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Mat. Zametki, 2016, Volume 99, Issue 5, Pages 715–731 (Mi mz11138)  

This article is cited in 1 scientific paper (total in 1 paper)

Reconstruction of the Potential of the Sturm–Liouville Operator from a Finite Set of Eigenvalues and Normalizing Constants

A. M. Savchuk

Lomonosov Moscow State University

Abstract: It is well known that the potential $q$ of the Sturm–Liouville operator
$$ Ly=-y"+q(x)y $$
on the finite interval $[0,\pi]$ can be uniquely reconstructed from the spectrum $\{\lambda_k\}_1^\infty$ and the normalizing numbers $\{\alpha_k\}_1^\infty$ of the operator $L_D$ with the Dirichlet conditions. For an arbitrary real-valued potential $q$ lying in the Sobolev space $W^\theta_2[0,\pi]$, $\theta>-1$, we construct a function $q_N$ providing a $2N$-approximation to the potential on the basis of the finite spectral data set $\{\lambda_k\}_1^N\cup\{\alpha_k\}_1^N$. The main result is that, for arbitrary $\tau$ in the interval $-1\le\tau <\theta$, the estimate
$$ \|q-q_N\|_\tau \le CN^{\tau-\theta} $$
is true, where $\|\cdot\|_\tau$ is the norm on the Sobolev space $W^\tau_2$. The constant $C$ depends solely on $\|q\|_\theta$.

Keywords: Sturm–Liouville operator, inverse problem, reconstruction of the potential, spectral data.

Funding Agency Grant Number
Russian Science Foundation 14-01-00754
This work was supported by the Russian Science Foundation under grant 14-01-00754.


DOI: https://doi.org/10.4213/mzm11138

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English version:
Mathematical Notes, 2016, 99:5, 715–728

Bibliographic databases:

UDC: 517.984.54
Received: 30.11.2015

Citation: A. M. Savchuk, “Reconstruction of the Potential of the Sturm–Liouville Operator from a Finite Set of Eigenvalues and Normalizing Constants”, Mat. Zametki, 99:5 (2016), 715–731; Math. Notes, 99:5 (2016), 715–728

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. N. F. Valeev, Ya. Sh. Il'yasov, “On an Inverse Optimization Spectral Problem and a Corresponding Nonlinear Boundary-Value Problem”, Math. Notes, 104:4 (2018), 601–605  mathnet  crossref  crossref  isi  elib
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