  Time-frequency concentration operators restrict the integral
analysis-synthesis formula for the short-time Fourier transform to a given
compact domain. We estimate how much the corresponding eigenvalue counting
function deviates from the Lebesgue measure of the time-frequency domain. For
window functions in the Gelfand-Shilov class, the bounds almost match known
asymptotics, with the advantage of being effective for concrete domains and
spectral thresholds. As such our estimates allow for applications where the
spectral threshold depends on the geometry of the time-frequency concentration
domain. We also consider window functions that decay only polynomially in time
and frequency.
