Reconstruction techniques for quantum trees

TL;DR

This paper introduces a numerical method for reconstructing potentials on quantum tree graphs from Weyl's matrix, combining leaf peeling and NSBF representations to localize and solve inverse problems efficiently.

Contribution

It proposes a novel numerical approach that integrates leaf peeling with NSBF representations for solving inverse problems on quantum trees.

Findings

The method successfully reconstructs potentials on quantum trees.

Numerical tests demonstrate the efficiency of the proposed algorithm.

The approach reduces complex inverse problems to linear algebra systems.

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.