General information
Latest issue
Forthcoming papers
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Izv. RAN. Ser. Mat.:

Personal entry:
Save password
Forgotten password?

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 $n\to\infty$.


Full text: PDF file (4066 kB)
References: PDF file   HTML file

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
\by V.~A.~Vinokurov, V.~A.~Sadovnichii
\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
\jour Izv. Math.
\yr 2000
\vol 64
\issue 4
\pages 695--754

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Vinokurov V.A., Sadovnichii V.A., “The uniform equiconvergence of the Fourier series in the eigenfunctions of the first boundary value problem and of the trigonometric Fourier series”, Doklady Mathematics, 64:2 (2001), 248–252  mathscinet  zmath  isi  elib
    2. Vinokurov V.A., Sadovnichii V.A., “The asymptotic behavior of eigenvalues and eigenfunctions and a trace formula for potentials containing delta-functions”, Doklady Mathematics, 63:1 (2001), 62–65  mathscinet  zmath  isi
    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
    5. S. I. Mitrokhin, “Spectral properties of a fourth-order differential operator with integrable coefficients”, Proc. Steklov Inst. Math., 270 (2010), 184–193  mathnet  crossref  mathscinet  zmath  isi  elib
    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
    7. S. I. Mitrokhin, “O spektralnykh svoistvakh odnogo differentsialnogo operatora s summiruemymi koeffitsientami s zapazdyvayuschim argumentom”, Ufimsk. matem. zhurn., 3:4 (2011), 95–115  mathnet  zmath
    8. A. Yu. Trynin, “Differentsialnye svoistva nulei sobstvennykh funktsii zadachi Shturma–Liuvillya”, Ufimsk. matem. zhurn., 3:4 (2011), 133–143  mathnet  zmath
    9. Mitrokhin S.I., “On the Spectral Properties of Odd-Order Differential Operators with Integrable Potential”, Differ Equ, 47:12 (2011), 1833–1836  crossref  mathscinet  zmath  isi  elib  elib  scopus
    10. Sergei A. Avdonin, Boris P. Belinskiy, “On controllability of a non-homogeneous elastic string with memory”, Journal of Mathematical Analysis and Applications, 2012  crossref  mathscinet  isi  scopus
    11. L.S.. Efremova, Gerhard Freiling, “Numerical solution of inverse spectral problems for Sturm-Liouville operators with discontinuous potentials”, centr.eur.j.math, 11:11 (2013), 2044  crossref  mathscinet  zmath  isi  elib  scopus
    12. A. Yu. Trynin, “On inverse nodal problem for Sturm-Liouville operator”, Ufa Math. J., 5:4 (2013), 112–124  mathnet  crossref  elib
    13. Avdonin S., Bell J., “Determining a Distributed Parameter in a Neural Cable Model via a Boundary Control Method”, J. Math. Biol., 67:1, SI (2013), 123–141  crossref  mathscinet  zmath  isi  elib  scopus
    14. Berglund N., Gentz B., “Sharp Estimates for Metastable Lifetimes in Parabolic SPDEs: Kramers' Law and Beyond”, Electron. J. Probab., 18 (2013), 1–58  crossref  mathscinet  isi  scopus
    15. L. S. Efremova, “Chislennoe reshenie obratnoi zadachi dlya operatora Shturma–Liuvillya s razryvnym potentsialom”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 14:3 (2014), 273–279  mathnet
    16. Avdonin S.A., Mikhaylov V.S., “Reconstructing the Potential For the One-Dimensional Schrodinger Equation From Boundary Measurements”, IMA J. Math. Control Inf., 31:1 (2014), 137–150  crossref  mathscinet  zmath  isi  scopus
    17. S. I. Mitrokhin, “Ob asimptotike sobstvennykh znachenii modelnoi kraevoi zadachi dlya semeistva differentsialnykh operatorov s summiruemym potentsialom”, Mezhdunar. nauch.-issled. zhurn., 2016, no. 10-2(52), 137–143  mathnet  crossref
    18. S. I. Mitrokhin, “Mnogotochechnye differentsialnye operatory: rasscheplenie kratnykh v glavnom sobstvennykh znachenii”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 17:1 (2017), 5–18  mathnet  crossref  elib
    19. S. I. Mitrokhin, “Ob effekte rasschepleniya dlya mnogotochechnykh differentsialnykh operatorov s summiruemym potentsialom”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 21:2 (2017), 249–270  mathnet  crossref  zmath  elib
    20. S. I. Mitrokhin, “Ob izuchenii spektra mnogotochechnoi kraevoi zadachi dlya differentsialnogo operatora nechetnogo poryadka s summiruemym potentsialom”, Matematicheskie zametki SVFU, 24:1 (2017), 26–42  mathnet  elib
    21. S. I. Mitrokhin, “Study of differential operator with summable potential and discontinuous weight function”, Ufa Math. J., 9:4 (2017), 72–84  mathnet  crossref  isi  elib
    22. S. I. Mitrokhin, “Periodicheskaya kraevaya zadacha dlya differentsialnogo operatora chetvertogo poryadka s summiruemym potentsialom”, Vladikavk. matem. zhurn., 19:4 (2017), 35–49  mathnet
    23. S. I. Mitrokhin, “Spectral properties of the family of even order differential operators with a summable potential”, Moscow University Mathematics Bulletin, 72:4 (2017), 137–148  mathnet  crossref  mathscinet  isi  elib
    24. S. I. Mitrokhin, “Asymptotics of spectrum of multipoint differential operators with summable potential”, J. Math. Sci., 231:2 (2018), 243–254  mathnet  crossref  crossref
    25. S. I. Mitrokhin, “Asymptotic of eigenvalues of differential operator with alternating weight function”, Russian Math. (Iz. VUZ), 62:6 (2018), 27–42  mathnet  crossref  isi
    26. S. I. Mitrokhin, “Asimptotika spektra periodicheskoi kraevoi zadachi dlya differentsialnogo operatora s summiruemym potentsialom”, Tr. IMM UrO RAN, 25, no. 1, 2019, 136–149  mathnet  crossref  elib
  •    .  Izvestiya: Mathematics
    Number of views:
    This page:705
    Full text:282
    First page:3

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019