Observable sets for the Schrödinger equation with potential $\lvert x\rvert$

In this paper, we study observable sets for the one-dimensional Schrödinger equation associated with the operator $H=-d^2/dx^2+|x|$ on $L^2(\mathbb R)$. We prove that a measurable set is observable at every positive time if and only if it is thick. For each fixed observation time, the observability