The Frisch–Parisi Formalism for Fluctuations of the Schrödinger Equation

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 \(\int \left|x\right|^{2\delta }\left|u(x,t)\right|^2\,dx\) , 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 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.