  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.
