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

TL;DR

This paper investigates the long-term observability of the Schrödinger equation with potentials, establishing asymptotic estimates and analyzing specific potential models, including wells and harmonic oscillators.

Contribution

It introduces a spectral theory-based framework for asymptotic observability in Schrödinger equations with potentials and characterizes the properties of the observability constant for small potentials.

Findings

Existence of a nonzero asymptotic observability constant for certain small potentials.

Explicit characterization of the observability constant and its properties.

Numerical analysis of potential wells and the modified harmonic oscillator.

Abstract

We study the problem of optimal observability and prove time asymptotic observability estimates for the Schr\"odinger equation with a potential in $L^{\infty}(\Omega)$, with $\Omega\subset \mathbb{R}^d$, 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.