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.

arXiv commentarXiv admin note: substantial text overlap with arXiv:2210.15536

References in corpus (9)

• Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions
• On an inverse problem for tree-like networks of elastic strings
• Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems
• A direct method for solving inverse Sturm-Liouville problems
• Spectrum completion and inverse Sturm-Liouville problems
• On the inverse problem of the two-velocity tree-like graph
• A practical method for recovering Sturm-Liouville problems from the Weyl function
• Recovery of a potential on a quantum star graph from Weyl's matrix
• Synthesis of a quantum tree Weyl matrix

Cited by in corpus (1)

• Representation of solutions of the one-dimensional Dirac equation in terms of Neumann series of Bessel functions