Neumann Series of Spherical Bessel Functions

• Neumann Series of Spherical Bessel Functions is a spectral representation that expands solutions to radial Sturm–Liouville problems in rapidly convergent spherical Bessel series.
• It employs recursive integration to compute explicit coefficients and uniform error bounds, ensuring high accuracy in modeling transmission eigenvalues even with variable refractive indices.
• The method underpins both forward and inverse eigenvalue problems, offering practical tools for radially symmetric scattering, transmission, and impedance-form challenges.

The Neumann Series of Spherical Bessel Functions (NSBF) is a spectral representation for solutions of Sturm–Liouville-type equations, especially those arising in radially symmetric scattering, transmission eigenvalue problems, and impedance-form partial differential equations. Mathematically, it expresses the solution as a rapidly convergent series in spherical Bessel functions with coefficient sequences computed via recursive one-dimensional integrals. This approach is directly applicable to problems involving variable refractive index, transmission eigenvalues (both real and complex), and generalized boundary conditions in both direct and inverse formulations. Recent works have established precise derivations, explicit recursions for NSBF coefficients, uniform error bounds independent of the spectral parameter, and algorithms for eigenvalue and spectral data recovery (Kravchenko et al., 22 Jul 2025, Márquez-Hernández et al., 8 Jan 2026, Kravchenko et al., 2017, Kravchenko et al., 2016, Kravchenko et al., 2016).

1. Mathematical Formulation and Liouville Transformation

Radially symmetric transmission and scattering problems typically lead to an ODE of the form

$y''(r) + k^2 n(r) y(r) = 0, \quad 0 < r < 1,$

with initial conditions at the origin and $k$-dependent boundary data at $r=1$ (Kravchenko et al., 22 Jul 2025). A Liouville change of variable,

$$\zeta(r) = \int_r^1 \sqrt{n(t)}\,dt, \qquad \delta = \zeta(0),$$

transforms this into the Sturm–Liouville form,

$- z''(\zeta) + p(\zeta) z(\zeta) = k^2 z(\zeta),$

where $$p(\zeta(r)) = n''(r)/[4 n(r)^2] - 5 n'(r)^2/[16 n(r)^3]$$. This sets the stage for spectral analysis and the construction of NSBF representations. The form admits a fundamental system $\{\phi(k,\zeta), S(k,\zeta)\}$ with normalized Cauchy data at $\zeta=0$.

2. NSBF Representations: Series Structure and Spherical Bessel Functions

The principal solutions of the transformed Sturm–Liouville equation can be expanded in NSBF as

$$S(\rho, \zeta) = \frac{\sin(\rho \zeta)}{\rho} + \frac{1}{\rho} \sum_{n=0}^\infty s_n(\zeta) j_{2n+1}(\rho \zeta), \ \phi(\rho, \zeta) = \cos(\rho \zeta) + \sum_{n=0}^\infty g_n(\zeta) j_{2n}(\rho \zeta),$$

where $\rho = k$, $j_\ell(x) = \sqrt{\frac{\pi}{2x}} J_{\ell+1/2}(x)$ is the spherical Bessel function (Kravchenko et al., 22 Jul 2025). The parity structure (even for $\phi$, odd for $S$) arises from the origin regularity. These NSBF expansions are fundamentally linked to Legendre polynomial decompositions of transmutation kernels (Kravchenko et al., 2016, Márquez-Hernández et al., 8 Jan 2026).

3. Recursive Integration for NSBF Coefficients

The coefficients $g_n(\zeta), s_n(\zeta)$ (or more generally, $\beta_n(x), \alpha_n(x)$ in other formulations) are computed via recursive integration procedures. For instance, the transmission problem uses

$$\sigma_{2n}(\zeta) = \zeta^{2n} g_n(\zeta) / 2, \quad \sigma_{2n+1}(\zeta) = \zeta^{2n+1} s_n(\zeta) / 2,$$

with base cases and integrals:

$$\sigma_{-1}(\zeta) = \frac{1}{2\zeta}, \quad \sigma_0(\zeta) = \frac{f(\zeta) - 1}{2},$$

$$\eta_n(\zeta) = \int_0^\zeta [t f'(t) + (n-1) f(t)] \sigma_{n-2}(t) dt,$$

$$\theta_n(\zeta) = \int_0^\zeta \frac{\eta_n(t) - t f(t) \sigma_{n - 2}(t)}{f^2(t)} dt,$$

and recurrence

$$\sigma_n(\zeta) = \frac{2n + 1}{2n - 3} [\zeta^2 \sigma_{n-2}(\zeta) + c_n f(\zeta) \theta_n(\zeta)],$$

with $c_n = 1$ for $n=1$, $c_n = 2(2n-1)$ otherwise (Kravchenko et al., 22 Jul 2025). Analogous recursions are formulated for Sturm–Liouville and impedance form problems (Márquez-Hernández et al., 8 Jan 2026, Kravchenko et al., 2016).

4. Convergence, Uniform Error Bounds, and Rate Estimates

NSBF series converge pointwise and uniformly in prescribed strips $|\operatorname{Im} \rho| \leq C$. For partial sums $S_N, \phi_N$ of length $N$, there exist functions $\varepsilon_N(\zeta) \to 0$ as $N \to \infty$ such that

$$| \rho S(\rho, \zeta) - \rho S_N(\rho, \zeta) | \leq \frac{\varepsilon_N(\zeta)}{C} \sinh(C\zeta),$$

$$| \phi(\rho, \zeta) - \phi_N(\rho, \zeta) | \leq \frac{\varepsilon_N(\zeta)}{C} \sinh(C\zeta),$$

and similarly for impedance-form and perturbed Bessel equations (Kravchenko et al., 22 Jul 2025, Kravchenko et al., 2016, Márquez-Hernández et al., 8 Jan 2026). The error estimates are independent of $\omega$ (spectral parameter) and allow precise control of truncation effects. For impedance-form Sturm–Liouville,

$$\sup_{0<x\leq L} \varepsilon_N(x) \leq c_0 \sqrt{L M_{\kappa, L} / N},$$

and smoothness of $\kappa$ enables $O(N^{-p - 1/2})$ decay for $C^{p+1}$ regularity (Márquez-Hernández et al., 8 Jan 2026).

5. Algorithmic Procedures and Eigenvalue Applications

The NSBF methodology enables direct calculation of transmission, spectral, and scattering eigenvalues, as well as eigenfunction reconstruction. For fixed $\delta$, the truncated characteristic function

$$D_{0,N}(k) = a(k)\left[ \cos(k\delta) + \sum_{n=0}^{N-1} g_n(\delta) j_{2n}(k\delta) \right] + b(k)\left[ \frac{\sin(k\delta)}{k} + \frac{1}{k} \sum_{n=0}^{N-1} s_n(\delta) j_{2n+1}(k\delta) \right],$$

is used for root-finding (Kravchenko et al., 22 Jul 2025). In eigenvalue problems for Dirichlet and Neumann boundary conditions, SBF coefficients provide the basis to compute high-index eigenvalues without accuracy degradation. For impedance problems, eigenvalues are zeros of $S_\kappa(\rho, L)$, with normalization and orthogonality ensured via explicit formulas (Márquez-Hernández et al., 8 Jan 2026).

The inverse transmission problem is solved by:

1. Recovering the transformed interval length $\delta$ from spectral data using NSBF-based algorithms.
2. Reconstructing $n(r)$ by solving a linear system in the first few NSBF coefficients.

A spectrum completion technique fills gaps when limited eigenvalue data are available (Kravchenko et al., 22 Jul 2025). All computational steps rely on stable forward recurrences and standard special-function libraries.

6. Relationship to Transmutation Operators, Legendre Series, and Bessel Theory

Transmutation operator theory underpins NSBF development. The integral kernels arising in these operators admit Fourier–Legendre expansions,

$$K(x, t) = \sum_{n=0}^\infty \frac{a_n(x)}{x} P_n(t/x),$$

with coefficients recovered as projections. Integration against $\exp(i\rho t)$ or $\cos(\rho t)$ yields the spherical Bessel series (Kravchenko et al., 2016, Márquez-Hernández et al., 8 Jan 2026). Classical Bessel functions $J_\nu(x)$ with half-integer $\nu$ are related by

$$j_\ell(x) = \sqrt{\frac{\pi}{2x}} J_{\ell + 1/2}(x),$$

and the Legendre polynomial structure determines the parity and convergence properties of the series (Kravchenko et al., 22 Jul 2025). The Paley–Wiener theorem guarantees compact support for inverse Fourier transforms and establishes exponential-type entire functions for NSBF differences (Kravchenko et al., 2016).

7. NSBF in Inverse and Spectral Problems: Numerical Results and Practical Aspects

NSBF approaches are robust across variable refractive indices, without requiring assumptions on the sign of contrast $1 - n(r)$ or the transformed interval length $\delta$ (Kravchenko et al., 22 Jul 2025). Numerical experiments demonstrate that even with $N$ in the range 10–20, transmission eigenvalues of both types are highly accurate. Eigenvalue spectral sequences are obtained without drift or precision loss as the index increases, even for hundreds of eigenvalues (Márquez-Hernández et al., 8 Jan 2026, Kravchenko et al., 2016). A posteriori indicators and Abel-integral-based identities offer practical selection of truncation order and error control.

The algorithms are implementable in concise code blocks, requiring only the solution of one-dimensional integrals and recurrence procedures. This ensures the wide applicability of NSBF representations for both direct and inverse spherically symmetric eigenvalue problems (Kravchenko et al., 22 Jul 2025, Márquez-Hernández et al., 8 Jan 2026, Kravchenko et al., 2017, Kravchenko et al., 2016, Kravchenko et al., 2016).