Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Curved Ingham inequalities and observability of the toroidal Schr{ö}dinger equation

Bernhard H Haak, Philippe Jaming, Ming Wang, Yunlei Wang

Abstract

We prove that solutions of the toroidal Schr{ö}dinger equation can be observed from suitably curved space-time trajectories, thus of zero Lebesgue measure. To do so, we establish new upper and lower bounds for certain trigonometric sums along curves, in the spirit of the celebrated Ingham inequality. In a second part, we establish observability properties over arbitrarily short curves of the low-and high-frequency components separately. For the low-frequency component, we establish strong restrictions on the zero sets of the trigonometric sums under consideration.

Curved Ingham inequalities and observability of the toroidal Schr{ö}dinger equation

Abstract

We prove that solutions of the toroidal Schr{ö}dinger equation can be observed from suitably curved space-time trajectories, thus of zero Lebesgue measure. To do so, we establish new upper and lower bounds for certain trigonometric sums along curves, in the spirit of the celebrated Ingham inequality. In a second part, we establish observability properties over arbitrarily short curves of the low-and high-frequency components separately. For the low-frequency component, we establish strong restrictions on the zero sets of the trigonometric sums under consideration.
Paper Structure (19 sections, 22 theorems, 250 equations, 4 figures, 1 table)

This paper contains 19 sections, 22 theorems, 250 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Observability of toroidal Schrödinger equations
  3. Vanishing of trigonometric sums
  4. Main results
  5. Curved Ingham Inequality
  6. A first technical result
  7. A Van der Corput type estimate
  8. A third technical lemma
  9. Proof of the curved Ingham inequality
  10. The minimal time condition in Theorem \ref{['thm:schroedinger']}
  11. Fractional Schrödinger equality: high frequencies
  12. A result that implies Theorem \ref{['thm:highfreq']}
  13. Observability for highly dispersive equation
  14. Fractional Schrödinger equality: low-frequencies
  15. Schrödinger equations with bounded potentials
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\gamma>0$ and $(\lambda_n)_{n\in\mathbb{Z}}$ be a sequence of real numbers with $\lambda_{n+1}-\lambda_n\geq \gamma$. Then, for every $T>0$, there exists $C(T)$ such that, for every $(c_n)_{n\in\mathbb{Z}}\in\ell^2$, Further, if $T>\dfrac{1}{\gamma}$, there exists $\tilde{C}(T)$ such that, for every $(c_n)_{n\in\mathbb{Z}}\in\ell^2$,

Figures (4)

  • Figure 1: Illustration of Assumption \ref{['ass:H-alpha']} in Example \ref{['example1']}. a) Left hand side: the function $p(t)=\dfrac{1}{3}\left(1+\frac{2}{\pi}\arctan t\right)t^3$ and $c \tfrac{t^3}{3}$ for $c=1,2$ in blue in comparison. b) Right hand side: $\eta(t)=\arctan t-2\arctan 2t+\arctan 4t$.
  • Figure 2: The sets $S^{+}_{\mathrm{good}}$ in red, $S^{-}_{\mathrm{good}}$ in blue and $S_{\mathrm{bad}}$ in gray, when $s=1.5$ and $\tau = 4$.
  • Figure 3: The curves $\Gamma_{q_\tau}^{0,T-\tau}$ and $\Gamma_p^{\tau,T}$.
  • Figure 4: The curve $C: |x|^{1.5}-|y|^{1.5}=-4(x-y),x\neq y$, line $x+y=0$, $x-y=0$ and three regions for the example $s=1.5$ with threshold $\tau=4$.

Theorems & Definitions (50)

  • Theorem 1.1: Ingham's Inequality
  • Example 1.2
  • Theorem 1.3: Curved Ingham Inequality
  • Theorem 1.4
  • Corollary 1.5: High-Frequency Curved Ingham Inequality
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • ...and 40 more