Coherent states of the asymmetric harmonic oscillator

We constructed formal coherent states for an asymmetric harmonic oscillator, where the asymmetry parameter is the square root of the ratio of spring constants. Although these states are constructed based on both Glauber's and Perelomov's approaches, in general they do not satisfy all the properties required for coherent states. Over time, the coherent states introduced in this way generally become incoherent. However, there are some specific parameters for the square root ratios of the spring constants $\frac{4k+1}{4l+1}$ or $\frac{4k+3}{4l+3}$. For these parameters it is possible to construct coherent states on the subspace of the Hilbert space of eigenstates. These coherent states keep their coherence during the time evolution. This case is also analyzed.