Classification of excited-state quantum phase transitions for arbitrary number of degrees of freedom

Classical stationary points of an analytic Hamiltonian induce singularities of the density of quantum energy levels and their flow with a control parameter in the system's infinite-size limit. We show that for a system with $f$ degrees of freedom, a non-degenerate stationary point with index $r$ causes a discontinuity (for $r$ even) or divergence ($r$ odd) of the $(f-1)$ th derivative of both density and flow of the spectrum. An increase of flatness for a degenerate stationary point shifts the singularity to lower derivatives. The findings are verified in an $f=3$ toy model.