TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Exponential coefficient decay gives a uniform partial-wave truncation bound

Statement

Assume Countable Choice and Dependent Choice. Let $b_{ℓ} \in H_{ℓ}$ for every integer $ℓ \geq 0$ , where $H_{ℓ}$ is defined in Spherical harmonics and partial-wave conventions in three dimensions. Suppose constants $C \geq 0$ and $ρ > 1$ satisfy $∥ b_{ℓ} ∥_{L^{2} (S^{2})} \leq C ρ^{- ℓ}$ for all $ℓ$ . Then $b (ω) : = \sum_{ℓ = 0}^{\infty} b_{ℓ} (ω)$ converges uniformly on $S^{2}$ , and for every integer $L \geq 0$ , $\sup_{ω \in S^{2}} ∣ b (ω) - \sum_{ℓ = 0}^{L} b_{ℓ} (ω) ∣ \leq C \sqrt{\frac{2 L + 3}{4 π}} \frac{ρ^{- (L + 1)}}{(1 - ρ^{- 1})^{2}} .$ This is a conditional angular-coefficient bound; it does not assert exponential decay for general scattering data.

Facts & Assumptions

Given: The harmonic spaces, constants $C \geq 0$ , $ρ > 1$ , and coefficient bounds above.

[F1]

Spherical harmonics form a complete basis with the addition formula: for $g \in H_{ℓ}$ , $∥ g ∥_{\infty} \leq \sqrt{(2 ℓ + 1) / (4 π)} ∥ g ∥_{2}$ .

Proof

Given: The data in the Statement.

1.1F1givenalgebra

By [F1], $∥ b_{ℓ} ∥_{\infty} \leq C \sqrt{(2 ℓ + 1) / (4 π)} ρ^{- ℓ}$ . The scalar series of these bounds converges because $ρ > 1$ , so the partial sums of $\sum b_{ℓ}$ are uniformly Cauchy and converge uniformly to a function $b$ .

2.1givenstep 1.1algebra∎

For $ℓ = L + 1 + n$ with $n \geq 0$ , $\sqrt{2 ℓ + 1} \leq (n + 1) \sqrt{2 L + 3}$ : squaring reduces this to $2 L + 3 + 2 n \leq (2 L + 3) (n + 1)^{2}$ , whose difference is nonnegative. Therefore the tail in step 1.1 is at most $C \sqrt{(2 L + 3) / (4 π)} ρ^{- (L + 1)} \sum_{n = 0}^{\infty} (n + 1) ρ^{- n}$ . For $t = ρ^{- 1} \in (0, 1)$ , multiplying the geometric series by itself or differentiating its partial sums gives $\sum_{n \geq 0} (n + 1) t^{n} = (1 - t)^{- 2}$ . This yields the claimed bound.

Depends on

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Sources

Pierre Capel, Introduction to Scattering Theory, TALENT lecture notes (standard reference, not scraped)
Maximilian Kreuzer, Quantum Theory: Lecture Notes, version July 15, 2009, Chapter 8 (standard reference, not scraped)