We extend the existing theory of continuous-wave lasers by systematically considering spontaneous emission. In a simple rate-equation approach, the laser eigenvalue, defined as the ratio of coherent photons coupled out of the resonator divided by the number of photons coupled in via spontaneous emission, emerges as the fundamental parameter describing a continuous-wave laser. We derive a general equation for the laser eigenvalue and confirm the point at which coherence manifests itself in the resonator. The theory describes all types of lasers of three-level, four-level, or any intermediate nature.
It is proven that in Levy's constrained search or Lieb's ensemble constrained search, if the minimum is achieved by wave functions that satisfy the steadiness condition for the current density kernel and have continuous second derivatives, the minimizing wave functions are eigenstates of some external potential. If the excited states of this potential can only achieve stationarity and cannot achieve a minimum in the constrained search, the minimizing wave functions are ground states and the corresponding electron density is
v-representable or ensemble
v-representable. This is also true for the ensemble system with a varying electron number. In time-dependent density-functional theory, if the stationarity of the action integral is achieved in the constrained search by a wave function that satisfies the continuity equation for the current density kernel and has a continuous first derivative with respect to the time coordinate and continuous second derivatives with respect to the space coordinates, the stationary wave function satisfies the time-dependent Schr\"odinger equation of some time-dependent external potential and the corresponding electron density is time-dependent
v-representable.