Quantitative observability for the Schrödinger equation with an anharmonic oscillator

In this paper, we study observability inequalities for the Schrödinger equation associated with an anharmonic oscillator $H=-\frac{\d^2}{\d~x^2}+|x|$. We build up an observability inequality over an arbitrarily short time interval $(0,T)$, with an explicit expression for the observation constant $C_{\rm~obs}$ 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 $H=-\frac{\d^2}{\d~x^2}+|x|^{2m}$ with $m\ge~1$.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.