  Avoided crossings are common in the quasienergy spectra of strongly driven
nonlinear quantum wells. In this paper we examine the sinusoidally driven
particle in a square potential well to show that avoided crossings can alter
the structure of Floquet states in this system. Two types of avoided crossings
are identified: on type leads only to temporary changes (as a function of
driving field strength) in Floquet state structure while the second type can
lead to permanent delocalization of the Floquet states. Radiation spectra from
these latter states show significant increase in high harmonic generation as
the system passes through the avoided crossing.
