Quantitative observability for the Schr\"{o}dinger equation with an anharmonic oscillator

TL;DR

This paper establishes observability inequalities for the Schrödinger equation with an anharmonic oscillator, providing explicit constants and geometric conditions for observable sets, and introduces new mathematical tools for analysis.

Contribution

It develops new observability inequalities with explicit constants for the Schrödinger equation with anharmonic potential and identifies geometric conditions for observable sets.

Findings

Explicit observability inequality over short time intervals

Counterexamples showing half-lines are not observable sets

Introduction of new spectral and Toeplitz matrix techniques

Abstract

This paper studies the observability inequalities for the Schr\"{o}dinger equation associated with an anharmonic oscillator $H=-\frac{\d^2}{\d x^2}+|x|$. 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 \cite{HWW}. 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 Schr\"{o}dinger operators $H=-\frac{\d^2}{\d x^2}+|x|^{2m}$ with $m\ge 1$. 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 \cite{Bour13}; third, the application of the Szeg\"{o}'s limit theorem from the theory of Toeplitz matrices, which provides a new mathematical tool for proving counterexamples of observability inequalities.