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arXiv · 2202.06645

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

S. Kumar·F. Ponce-Vanegas·L. Roncal·L. Vega
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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 $R$.

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S. Kumar, F. Ponce-Vanegas, L. Roncal, L. Vega. 2022-02-14. The Frisch--Parisi formalism for fluctuations of the Schr\"odinger equation. https://arxiv.org/abs/2202.06645

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