We consider the
solution
to mass-critical
NLS
. We prove
that in dimensions
,
if the solution is spherically symmetric and is
almost periodic modulo scaling, then it must
lie in
for
some
.
Moreover, the kinetic energy of the solution is localized uniformly in time.
One important application of the theorem is a simplified proof of the
scattering conjecture for mass-critical NLS without reducing to three enemies.
As another important application, we establish a Liouville type result for
initial data with ground state mass. We prove that if a radial
solution
to focusing mass-critical problem has the ground state mass and does not scatter in
both time directions, then it must be global and coincide with the solitary wave up to
symmetries. Here the ground state is the unique, positive, radial solution to elliptic
equation
.
This is the first rigidity type result in scale invariant space
.