Asymptotics for optimal design problems for the Schr\"odinger equation with a potential

We study the problem of optimal observability and prove time asymptotic observability estimates for the Schr"odinger equation with a potential in $L^{\mathrm{\infty }}(\mathrm{\Omega })$, with $\mathrm{\Omega }\subset {\mathbb{R}}^d$, using spectral theory. An elegant way to model the problem using a time asymptotic observability constant is presented. For certain small potentials, we demonstrate the existence of a nonzero asymptotic observability constant under given conditions and describe its explicit properties and optimal values. Moreover, we give a precise description of numerical models to analyze the properties of important examples of potentials wells, including that of the modified harmonic oscillator.