  We present a closed form solution to the eigenvalue problem of a class of
master equations that describe open quantum system with loss and dephasing but
without gain. The method relies on the existence of a conserved number of
excitation in the Hamiltonian part and that none of the Lindblad operators
describe an excitation of the system. In the absence of dephasing Lindblad
operators, the eigensystem of the Liouville operator can be constructed from
the eigenvalues and eigenvectors of the effective non-Hermitian Hamiltonian
used in the quantum jump approach. Open versions of spin chains, the
Tavis-Cummings model and coupled Harmonic oscillators without gain can be
solved using this technique.
