Uncertainty Principle and Geometric Condition for the Observability of Schrödinger Equations
Longben Wei, Zhiwen Duan, Hui Xu
TL;DR
This work characterizes exact observability for the Schrödinger equation with inverse-square potentials on the half-line, showing that a simple geometric thick-set condition on the observation domain is necessary and sufficient. The authors develop a Hankel-transform–based Logvinenko-Sereda–type uncertainty principle, proving that frequency support in a fixed interval yields a spectral inequality uniformly in interval location, provided the observing set is thick. By proving these Hankel LS-type results and exploiting the Hankel diagonalization of the Hamiltonian, they extend sharp observability results known for free and harmonic Schrödinger equations to a critical singular-potential setting, and connect observability to internal controllability on non-compact domains. The results rely on novel adaptations of LS-type inequalities to Hankel and modified Hankel transforms, as well as elementary lemmas on thick sets and frequency localization, yielding a robust geometric criterion with potential extensions to broader potentials.
Abstract
We provide necessary and sufficient geometric conditions for the exact observability of the Schrödinger 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 Schrödinger 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 $Ω$, 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 $Ω$, with a constant independent of the position of the interval.