  We characterize diagonals of unbounded self-adjoint operators on a Hilbert
space H that have only discrete spectrum, i.e., with empty essential spectrum.
Our result extends the Schur-Horn theorem from a finite dimensional setting to
an infinite dimensional Hilbert space, analogous to Kadison's theorem for
orthogonal projections, Kaftal and Weiss' results for positive compact
operators, and Bownik and Jasper's characterization for operators with finite
spectrum. Furthermore, we show that if a symmetric unbounded operator E on H
has a nondecreasing unbounded diagonal, then any sequence that weakly majorizes
this diagonal is also a diagonal of E.
