Under regularity assumptions, we establish a sharp large
deviation principle for Hermitian quadratic forms of
stationary Gaussian processes. Our result is similar to
the well-known Bahadur-Rao theorem [2] on the sample
mean. We also provide several examples of application
such as the sharp large deviation properties of
the Neyman-Pearson likelihood ratio test, of the sum of squares,
of the Yule-Walker
estimator of the parameter of a stable autoregressive Gaussian process,
and finally of the empirical spectral repartition function.