  Spectroscopic methods involving the sudden injection or ejection of electrons
in materials are a powerful probe of electronic structure and interactions.
These techniques, such as photoemission and tunneling, yield measurements of
the "single particle" density of states (SPDOS) spectrum of a system. The SPDOS
is proportional to the probability of successfully injecting or ejecting an
electron in these experiments. It is equal to the number of electronic states
in the system able to accept an injected electron as a function of its energy
and is among the most fundamental and directly calculable quantities in
theories of highly interacting systems. However, the two-dimensional electron
system (2DES), host to remarkable correlated electron states such as the
fractional quantum Hall effect, has proven difficult to probe
spectroscopically. Here we present an improved version of time domain
capacitance spectroscopy (TDCS) that now allows us to measure the SPDOS of a
2DES with unprecedented fidelity and resolution. Using TDCS, we perform
measurements of a cold 2DES, providing the first direct measurements of the
single-particle exchange-enhanced spin gap and single particle lifetimes in the
quantum Hall system, as well as the first observations of exchange splitting of
Landau levels not at the Fermi surface. The measurements reveal the difficult
to reach and beautiful structure present in this highly correlated system far
from the Fermi surface.
