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Reconstruction techniques for complex potentials

Vladislav V. Kravchenko

Abstract

An approach for solving a variety of inverse coefficient problems for the Sturm-Liouville equation -y''+q(x)y=λy with a complex valued potential q(x) is presented. It is based on Neumann series of Bessel functions representations for solutions. With their aid the problem is reduced to a system of linear algebraic equations for the coefficients of the representations. The potential is recovered from an arithmetic combination of the first two coefficients. Special cases of the considered problems include the recovery of the potential from a Weyl function, inverse two-spectra Sturm-Liouville problems, as well as the inverse scattering problem on a finite interval. The approach leads to efficient numerical algorithms for solving coefficient inverse problems. Numerical efficiency is illustrated by several examples.

Reconstruction techniques for complex potentials

Abstract

An approach for solving a variety of inverse coefficient problems for the Sturm-Liouville equation -y''+q(x)y=λy with a complex valued potential q(x) is presented. It is based on Neumann series of Bessel functions representations for solutions. With their aid the problem is reduced to a system of linear algebraic equations for the coefficients of the representations. The potential is recovered from an arithmetic combination of the first two coefficients. Special cases of the considered problems include the recovery of the potential from a Weyl function, inverse two-spectra Sturm-Liouville problems, as well as the inverse scattering problem on a finite interval. The approach leads to efficient numerical algorithms for solving coefficient inverse problems. Numerical efficiency is illustrated by several examples.
Paper Structure (9 sections, 3 theorems, 73 equations, 4 figures)

This paper contains 9 sections, 3 theorems, 73 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Neumann series of Bessel functions representations for solutions
  3. Computation of $\varphi(\rho,L)$ and $S(\rho,L)$ from the input data (\ref{['inpdata']})
  4. Recovery of $q(x)$ from $\varphi(\rho,L)$ and $S(\rho,L)$
  5. Main system of linear algebraic equations
  6. Final step: recovery of $q(x)$
  7. Summary of the method
  8. Numerical results
  9. Conclusions

Key Result

Theorem 2.1

Let $q\in\mathcal{L}_{2}(0,L)$. The solutions $\varphi(\rho,x)$ and $S(\rho,x)$ admit the following series representations where $\mathbf{j}_{k}(z)$ stands for the spherical Bessel function of order $k$ (see, e.g., AbramowitzStegunSpF). The coefficients $g_{n}(x)$ and $s_{n}(x)$ can be calculated following a simple recurrent integration procedure (see KNT or KrBook2020), starting with For every

Figures (4)

  • Figure 1: Real and imaginary parts of the potential from Example 1, recovered from 10 eigenpairs. The maximum absolute error of the recovered potential is $0.056$.
  • Figure 2: Real and imaginary parts of the recovered potential from Example 2.
  • Figure 3: Real and imaginary parts of the recovered potential from Example 3.
  • Figure 4: Real and imaginary parts of the recovered potential from Example 4.

Theorems & Definitions (8)

  • Theorem 2.1: KNT
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1
  • Remark 4.2