Physlib.SpaceAndTime.Space.Norm.IteratedLaplacian

Iterated Laplacians of norm distributions

i. Overview

This file proves the distributional identity corresponding to the classical odd-dimensional formula that, in dimension `2 * m + 1`, applying the Laplacian `m + 1` times to the norm gives a nonzero constant multiple of the Dirac delta at the origin.

ii. Key results

- `iterated_distLaplacian_norm_zpow_odd_eq_smul_diracDelta` : The `(m + 1)`-fold Laplacian of the norm in dimension `2 * m + 1` is a nonzero multiple of the Dirac delta.

iii. Table of contents

A. The odd-dimensional iterated Laplacian of the norm

iv. References

A. The odd-dimensional iterated Laplacian of the norm

3 declarations

definition

Scalar factor $C_m$ for the iterated Laplacian $\Delta^{m+1} \|x\|$ in dimension $2m+1$

#oddNormIteratedLaplacianCoeff

For a natural number $m$ , let $d = 2m + 1$ be the dimension of the Euclidean space. The scalar factor $C_m$ is defined as: $C_m = \left( \prod_{k=0}^{m-1} (1 - 2k)(2m - 2k) \right) \cdot (1 - 2m) \cdot (2m + 1) \cdot \text{Vol}(B_d(0, 1))$ where $\text{Vol}(B_d(0, 1))$ is the volume of the unit ball in $\mathbb{R}^d$ . This constant is the coefficient appearing in the distributional identity where the Laplacian operator $\Delta$ applied $m+1$ times to the norm $\|x\|$ yields a multiple of the Dirac delta distribution $\delta(x)$ at the origin.

theorem

The Scalar Factor $C_m$ for the Iterated Laplacian $\Delta^{m+1} \|x\|$ is Nonzero

#oddNormIteratedLaplacianCoeff_ne_zero

For any natural number $m$ , let $C_m$ be the scalar factor associated with the $(m+1)$ -fold Laplacian of the norm $\|x\|$ in dimension $d = 2m + 1$ , defined as: $C_m = \left( \prod_{k=0}^{m-1} (1 - 2k)(2m - 2k) \right) \cdot (1 - 2m) \cdot (2m + 1) \cdot \text{Vol}(B_d(0, 1))$ where $\text{Vol}(B_d(0, 1))$ is the volume of the unit ball in $\mathbb{R}^d$ . Then $C_m \neq 0$ .

theorem

$\Delta^{m+1} \|x\| = C_m \delta_0$ in dimension $2m + 1$

#iterated_distLaplacian_norm_zpow_odd_eq_smul_diracDelta

For any natural number $m$ , let $d = 2m + 1$ be the dimension of the Euclidean space. The $(m+1)$ -fold application of the distributional Laplacian operator $\Delta$ to the distribution induced by the norm function $x \mapsto \|x\|$ satisfies the identity: $\Delta^{m+1} \|x\| = C_m \delta_0$ where $\delta_0$ is the Dirac delta distribution at the origin and $C_m$ is the scalar factor defined by `oddNormIteratedLaplacianCoeff m`.