Boundary spectral inverse problem on a class of graphs (trees) by the BC method

@article{Belishev2004BoundarySI,
  title={Boundary spectral inverse problem on a class of graphs (trees) by the BC method},
  author={Mikhail I. Belishev},
  journal={Inverse Problems},
  year={2004},
  volume={20},
  pages={647-672}
}
  • M. Belishev
  • Published 1 June 2004
  • Mathematics
  • Inverse Problems
A planar graph consisting of strings of variable densities is considered. The spectrum of the Dirichlet problem on the graph and the values of derivatives of the (normalized) eigenfunctions at the boundary vertices form the spectral data. We show that the graph without cycles (tree) and the densities of its edges are determined by the spectral data uniquely up to a natural isometry in the plane. In the framework of our approach (boundary control method; Belishev 1986) we study the boundary… 

Figures from this paper

Inverse problems on graphs: recovering the tree of strings by the BC-method
A planar graph consisting of strings of variable densities is considered. The spectrum of the Dirichlet problem on the graph and the values of derivatives of the normalized eigenfunctions at the
Inverse problems for discrete heat equations and random walks
We study the inverse problem of determining a finite weighted graph (X,E) from the source-to-solution map on a vertex subset B ⊂ X for heat equations on graphs, where the time variable can be either
Inverse eigenvalue problem for a simple star graph
A Schrödinger operator and associated spectra may be de ned for a graph by identifying edges with intervals of R, on which coe cient functions are de ned, imposing appropriate matching conditions at
Inverse problem for a differential operator on a star-shaped graph with nonlocal matching condition
In this paper, we develop two approaches to investigation of inverse spectral problems for a new class of nonlocal operators on metric graphs. The Laplace differential operator is considered on a
An inverse problem for differential pencils on graphs with a cycle
Abstract We study boundary value problems on compact graphs with a cycle for second-order ordinary differential equations with nonlinear dependence on the spectral parameter. We establish properties
Self-adjoint restrictions of maximal operator on graph
. In the work we study differential operators on arbitrary geometric graphs without loops. We extend the known results for differential operators on an interval to the differential operators on the
On the inverse problem of the two‐velocity tree‐like graph
In this article the authors continue the discussion in about inverse problems for second order elliptic and hyperbolic equations on metric trees from boundary measurements. In the present paper we
Dynamical inverse problem on a metric tree
We consider the problem of reconstruction of the potential for the wave equation on a specific star graph using the dynamical Dirichlet-to-Neumann map. Our algorithm is based on the boundary control
Inverse problems for quantum trees
Three different inverse problems for the Schrodinger operator on a metric tree are considered, so far with standard boundary conditions at the vertices. These inverse problems are connected with
...
...

References

SHOWING 1-10 OF 23 REFERENCES
Inverse Problem for the Sturm-Liouville Equation on a Simple Graph
TLDR
It is shown that the eigenvalues of the spectra interlace in some sense; thus an analogue of Sturm theorem is established.
On relations between spectral and dynamical inverse data
Abstract - As is known, boundary spectral data of the compact Riemannian manifold Ω (spectrum of Laplacian with zero Dirichlet boundary condition plus traces of normal derivatives of eigenfunctions
On the inverse scattering problem on branching graphs
The inverse scattering problem on branching graphs is studied. The definition of the Schrodinger operator on such graphs is discussed. The operator is defined with real potentials with finite first
Solving the dynamical inverse problem for the Schrödinger equation by the boundary control method
We consider the inverse problem of determining the potential in the one-dimensional Schrodinger equation from dynamical boundary observations, which are the range values of the Neumann-to-Dirichlet
A problem of M. Kac concerning recovery of the shape of a domain from the spectrum of a Dirichlet problem
We consider an inverse problem of M. Kac, consisting in the recovery of a domain from the spectrum of a homogeneous Dirichlet boundary value problem. We describe a recovery procedure for a
Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems
From the Publisher: This book presents the newly developed theory of non-harmonic Fourier series and its applications to the control of distributed parameter systems. The book begins with the modern
The Calderon Problem for Two-Dimensional Manifolds by the BC-Method
TLDR
The smooth two-dimensional compact orientable Riemann manifold with the boundary is uniquely determined with respect to the Eulerian manifold.
Inverse problems of wave processes
Part 1 One-dimensional inverse problems: setting of a problem for string equation peculiarities of solution - formulation of the direct problem the first method of solution of inverse problem method
Spectral Theory of Self-Adjoint Operators in Hilbert Space
1. Preliminaries.- 1. Metric Spaces. Normed Spaces.- 2. Algebras and ?-Algebras of Sets.- 3. Countably Additive Functions and Measures.- 4. Measurable Functions.- 5. Integration.- 6. Function
Scattering problems on noncompact graphs
A Sturm-Liouville problem on compact graphs is formulated and analyzed. The scattering problem for the Schroedinger equation on noncompact graphs is also formulated and analyzed.
...
...