Spherically Decomposed Green's Function

• Spherically decomposed Green's functions are representations that express PDE solutions on spherical domains via spherical harmonics, preserving intrinsic geometry.
• They allow explicit control of rotational equivariance and anisotropy by decoupling spectral coefficients, leading to efficient spectral algorithms free from grid artifacts.
• Applications include neural operator learning, geophysical modeling, and wave propagation, where fast harmonic transforms and spectral multipliers enhance both theory and computation.

A spherically decomposed Green's function is a representation of the Green's function for a linear partial differential operator defined on a spherical domain, decomposed explicitly in spherical harmonics. This formalism is crucial for the analysis and design of integral and convolutional operators on spherical manifolds S² and higher, and is foundational in areas ranging from neural operator learning and computational physics to geophysical modeling and wave propagation. Spherical decomposition preserves the intrinsic geometry of the domain, allows explicit control of rotational equivariance and anisotropy, and enables efficient spectral algorithms without grid artifacts.

1. Definition and Operator-Theoretic Setting

A Green's function $G(\omega, \omega')$ for a linear differential operator $D$ on the sphere S² satisfies

$$D_{\omega} G(\omega, \omega') = \delta(\omega, \omega')$$

where $\delta$ is the Dirac delta on S². For $D = -\Delta_{\mathrm{LB}} + \alpha$ ($\Delta_{\mathrm{LB}}$: Laplace–Beltrami operator), the Green's function characterizes the solution to $Dg = f$ via the integral operator

$$g(\omega) = \int_{S^2} G(\omega, \omega') f(\omega')\, d\omega'$$

This formulation generalizes to operators diagonalizable on the spherical harmonics basis $Y_\ell^m$ (Tang et al., 11 Dec 2025). The Green's function can be viewed as the kernel of a convolution operator, defined intrinsically on the sphere, which is essential for building operator-theoretic foundations in spherical learning and physics.

2. Spherical Harmonic Spectral Decomposition

Owing to the orthonormality and completeness of $\{Y_\ell^m\}_{\ell \geq 0, |m| \leq \ell}$ in $L^2(S^2)$, any Green's function (modulo singularity at coincidence) admits the expansion

$$G(\omega, \omega') = \sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} g_\ell Y_\ell^m(\omega) Y_\ell^{m*}(\omega')$$

where the coefficients $g_\ell$ encode the spectral response of the operator: $$D Y_\ell^m = \mu_\ell Y_\ell^m \implies g_\ell = \frac{1}{\mu_\ell}$$ For $D = -\Delta_{\mathrm{LB}} + \alpha$, $\mu_\ell = \ell(\ell+1) + \alpha$.

If $\alpha = 0$ (Laplace), the expansion reduces to that of the Newtonian potential. Incorporating anisotropy or system-dependent effects, one allows $g_{\ell m}$ dependence, enabling fine-grained spectral control (Tang et al., 11 Dec 2025).

3. Absolute versus Relative Kernel Decompositions

Standard equivariant spherical convolutions employ relative-position kernels $G(\omega, \omega') = G(R^{-1}\omega)$, depending only on geodesic separation, ensuring SO(3)-equivariance. However, complex physical systems frequently demand explicit symmetry breaking, e.g., for boundaries or inhomogeneous media.

The generalized framework expresses the Green's function as

$$G(R, \omega) = \sum_{\ell', m'} G^{(1)}_{\ell' m'} Y_{\ell'}^{m'}(R^{-1}\omega) + \sum_{\ell', m'} G^{(2)}_{\ell' m'} Y_{\ell'}^{m'}(\omega)$$

• Relative (equivariant) term: depends on $R^{-1}\omega$ (geodesic distance from reference $R$), imparting equivariant responses.
• Absolute (invariant/correction) term: depends directly on $\omega$, allowing encoding of position-dependent or boundary-related effects.

In harmonic space, this duality yields for an input $f$: $$\widehat{g}_{\ell m} = G_{0}(\ell) \widehat{f}_{\ell m} + G_{1}(\ell, m) C_f$$ where $C_f$ is a global summary of $f$ (such as the integral over $S^2$), and the $G_0$, $G_1$ are spectral weights—possibly parameterized or learned in data-driven frameworks (Tang et al., 11 Dec 2025).

4. Computational Methods and Theoretical Underpinnings

The efficient application of spherically decomposed Green's-function operators leverages the harmonic representation:

• Fast transforms: Spherical harmonic transforms (SHT) map between spatial and spectral domains efficiently, with complexity $\mathcal{O}(L^2)$ for maximum degree $L$.
• Spectral multipliers: Operations reduce to diagonal or block-diagonal multiplication of spherical harmonic coefficients, vastly outperforming brute-force quadrature or sliding-window convolutions in both efficiency and accuracy on S².

Key mathematical theorems utilized include:

• Spherical convolution theorem: For zonal (isotropic) kernels, convolution in real space becomes pointwise multiplicative modulation in harmonic space (Driscoll–Healy theorem).
• Wigner D-matrix relations: Under rotation, $Y_\ell^m$ transform via Wigner D-matrices, fundamental for constructing equivariant kernels.
• Orthonormality: $$\int_{S^2} Y_\ell^m(\omega) Y_{\ell'}^{m'*}(\omega) d\omega = \delta_{\ell\ell'}\delta_{mm'}$$ underpins frequency-space decoupling.

5. Applications: Expressivity, Anisotropy, and Real-World Modeling

Spherically decomposed Green's functions admit efficient, physically grounded, and expressive operator designs across diverse domains:

• Neural Operators: Green's-function Spherical Neural Operators (GSNO) utilize learned or analytically constructed $g_{\ell m}$ to encode fine anisotropy, spatial inhomogeneity, and real-world constraints, enabling models that bridge theoretical rigor and empirical performance (Tang et al., 11 Dec 2025).
• Geophysical and Fluid Dynamics: Screened-Poisson and Laplace–Beltrami Green's functions, expanded spherically, are essential for solving inverse problems on the sphere, such as vorticity-to-streamfunction inversion in shallow water and meteorological models (Tanios et al., 2019).
• Boundary and Constraint Handling: The absolute term in the generalized Green's kernel enables explicit imposition of boundary layers or domain-specific corrections, particularly valuable in climatology, remote sensing, or any setting where the ideal symmetry of S² is broken.

In all cases, allowing $m$-dependence (i.e., non-zonal $g_{\ell m}$) provides controlled anisotropy—directional sensitivity critical for parameterizing non-isotropic effects such as terrain-following flows or stress distributions.

6. Significance in Modern Spherical Operator Learning

By decomposing Green's functions harmonically, one unifies mathematical rigor with data-driven flexibility:

• Spectral efficiency: Spherical multiplication in the harmonic domain is immune to grid aliasing and supports globally correlated, long-range interaction modeling at minimal computational cost.
• Expressive kernels: Direct manipulation of $g_{\ell}, g_{\ell m}$ enables seamless trade-offs between rotational invariance (physics-derived symmetries) and learned invariance-breaking (data-driven corrections).
• Operator-theoretical grounding: The framework provides a principled design space for spherical convolutional and integral operators with clear connections to PDE theory and spectral analysis.

In conclusion, spherically decomposed Green's functions are central objects in both analytical and machine-learning formulations of PDEs and operator equations on spherical domains. They enable rigorous, efficient, and flexible modeling of complex, anisotropic, or non-equivariant phenomena, and serve as the mathematical backbone for state-of-the-art neural operator architectures and advanced spectral solvers on spherical manifolds (Tang et al., 11 Dec 2025).