  We study the non-integrable Dicke model, and its integrable approximation,
the Tavis-Cummings model, as functions of both the coupling constant and the
excitation energy. Excited-state quantum phase transitions (ESQPT) are found
analyzing the density of states in the semi-classical limit and comparing it
with numerical results for the quantum case in large Hilbert spaces, taking
advantage of efficient methods recently developed. Two different ESQPTs are
identified in both models, which are signaled as singularities in the
semi-classical density of states, one {\em static} ESQPT occurs for any
coupling, whereas a dynamic ESQPT is observed only in the superradiant phase.
The role of the unstable fixed points of the Hamiltonian semi-classical flux in
the occurrence of the ESQPTs is discussed and determined. Numerical evidence is
provided that shows that the semi-classical result describes very well the
tendency of the quantum energy spectrum for any coupling in both models.
Therefore the semi-classical density of states can be used to study the
statistical properties of the fluctuation in the spectra, a study that is
presented in a companion paper.
