Time-dependent-harmonic plus inverse-harmonic potential in quantum mechanics

For a time-dependent system characterized by the potential V(x,t)=${\mathrm{a}}_2$(t)${\mathrm{x}}^2$+${\mathrm{a}}_1$/${\mathrm{x}}^2$, we first demonstrate the construction of classical and quantum invariants and then obtain an exact solution of the Schrödinger equation for this potential. The role of the invariant so constructed is further discussed in the context of studies of the coherent states.