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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.

DOI: https://doi.org/10.4213/mzm3363

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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
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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
    2. Yu. V. Pokornyi, M. B. Zvereva, S. A. Shabrov, “Sturm–Liouville oscillation theory for impulsive problems”, Russian Math. Surveys, 63:1 (2008), 109–153  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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
    7. Aceto L., Ghelardoni P., Marletta M., “Numerical solution of forward and inverse Sturm-Liouville problems with an angular momentum singularity”, Inverse Problems, 24:1 (2008), 015001, 21 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    8. Djakov P., Mityagin B., “Spectral gap asymptotics of one-dimensional Schrodinger operators with singular periodic potentials”, Integral Transforms Spec. Funct., 20:3-4 (2009), 265–273  crossref  mathscinet  zmath  isi  elib  scopus
    9. Makhnei A. V., Tatsii R. M., “Asymptotics of the eigenvalues of a boundary value problem for a vector singular differential equation”, Differ. Equ., 45:6 (2009), 805–813  crossref  mathscinet  zmath  isi  elib  scopus
    10. Djakov P., Mityagin B., “Spectral gaps of Schrodinger operators with periodic singular potentials”, Dyn. Partial Differ. Equ., 6:2 (2009), 95–165  crossref  mathscinet  zmath  isi  elib  scopus
    11. 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
    12. I. V. Sadovnichaya, “Equiconvergence of eigenfunction expansions for Sturm-Liouville operators with a distributional potential”, Sb. Math., 201:9 (2010), 1307–1322  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    13. I. V. Sadovnichaya, “Equiconvergence theorems for Sturm–Lioville operators with singular potentials (rate of equiconvergence in $W_2^\theta$-norm)”, Eurasian Math. J., 1:1 (2010), 137–146  mathnet  mathscinet  zmath
    14. A. M. Savchuk, A. A. Shkalikov, “Inverse Problems for Sturm–Liouville Operators with Potentials in Sobolev Spaces: Uniform Stability”, Funct. Anal. Appl., 44:4 (2010), 270–285  mathnet  crossref  crossref  mathscinet  zmath  isi
    15. Djakov P. Mityagin B., “Fourier Method for One-Dimensional Schrodinger Operators with Singular Periodic Potentials”, Topics in Operator Theory, Vol 2: Systems and Mathematical Physics, Operator Theory Advances and Applications, 203, ed. Ball J. Bolotnikov V. Helton J. Rodman L. Spitkovsky I., Birkhauser Verlag Ag, 2010, 195–236  crossref  mathscinet  isi
    16. Wei G., Wang Ya., “Asymptotic behavior for differences of eigenvalues of two Sturm-Liouville problems with smooth potentials”, J. Math. Anal. Appl., 377:2 (2011), 659–669  crossref  mathscinet  zmath  isi  elib  scopus
    17. È. F. Akhmerova, “Asymptotics of the Spectrum of Nonsmooth Perturbations of Differential Operators of Order $2m$”, Math. Notes, 90:6 (2011), 813–823  mathnet  crossref  crossref  mathscinet  isi
    18. A. Yu. Trynin, “Differentsialnye svoistva nulei sobstvennykh funktsii zadachi Shturma–Liuvillya”, Ufimsk. matem. zhurn., 3:4 (2011), 133–143  mathnet  zmath
    19. Sadovnichaya I.V., “Equiconvergence theorems in Sobolev and Hölder spaces of eigenfunction expansions for Sturm-Liouville operators with singular potentials”, Dokl. Math., 83:2 (2011), 169–170  crossref  mathscinet  zmath  isi  elib  elib  scopus
    20. Andrew A.L., “Finite Difference Methods for Half Inverse Sturm-Liouville Problems”, Appl. Math. Comput., 218:2 (2011), 445–457  crossref  mathscinet  zmath  isi  elib  scopus
    21. Hryniv R.O., “Analyticity and Uniform Stability in the Inverse Singular Sturm-Liouville Spectral Problem”, Inverse Probl., 27:6 (2011), 065011  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    22. Sadovnichaya I.V., “Equiconvergence of Expansions in Eigenfunctions of Sturm-Liouville Operators with Distributional Potentials in Holder Spaces”, Differ. Equ., 48:5 (2012), 681–692  crossref  mathscinet  zmath  isi  elib  elib  scopus
    23. Hryniv R., Pronska N., “Inverse Spectral Problems for Energy-Dependent Sturm-Liouville Equations”, Inverse Probl., 28:8 (2012), 085008  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    24. Kappeler T., Schaad B., Topalov P., “Asymptotics of Spectral Quantities of Schrodinger Operators”, Spectral Geometry, Proceedings of Symposia in Pure Mathematics, 84, eds. Barnett A., Gordon C., Perry P., Uribe A., Amer Mathematical Soc, 2012, 243–284  crossref  mathscinet  zmath  isi
    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
    26. A. M. Savchuk, A. A. Shkalikov, “Uniform stability of the inverse Sturm–Liouville problem with respect to the spectral function in the scale of Sobolev spaces”, Proc. Steklov Inst. Math., 283 (2013), 181–196  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    27. A. Yu. Trynin, “On inverse nodal problem for Sturm-Liouville operator”, Ufa Math. J., 5:4 (2013), 112–124  mathnet  crossref  elib
    28. Kappeler T., Schaad B., Topalov P., “Qualitative Features of Periodic Solutions of KdV”, Commun. Partial Differ. Equ., 38:9 (2013), 1626–1673  crossref  mathscinet  zmath  isi  elib  scopus
    29. Eckhardt J. Teschl G., “Sturm-Liouville Operators with Measure-Valued Coefficients”, J. Anal. Math., 120 (2013), 151–224  crossref  mathscinet  zmath  isi  elib  scopus
    30. Pronska N., “Reconstruction of Energy-Dependent Sturm-Liouville Equations From Two Spectra”, Integr. Equ. Oper. Theory, 76:3 (2013), 403–419  crossref  mathscinet  zmath  isi  elib  scopus
    31. Djakov P. Mityagin B., “Equiconvergence of Spectral Decompositions of Hill-Schrodinger Operators”, J. Differ. Equ., 255:10 (2013), 3233–3283  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    32. Wei G., Xu H.-K., “Inverse Spectral Analysis for the Transmission Eigenvalue Problem”, Inverse Probl., 29:11 (2013), 115012  crossref  mathscinet  zmath  adsnasa  isi  scopus
    33. Davies E.B., “Singular Schrodinger Operators in One Dimension”, Mathematika, 59:1 (2013), 141–159  crossref  mathscinet  zmath  isi  elib  scopus
    34. Eckhardt J., Gesztesy F., Nichols R., Teschl G., “Inverse Spectral Theory for Sturm-Liouville Operators with Distributional Potentials”, J. Lond. Math. Soc.-Second Ser., 88:3 (2013), 801–828  crossref  mathscinet  zmath  isi  scopus
    35. T. Kappeler, A. M. Savchuk, P. Topalov, A. A. Shkalikov, “Interpolation of Nonlinear Maps”, Math. Notes, 96:6 (2014), 957–964  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    36. Eckhardt J. Gesztesy F. Nichols R. Teschl G., “Supersymmetry and Schrodinger-Type Operators With Distributional Matrix-Valued Potentials”, J. Spectr. Theory, 4:4 (2014), 715–768  crossref  mathscinet  zmath  isi  scopus
    37. Qi J., Chen Sh., Xie B., “Instability of Plane Shear Flows”, Nonlinear Anal.-Theory Methods Appl., 109 (2014), 23–32  crossref  zmath  isi  scopus
    38. Baev A.D. Zvereva M.B. Shabrov S.A., “Stieltjes Differential in Nonlinear Momentum Problems”, Dokl. Math., 90:2 (2014), 613–615  crossref  mathscinet  zmath  isi  scopus
    39. Kappeler T., Topalov P., “on Nonlinear Interpolation”, Proc. Amer. Math. Soc., 143:8 (2015), 3421–3428  crossref  mathscinet  zmath  isi  elib  scopus
    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
    41. Qi J., Chen Sh., “Extremal norms of the potentials recovered from inverse Dirichlet problems”, Inverse Probl., 32:3 (2016), 035007  crossref  mathscinet  zmath  isi  elib  scopus
    42. A. G. Baskakov, D. M. Polyakov, “The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential”, Sb. Math., 208:1 (2017), 1–43  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    43. D. M. Polyakov, “A one-dimensional Schrödinger operator with square-integrable potential”, Siberian Math. J., 59:3 (2018), 470–485  mathnet  crossref  crossref  isi  elib
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