On quantum Floquet theorem
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866929309141172224 |
|---|---|
| author | Treschev, Dmitry |
| author_facet | Treschev, Dmitry |
| contents | We consider the Schrödinger equation $ih\partial_tψ= Hψ$, $ψ=ψ(\cdot,t)\in L^2({\mathbb T})$. The operator $H = -\partial^2_x + V(x,t)$ includes smooth potential $V$, which is assumed to be time $T$-periodic. Let $W=W(t)$ be the fundamental solution of this linear ODE system on $L^2({\mathbb T})$. Then according to terminology from Lyapunov-Floquet theory, ${\cal M}=W(T)$ is the monodromy operator. We prove that ${\cal M}$ is unitarily conjugated to $\exp\big(-\frac{T}{ih} \partial^2_x\big) + {\cal C}$, where ${\cal C}$ is a compact operator with an arbitrarily small norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06999 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On quantum Floquet theorem Treschev, Dmitry Dynamical Systems 37K55 We consider the Schrödinger equation $ih\partial_tψ= Hψ$, $ψ=ψ(\cdot,t)\in L^2({\mathbb T})$. The operator $H = -\partial^2_x + V(x,t)$ includes smooth potential $V$, which is assumed to be time $T$-periodic. Let $W=W(t)$ be the fundamental solution of this linear ODE system on $L^2({\mathbb T})$. Then according to terminology from Lyapunov-Floquet theory, ${\cal M}=W(T)$ is the monodromy operator. We prove that ${\cal M}$ is unitarily conjugated to $\exp\big(-\frac{T}{ih} \partial^2_x\big) + {\cal C}$, where ${\cal C}$ is a compact operator with an arbitrarily small norm. |
| title | On quantum Floquet theorem |
| topic | Dynamical Systems 37K55 |
| url | https://arxiv.org/abs/2404.06999 |