Quantum states of the Kapitza pendulum

The quantum states of the Kapitza pendulum are described within the effective potential obtained by the method of averaging over the fast oscillations. An analytical estimate of the energy spectrum of stabilized states is given using approximate model potential. For the lowest states of an inverted pendulum, the spectrum is repeduced by the energies of a harmonic oscillator with perturbation theory corrections. Tunneling effect contribution to the energies of resonance states in the double-well effective potential is estimated. The results of numerical simulations of vibrational and rotational spectra of the Kapitza pendulum by the semiclassical method and by the Numerov algorithm are compared.