A Structured Inverse Spectrum Problem for Infinite Graphs and Unbounded Operators

TL;DR

This paper proves that for any infinite graph and any closed, infinite set of real numbers, there exists an unbounded self-adjoint operator with the graph structure and spectrum matching these sets.

Contribution

It establishes the existence of unbounded self-adjoint operators with prescribed graph structures and spectra for infinite graphs, solving a structured inverse spectrum problem.

Findings

Existence of unbounded self-adjoint operators with given graph and spectrum

Construction method for operators on infinite graphs

Extension of inverse spectral theory to unbounded operators

Abstract

Given an infinite graph $G$ on countably many vertices, and a closed, infinite set $\Lambda$ of real numbers, we prove the existence of an unbounded self-adjoint operator whose graph is $G$ and whose spectrum is $\Lambda$.