Quantitative observability for one-dimensional Schr\"odinger equations with potentials

TL;DR

This paper establishes explicit control costs for the 1D Schrödinger equation with potentials, extending observability and spectral inequalities to broader classes of potentials and shorter times.

Contribution

It provides the first quantitative observability results with explicit control costs for 1D Schrödinger equations with bounded continuous potentials over thick sets.

Findings

Explicit control cost for 1D Schrödinger with potentials

Extension of large time observability to short times

Spectral inequality for bounded continuous potentials

Abstract

In this note, we prove the quantitative observability with an explicit control cost for the 1D Schr\"odinger equation over $\mathbb{R}$ with real-valued, bounded continuous potential on thick sets. Our proof relies on different techniques for low-frequency and high-frequency estimates. In particular, we extend the large time observability result for the 1D free Schrodinger equation in Theorem 1.1 of Huang-Wang-Wang [20] to any short time. As another byproduct, we extend the spectral inequality of Lebeau-Moyano [27] for real-analytic potentials to bounded continuous potentials in the one-dimensional case.