Inverse dynamic and spectral problems for the one-dimensional Dirac system on a finite tree
-
Alexander Mikhaylov
,
Victor S. Mikhaylov
und
Gulden Murzabekova
Abstract
We consider inverse dynamic and spectral problems for the one-dimensional Dirac system on a finite tree. Our aim will be to recover the topology of a tree (lengths and connectivity of edges) as well as the matrix potentials on each edge. As inverse data we use the Weyl–Titchmarsh matrix function or the dynamic response operator.
Funding source: Russian Foundation for Basic Research
Award Identifier / Grant number: 17-01-00529-a
Award Identifier / Grant number: 17-01-00099-a
Funding statement: The research of Victor Mikhaylov was supported in part by RFBR 17-01-00529. Alexandr Mikhaylov was supported by RFBR 17-01-00099; A. S. Mikhaylov and V. S. Mikhaylov were partly supported by the VW Foundation program “Modeling, Analysis, and Approximation Theory toward application in tomography and inverse problems”. Gulden Murzabekova and Victor Mikhaylov were also partly supported by the Ministry of Education and Science of Republic of Kazakhstan, grant no. 4290/GF4.
References
[1] S. A. Avdonin, A. S. Blagovestchensky, A. Choque Rivero and V. S. Mikhaylov, Dynamic inverse problem for two-velocity systems on finite trees, Days on Diffraction (St. Petersburg 2016), IEEE Press, Piscataway (2016), 25–31. 10.1109/DD.2016.7756807Suche in Google Scholar
[2] S. A. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Probl. Imaging 2 (2008), no. 1, 1–21. 10.3934/ipi.2008.2.1Suche in Google Scholar
[3] S. A. Avdonin, A. S. Mikhaylov and V. S. Mikhaylov, On some applications of the boundary control method to spectral estimation and inverse problems, Nanosyst. Phys. Chem. Math. 6 (2015), no. 1, 63–78. 10.17586/2220-8054-2015-6-1-63-78Suche in Google Scholar
[4] S. A. Avdonin, V. S. Mikhaylov and K. B. Nurtazina, On inverse dynamical and spectral problems for the wave and Schrödinger equations on finite trees. The leaf peeling method, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 438 (2015), no. 45, 7–21. Suche in Google Scholar
[5] S. A. Avdonin, G. Y. Murzabekova and K. B. Nurtazina, Source identification for the differential equation with memory, New Trends in Analysis and Interdisciplinary Applications, Trends Math. Res. Perspect., Birkhäuser/Springer, Cham (2017), 111–120. 10.1007/978-3-319-48812-7_15Suche in Google Scholar
[6] S. A. Avdonin, C. Rivero Abdon, G. Leugering and V. Mikhaylov, On the inverse problem of the two-velocity tree-like graph, ZAMM Z. Angew. Math. Mech. 95 (2015), no. 12, 1490–1500. 10.1002/zamm.201400126Suche in Google Scholar
[7] M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems 20 (2004), no. 3, 647–672. 10.1088/0266-5611/20/3/002Suche in Google Scholar
[8] M. I. Belishev, Recent progress in the boundary control method, Inverse Problems 23 (2007), no. 5, R1–R67 10.1088/0266-5611/23/5/R01Suche in Google Scholar
[9] M. I. Belishev and V. S. Mikhailov, Inverse problem for a one-dimensional dynamical Dirac system (BC-method), Inverse Problems 30 (2014), no. 12, Article ID 125013. 10.1088/0266-5611/30/12/125013Suche in Google Scholar
[10] M. I. Belishev and A. F. Vakulenko, Inverse problems on graphs: recovering the tree of strings by the BC-method, J. Inverse Ill-Posed Probl. 14 (2006), no. 1, 29–46. 10.1515/156939406776237474Suche in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
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Artikel in diesem Heft
- Frontmatter
- Identification of mathematical model of bacteria population under the antibiotic influence
- On the determination of differential pencils with nonlocal conditions
- An inverse problem in elastography involving Lamé systems
- Solution of the inverse seismic problem in a layered elastic medium by means of the τ-p Radon transform
- An adaptive multigrid conjugate gradient method for the inversion of a nonlinear convection-diffusion equation
- Ambarzumyan-type theorems on a time scale
- A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations
- Lipschitz stability estimates in inverse source problems for a fractional diffusion equation of half order in time by Carleman estimates
- Inverse dynamic and spectral problems for the one-dimensional Dirac system on a finite tree
- Phaseless inverse problems with interference waves
- Quasi-solution of linear inverse problems in non-reflexive Banach spaces
