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Asymptotic fractional uncertainty principle for the Helmholtz equation with periodic scattering data

Javier Canto, Nico Michele Schiavone, Luis Vega

TL;DR

The paper develops a fractional-dispersion analysis for the Helmholtz equation with periodic scattering data, proving an Asymptotic Fractional Uncertainty Principle that links large-height dispersion to the scattering data via a precise integral identity. It shows that, for periodic data, the dispersion splits into a singular part that blows up and a regular part that encodes inter-frequency interactions, with the latter converging to a Schrödinger-type periodic dispersion in the Dirac-comb limit $k\to\infty$. The analysis relies on a careful representation of the dispersion through the scattering data, a tempered-distribution framework for the fractional dispersion $h_b$, and a decomposition into singular and regular components. In the $k\to\infty$ regime, the Helmholtz-periodic dispersion reproduces the Talbot-like periodic structure of the Schrödinger equation, with a delta-comb Fourier transform and explicit coefficients $\alpha_b(r)$, establishing a rigorous bridge between Helmholtz and Schrödinger fractional-dispersion phenomena.

Abstract

We investigate the fractional dispersion of solutions to the Helmholtz equation with periodic scattering data. We show that, under appropriate rescaling, the interaction between the different frequencies exhibits the same fluctuating behavior as for the Schrödinger equation. To achieve this, we first establish an asymptotic fractional uncertainty principle for solutions to the Helmholtz equation.

Asymptotic fractional uncertainty principle for the Helmholtz equation with periodic scattering data

TL;DR

The paper develops a fractional-dispersion analysis for the Helmholtz equation with periodic scattering data, proving an Asymptotic Fractional Uncertainty Principle that links large-height dispersion to the scattering data via a precise integral identity. It shows that, for periodic data, the dispersion splits into a singular part that blows up and a regular part that encodes inter-frequency interactions, with the latter converging to a Schrödinger-type periodic dispersion in the Dirac-comb limit . The analysis relies on a careful representation of the dispersion through the scattering data, a tempered-distribution framework for the fractional dispersion , and a decomposition into singular and regular components. In the regime, the Helmholtz-periodic dispersion reproduces the Talbot-like periodic structure of the Schrödinger equation, with a delta-comb Fourier transform and explicit coefficients , establishing a rigorous bridge between Helmholtz and Schrödinger fractional-dispersion phenomena.

Abstract

We investigate the fractional dispersion of solutions to the Helmholtz equation with periodic scattering data. We show that, under appropriate rescaling, the interaction between the different frequencies exhibits the same fluctuating behavior as for the Schrödinger equation. To achieve this, we first establish an asymptotic fractional uncertainty principle for solutions to the Helmholtz equation.

Paper Structure

This paper contains 8 sections, 14 theorems, 132 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Notation and preliminaries
  3. Non-fractional analysis
  4. Dispersion
  5. Periodic scattering data for h1
  6. Fractional dispersion
  7. Periodic scattering data for b<1
  8. Limit as k tends to infinity

Key Result

Theorem 1.2

Let $0<b<1$ and $f\in L^2(\mathbb{R})$, then

Figures (1)

  • Figure 1: Plot of the distribution $\mathsf h_{b,\text{per}}(2t)$ for $b=0.25$. The Dirac deltas at rational points appear as segments of lengths proportional to their coefficients.

Theorems & Definitions (31)

  • Definition 1.1: Fractional derivatives, NahasPonce2009
  • Theorem 1.2: Stein Stein1961Stein1970-b
  • Proposition 1.3: Fractional Leibniz rule, Proposition 1 in NahasPonce2009
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Theorem 3.1: Asymptotic fractional uncertainty principle
  • Theorem 3.2
  • ...and 21 more