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Quantitative observability for the Schrödinger equation with an anharmonic oscillator

Shanlin Huang, Gengsheng Wang, Ming Wang

TL;DR

This work analyzes observability for the Schrödinger equation with an anharmonic oscillator $H=-\frac{d^2}{dx^2}+|x|$ on $\mathbb{R}$. It introduces an $\alpha$-thin complement framework and develops an Ingham-type spectral inequality, a quantitative unique-continuation argument, and Toeplitz-/Szegö-based tools to obtain observability on arbitrarily short times with explicit $C_{obs}(T,E)$ for suitable $E$. It proves that half-lines are not observable and characterizes observability at some time via thickness-type conditions, highlighting new geometric phenomena compared to quadratic or higher-order potentials. The results remain robust under $L^{\infty}$ perturbations of the potential and showcase a novel combination of spectral inequalities, compactness arguments, and Toeplitz matrix theory in the Schrödinger observability context.

Abstract

This paper studies the observability inequalities for the Schrödinger 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 Schrödinger 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 Szegö's limit theorem from the theory of Toeplitz matrices, which provides a new mathematical tool for proving counterexamples of observability inequalities.

Quantitative observability for the Schrödinger equation with an anharmonic oscillator

TL;DR

This work analyzes observability for the Schrödinger equation with an anharmonic oscillator on . It introduces an -thin complement framework and develops an Ingham-type spectral inequality, a quantitative unique-continuation argument, and Toeplitz-/Szegö-based tools to obtain observability on arbitrarily short times with explicit for suitable . It proves that half-lines are not observable and characterizes observability at some time via thickness-type conditions, highlighting new geometric phenomena compared to quadratic or higher-order potentials. The results remain robust under perturbations of the potential and showcase a novel combination of spectral inequalities, compactness arguments, and Toeplitz matrix theory in the Schrödinger observability context.

Abstract

This paper studies the observability inequalities for the Schrödinger equation associated with an anharmonic oscillator . We build up the observability inequality over an arbitrarily short time interval , with an explicit expression for the observation constant in terms of , 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 Schrödinger operators with . 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 Szegö's limit theorem from the theory of Toeplitz matrices, which provides a new mathematical tool for proving counterexamples of observability inequalities.
Paper Structure (21 sections, 14 theorems, 229 equations, 1 figure)

This paper contains 21 sections, 14 theorems, 229 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem, aim and motivation.
  3. Main results.
  4. Strategy and further comments.
  5. Strategy to prove Theorem \ref{['thm1.1']}.
  6. Idea to prove Theorem \ref{['thm1.2']}.
  7. Comparison with related works
  8. Plan of the paper
  9. Auxiliary results and tools
  10. Spectral properties of the operator $H=-\frac{{\rm d}^2}{{\rm d} x^2}+|x|$
  11. Spectral property of Hermitian Toeplitz matrix and Szegö's theorem
  12. Salem and Nazarov inequality
  13. Proof of Theorem \ref{['thm1.1']}
  14. Ingham-type spectral inequality
  15. Relaxed observability inequality
  16. ...and 6 more sections

Key Result

Theorem 1.3

Let $E\subset \mathbb{R}$ be a measurable set. Suppose that $E^c$ is $\alpha$-thin with $\alpha>\frac{1}{2}$. Then $E$ is an observable set at any time for equ1.1. Moreover, for each $T\in(0,\frac{1}{2})$, we can choose where $C>0$ is a constant depending only on $E$ and $\alpha$.

Figures (1)

  • Figure 1: The symbol $f(\theta)$

Theorems & Definitions (27)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2: Szegö's limit theorem
  • Lemma 2.3: Salem inequality
  • Lemma 2.4: Nazarov inequality Nazarov
  • Proposition 3.1
  • ...and 17 more