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Mat. Zametki, 2006, Volume 80, Issue 6, Pages 864–884 (Mi mz3363)  

This article is cited in 43 scientific papers (total in 43 papers)

On the eigenvalues of the Sturm–Liouville operator with potentials from Sobolev spaces

A. M. Savchuk, A. A. Shkalikov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study the asymptotic behavior of the eigenvalues the Sturm–Liouville operator $Ly= -y" +q(x)y$ with potentials from the Sobolev space $W_2^{\theta-1}$, $\theta\ge0$, including the nonclassical case $\theta\in[0,1)$ in which the potential is a distribution. The results are obtained in new terms. Let $s_{2k}(q)=\lambda_{k}^{1/2}(q)-k$, $s_{2k-1}(q)=\mu_{k}^{1/2}(q)-k-1/2$, where $\{\lambda_k\}_1^{\infty}$ and $\{\mu_k\}_1^{\infty}$ are the sequences of eigenvalues of the operator $L$ generated by the Dirichlet and Dirichlet–Neumann boundary conditions, respectively. We construct special Hilbert spaces $\hat\ell_2^{ \theta}$ such that the mapping $F\colon W^{\theta-1}_2\to\hat\ell_2^{ \theta}$ defined by the equality $F(q)=\{s_n\}_1^{\infty}$ is well defined for all $\theta\ge0$. The main result is as follows: for $\theta>0$, the mapping $F$ is weakly nonlinear, i.e., can be expressed as $F(q)=Uq+\Phi(q)$, where $U$ is the isomorphism of the spaces $W^{\theta-1}_2$ and $\hat\ell_2^{ \theta}$, and $\Phi(q)$ is a compact mapping. Moreover, we prove the estimate $\|\Phi(q)\|_{\tau}\le C\|q\|_{\theta-1}$, where the exact value of $\tau=\tau(\theta)>\theta-1$ is given and the constant $C$ depends only on the radius of the ball $\|q\|_{\theta-1}\le R$, but is independent of the function $q$ varying in this ball.


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English version:
Mathematical Notes, 2006, 80:6, 814–832

Bibliographic databases:

UDC: 517.984
Received: 28.06.2006
Revised: 18.07.2006

Citation: A. M. Savchuk, A. A. Shkalikov, “On the eigenvalues of the Sturm–Liouville operator with potentials from Sobolev spaces”, Mat. Zametki, 80:6 (2006), 864–884; Math. Notes, 80:6 (2006), 814–832

Citation in format AMSBIB
\by A.~M.~Savchuk, A.~A.~Shkalikov
\paper On the eigenvalues of the Sturm--Liouville operator with potentials from Sobolev spaces
\jour Mat. Zametki
\yr 2006
\vol 80
\issue 6
\pages 864--884
\jour Math. Notes
\yr 2006
\vol 80
\issue 6
\pages 814--832

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    This publication is cited in the following articles:
    1. Savchuk A.M., Shkalikov A.A., “Inverse problem for Sturm-Liouville operators with distribution potentials: Reconstruction from two spectra”, Russ. J. Math. Phys., 12:4 (2005), 507–514  mathscinet  zmath  isi  elib
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    3. A. M. Savchuk, A. A. Shkalikov, “On the Properties of Maps Connected with Inverse Sturm–Liouville Problems”, Proc. Steklov Inst. Math., 260 (2008), 218–237  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    4. A. M. Savchuk, “A Mapping Method in Inverse Sturm–Liouville Problems with Singular Potentials”, Proc. Steklov Inst. Math., 261 (2008), 237–242  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    5. I. V. Sadovnichaya, “Equiconvergence of the Trigonometric Fourier Series and the Expansion in Eigenfunctions of the Sturm–Liouville Operator with a Distribution Potential”, Proc. Steklov Inst. Math., 261 (2008), 243–252  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    6. Sadovnichaya I. V., “On the equiconvergence rate of trigonometric series expansions and eigenfunction expansions for the Sturm-Liouville operator with a distributional potential”, Differ. Equ., 44:5 (2008), 675–684  crossref  mathscinet  zmath  isi  elib  elib  scopus
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    25. A. M. Savchuk, A. A. Shkalikov, “On the Interpolation of Analytic Mappings”, Math. Notes, 94:4 (2013), 547–550  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
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    40. Chirilus-Bruckner M., Wayne C.E., “Inverse Spectral Theory For Uniformly Open Gaps in a Weighted Sturm-Liouville Problem”, J. Math. Anal. Appl., 427:2 (2015), 1168–1189  crossref  mathscinet  zmath  isi  scopus
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