Quantitative observability for one-dimensional Schrödinger equations with potentials

Abstract

In this note, we prove the quantitative observability with an explicit control cost for the 1D Schrödinger 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.

In this post-published version, we corrected several minor mistakes in the published version: J. Funct. Analysis (2025), Vol.288, 2. More precisely, the statement of Corollary 4.4 has been corrected, and the Proposition 5.1 has been modified accordingly. The final resut in Theorem 1.2 with the desired control cost remains valid