Spectral localization estimates for abstract linear Schrödinger equations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917777686659072 |
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| author | Zhang, Jingxuan |
| author_facet | Zhang, Jingxuan |
| contents | We study the propagation properties of abstract linear Schrödinger equations of the form $i\partial_tψ= H_0ψ+V(t)ψ$, where $H_0$ is a self-adjoint operator and $V(t)$ a time-dependent potential. We present explicit sufficient conditions ensuring that if the initial state $ψ_0$ has spectral support in $(-\infty,0]$ with respect to a reference self-adjoint operator $ϕ$, then, for some $c>0$ independent of $ψ_0$ and all $t\ne0$, the solution $ψ_t$ remains spectrally supported in $(-\infty,c|t|]$ with respect to $ϕ$, up to an $O(|t|^{-n})$ remainder in norm. The main condition is that the multiple commutators of $H_0$ and $ϕ$ are uniformly bounded in operator norm up to the $(n+1)$-th order. We then apply the abstract theory to a class of nonlocal Schrödinger equations on $\mathbb{R}^d$, proving that any solution with compactly supported initial state remains approximately supported, up to a polynomially suppressed tail in $L^2$-norm, inside a linearly spreading region around the initial support for all $t\ne0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10873 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral localization estimates for abstract linear Schrödinger equations Zhang, Jingxuan Analysis of PDEs Dynamical Systems We study the propagation properties of abstract linear Schrödinger equations of the form $i\partial_tψ= H_0ψ+V(t)ψ$, where $H_0$ is a self-adjoint operator and $V(t)$ a time-dependent potential. We present explicit sufficient conditions ensuring that if the initial state $ψ_0$ has spectral support in $(-\infty,0]$ with respect to a reference self-adjoint operator $ϕ$, then, for some $c>0$ independent of $ψ_0$ and all $t\ne0$, the solution $ψ_t$ remains spectrally supported in $(-\infty,c|t|]$ with respect to $ϕ$, up to an $O(|t|^{-n})$ remainder in norm. The main condition is that the multiple commutators of $H_0$ and $ϕ$ are uniformly bounded in operator norm up to the $(n+1)$-th order. We then apply the abstract theory to a class of nonlocal Schrödinger equations on $\mathbb{R}^d$, proving that any solution with compactly supported initial state remains approximately supported, up to a polynomially suppressed tail in $L^2$-norm, inside a linearly spreading region around the initial support for all $t\ne0$. |
| title | Spectral localization estimates for abstract linear Schrödinger equations |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2409.10873 |