A Structured Inverse Spectrum Problem For Infinite Graphs

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