[F1]

Stationary outgoing integral and flux. Assume Countable Choice. Let $R > 0$ , real $V_{0} \in L^{\infty}$ with support in $B_{R}$ and $k > 0$ ; with $K_{k}$ built from $V_{0}$ and $G_{k}$ as above (with $V_{0}$ in place of $W$ ), the operator $I + K_{k}$ is boundedly invertible, $u = (I + K_{k})^{- 1} e_{i n c}$ , $Ψ = e_{i n c} - \int G_{k} (\cdot - y) V_{0} (y) u (y) d y$ satisfies $Ψ \in H_{l o c}^{2}$ , $(- Δ + V_{0} - k^{2}) Ψ = 0$ distributionally and $Ψ ∣_{B_{R}} = u$ ; the amplitude $f_{k} (Ω) = - (4 π)^{- 1} \int e^{- i k Ω \cdot y} V_{0} (y) u (y) d y$ is smooth with the displayed uniform far-field and derivative bounds, and the scattered radial current and cross-section displays hold (Stationary outgoing integral and flux clauses (a), (b), (c)).

[F2]

Fredholm invertibility. Assume Countable Choice. For real bounded compactly supported $V_{0}$ and every $k > 0$ , the operator $I + K_{k}$ of [F1] is boundedly invertible (Outgoing Fredholm invertibility).

[F3]

Scaling correspondence. With $c = ℏ^{2} / (2 m)$ , the model's energy operator satisfies $H = c A$ , $A = - Δ_{w} + M_{W}$ with $W = 2 m V / ℏ^{2}$ ; the physical energy $E$ and the energy-divided spectral value $k^{2}$ in $m^{- 2}$ correspond by $E = c k^{2}$ , i.e. $E = ℏ^{2} k^{2} / (2 m)$ (Unit, time and spectral correspondence between the physical scattering operators and the nondimensional Laplacian clauses (a), (b)).

[F4]

Unit-shape and physical current readings. The mathematical current functional is $j^{u n i t} [ψ] = (ℏ / m) Im (\overline{ψ} \nabla ψ)$ . For the unit incident shape its value is $(ℏ k / m) d$ , and for the exact outgoing shape $f e^{i k r} / r$ its radial value is $(ℏ k / m) ∣ f ∣^{2} / r^{2}$ (The three-dimensional continuity identity, the vanishing stationary divergence and the incident and outgoing current readings clause (c)). For the declared constant $n_{i n c} > 0$ , the beam model reads the physical field as $\sqrt{n_{i n c}} ψ$ , so the quadratic functional gives $j [\sqrt{n_{i n c}} ψ] = n_{i n c} j^{u n i t} [ψ]$ , and its cross sections are ratios with $J_{i n c} = n_{i n c} ℏ k / m$ (The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections (b)-(d)).

[F5]

The beam model's solubility vocabulary. A generalized stationary solution belongs to $H_{l o c}^{2} (R^{3})$ , has $V Ψ \in L_{l o c}^{2}$ , satisfies the stationary equation distributionally on $R^{3}$ , and is $C^{2}$ on the complement of the essential closed support of $V$ . On this exterior it satisfies the free equation $- ℏ^{2} Δ Ψ / (2 m) = E Ψ$ pointwise; the equation with the declared $V$ holds almost everywhere, or pointwise after replacing $V$ by its zero exterior representative. Its outgoing reading has the specified amplitude and remainder bounds (The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections clause (c)).

Proof

technique · direct; the physical stationary equation divided by $c=\hbar^{2}/(2m)$ is the energy-divided equation in physical coordinates with potential $W=W_x$, so the cited mathematical stationary theorem applies verbatim and its amplitude and flux displays are exactly the model's readings

Given: the data and hypotheses of the statement and [F1]-[F5].

1.1F3givenalgebra

Since $V$ is real, bounded and supported in $B_{R}$ , so is $W = 2 m V / ℏ^{2}$ , which is read in $m^{- 2}$ ; dividing the physical stationary equation $- \frac{ℏ^{2}}{2 m} Δ Ψ + V Ψ = E Ψ$ by $c = ℏ^{2} / (2 m)$ gives $- Δ Ψ + W Ψ = (E / c) Ψ$ , and $E / c = ℏ^{2} k^{2} / (2 m) \cdot 2 m / ℏ^{2} = k^{2}$ by [F3] with $E = ℏ^{2} k^{2} / (2 m)$ .

2.1F2F1step 1.1algebra

The kernel $G_{k} (x - y) W (y)$ on $B_{R} \times B_{R}$ is square-integrable because $∣ x - y ∣^{- 2}$ is locally integrable in three dimensions and $B_{R}$ is bounded, so $K_{k}$ is Hilbert-Schmidt and [F2] gives bounded invertibility of $I + K_{k}$ ; [F1] then gives $Ψ \in H_{l o c}^{2}$ , the distributional equation $(- Δ + W - k^{2}) Ψ = 0$ and $Ψ ∣_{B_{R}} = u$ , which is (a).

3.1F1step 2.1

By [F1] the smooth amplitude $f_{k} (Ω) = - (4 π)^{- 1} \int e^{- i k Ω \cdot y} W (y) u (y) d y$ and the uniform expansion with the $r^{- 2}$ remainder and its tangential and radial derivative bounds hold; these are exactly the displays of (b), with the constant depending on $k, R$ and $∥ W u ∥_{1}$ .

4.1F1F4step 3.1algebra

By [F1] the unit-shape scattered field $Ψ_{s c} = Ψ - e^{i k d \cdot x}$ has radial mathematical current $(ℏ k / m) ∣ f_{k} ∣^{2} / r^{2} + O (r^{- 3})$ uniformly. Write $Ψ_{s c} = u + i v$ ; the physical field has real and imaginary parts $\sqrt{n_{i n c}} u, \sqrt{n_{i n c}} v$ . Since the scale is constant, its current is $(ℏ / m) [(\sqrt{n_{i n c}} u) \nabla (\sqrt{n_{i n c}} v) - (\sqrt{n_{i n c}} v) \nabla (\sqrt{n_{i n c}} u)] = n_{i n c} (ℏ / m) (u \nabla v - v \nabla u)$ by [F4]. Thus the physical scattered current is the display in (c), and the same calculation on the unit incident plane wave gives $J_{i n c} = n_{i n c} ℏ k / m > 0$ . Dividing yields $\lim_{r \to \infty} r^{2} j_{r}^{s c} / J_{i n c} = ∣ f_{k} ∣^{2}$ . Smoothness and boundedness of $f_{k}$ on the sphere, with [F1]'s uniform error, give the finite total cross section $σ = \int_{S^{2}} ∣ f_{k} ∣^{2} d Ω$ . The unit-shape amplitude stays in metres and the density cancels from both cross-section readings. This is (c).

5.1F3F5step 2.1step 3.1algebra∎

By step 2.1 the constructed $Ψ$ lies in $H_{l o c}^{2}$ and satisfies the global distributional stationary equation after the scaling of [F3]; boundedness of $V$ gives $V Ψ \in L_{l o c}^{2}$ . On the complement of its essential closed support, the convolution integrates the compactly supported $L^{1}$ density $W u$ at a positive distance from the kernel singularity locally, so differentiation under the integral gives a smooth field. Here $V = 0$ almost everywhere; thus the continuous free-equation residual vanishes distributionally and hence pointwise, as in [F5]. The equation with the declared representative $V$ therefore holds almost everywhere there, and pointwise for its zero exterior representative. Step 3.1 gives the outgoing reading and all required remainder bounds. These facts verify the beam model's generalized stationary solution conditions. The invertible integral problem uniquely fixes the constructed $u$ and hence $Ψ$ , which proves the construction-specific uniqueness in (d) and completes the proof.

Remarks

What is read, and what is not. The amplitude and cross sections are computed from the explicitly constructed stationary field; the pair asserts nothing about a preparation that would realize the incident plane-wave reading, and no measured count rate is predicted. Converting these model quantities into an expected count rate requires a declared target, fluence and detector model with its own uncertainties.
Normalization. With physical coordinates $x$ in metres and $k$ in inverse metres, the unit-incident shape $Ψ$ is dimensionless, the reduced potential $W = (2 m / ℏ^{2}) V$ is in $m^{- 2}$ , and the displayed integral for $f_{k}$ has dimension length. The physical beam field $\sqrt{n_{i n c}} Ψ$ is in $m^{- 3 / 2}$ ; its number currents are in $m^{- 2} s^{- 1}$ . The common density factor cancels, leaving $d σ / d Ω$ in $m^{2} s r^{- 1}$ independently of its value.

The model's stationary scattering wave, its outgoing amplitude and the differential and total cross-section readings

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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