Exact analytical eigenfunctions for the quantum theory of the laser

We present an exact solution of the quantum laser master equation using generating functions and confluent Heun equations. By reformulating the evolution of both diagonal and off-diagonal elements of the reduced density matrix, we derive closed-form expressions for the generating functions and identify the full spectrum of decay rates from a two-point connection problem for the confluent Heun equation. This method captures both transient dynamics and steady-state properties without truncation. In the thermodynamic limit of vanishing saturation, the spectrum separates into two analytically tractable eigenvalue families; their intersections mark exceptional points, reflecting nontrivial structure in the system's Liouvillian dynamics. At the lasing threshold, we observe a Liouvillian spectral collapse, signaling a nonequilibrium phase transition. Our results provide a unified, nonperturbative description of laser dynamics across different regimes.