Laplacian

The Laplacian on \(\mathbb{R}^3 \), \[ \Delta = \left( \frac{\partial^2}{\partial x^2},\ \frac{\partial^2}{\partial y^2},\ \frac{\partial^2}{\partial z^2 } \right), \] is a second-order differential operator. When this operator is expressed in spherical polar coordinates in \( \mathbb{R}^3 \), where \[ x = r \sin \theta \cos \varphi,\ y = r \sin \theta \sin\varphi,\ z = r \cos \theta,\] and restricted to \(r = 1 \), it becomes what is called the Laplacian on the sphere \( S^2 \), denoted \( \Delta S^2 \).

Simplifying the Calculation of \( \Delta S^2 \)

Deriving \( \Delta S^2 \) by directly converting to spherical coordinates and calculating can be quite complicated. A simpler approach is to first convert using \[ x = \rho \cos \varphi,\ y = \rho \sin \varphi,\ z = z, \] and then perform a 2-dimensional spherical polar coordinate transformation as \[ \rho = r \sin \theta,\ z = r \cos \theta,\ \varphi = \varphi. \] This significantly simplifies the calculation.

Harmonic Polynomials and Their Relationship with the Laplacian

Let \( Vn \) denote the space of complex-coefficient homogeneous polynomials of degree \( n \) on \( \mathbb{R}^3 \). This space forms a complex vector space. The Laplacian \( \Delta \), when restricted to \(V_n \), maps \( V_n \) to \( V_{n−2} \), as applying \( \Delta \) reduces the degree of polynomials by \( 2 \). This is expressed as: \( \Delta | V_n : V_n \to V_{n−2}\). The kernel of this restriction, denoted \( U_n \), consists of the polynomials that are annihilated by \( \Delta \), meaning they satisfy \( \Delta F = 0\). These polynomials are called harmonic polynomials of degree \( n \).

Spherical Harmonics: Restriction to the Sphere

When harmonic polynomials in \( U_n \) are restricted to the surface of the sphere \( S^2 \), they become what are known as “spherical harmonic polynomials (in Japanese)” of degree \( n \). A key point to note is that while these polynomials satisfy \( \Delta F = 0\) in \( \mathbb{R}^3 \), they do not satisfy \( \Delta S^2 F(1, \theta, \varphi ) = 0 \) on the sphere. Instead, spherical harmonics satisfy a specific eigenvalue equation with respect to \(\Delta S^2 \), given by: \( \Delta S^2 F = −n(n+1)F \). This shows that spherical harmonic polynomials are not annihilated by the spherical Laplacian but are instead eigenfunctions of \( \Delta S^2 \), with eigenvalue \( −n(n+1) \). These polynomials are called “spherical harmonics (in English)” of degree \( n \).

Clarifying the Term “Spherical Harmonics”

While the term spherical harmonics might suggest that these polynomials satisfy \( \Delta S^2 F=0 \), this is not the case. They are actually eigenfunctions of the spherical Laplacian with a non-zero eigenvalue. Therefore, though the term “spherical harmonic polynomials” is a common translation from English, it may cause confusion. It might be more appropriate to simply use the term “spherical harmonics” in English, as it more accurately conveys the mathematical meaning. 🙁