A Structured Inverse Spectrum Problem For Infinite Graphs

TL;DR

This paper demonstrates that for any infinite graph and a specified compact spectrum, one can construct a real symmetric matrix with that graph and spectrum, linking spectral properties to graph structure.

Contribution

It introduces a method to realize any compact spectrum on an infinite graph as the spectrum of a symmetric matrix, with detailed spectral properties.

Findings

Constructed matrices have spectra matching the given set

Limit points of spectrum correspond to the essential spectrum

Isolated spectrum points are simple eigenvalues

Abstract

It is shown that for a given infinite graph $G$ on countably many vertices, and a compact, infinite set of real numbers $\Lambda$ there is a real symmetric matrix $A$ whose graph is $G$ and its spectrum is $\Lambda$. Moreover, the set of limit points of $\Lambda$ equals the essential spectrum of $A$, and the isolated points of $\Lambda$ are eigenvalues of $A$ with multiplicity one. It is also shown that any two such matrices constructed by our method are approximately unitarily equivalent.