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 $\int\lvert x \rvert^{2\delta}\lvert u(x,t) \rvert^2\,dx$, for $0 < \delta < 1$, after removing a smooth backgro...
| Autores: | , , , |
|---|---|
| Tipo de recurso: | artículo |
| Estado: | Versión enviada para evaluación y publicación |
| Fecha de publicación: | 2022 |
| País: | España |
| Institución: | Basque Center for Applied Mathematics (BCAM) |
| Repositorio: | BIRD. BCAM's Institutional Repository Data |
| OAI Identifier: | oai:bird.bcamath.org:20.500.11824/1429 |
| Acceso en línea: | http://hdl.handle.net/20.500.11824/1429 |
| Access Level: | acceso abierto |
| Palabra clave: | Schrödinger equation Vortex filament equation Talbot effect Frisch--Parisi formalism Multifractals |
| Sumario: | 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 $\int\lvert x \rvert^{2\delta}\lvert u(x,t) \rvert^2\,dx$, 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 $R$. |
|---|