Inverse scattering problem on a noncompact graph

Gerasimenko, N. I.

doi:10.1007/bf01017484

Cited by 83 publications

(76 citation statements)

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“…For noncompact graphs the theory is only at the beginning. Some aspects of the inverse problem theory for noncompact graphs were considered in [5,7,10,11,18,34]. We note that inverse spectral problems for second-order and for higher-order ordinary differential operators on an interval (finite or infinite) have been studied by many authors (see the monographs [2,6,8,15,17,24,26,27] and references therein).…”

Section: Introductionmentioning

confidence: 99%

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

Yurko

2013

Open Mathematics

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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. MSC:34A55, 34L05, 47E05

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Section: Introductionmentioning

confidence: 99%

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

Yurko

2013

Open Mathematics

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“…By quantum graphs, we understand differential operators on geometric graphs coupled by certain vertex matching conditions. B. S. Pavlov and his former student N. I. Gerasimenko were among the first researchers to develop a mathematically rigorous approach to differential operators on metric graphs in the 1980's [1,2].…”

Section: Introductionmentioning

confidence: 99%

Control and inverse problems for networks of vibrating strings with attached masses

Al-Musallam¹,

Avdonin²,

Avdonina³

et al. 2016

Nanosystems: Phys. Chem. Math.

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We consider the control and inverse problems for serially connected and tree-like networks of strings with point masses loaded at the internal vertices. We prove boundary controllability of the systems and the identifiability of varying coefficients of the string equations along with the complete information on the graph, i.e. the loaded masses, the lengths of the edges and the topology (connectivity) of the graph. The results are achieved using the Titchmarch-Weyl function for the spectral problem and the Steklov-Poincaré operator for the dynamic wave equation on the tree. The general result is obtained by the leaf peeling method which reduces the inverse problem layer-by-layer from the leaves to the fixed root of the tree.

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“…For Sturm-Liouville operators on compact graphs inverse problems were studied in [7]- [14] and other works. Noncompact case for Sturm-Liouville operators was considered in [15]- [17]. Bessel operators on graphs have not been studied yet.…”

Section: Introductionmentioning

confidence: 99%

Recovering singular differential operators on noncompact star-type graphs from Weyl functions

Yurko

2011

Tamkang J. Math.

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Inverse scattering problem on a noncompact graph

Cited by 83 publications

References 1 publication

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

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

Control and inverse problems for networks of vibrating strings with attached masses

Recovering singular differential operators on noncompact star-type graphs from Weyl functions

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