# When does the Weyl-von Neumann Theorem hold?

## TL;DR

This paper characterizes the specific closed subsets of the real line for which the Weyl-von Neumann Theorem extends to unbounded self-adjoint operators, identifying when such operators with a given essential spectrum are unitarily equivalent modulo the compacts.

## Contribution

It precisely determines the subsets of the real line for which the Weyl-von Neumann Theorem holds for unbounded self-adjoint operators.

## Key findings

- Identifies the class of subsets M where the theorem applies
- Provides a complete characterization of these subsets
- Extends the classical theorem to certain unbounded operators

## Abstract

A famous theorem due to Weyl and von Neumann asserts that two bounded self-adjoint operators are unitarily equivalent modulo the compacts, if and only if their essential spectrum agree. The above theorem does not hold for unbounded operators. Nevertheless, there exist closed subsets $M$ of $\mathbb{R}$ on which the Weyl--von Neumann Theorem hold: all (not necessarily bounded) self-adjoint operators with essential spectrum $M$ are unitarily equivalent modulo the compacts. In this paper, we determine exactly which $M$ satisfies this property.