On inverse dynamical and spectral problems for the wave and Schr\"odinger equations on finite trees. The leaf peeling method

TL;DR

This paper discusses a leaf peeling method for solving inverse spectral and dynamical problems on finite trees, enabling efficient data recalculation by iteratively removing leaves, with potential applications in nano-electronics and quantum waveguides.

Contribution

It describes the main step of a new peeling algorithm for inverse problems on trees, extending previous work to spectral and dynamical cases.

Findings

Efficient leaf peeling algorithm for inverse data recalculation.

Application potential in nano-electronics and quantum waveguides.

Extension of inverse problem solutions to spectral and dynamical settings.

Abstract

Interest in inverse dynamical, spectral and scattering problems for differential equations on graphs is motivated by possible applications to nano-electronics and quantum waveguides and by a variety of other classical and quantum applications. Recently a new effective leaf peeling method has been proposed by S. Avdonin and P. Kurasov \cite{AK} for solving inverse problems on trees (graphs without cycles). It allows recalculating efficiently the inverse data from the original tree to the smaller trees, `removing' leaves step by step up to the rooted edge. In this paper we describe the main step of the spectral and dynamical versions of the peeling algorithm -- recalculating the inverse data for the `peeled tree'.