A structured inverse spectrum problem for infinite graphs

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