The Frisch--Parisi formalism for fluctuations of the Schr\"odinger equation

TL;DR

This paper investigates the fluctuations in the Schr"odinger equation's solutions with initial data approaching a Dirac comb, demonstrating the applicability of the Frisch--Parisi formalism to a specific fluctuation measure related to vortex filament evolution.

Contribution

It proves that the Frisch--Parisi formalism applies to a particular fluctuation process in the Schr"odinger equation with singular initial data, linking it to the behavior of Riemann's non-differentiable curve.

Findings

Frisch--Parisi formalism holds for the fluctuation process

The fluctuation process relates to Riemann's non-differentiable curve

Application to vortex filament evolution

Abstract

We consider the solution of the Schr\"odinger equation $u$ in $\mathbb{R}$ when the initial datum tends to the Dirac comb. Let $h_{\text{p}, \delta}(t)$ be the fluctuations in time of $\int\lvert x\rvert^{2\delta}\lvert u(x,t)\rvert^2\,dx$, for $0 < \delta < 1$, after removing a smooth background. We prove that the Frisch--Parisi formalism holds for $H_\delta(t) = \int_{[0,t]}h_{\text{p}, \delta}(2s)\,ds$, which is morally a simplification of the Riemann's non-differentiable curve $R$. Our motivation is to understand the evolution of the vortex filament equation of polygonal filaments, which are related to $R$.