`BookProof.ChapterRadialLaplacian.helmholtz_sbessel_zero` [FiniteDimensional ℝ E] (h3 : Module.finrank ℝ E = 3) {p : ℝ} {x : E} (hp : p ≠ 0) (hx : x ≠ 0) : -(Δ fun y : E => sbessel

Proved

BookProof.ChapterRadialLaplacian.helmholtz_sbessel_zero

by leonardopedro · Oct 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterRadialLaplacian.

BookProof.ChapterRadialLaplacian.helmholtz_sbessel_zero [FiniteDimensional ℝ E] (h3 : Module.finrank ℝ E = 3) {p : ℝ} {x : E} (hp : p ≠ 0) (hx : x ≠ 0) : -(Δ fun y : E => sbessel 0 (p * ‖y‖)) x = p ^ 2 * sbessel 0 (p * ‖x‖)

Formalization note: Lean 4 identifier BookProof.ChapterRadialLaplacian.helmholtz_sbessel_zero.

Preamble

-- Generated from ChapterRadialLaplacian.lean — theorem BookProof.ChapterRadialLaplacian.helmholtz_sbessel_zero
import Mathlib
import Definitions.Def_ChapterRadialLaplacian
import Definitions.Def_ChapterSphericalBessel
open BookProof.ChapterSphericalBessel
open BookProof.ChapterRadialLaplacian



open Filter Laplacian InnerProductSpace
open scoped InnerProductSpace RealInnerProductSpace

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]

Formal statement

theorem BookProof.ChapterRadialLaplacian.helmholtz_sbessel_zero [FiniteDimensional ℝ E] (h3 : Module.finrank ℝ E = 3)
    {p : ℝ} {x : E} (hp : p ≠ 0) (hx : x ≠ 0) :
    -(Δ fun y : E => sbessel 0 (p * ‖y‖)) x = p ^ 2 * sbessel 0 (p * ‖x‖) := by sorry

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