Quantitative observability for the Schrödinger equation with an anharmonic oscillator

2025/01/02 by Huang, Shanlin, Wang, Gengsheng, Wang, Ming · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2501.01258

Abstract

This paper studies the observability inequalities for the Schrödinger equation associated with an anharmonic oscillator H=-(\d²)/(\d x²)+|x|. We build up the observability inequality over an arbitrarily short time interval (0,T), with an explicit expression for the observation constant C_obs in terms of T, for some observable set that has a different geometric structure compared to those discussed in \citeHWW. We obtain the sufficient conditions and the necessary conditions for observable sets, respectively. We also present counterexamples to demonstrate that half-lines are not observable sets, highlighting a major difference in the geometric properties of observable sets compared to those of Schrödinger operators H=-(\d²)/(\d x²)+|x|^2m with m≥ 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-Burq-Zworski \citeBour13; third, the application of the Szegö's limit theorem from the theory of Toeplitz matrices, which provides a new mathematical tool for proving counterexamples of observability inequalities.

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