Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings

TL;DR

This paper develops constructive algorithms to solve inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, enabling the recovery of mass distributions from spectral data.

Contribution

It introduces new algorithms for reconstructing mass distributions on star graphs from spectral data, addressing both central and pendant root cases with partial uniqueness results.

Findings

Spectral data determine the main string parameters if the root is a pendant vertex.

Mass distribution on other edges may not be uniquely determined.

The results relate to inverse problems for tree-patterned matrices.

Abstract

We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging problem, a pendant vertex of the star graph. At all other pendant vertices Dirichlet conditions are imposed; at the central vertex, at which a mass may be placed, continuity and Kirchhoff conditions are assumed. We derive conditions on two sets of real numbers to be the spectra of the above Dirichlet and Neumann problems. Our solution for the inverse problems is constructive: we establish algorithms to recover the mass distribution on the star graph (i.e. the point masses and lengths of subintervals between them) from these two spectra and from the lengths of the separate strings. If the root is a pendant vertex, the two spectra uniquely determine the parameters on the main string (i.e. the string incident to the root) if the length of the main string is known. The mass distribution on the other edges need not be unique; the reason for this is the non-uniqueness caused by the non-strict interlacing of the given data in the case when the root is the central vertex. Finally, we relate of our results to tree-patterned matrix inverse problems.