  Optical transitions in a semiconductor quantum dot are theoretically
investigated, with emphasis on the coupling to longitudinal optical phonons,
and including excitonic effects. When limiting to a finite number of $m$
electron and $n$ hole levels in the dot, the model can be solved exactly within
numerical accuracy. Crucial for this to work is the absence of dispersion of
the phonons. A suitable orthogonalization procedure leaves only
$m(m+1)/2+n(n+1)/2-2$ phonon modes to be coupled to the electronic system. We
calculate the linear optical polarization following a delta pulse excitation,
and by a subsequent Fourier transformation the resulting optical absorption.
This strict result is compared with a frequently used approximation modeling
the absorption as a convolution between spectral functions of electron and
hole, which tends to overestimate the effect of the phonon coupling. Numerical
results are given for two electron and three hole states in a quantum dot made
from the polar material CdSe. Parameter values are chosen such that a quantum
dot with a resonant sublevel distance can be compared with a nonresonant one.
