Extended Spherical Scattering Operators

• Extended Spherical Scattering Operators are frameworks that extend classical spherical scattering theory to accommodate complex media, geometries, and symmetry classes.
• They decouple and model physical phenomena ranging from multilayer electromagnetic interactions in implantable antennas to signal processing in spherical CNNs.
• The formulations utilize spherical harmonics, wavelets, and multipole expansions to separate environmental effects from intrinsic signals, enhancing computational efficiency.

Extended spherical scattering operators (SSOs) denote operator constructions that extend classical sphere-based scattering relations beyond the simplest isotropic, fixed-geometry setting. In the literature considered here, the exact term is used for the block operators that model multilayered spherical environments in a generalized scattering-matrix framework for implantable antennas, while closely related constructions appear in spherical scattering networks on $S^2$, unitary scattering maps between boundary degenerations of real spherical varieties, surface-density generalizations of multipole expansions, and singular radial or transport operators written in spherical-harmonic form (Shi et al., 17 Jul 2025, McEwen et al., 2021, Delorme, 2020). The literature therefore suggests that SSOs are best understood not as a single standardized object, but as a family of operator formalisms whose common role is to encode scattering, propagation, or multiscale transfer in representations adapted to spherical symmetry or spherical geometry.

1. Scope and domain-specific meanings

The expression has a narrow and a broad use. In the narrow sense, it names the shell operators $\mathbf t$, $\mathbf \Phi$, $\mathbf \Psi$, and $\boldsymbol\rho$ in multilayered spherical electromagnetics. In the broader sense, it covers operator families that generalize spherical scattering to native spherical data, boundary degenerations, arbitrary enclosing surfaces, bianisotropic media, singular angular-momentum channels, or arbitrary spherical-harmonic truncation orders.

Setting	Operator form	Functional role
Multilayered spherical media	$$\begin{bmatrix}\mathbf f^{\mathrm f}\ \mathbf a^{\mathrm b}\end{bmatrix}=\begin{bmatrix}\mathbf t & \mathbf \Psi\ \mathbf \Phi & \boldsymbol\rho\end{bmatrix}\begin{bmatrix}\mathbf a^{\mathrm f}\ \mathbf f^{\mathrm b}\end{bmatrix}$$	Decouples antenna GSM from the spherical environment
Spherical scattering on $S^2$	$S[p]f=\Phi U[p]f$	Rotation-equivariant, stable feature extraction
Real spherical varieties	$S_w:L^2(X_I)_{td}\to L^2(X_J)_{td}$	Relates boundary degenerations in the Plancherel theory
Surface-integral multiple scattering	$$\mathcal T^{\Gamma^-}_{\rm loc}:H^{1/2}(\Gamma^-)\to H^{-1/2}(\Gamma^-)$$	Replaces sphere-based multipole coefficients by boundary densities

A recurrent structural feature is that the operator acts on a spherical mode space, a boundary trace space, or a cascade of spherical wavelet responses, and maps that input to an outgoing, averaged, or asymptotic representation. A plausible implication is that the adjective “extended” usually signals one of five enlargements: more general media, more general geometry, more general symmetry classes, more singular radial operators, or more general operator calculus on spherical harmonics (Shi et al., 17 Jul 2025, Felbacq et al., 2023).

2. Block SSOs for multilayered spherical electromagnetic media

In the exact terminology of "Generalized Scattering Matrix Framework for Modeling Implantable Antennas in Multilayered Spherical Media" (Shi et al., 17 Jul 2025), extended spherical scattering operators rigorously represent the electromagnetic interaction of a spherical shell with incoming and outgoing spherical waves. The canonical geometry consists of an antenna inside a homogeneous spherical bubble of radius $\mathbf t$0, surrounded by a shell extending to $\mathbf t$1, and free space outside. The bubble and outer region are homogeneous and isotropic, while the shell may be radially inhomogeneous and uniaxially anisotropic.

Inside the shell, the fields are expanded in a generalized anisotropic spherical basis built from radial functions $\mathbf t$2 and $\mathbf t$3, with constitutive tensors

$\mathbf t$4

The radial functions satisfy decoupled ODEs,

$\mathbf t$5

so the shell problem reduces to radial propagation and tangential field matching.

The SSO itself is the block relation

$\mathbf t$6

Here $\mathbf t$7 is the transition matrix from outer free space into free-space scattering response, $\mathbf t$8 is the inward transmission operator, $\mathbf t$9 is the outward transmission operator, and $\mathbf \Phi$0 is the internal reflection operator seen from the bubble. When $\mathbf \Phi$1, the relation reduces to $\mathbf \Phi$2, namely the classical T-matrix case.

The decisive architectural step is the coupling of this shell operator with the antenna’s free-space generalized scattering matrix,

$\mathbf \Phi$3

After substitution of $\mathbf \Phi$4, the complete antenna-plus-shell system is expressed by an effective GSM in which the shell appears only through $\mathbf \Phi$5, $\mathbf \Phi$6, $\mathbf \Phi$7, and $\mathbf \Phi$8, and the antenna appears only through $\mathbf \Phi$9, $\mathbf \Psi$0, $\mathbf \Psi$1, and $\mathbf \Psi$2. This decoupling is the central meaning of SSOs in that paper.

For homogeneous isotropic shells, the radial ODEs admit analytic Riccati–Bessel solutions and the operators are diagonal in $\mathbf \Psi$3 because of spherical symmetry. For piecewise homogeneous shells, the operators are obtained by cascading interface solutions. For radially continuous inhomogeneous profiles, the radial ODEs are solved numerically with MATLAB’s ode45. The reported timings emphasize the parametric-sweep advantage: one-time antenna GSM precomputation takes about $\mathbf \Psi$4, each new shell evaluation via the proposed framework about $\mathbf \Psi$5, and DGF-based MoM about $\mathbf \Psi$6 per case. Validation against FEKO and DGF-based solutions is reported for homogeneous isotropic shells, homogeneous uniaxial anisotropic shells, two-layer isotropic and anisotropic shells, and continuously varying anisotropic shells (Shi et al., 17 Jul 2025).

3. Scattering operators on the sphere in signal processing and machine learning

A second major meaning arises in spherical representation learning, where scattering operators are built natively on $\mathbf \Psi$7 from spherical wavelets, pointwise modulus nonlinearities, and low-pass projection. In "Scattering Networks on the Sphere for Scalable and Rotationally Equivariant Spherical CNNs" (McEwen et al., 2021), the basic propagator at scale $\mathbf \Psi$8 is

$\mathbf \Psi$9

and a path $\boldsymbol\rho$0 defines

$\boldsymbol\rho$1

The scattering coefficients are

$\boldsymbol\rho$2

with $\boldsymbol\rho$3. The operators use the axisymmetric scale-discretized wavelet transform on the sphere, implemented via spherical harmonics and spherical convolutions.

This construction is rotationally equivariant because the spherical wavelet transform is rotationally equivariant and the modulus is pointwise. It is invariant to isometries up to a scale controlled by $\boldsymbol\rho$4, and stable to diffeomorphisms. The paper also emphasizes descending paths $\boldsymbol\rho$5 to move information from high to low frequencies efficiently. Computationally, the axisymmetric spherical wavelet transform and spherical scattering transform scale as $\boldsymbol\rho$6, whereas efficient generalized spherical CNNs are described as typically scaling as $\boldsymbol\rho$7. The resulting architecture uses scattering as an initial preprocessing layer: high-resolution input is converted into low-resolution additional channels before the learned spherical CNN layers operate. The stated application regime reaches spherical signals of many tens of megapixels and beyond (McEwen et al., 2021).

A related but distinct spherical formulation appears in "Scattering transforms on the sphere, application to large scale structure modelling" (Mousset et al., 2024). There the representation is built from directional spherical wavelets and scattering covariance statistics. The directional convolution is

$\boldsymbol\rho$8

with $\boldsymbol\rho$9. The summary statistics include first-order moments and higher-order covariances such as

$$\begin{bmatrix}\mathbf f^{\mathrm f}\ \mathbf a^{\mathrm b}\end{bmatrix}=\begin{bmatrix}\mathbf t & \mathbf \Psi\ \mathbf \Phi & \boldsymbol\rho\end{bmatrix}\begin{bmatrix}\mathbf a^{\mathrm f}\ \mathbf f^{\mathrm b}\end{bmatrix}$$0

$$\begin{bmatrix}\mathbf f^{\mathrm f}\ \mathbf a^{\mathrm b}\end{bmatrix}=\begin{bmatrix}\mathbf t & \mathbf \Psi\ \mathbf \Phi & \boldsymbol\rho\end{bmatrix}\begin{bmatrix}\mathbf a^{\mathrm f}\ \mathbf f^{\mathrm b}\end{bmatrix}$$1

These statistics are then used in a maximum-entropy microcanonical generative model defined by

$$\begin{bmatrix}\mathbf f^{\mathrm f}\ \mathbf a^{\mathrm b}\end{bmatrix}=\begin{bmatrix}\mathbf t & \mathbf \Psi\ \mathbf \Phi & \boldsymbol\rho\end{bmatrix}\begin{bmatrix}\mathbf a^{\mathrm f}\ \mathbf f^{\mathrm b}\end{bmatrix}$$2

The reported cosmological application uses a full-sky weak-lensing field from CosmoGrid, with about $$\begin{bmatrix}\mathbf f^{\mathrm f}\ \mathbf a^{\mathrm b}\end{bmatrix}=\begin{bmatrix}\mathbf t & \mathbf \Psi\ \mathbf \Phi & \boldsymbol\rho\end{bmatrix}\begin{bmatrix}\mathbf a^{\mathrm f}\ \mathbf f^{\mathrm b}\end{bmatrix}$$3 gradient-descent iterations typically sufficient and an $$\begin{bmatrix}\mathbf f^{\mathrm f}\ \mathbf a^{\mathrm b}\end{bmatrix}=\begin{bmatrix}\mathbf t & \mathbf \Psi\ \mathbf \Phi & \boldsymbol\rho\end{bmatrix}\begin{bmatrix}\mathbf a^{\mathrm f}\ \mathbf f^{\mathrm b}\end{bmatrix}$$4 image generated in about $$\begin{bmatrix}\mathbf f^{\mathrm f}\ \mathbf a^{\mathrm b}\end{bmatrix}=\begin{bmatrix}\mathbf t & \mathbf \Psi\ \mathbf \Phi & \boldsymbol\rho\end{bmatrix}\begin{bmatrix}\mathbf a^{\mathrm f}\ \mathbf f^{\mathrm b}\end{bmatrix}$$5 seconds. The generated ensemble is described as reproducing the PDF well over five orders of magnitude and the angular power spectrum well over all scales, though with small oscillations attributed to the wavelet frequency bands (Mousset et al., 2024).

4. Unitary scattering operators on real spherical varieties

In harmonic analysis, spherical scattering operators arise in a different but mathematically stringent sense. "Scattering and a Plancherel formula of spherical varieties for real reductive split groups" (Delorme, 2020) studies a unimodular real spherical $$\begin{bmatrix}\mathbf f^{\mathrm f}\ \mathbf a^{\mathrm b}\end{bmatrix}=\begin{bmatrix}\mathbf t & \mathbf \Psi\ \mathbf \Phi & \boldsymbol\rho\end{bmatrix}\begin{bmatrix}\mathbf a^{\mathrm f}\ \mathbf f^{\mathrm b}\end{bmatrix}$$6-variety $$\begin{bmatrix}\mathbf f^{\mathrm f}\ \mathbf a^{\mathrm b}\end{bmatrix}=\begin{bmatrix}\mathbf t & \mathbf \Psi\ \mathbf \Phi & \boldsymbol\rho\end{bmatrix}\begin{bmatrix}\mathbf a^{\mathrm f}\ \mathbf f^{\mathrm b}\end{bmatrix}$$7, with $$\begin{bmatrix}\mathbf f^{\mathrm f}\ \mathbf a^{\mathrm b}\end{bmatrix}=\begin{bmatrix}\mathbf t & \mathbf \Psi\ \mathbf \Phi & \boldsymbol\rho\end{bmatrix}\begin{bmatrix}\mathbf a^{\mathrm f}\ \mathbf f^{\mathrm b}\end{bmatrix}$$8 the group of real points of a split connected reductive algebraic group and a minimal parabolic $$\begin{bmatrix}\mathbf f^{\mathrm f}\ \mathbf a^{\mathrm b}\end{bmatrix}=\begin{bmatrix}\mathbf t & \mathbf \Psi\ \mathbf \Phi & \boldsymbol\rho\end{bmatrix}\begin{bmatrix}\mathbf a^{\mathrm f}\ \mathbf f^{\mathrm b}\end{bmatrix}$$9 having an open orbit. For each subset $S^2$0 of simple spherical roots, there is a boundary degeneration $S^2$1, carrying a commuting right action of $S^2$2.

The key operators are the scattering maps

$S^2$3

between twisted discrete spectral pieces of the boundary degenerations. They satisfy four defining relations: $S^2$4

$S^2$5

$S^2$6

and each $S^2$7 is unitary. The corresponding Plancherel isomorphism identifies $S^2$8 with the subspace of tuples $S^2$9 satisfying the scattering relations $S[p]f=\Phi U[p]f$0.

These operators depend on several real-analytic ingredients absent from the $S[p]f=\Phi U[p]f$1-adic setting: special coverings, approximate partitions, uniform asymptotic estimates, and spectral projections. A structural input is Knop’s Harish-Chandra homomorphism, identified in the paper by

$S[p]f=\Phi U[p]f$2

which controls the $S[p]f=\Phi U[p]f$3- and invariant-differential-operator eigenstructure. Constant terms commute with invariant differential operators according to

$S[p]f=\Phi U[p]f$4

In this setting, scattering operators do not average or compress data; rather, they compare asymptotic channels at infinity and furnish the bridge between $S[p]f=\Phi U[p]f$5 and the twisted discrete spectra of the $S[p]f=\Phi U[p]f$6. The real-case extension is the passage from the Sakellaridis–Venkatesh $S[p]f=\Phi U[p]f$7-adic theorem to real spherical varieties (Delorme, 2020).

5. Generalizations of sphere-based multipole scattering

Classical sphere-based scattering expansions can also be extended away from exact spherical boundaries. "A single layer representation of the scattered field for multiple scattering problems" (Felbacq et al., 2023) proves that the scattered field of a set of scatterers can be represented as an integral on any smooth closed surface $S[p]f=\Phi U[p]f$8 enclosing them: $S[p]f=\Phi U[p]f$9 provided that $S_w:L^2(X_I)_{td}\to L^2(X_J)_{td}$0 is not a Dirichlet eigenvalue of $S_w:L^2(X_I)_{td}\to L^2(X_J)_{td}$1 in $S_w:L^2(X_I)_{td}\to L^2(X_J)_{td}$2. The density is unique and equals the jump $S_w:L^2(X_I)_{td}\to L^2(X_J)_{td}$3. For subsets of scatterers, the induced local operator

$S_w:L^2(X_I)_{td}\to L^2(X_J)_{td}$4

maps the boundary trace of the local regular field to the single-layer density representing the local scattered field. When $S_w:L^2(X_I)_{td}\to L^2(X_J)_{td}$5 is a sphere, the density can be expanded in spherical harmonics and the formulation recovers the usual multipole structure. This is therefore a coordinate-free boundary-integral generalization of the classical spherical scattering representation. The computational purpose is to extend Fast Multipole-like coupling to subsets that are not enclosed by non-intersecting balls (Felbacq et al., 2023).

An electromagnetic generalization of the multipole operator itself appears in "Electromagnetic Scattering by Bianisotropic Spheres" (Durach, 2023). There the constitutive law is written in full $S_w:L^2(X_I)_{td}\to L^2(X_J)_{td}$6 block form,

$S_w:L^2(X_I)_{td}\to L^2(X_J)_{td}$7

and the interior fields are expanded in “bianisotropic orbitals,” namely linear combinations of vector spherical harmonics that solve the coupled Maxwell system in the sphere. After boundary matching, the incident multipole coefficients $S_w:L^2(X_I)_{td}\to L^2(X_J)_{td}$8 are mapped to the scattered coefficients $S_w:L^2(X_I)_{td}\to L^2(X_J)_{td}$9 by a T-matrix

$$\mathcal T^{\Gamma^-}_{\rm loc}:H^{1/2}(\Gamma^-)\to H^{-1/2}(\Gamma^-)$$0

This is an SSO-like operator in the precise sense that it remains a linear map on spherical multipole coefficients, but it is no longer block diagonal in the standard electric/magnetic basis because the interior eigenmodes mix them. In the Rayleigh limit, the theory yields the generalized polarizability

$$\mathcal T^{\Gamma^-}_{\rm loc}:H^{1/2}(\Gamma^-)\to H^{-1/2}(\Gamma^-)$$1

in agreement with earlier electrostatic results (Durach, 2023).

6. Singular radial operators and operator calculus in spherical harmonics

A further extension occurs in radial quantum scattering. "Spectral Asymptotics for Perturbed Spherical Schrödinger Operators and Applications to Quantum Scattering" (Kostenko et al., 2012) studies

$$\mathcal T^{\Gamma^-}_{\rm loc}:H^{1/2}(\Gamma^-)\to H^{-1/2}(\Gamma^-)$$2

with a singular endpoint at $$\mathcal T^{\Gamma^-}_{\rm loc}:H^{1/2}(\Gamma^-)\to H^{-1/2}(\Gamma^-)$$3. The singular Weyl solution is

$$\mathcal T^{\Gamma^-}_{\rm loc}:H^{1/2}(\Gamma^-)\to H^{-1/2}(\Gamma^-)$$4

and the $$\mathcal T^{\Gamma^-}_{\rm loc}:H^{1/2}(\Gamma^-)\to H^{-1/2}(\Gamma^-)$$5-function admits the high-energy asymptotic

$$\mathcal T^{\Gamma^-}_{\rm loc}:H^{1/2}(\Gamma^-)\to H^{-1/2}(\Gamma^-)$$6

with leading term equal to the unperturbed Bessel/Weyl asymptotic. The scattering quantities are tied to the Jost solution $$\mathcal T^{\Gamma^-}_{\rm loc}:H^{1/2}(\Gamma^-)\to H^{-1/2}(\Gamma^-)$$7, the Jost function $$\mathcal T^{\Gamma^-}_{\rm loc}:H^{1/2}(\Gamma^-)\to H^{-1/2}(\Gamma^-)$$8, and the companion coefficient $$\mathcal T^{\Gamma^-}_{\rm loc}:H^{1/2}(\Gamma^-)\to H^{-1/2}(\Gamma^-)$$9 through

$\mathbf t$00

The paper proves uniqueness results showing that $\mathbf t$01, together with eigenvalues and norming constants, uniquely determines $\mathbf t$02 and $\mathbf t$03, and that under stronger assumptions the phase shift $\mathbf t$04 likewise determines $\mathbf t$05. The extension here is from the classical $\mathbf t$06 case to arbitrary, including noninteger, angular momentum (Kostenko et al., 2012).

The transformation-operator foundations for such radial scattering are developed in "Transformation operators for spherical Schrödinger operators" (Holzleitner, 2018). Near the origin, the regular solution is related to the unperturbed one by

$\mathbf t$07

while near infinity the Jost solution satisfies

$\mathbf t$08

The paper derives PDEs for $\mathbf t$09 and $\mathbf t$10, converts them to Volterra-type integral equations, and proves kernel estimates that depend explicitly on $\mathbf t$11, including the critical regime $\mathbf t$12. These kernels are not themselves named SSOs, but they supply the analytic machinery needed to compare perturbed and unperturbed spherical scattering dynamics (Holzleitner, 2018).

An operator-matrix extension of spherical scattering appears in kinetic theory. "Using Spherical Harmonics to solve the Boltzmann equation: an operator based approach" (Schween et al., 2024) expands the phase-space density as

$\mathbf t$13

and represents each angular operator by

$\mathbf t$14

The isotropic small-angle collision operator becomes diagonal because

$\mathbf t$15

yielding a diagonal matrix representation $\mathbf t$16. Direction operators and angular-momentum operators become sparse coupling matrices, products of operators become matrix products, and rotations act by similarity transforms through Wigner-$\mathbf t$17 matrices. The eigenvalues of the truncated direction matrices are given by the roots of associated Legendre polynomials. This is an operator-based spherical scattering formalism in coefficient space, extended to arbitrary truncation order (Schween et al., 2024).

7. Shared principles and persistent distinctions

Across these literatures, the operator input-output pair changes substantially. In multilayer electromagnetics, the operator maps spherical-wave amplitudes at two shell boundaries to reflected and transmitted amplitudes. In spherical deep learning, it maps a signal on $\mathbf t$18 to a hierarchy of wavelet-modulus-averaged coefficients. In real spherical harmonic analysis, it maps one twisted discrete boundary spectrum to another. In boundary-integral scattering, it maps a trace to a density on an arbitrary enclosing surface. In radial Schrödinger theory, the relevant objects map free solutions to perturbed ones or connect Jost data to phase shifts. In Boltzmann transport, the operator acts on spherical-harmonic coefficient vectors.

The shared principles are more stable than the notation. Each framework uses a basis or representation adapted to spherical geometry: spherical waves, spherical harmonics, vector spherical harmonics, scale-discretized spherical wavelets, or boundary degenerations indexed by spherical roots. Each framework also exploits a form of structural simplification: rotational equivariance, diagonalization in modal index, unitarity, Volterra causality, or block sparsity. The practical purpose is likewise recurrent: separate a difficult environment from an intrinsic object, convert high-resolution or high-frequency content into a more tractable representation, or make asymptotic channels explicit.

A common misconception is that “extended spherical scattering operators” name a single canonical operator. The literature considered here suggests the opposite. The exact phrase is specific to multilayered spherical media (Shi et al., 17 Jul 2025), whereas nearby fields employ the same underlying idea under different names: spherical scattering networks (McEwen et al., 2021), scattering operators $\mathbf t$19 (Delorme, 2020), local surface scattering maps (Felbacq et al., 2023), generalized T-matrices (Durach, 2023), or transformation operators (Holzleitner, 2018). The unifying feature is therefore conceptual rather than terminological: an SSO is an operatorial extension of spherical scattering theory to richer geometries, richer media, richer symmetry classes, or richer representation spaces.