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Izv. RAN. Ser. Mat., 2000, Volume 64, Issue 4, Pages 47–108 (Mi izv295)  

This article is cited in 26 scientific papers (total in 26 papers)

Asymptotics of any order for the eigenvalues and eigenfunctions of the Sturm–Liouville boundary-value problem on a segment with a summable potential

V. A. Vinokurov, V. A. Sadovnichiia

a M. V. Lomonosov Moscow State University

Abstract: For the Sturm–Liouville boundary-value problem on a segment we construct asymptotics for $s_n=\sqrt{\lambda_n}$, where $\lambda_n$ are the eigenvalues, and for the normalized eigenfunctions $y_n(x)$ of the form
$$ s_n=s_{n,m}(q)+\psi_{n,m}, \qquad y_n(x)=y_{n,m}(q,x)+\Delta y_{n,m}(x) $$
for any $m=0,1,2,…$, where $s_{n,m}(q)$ and $y_{n,m}(q,x)$ are expressed explicitly in terms of the potential $q(x)$. Under the assumption that $q(x)$ is a real summable function, the terms $\psi_{n,m}$ and $\Delta y_{n,m}(x)$ are $O(\dfrac1{n^{m+1}})$ as $n\to\infty$.

DOI: https://doi.org/10.4213/im295

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English version:
Izvestiya: Mathematics, 2000, 64:4, 695–754

Bibliographic databases:

MSC: 47E05, 34L99, 47A70, 34E99
Received: 24.12.1998

Citation: V. A. Vinokurov, V. A. Sadovnichii, “Asymptotics of any order for the eigenvalues and eigenfunctions of the Sturm–Liouville boundary-value problem on a segment with a summable potential”, Izv. RAN. Ser. Mat., 64:4 (2000), 47–108; Izv. Math., 64:4 (2000), 695–754

Citation in format AMSBIB
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\paper Asymptotics of any order for the eigenvalues and eigenfunctions of the Sturm--Liouville boundary-value problem on a~segment with a~summable potential
\jour Izv. RAN. Ser. Mat.
\yr 2000
\vol 64
\issue 4
\pages 47--108
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
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    3. Chernyatin V.A., “Higher-order spectral asymptotics for the Sturm-Liouville operator”, Differential Equations, 38:2 (2002), 217–227  mathnet  crossref  mathscinet  zmath  isi  elib  scopus
    4. A. Yu. Trynin, “Asymptotic behavior of the solutions and nodal points of Sturm–Liouville differential expressions”, Siberian Math. J., 51:3 (2010), 525–536  mathnet  crossref  mathscinet  zmath  isi  elib  elib
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    6. Mitrokhin S.I., “Spectral properties of boundary value problems for functional-differential equations with integrable coefficients”, Differential Equations, 46:8 (2010), 1095–1103  crossref  mathscinet  zmath  isi  elib  scopus
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