Find the eigenvalues of Liouvillian

[emind]

The master equation of the damped harmonic oscillator is
$\frac{d}{dt}\rho_S(t) = -i\omega_0 [a^\dagger a,\rho_S(t)] + \gamma_0(\bar n+1) \{ a\rho_S(t) a^\dagger -\frac{1}{2} a^\dagger a \rho_S(t) -\frac{1}{2} \rho_S(t) a^\dagger a \} + \gamma_0\bar n \{ a^\dagger \rho_S(t) a -\frac{1}{2} a a^\dagger \rho_S(t) -\frac{1}{2} \rho_S(t) a a^\dagger \} \equiv \mathcal{L}\rho_S(t).$

Is there any method to analytically find out some eigenvalues, like the smallest three eigenvalues, of the liouvillian?

I am an engineering graduate, but now a rookie in physics.
Thanks for any advices.

 

[azntoon]

Unfortunately, there is no general method for finding the eigenvalues of a Liouvillian operator. In general, the only way to find the eigenvalues is to solve the master equation numerically, which can be quite difficult. However, in some cases, it may be possible to find approximate analytical solutions. For example, in the case of the damped harmonic oscillator, it may be possible to find approximate analytical solutions using perturbation theory or the Wigner-Weisskopf approximation.