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Sinc method in spectrum completion and inverse Sturm-Liouville problems

Vladislav V. Kravchenko, L. Estefania Murcia-Lozano

Abstract

Cardinal series representations for solutions of the Sturm-Liouville equation $-y''+q(x)y=ρ^{2}y$, $x\in(0,L)$ with a complex valued potential $q(x)$ are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to $ρ$ in any strip $\left|\text{Im}ρ\right|<C$ of the complex plane. This property of the obtained series representations leads to their applications in a variety of spectral problems. In particular, we show their applicability to the spectrum completion problem, consisting in computing large sets of the eigenvalues from a reduced finite set of known eigenvalues, without any information on the potential $q(x)$ as well as on the constants from boundary conditions. Among other applications this leads to an efficient numerical method for computing a Weyl function from two finite sets of the eigenvalues. This possibility is explored in the present work and illustrated by numerical tests. Finally, based on the cardinal series representations obtained, we develop a method for the numerical solution of the inverse two-spectra Sturm-Liouville problem and show its numerical efficiency.

Sinc method in spectrum completion and inverse Sturm-Liouville problems

Abstract

Cardinal series representations for solutions of the Sturm-Liouville equation , with a complex valued potential are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to in any strip of the complex plane. This property of the obtained series representations leads to their applications in a variety of spectral problems. In particular, we show their applicability to the spectrum completion problem, consisting in computing large sets of the eigenvalues from a reduced finite set of known eigenvalues, without any information on the potential as well as on the constants from boundary conditions. Among other applications this leads to an efficient numerical method for computing a Weyl function from two finite sets of the eigenvalues. This possibility is explored in the present work and illustrated by numerical tests. Finally, based on the cardinal series representations obtained, we develop a method for the numerical solution of the inverse two-spectra Sturm-Liouville problem and show its numerical efficiency.
Paper Structure (15 sections, 14 theorems, 138 equations, 24 figures, 5 tables)

This paper contains 15 sections, 14 theorems, 138 equations, 24 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Neumann series of Bessel functions representations
  4. Cardinal series representations for solutions
  5. Cardinal series representation for solutions of (\ref{['eq:PrincipalEq']}) with $q\in\mathcal{L}_{2}[0,L]$
  6. Representation for derivatives of solutions
  7. Cardinal series representation for solutions when $q\in W_{2}^{1}\left[0,L\right]$
  8. Remainder estimates
  9. Spectrum completion
  10. Dirichlet-Dirichlet spectrum completion
  11. Neumann-Dirichlet spectrum
  12. Other spectra
  13. Computation of a Weyl function from two finite sets of eigenvalues
  14. Inverse problem
  15. Conclusions

Key Result

Theorem 1

Let $q\in\mathcal{L}_{2}\left[0,L\right]$. There exist functions $\boldsymbol{\mathcal{S}}(x,t)$ and $\boldsymbol{\mathcal{G}}(x,t)$ defined in the domain $0\leq t\leq x\leq L,$ such that for all $\rho\in\mathbb{C}$. The functions $\boldsymbol{\mathcal{S}}(x,\cdot)$ and $\boldsymbol{\mathcal{G}}(x,\cdot)$ possess the same regularity as $\int_{0}^{x}q(t)dt$, see FreYurko. In particular, $\boldsymb

Figures (24)

  • Figure 5.1: Example \ref{['ExamplePAine']}. Absolute errors of the Dirichlet-Dirichlet spectrum completed from 15, 10 and 5 eigenvalues (columns in the figure). In the top row the exact eigenvalues were used as the input data, while in the other two rows the spectrum is completed from randomly noisy data. The maximum noise level is indicated on the top of each figure.
  • Figure 5.2: Example \ref{['ExamplePAine']}. Absolute errors of the Dirichlet-Dirichlet spectrum completed from given 15 eigenvalues and considering overdetermined systems in the spectrum completion procedure. The maximum noise level is indicated on the top of each figure.
  • Figure 5.3: Example \ref{['ExampleC1Pot']}. Absolute errors of the Dirichlet-Dirichlet spectrum completed from given 15, 10 and 5 eigenvalues (columns in the figure). In the top row the exact eigenvalues were used as the input data, while in the other two rows the spectrum is completed from randomly noisy data. The maximum noise level is indicated on the top of each figure.
  • Figure 5.4: Example \ref{['ExampleC1Pot']}. Absolute errors of the Dirichlet-Dirichlet spectrum completed given 15 eigenvalues and considering overdetermined systems in the spectrum completion procedure. The maximum noise level is indicated on the top of each figure.
  • Figure 5.5: Example \ref{['Example-2.-Razavy']}. Abs. and rel. errors of $\rho_{k}$ computed from 5 eigenvalues (in (A)), from 15 eigenvalues (in (B)), from 25 eigenvalues (in (C)) and from 35 eigenvalues (in (D)).
  • ...and 19 more figures

Theorems & Definitions (41)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • Remark 4
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Remark 7
  • ...and 31 more