In this paper, we study observability inequalities for the Schrödinger equation associated with an anharmonic oscillator \(H=-{d^{2}\over{dx^{2}}}+\vert x\vert\) . We build up an observability inequality over an arbitrarily short time interval (0, T), with an explicit expression for the observation constant Cobs in terms of T, for some observable sets with novel geometric features. We derive sufficient conditions and necessary conditions for observable sets, respectively. Furthermore, we present counterexamples to demonstrate that half-lines are not observable sets, highlighting a major difference in the geometric properties of observable sets compared with those of Schrödinger operators \(H=-{d^{2}\over{dx^{2}}}+\vert x\vert^{2m}\) with m ⩾ 1. Our approach is based on the following ingredients: first, the use of an Ingham-type spectral inequality constructed in this paper; second, the adaptation of a quantitative unique compactness argument, inspired by the work of Bourgain et al. (2013); third, the application of Szegö’s limit theorem from the theory of Toeplitz matrices, which provides a new mathematical tool for constructing counterexamples of observability inequalities.
