  In this study, we develop a semi-analytical framework to solve generalized
Jaynes-Tavis-Cummings Hamiltonians describing multi-qudit systems coupled via
EM resonators. Besides the multi-level generalization we allow for an arbitrary
number of resonators and/or modes, with nonidentical couplings to the qudits.
Our method is based on generic excitation-number operators which commute with
the respective Hamiltonians in the rotating wave approximation (RWA). The
validity of the RWA is assessed explicitly. The formalism enables the study of
eigenstates, eigenenergies and corresponding time evolutions of such coupled
multi-qudit systems. The technique can be applied in cavity quantum
electrodynamics and circuit quantum electrodynamics. It is also applicable to
atomic physics, describing the coupling of a single-mode photon to an atom. As
an example, we solve the Schr\"odinger equation for a two-qubit-one-resonator
system, in principle to arbitrary high excitations. We also solve the
Tavis-Cummings Hamiltonian in the one-excitation subspace for an arbitrary
number of identical qubits resonantly coupled to one resonator. As a final
example, we calculate of the low-excitation spectrum of a coupled two-transmon
system.
