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Reconstruction techniques for quantum trees

Sergei A. Avdonin, Kira V. Khmelnytskaya, Vladislav V. Kravchenko

Abstract

The inverse problem of recovery of a potential on a quantum tree graph from Weyl's matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm-Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two-spectra inverse problems on separate edges and reduce them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests.

Reconstruction techniques for quantum trees

Abstract

The inverse problem of recovery of a potential on a quantum tree graph from Weyl's matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm-Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two-spectra inverse problems on separate edges and reduce them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests.
Paper Structure (12 sections, 2 theorems, 74 equations, 4 figures)

This paper contains 12 sections, 2 theorems, 74 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Fundamental system of solutions and the Weyl solutions
  4. Leaf peeling
  5. Solution of local problem
  6. Calculation of coefficients $\left\{ g_{i,n}(L_{i})\right\} _{n=0}^{N}$ and $\left\{ s_{i,n}(L_{i})\right\} _{n=0}^{N}$
  7. Reduction to the two-spectra inverse problem on the edge
  8. Solution of two-spectra inverse problem
  9. Computation of functions (\ref{['phiS']})
  10. Summary of the method
  11. Numerical examples
  12. Conclusions

Key Result

Theorem 3.1

The solutions $\varphi_{i}(\rho,x)$ and $S_{i}(\rho,x)$ of (Schri) and their derivatives with respect to $x$ admit the following series representations where $\mathbf{j}_{k}(z)$ stands for the spherical Bessel function of order $k$, $\mathbf{j}_{k}(z):=\sqrt{\frac{\pi}{2z}}J_{k+\frac{1}{2}}(z)$ (see, e.g., AbramowitzStegunSpF). The coefficients $g_{i,n}(x)$, $s_{i,n}(x)$, $\gamma_{i,n}(x)$ and $\

Figures (4)

  • Figure 1: An example of a tree graph with two sheaves $S_{1}$ and $S_{2}$ highlighted in bold lines. Here $v_{0}$ is the abscission vertex and $e_{0}$ is the stem edge of $S_{1}$.
  • Figure 2: The tree graph considered in numerical tests.
  • Figure 3: The potential of the quantum graph from Example 1, recovered from the Weyl matrix given at 180 points, with $N=9$.
  • Figure 4: The potential of the quantum graph from Example 2, recovered from the Weyl matrix given at 180 points, with $N=9$.

Theorems & Definitions (4)

  • Definition 2.1
  • Theorem 3.1: KNT
  • Theorem 3.2
  • Definition 4.1