Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The resonance model's phase-shift law, unitarity and channel cross section

Physical scope

The declared single-channel resonance model with every real resonance-centre parameter E_r and positive width Gamma: unitarity and a nonsingular phase law at positive physical energies, the full-real-axis auxiliary Lorentzian factor and its exact half-height width, and the positive-energy channel cross section with its resonance-centre value when E_r>0. Half-height locations have a physical reading only when positive. Conditional algebra of a declared model; no producing potential, decay law, lifetime or measured width or count is asserted.

Statement

Adopt the declared single-channel resonance model The declared single-channel resonance model: pole-shaped channel factor and its cross-section reading with channel label $l$ , mass $m > 0$ , action constant $ℏ > 0$ and parameters $E_{r} \in R$ , $Γ > 0$ , with the phase shift read modulo $π$ from $S_{l} = e^{2 i δ_{l}}$ and the channel cross-section reading $σ_{l} (E)$ of that definition. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Put $y : = Γ / 2 > 0$ . For every real $E$ define the auxiliary real-axis Lorentzian factor $L (E) : = \frac{y^{2}}{(E - E_{r})^{2} + y^{2}} .$ This is a mathematical extension of the positive-energy phase-shift factor; physical phase shifts, $k (E)$ and cross sections remain defined only for $E > 0$ . Then:

(a) for every $E > 0$ , $∣ S_{l} (E) ∣ = 1$ and $δ_{l} (E) = - Arg (E - E_{r} + i y) (m o d u l o π),$ where for $z = x + i y$ with $y > 0$ , $Arg z$ is the unique angle $θ \in (0, π)$ with $\cos θ = x / \sqrt{x^{2} + y^{2}}$ and $\sin θ = y / \sqrt{x^{2} + y^{2}}$ . Equivalently, for $E \neq E_{r}$ one may use $δ_{l} (E) = - \arctan (y / (E - E_{r}))$ modulo $π$ ; if $E = E_{r} > 0$ , $δ_{l} (E) = - π / 2$ modulo $π$ ;

(b) for every $E > 0$ , $\sin^{2} δ_{l} (E) = L (E)$ . On the entire real axis, $L (E)$ has value $1$ exactly at $E = E_{r}$ and value $1 / 2$ exactly at $E = E_{r} \pm Γ / 2$ . Thus the full-real-axis half-height width of this factor is $Γ$ for every real $E_{r}$ . A peak or half-height location is in the physical energy range only when that location is positive; the width statement itself concerns $L$ on $R$ ;

(c) for every $E > 0$ , the channel cross-section reading is $σ_{l} (E) = \frac{4 π}{k (E)^{2}} (2 l + 1) L (E), k (E) = \frac{\sqrt{2 m E}}{ℏ},$ read in $m^{2}$ . If $E_{r} > 0$ , its value at that physical energy is $σ_{l} (E_{r}) = \frac{4 π ℏ^{2}}{2 m E_{r}} (2 l + 1) .$ If $E_{r} \leq 0$ , this model defines no physical cross section at $E_{r}$ . No maximum statement at $E_{r}$ is made for $σ_{l}$ itself, whose energy dependence also contains the factor $k (E)^{- 2}$ ; no slow-variation assumption on that factor is required.

Every display is a conditional algebraic statement about the declared model; no potential producing $S_{l}$ , no decay law, no lifetime and no measured quantity is asserted.

Facts & Assumptions

Given: the declared single-channel resonance model with channel label $l$ , parameters $E_{r}$ , $Γ > 0$ , the phase-shift and cross-section readings, and the choice hypotheses.

[F1]

The declared channel factor. For $E > 0$ , $S_{l} (E) = \frac{E - E_{r} - i Γ / 2}{E - E_{r} + i Γ / 2}$ and $S_{l} = e^{2 i δ_{l}}$ modulo $π$ , with $σ_{l} (E) = \frac{4 π}{k (E)^{2}} (2 l + 1) \sin^{2} δ_{l} (E)$ and $k (E) = ℏ^{- 1} \sqrt{2 m E}$ (The declared single-channel resonance model: pole-shaped channel factor and its cross-section reading clause (b)).

[F2]

Resonance model algebra in the cited mathematics. For the single-channel model $S (E) = \frac{E - E_{r} - i Γ / 2}{E - E_{r} + i Γ / 2}$ with $Γ > 0$ , $∣ S (E) ∣ = 1$ for real $E$ and $\sin^{2} δ = (Γ / 2)^{2} / ((E - E_{r})^{2} + (Γ / 2)^{2})$ , and substituting into the phase-shift series gives the corresponding single-channel cross section (Partial-wave cross-section convergence clause (c)).

[F3]

The pair's channel reading. The channel cross-section reading of the pair is the summand of the phase-shift series of the partial-wave theorem, with $l$ , $δ_{l}$ and $k$ read as in the resonance model (The model's partial-wave and phase-shift vocabulary for a declared central potential clause (c), Partial-wave expansion, unit-modulus channel factors and the phase-shift cross-section series clause (b)).

Proof

technique · direct; compute $|S_{l}|$ and the phase of the declared factor, then substitute the declared function into the model's channel reading

Given: the data and hypotheses of the statement and [F1]-[F3].

1.1F1givenalgebra

For any real $E$ , write $x : = E - E_{r}$ and $y : = Γ / 2 > 0$ . Then $x + i y \neq 0$ and the purely algebraic ratio $R (E) : = (x - i y) / (x + i y)$ has modulus one because its numerator is the conjugate of its denominator. On the physical range $E > 0$ , [F1] identifies $R (E) = S_{l} (E)$ , proving the first part of (a). Defining $R$ as an auxiliary real-axis rational formula does not assign $k$ or a physical cross section outside the range $E > 0$ .

2.1F1step 1.1algebra

With $r : = \sqrt{x^{2} + y^{2}} > 0$ and $θ : = Arg (x + i y) \in (0, π)$ , one has $x + i y = r e^{i θ}$ and $x - i y = r e^{- i θ}$ , hence $R (E) = e^{- 2 i θ}$ . For $E > 0$ the phase $δ_{l} = - θ$ modulo $π$ therefore matches [F1]. When $x \neq 0$ , $\tan θ = y / x$ and the principal arctangent gives the same angle modulo $π$ ; when $x = 0$ , $θ = π / 2$ , so the phase is $- π / 2$ with no division by zero. This proves all of (a).

3.1F2step 1.1step 2.1algebra

For $E > 0$ , $\sin^{2} δ_{l} = \sin^{2} θ = y^{2} / (x^{2} + y^{2}) = L (E)$ , in agreement with [F2]. Independently, the real-axis formula has positive denominator for every real $E$ and satisfies $0 < L (E) \leq 1$ , with equality $L (E) = 1$ exactly when $x = 0$ . Also $L (E) = 1 / 2$ if and only if $x^{2} = y^{2}$ , so its two real half-height points are $E_{r} - y$ and $E_{r} + y$ , separated by $2 y = Γ$ . Restricting a point to the physical range requires that point to be positive; the mathematical peak and width identities impose no condition such as $E_{r} > y$ . This proves (b).

4.1F1F3step 3.1algebra∎

For $E > 0$ , substituting step 3.1 into [F1], [F3] gives $σ_{l} (E) = 4 π (2 l + 1) L (E) / k (E)^{2}$ . If $E_{r} > 0$ , this formula is defined at $E = E_{r}$ , where $L (E_{r}) = 1$ and $k (E_{r})^{2} = 2 m E_{r} / ℏ^{2}$ , yielding the displayed resonance-centre value. If $E_{r} \leq 0$ , no such physical evaluation is in the model's domain and the auxiliary $L (E_{r}) = 1$ does not create one. The extra factor $k (E)^{- 2}$ remains energy dependent, so no location-of-maximum or slow-variation assertion is used. This proves (c), retaining every real $E_{r}$ and positive $Γ$ .

Remarks

Why no peak claim. The auxiliary real-axis Lorentzian factor peaks at $E_{r}$ ; that peak is physical only if $E_{r} > 0$ . The full channel reading carries the factor $k (E)^{- 2}$ as well; the pair states the full-real-axis half-height width of the Lorentzian factor, which is exactly $Γ$ , without asserting where the maximum of $σ_{l}$ lies.
Modulo $π$ . Both $\sin^{2}$ and the cross-section reading are insensitive to the modulo- $π$ ambiguity of $δ_{l}$ .

The resonance model's phase-shift law, unitarity and channel cross section

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources