  The fundamental question of how Bose-Einstein condensates tunnel into a
barrier is addressed. The cubic nonlinear Schrodinger equation with a finite
square well potential, which models a Bose-Einstein condensate in a
quasi-one-dimensional trap of finite depth, is solved for the complete set of
localized and partially localized stationary states, which the former evolve
into when the nonlinearity is increased. An immediate application of these
different solution types is tunable tunneling. Magnetically tunable Feshbach
resonances can change the scattering length of certain Bose-condensed atoms,
such as $^{85}$Rb, by several orders of magnitude, including the sign, and
thereby also change the mean field nonlinearity term of the equation and the
tunneling of the wavefunction. We find both linear-type localized solutions and
uniquely nonlinear partially localized solutions where the tails of the
wavefunction become nonzero at infinity when the nonlinearity increases. The
tunneling of the wavefunction into the non-classical regime and thus its
localization therefore becomes an external experimentally controllable
parameter.
