Roles of phase cancellation removing spuriously quantized states in semiclassical mechanics

Two types of destructive interference among quantum phases are discussed that play an essential role in canceling out the unquantized states and thereby leaving only the correct projection of true eigenstates behind. Without these mechanisms, the Fourier spectra of the autocorrelation function, from which the energy spectra can be deduced numerically, can suffer from spurious features. We show two such examples: One is a spurious spectrum that appears in the negative energy domain of the Fourier frequencies for a positive potential. The other is a family of harmonic and subharmonic frequencies that are born from the "quantizing orbits," which are responsible for forming eigenstates Like thr periodic orbits or those satisfying the Einstein-Brillouin-Keller condition. The mechanisms for the spurious spectra to appear are analyzed, emphasizing their relationship to the quality of semiclassical wave functions.