Asymptotical quasinormal mode spectrum for piecewise approximate effective potential

It was pointed out that the black hole quasinormal modes resulting from a piecewise approximate potential are drastically distinct those pertaining to original metric. In particular, instead of lining up parallel imaginary axis, spectrum is found stretch along real axis. this work, we prove if there single discontinuity in effective potential, no matter how insignificant it is, asymptotical behavior will be appreciably modified. Besides showing numerical evidence, give analytical derivations support above assertion even when located significantly further away maximum and/or size step arbitrarily small. Moreover, discuss astrophysical significance implications terms present findings.