We provide necessary and sufficient geometric conditions for the exact observability of the Schr\"odinger equation with inverse-square potentials on the half-line. These conditions are derived from a Logvinenko-Sereda type theorem for generalized Fourier transform. Specifically, the generalized Fourier transform associated with the Schr\"odinger operator with inverse-square potentials on the half-line is the well-known Hankel transform. We present a necessary and sufficient condition for a subset $\Omega$, such that a function whose Hankel transform is supported in a given interval can be bounded, in the $L^2$-norm, from above by its restriction to $\Omega$, with a constant independent of the position of the interval.
Categories: math.AP math-ph math.MP