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Uspekhi Mat. Nauk, 2004, Volume 59, Issue 3(357), Pages 115–150 (Mi umn738)  

This article is cited in 22 scientific papers (total in 22 papers)

Some problems of the qualitative Sturm–Liouville theory on a spatial network

Yu. V. Pokornyi, V. L. Pryadiev

Voronezh State University

Abstract: An analogue of the Sturm oscillation theory of the distribution of the zeros of eigenfunctions is constructed for the problem
\begin{equation} Lu\overset{def}{=}-\frac d{d\Gamma}(pu')+qu=\lambda mu, \qquad u|_{\partial\Gamma}=0 \tag{1} \end{equation}
on a spatial network $\Gamma$ (in other terms, $\Gamma$ is a metric graph, a CW complex, a stratified locally one-dimensional manifold, a branching space, a quantum graph, and so on), where $\partial\Gamma$ is the family of boundary vertices of $\Gamma$. At interior points of the edges of $\Gamma$ the quasi-derivative $\displaystyle\frac d{d\Gamma}(pu')$ has the classical form $(pu')'$, and at interior nodes it is assumed that
$$ \frac d{d\Gamma}(pu')=-\sum_\gamma\alpha_\gamma(a)u'_\gamma(a), $$
where the summation is taken over the edges $\gamma$ incident to the node $a$ and, for an edge $\gamma$, $u'_\gamma (a)$ stands for the ‘endpoint’ derivative of the restriction $u_\gamma (x)$ of the function $u\colon\Gamma\to\mathbb R$ to $\gamma$. Despite the branching argument, which is a kind of intermediate type between the one-dimensional and multidimensional cases, the outward form of the results turns out to be quite classical. The classical nature of the operator $L$ is clarified, and exact analogues of the maximum principle and of the Sturm theorem on alternation of zeros are established, together with the sign-regular oscillation properties of the spectrum of the problem (1) (including the simplicity and positivity of the points of the spectrum and also the number of zeros and their alternation for the eigenfunctions).

DOI: https://doi.org/10.4213/rm738

Full text: PDF file (480 kB)
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English version:
Russian Mathematical Surveys, 2004, 59:3, 515–552

Bibliographic databases:

UDC: 517.927
MSC: Primary 34B24, 34B45; Secondary 34B10, 05C99, 35Q99
Received: 07.04.2002

Citation: Yu. V. Pokornyi, V. L. Pryadiev, “Some problems of the qualitative Sturm–Liouville theory on a spatial network”, Uspekhi Mat. Nauk, 59:3(357) (2004), 115–150; Russian Math. Surveys, 59:3 (2004), 515–552

Citation in format AMSBIB
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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:
    1. Neuberger J.M., “Nonlinear elliptic partial difference equations on graphs”, Experiment. Math., 15:1 (2006), 91–107  crossref  mathscinet  zmath  isi
    2. V. A. Yurko, “On recovering Sturm–Liouville operators on graphs”, Math. Notes, 79:4 (2006), 572–582  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. Oren I., “Nodal domain counts and the chromatic number of graphs”, J. Phys. A, 40:32 (2007), 9825–9832  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Currie S., Watson B.A., “Boundary estimates for solutions of non-homogeneous boundary value problems on graphs”, Applied Mathematics for Science and Engineering, 2007, 37–42  mathscinet  isi
    5. 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
    6. V. A. Yurko, “Inverse spectral problem for differential operator pencils on noncompact spatial networks”, Differ. Equ., 44:12 (2008), 1721–1729  crossref  mathscinet  mathscinet  zmath  isi  elib  elib
    7. Currie S., Watson B.A., “Green's functions and regularized traces of Sturm-Liouville operators on graphs”, Proc. Edinb. Math. Soc. (2), 51:2 (2008), 315–335  crossref  mathscinet  zmath  isi  elib
    8. Berkolaiko G., “A lower bound for nodal count on discrete and metric graphs”, Comm. Math. Phys., 278:3 (2008), 803–819  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. Band R., Oren I., Smilansky U., “Nodal domains on graphs - How to count them and why?”, Analysis on Graphs and its Applications, Proceedings of Symposia in Pure Mathematics, 77, 2008, 5–27  crossref  mathscinet  zmath  isi
    10. V. A. Yurko, “Recovering Sturm-Liouville operators from spectra on a graph with a cycle”, Sb. Math., 200:9 (2009), 1403–1415  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. Currie S., Watson B.A., “The $M$-matrix inverse problem for the Sturm-Liouville equation on graphs”, Proc. Roy. Soc. Edinburgh Sect. A, 139:4 (2009), 775–796  crossref  mathscinet  zmath  isi  elib
    12. 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
    13. V. A. Yurko, “Inverse Problem for Sturm–Liouville Operators on Hedgehog-Type Graphs”, Math. Notes, 89:3 (2011), 438–449  mathnet  crossref  crossref  mathscinet  isi
    14. A. Yu. Trynin, “Differentsialnye svoistva nulei sobstvennykh funktsii zadachi Shturma–Liuvillya”, Ufimsk. matem. zhurn., 3:4 (2011), 133–143  mathnet  zmath
    15. Currie S., Watson B.A., “Indefinite Boundary Value Problems on Graphs”, Oper Matrices, 5:4 (2011), 565–584  crossref  mathscinet  zmath  isi  elib
    16. Yurko V.A., “Reconstruction of Sturm–Liouville Differential Operators on A-Graphs”, Differ Equ, 47:1 (2011), 50–59  crossref  mathscinet  zmath  isi  elib  elib
    17. Band R., Berkolaiko G., Smilansky U., “Dynamics of Nodal Points and the Nodal Count on a Family of Quantum Graphs”, Ann Henri Poincaré, 13:1 (2012), 145–184  crossref  mathscinet  zmath  adsnasa  isi
    18. A. Yu. Trynin, “On inverse nodal problem for Sturm-Liouville operator”, Ufa Math. J., 5:4 (2013), 112–124  mathnet  crossref  elib
    19. Currie S., “Self-Adjoint Boundary Conditions and Interlacing of Eigenvalues For the Sturm-Liouville Equation on Graphs”, Oper. Matrices, 8:2 (2014), 467–483  crossref  mathscinet  zmath  isi
    20. L. K. Zhapsarbayeva, B. E. Kanguzhin, M. N. Konyrkulzhayeva, “Self-adjoint restrictions of maximal operator on graph”, Ufa Math. J., 9:4 (2017), 35–43  mathnet  crossref  isi  elib
    21. Akduman S., Pankov A., “Schrodinger Operators With Locally Integrable Potentials on Infinite Metric Graphs”, Appl. Anal., 96:12 (2017), 2149–2161  crossref  mathscinet  zmath  isi
    22. Akduman S., Pankov A., “Exponential Estimates For Quantum Graphs”, Electron. J. Differ. Equ., 2018, 162  zmath  isi
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