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Neumann series of Bessel functions for the solutions of the Sturm-Liouville equation in impedance form and related boundary value problems

Abigail G. Márquez-Hernández, Víctor A. Vicente-Benítez

TL;DR

The paper addresses solving Sturm--Liouville equations in impedance form with low-regularity conductivities κ∈W^{1,2}(0,L) by constructing a Neumann series of spherical Bessel functions (NSBF) representation for the solution e_{κ}(ρ,x). It develops a transmutation-operator-based framework that yields a Fourier–Legendre expansion of the transmutation kernel, from which explicit NSBF coefficients α_{n}(x) are obtained via a simple recursive integration scheme, with rigorous uniform truncation error bounds for |Im ρ|≤C. The NSBF forms for e_{κ}, C_{κ}, and S_{κ} enable direct, accurate computation of Dirichlet spectral data, Weyl functions, and, through a Darboux transform, Neumann data; the method also provides a numerically stable pathway to eigenvalues and eigenfunctions without spectral-shift requirements. Numerical experiments demonstrate high-precision eigenvalues even for high-index modes and non-smooth κ, highlighting the approach’s robustness for applications in acoustics, geophysics, and wave propagation in inhomogeneous media.

Abstract

We present a Neumann series of spherical Bessel functions representation for solutions of the Sturm--Liouville equation in impedance form \[ (κ(x)u')' + λκ(x)u = 0,\quad 0 < x < L, \] in the case where $κ\in W^{1,2}(0,L)$ and has no zeros on the interval of interest. The $x$-dependent coefficients of this representation can be constructed explicitly by means of a simple recursive integration procedure. Moreover, we derive bounds for the truncation error, which are uniform whenever the spectral parameter $ρ=\sqrtλ$ satisfies a condition of the form $|\operatorname{Im}ρ|\leq C$. Based on these representations, we develop a numerical method for solving spectral problems that enables the computation of eigenvalues with non-deteriorating accuracy.

Neumann series of Bessel functions for the solutions of the Sturm-Liouville equation in impedance form and related boundary value problems

TL;DR

The paper addresses solving Sturm--Liouville equations in impedance form with low-regularity conductivities κ∈W^{1,2}(0,L) by constructing a Neumann series of spherical Bessel functions (NSBF) representation for the solution e_{κ}(ρ,x). It develops a transmutation-operator-based framework that yields a Fourier–Legendre expansion of the transmutation kernel, from which explicit NSBF coefficients α_{n}(x) are obtained via a simple recursive integration scheme, with rigorous uniform truncation error bounds for |Im ρ|≤C. The NSBF forms for e_{κ}, C_{κ}, and S_{κ} enable direct, accurate computation of Dirichlet spectral data, Weyl functions, and, through a Darboux transform, Neumann data; the method also provides a numerically stable pathway to eigenvalues and eigenfunctions without spectral-shift requirements. Numerical experiments demonstrate high-precision eigenvalues even for high-index modes and non-smooth κ, highlighting the approach’s robustness for applications in acoustics, geophysics, and wave propagation in inhomogeneous media.

Abstract

We present a Neumann series of spherical Bessel functions representation for solutions of the Sturm--Liouville equation in impedance form in the case where and has no zeros on the interval of interest. The -dependent coefficients of this representation can be constructed explicitly by means of a simple recursive integration procedure. Moreover, we derive bounds for the truncation error, which are uniform whenever the spectral parameter satisfies a condition of the form . Based on these representations, we develop a numerical method for solving spectral problems that enables the computation of eigenvalues with non-deteriorating accuracy.
Paper Structure (14 sections, 16 theorems, 119 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 16 theorems, 119 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background on transmutation operators for the SLEIF
  3. Neumann series of Bessel functions representation
  4. The Fourier-Legendre representation for the kernel
  5. NSBF representation for the solutions
  6. Recursive relations for the NSBF coefficients
  7. Solution of boundary valued problems
  8. The Dirichlet problem
  9. The Weyl function
  10. Relation with the Neumann problem
  11. Numerical solution
  12. Implementation details
  13. Numerical examples
  14. Discussion and summary

Key Result

Theorem 1

The solution $e_{\kappa}(\rho,x)$ admits the SPPS representation This series converges uniformly and absolutely with respect to $x$ on $[0,L]$, as does the series obtained by termwise differentiation once. The series of second derivatives converges in $L^2(0,L)$. The series converges uniformly and absolutely with respect to $\rho$ on compact subsets of the comple

Figures (7)

  • Figure 1: The domain $\mathcal{T}_L$. The domain of integration $-x<t<x$ is indicated by the blue dotted line.
  • Figure 2: Absolute and relative errors between the exact eigenvalues and those computed with the NSBF method (See Table \ref{['Tab:ejemplo1']}). A logarithmic scale is used on the vertical axis to highlight the error magnitude across several orders.
  • Figure 3: Eigenfunctions associated with the eigenvalues listed in Table \ref{['Tab:ejemplo1']}, computed using the NSBF method.
  • Figure 4: Graph of the real and imaginary part of Weyl function of the Dirichlet problem for impedance $a(x)=(1+x)^2$ in the $\lambda$ complex plane.
  • Figure 5: The graph of the triangular conductivity.
  • ...and 2 more figures

Theorems & Definitions (25)

  • Theorem 1: sppscamposspps
  • Theorem 2
  • Theorem 3: mineimpedance3mineshrodinger
  • Proposition 4
  • Remark 5
  • Proposition 6: mineimpedance1, Prop. 37
  • Lemma 7
  • Theorem 8
  • Proposition 9
  • Proposition 10
  • ...and 15 more