The Frisch--Parisi formalism for fluctuations of the Schrödinger equation

Abstract

We consider the solution of the Schrödinger 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 $\int |x{|}^{2\delta }|u(x,t){|}^{2}dx$, 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 $R$.

17 pages, 4 figures