Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2014 | 12 | 3 | 483-499
Tytuł artykułu

Inverse problems on star-type graphs: differential operators of different orders on different edges

Treść / Zawartość
Warianty tytułu
Języki publikacji
We study inverse spectral problems for ordinary differential equations on compact star-type graphs when differential equations have different orders on different edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.
  • [1] Avdonin S., Kurasov P., Inverse problems for quantum trees, Inverse Probl. Imaging, 2008, 2(1), 1–21
  • [2] Beals R., Deift P., Tomei C., Direct and Inverse Scattering on the Line, Math. Surveys Monogr., 28, American Mathematical Society, Providence, 1988
  • [3] Belishev M.I., Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems, 2004, 20(3), 647–672
  • [4] Brown B.M., Weikard R., A Borg-Levinson theorem for trees, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2005, 461(2062), 3231–3243
  • [5] Buterin S.A., Freiling G., Inverse scattering problem for the Sturm-Liouville operator on a noncompact star-type graph, Schriftenreihe des Instituts für Mathematik, SM-DU-725, Universität-Duisburg-Essen, 2011, 1–17
  • [6] Chadan K., Colton D., Päivärinta L., Rundell W., An Introduction to Inverse Scattering and Inverse Spectral Problems SIAM Monogr. Math. Model. Comput., SIAM, Philadelphia, 1997
  • [7] Freiling G., Ignatyev M., Spectral analysis for Sturm-Liouville operator on sun-type graphs, Inverse Problems, 2011, 27(9), #095003
  • [8] Freiling G., Yurko V., Inverse Sturm-Liouville Problems and their Applications, NOVA Science, Huntington, 2001
  • [9] Freiling G., Yurko V., Inverse problems for differential operators on trees with general matching conditions, Appl. Anal., 2007, 86(6), 653–667
  • [10] Freiling G., Yurko V., Inverse problems for Sturm-Liouville operators on noncompact trees, Results Math., 2007, 50(3–4), 195–212
  • [11] Gerasimenko N.I., The inverse scattering problem on a noncompact graph, Theoret. and Math. Phys., 1988, 75(2), 460–470
  • [12] Kottos T., Smilansky U., Quantum chaos on graphs, Phys. Rev. Lett., 1997, 79(24), 4794–4797
  • [13] Kuchment P., Quantum graphs: I. Some basic structures, Waves in Random and Complex Media, 2004, 14(1), S107–S128
  • [14] Langese J., Leugering G., Schmidt E.J.P.G., Modeling Analysis and Control of Dynamic Elastic Multi-Link Structures, Systems Control Found. Appl., Birkhäuser, Boston, 1994
  • [15] Levitan B.M., Inverse Sturm-Liouville Problems, VSP, Zeist, 1987
  • [16] Levitan B.M., Sargsyan I.S., Introduction to Spectral Theory, Transl. Math. Monogr., 39, American Mathematical Society, Providence, 1975
  • [17] Marchenko V.A., Sturm-Liouville Operators and Applications, Oper. Theory Adv. Appl., 22, Birkhäuser, Basel, 1986
  • [18] Marchenko V., Mochizuki K., Trooshin I., Inverse scattering on a graph containing circle, In: Analytic Methods of Analysis and Differential Equations, Minsk, September 13–19, 2006, Camb. Sci. Publ., Cambridge, 2008, 237–243
  • [19] Montroll E.W., Quantum theory on a network. I. A solvable model whose wavefunctions are elementary functions, J. Math. Phys., 1970, 11(2), 635–648
  • [20] Naimark M.A., Linear Differential Operators, 2nd ed., Nauka, Moscow, 1969 (in Russian)
  • [21] Pokornyi Yu.V., Beloglazova T.V., Dikareva E.V., Perlovskaya T.V., Green function for a locally interacting system of ordinary equations of different orders, Math. Notes, 2003, 74(1–2), 141–143
  • [22] Pokornyi Yu.V., Borovskikh A.V., Differential equations on networks (geometric graphs), J. Math. Sci. (N.Y.), 2004, 119(6), 691–718
  • [23] Pokornyi Yu.V., Pryadiev V.L., The qualitative Sturm-Liouville theory on spatial networks, J. Math. Sci. (N.Y.), 2004, 119(6), 788–835
  • [24] Ramm A.G., Inverse Problems, Math. Anal. Tech. Appl. Eng., Springer, New York, 2005
  • [25] Yang C.-F., Inverse spectral problems for the Sturm-Liouville operators on a d-star graph, J. Math. Anal. Appl., 2010, 365(2), 742–749
  • [26] Yurko V.A., Inverse Spectral Problems for Differential Operators and their Applications, Anal. Methods Spec. Funct., 2, Gordon and Breach, Amsterdam, 2000
  • [27] Yurko V., Method of Spectral Mappings in the Inverse Problem Theory, Inverse Ill-Posed Probl. Ser., VSP, Utrecht, 2002
  • [28] Yurko V., Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems, 2005, 21(3), 1075–1086
  • [29] Yurko V.A., An inverse problem for higher order differential operators on star-type graphs, Inverse Problems, 2007, 23(3), 893–903
  • [30] Yurko V.A., Inverse problems for differential of any order on trees, Math. Notes, 2008, 83(1–2), 125–137
  • [31] Yurko V., Inverse problems for Sturm-Liouville operators on bush-type graphs, Inverse Problems, 2009, 25(10), #105008
  • [32] Yurko V., An inverse problem for Sturm-Liouville differential operators on A-graphs, Appl. Math. Lett., 2010, 23(8), 875–879
  • [33] Yurko V.A., Inverse spectral problems for differential operators on arbitrary compact graphs, J. Inverse Ill-Posed Probl., 2010, 18(3), 245–261
  • [34] Yurko V.A., Inverse spectral problems for arbitrary order differential operators on noncompact trees, J. Inverse Ill-Posed Probl., 2012, 20(1), 111–131
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.