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A smoothing effect for the fractional Schrödinger equations on the circle and observability

Paul Alphonse, Nikolay Tzvetkov

TL;DR

The paper proves a smoothing phenomenon for the fractional Schrödinger flow on the circle, showing that after a renormalization the squared modulus of the evolved data has a distributional limit with negative Sobolev regularity. It then derives observability estimates with rough $L^1$-type controls by combining a time–frequency smoothing analysis, a Strichartz-type bound, and a Bardos–Lebeau–Rauch-type unique continuation argument. The results generalize known observability for the classical Schrödinger equation to the fractional regime $\alpha>1$ and provide sharp regularity thresholds for the renormalized quadratic expressions. The methods interweave bilinear smoothing, ergodic projections, and energy-regularity arguments to achieve sharp optimality results and control with minimal regularity assumptions. This contributes to understanding dispersion-quantified smoothing and controllability for fractional dispersive equations on compact manifolds, with potential extensions to higher-dimensional tori.

Abstract

We show that, after a renormalisation, one can define the square of the modulus of the solution of the fractional Schrödinger equations on the circle with data in Sobolev spaces of arbitrary negative index. As an application, we obtain observability estimates with rough controls.

A smoothing effect for the fractional Schrödinger equations on the circle and observability

TL;DR

The paper proves a smoothing phenomenon for the fractional Schrödinger flow on the circle, showing that after a renormalization the squared modulus of the evolved data has a distributional limit with negative Sobolev regularity. It then derives observability estimates with rough -type controls by combining a time–frequency smoothing analysis, a Strichartz-type bound, and a Bardos–Lebeau–Rauch-type unique continuation argument. The results generalize known observability for the classical Schrödinger equation to the fractional regime and provide sharp regularity thresholds for the renormalized quadratic expressions. The methods interweave bilinear smoothing, ergodic projections, and energy-regularity arguments to achieve sharp optimality results and control with minimal regularity assumptions. This contributes to understanding dispersion-quantified smoothing and controllability for fractional dispersive equations on compact manifolds, with potential extensions to higher-dimensional tori.

Abstract

We show that, after a renormalisation, one can define the square of the modulus of the solution of the fractional Schrödinger equations on the circle with data in Sobolev spaces of arbitrary negative index. As an application, we obtain observability estimates with rough controls.
Paper Structure (12 sections, 8 theorems, 145 equations)

This paper contains 12 sections, 8 theorems, 145 equations.

Table of Contents

  1. Introduction and main results
  2. Smoothing effect
  3. Observability
  4. Smoothing estimates
  5. Prolegomena
  6. Bilinear estimates
  7. Optimality
  8. Observability estimates
  9. Step 1: Strichartz type estimate
  10. Step 2: estimate with error term
  11. Step 3: unique continuation
  12. Step 4: removing the $H^{-\alpha}$-norm

Key Result

Theorem 1.1

Let $\alpha>1$ be a positive real number and $u_0\in L^2(\mathbb T)$. For every $\sigma>0$, the following limit exists in $\mathscr S'(\mathbb R\times\mathbb T)$ Moreover, with where $(\frac{1}{2}-\frac{2\sigma}{\alpha-1})_+$ stands for any positive number $\lambda>0$ such that $\lambda>\frac{1}{2}-\frac{2\sigma}{\alpha-1}$, and the above convergence actually holds in the space $W^{-s_1,\infty}(

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 4 more