  We provide necessary and sufficient conditions for the exact observability of
the Schrodinger equations with point interaction and inverse-square potentials
on half-line. The necessary and sufficient condition for these two cases are
derived from two Logvinenko-Sereda type theorems for generalized Fourier
transform. Specifically, the generalized Fourier transform associated to the
Schr\"{o}dinger operators with inverse-square potentials on half-line are the
well known Hankel transforms. We provide a necessary and sufficient condition
for a subset $\Omega$ such that a function with its Hankel transform supporting
in a given interval can be bounded, in $L^{2}$-norm, from above by its
restriction to the set $\Omega$, with constant independent of the position of
the interval
