Weakly localized states of one dimensional Schrodinger equations have localized energy
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908733469097984 |
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| author | Stewart, Gavin Soffer, Avy |
| author_facet | Stewart, Gavin Soffer, Avy |
| contents | We study the asymptotics of the Schrödinger equation with time-dependent potential in dimension one. Assuming that the potential decays sufficiently rapidly as $|x| \to \infty$, we prove that the solution can be written as the sum of a free wave $e^{-itΔ} u_+$ and a weakly bound component $u_{\text{wb}}(t)$. Moreover, we show that the weakly bound part decomposes as $u_{\text{wb}}(t) = u_{\text{loc}}(t) + o_{\dot{H}^1}(1)$, where $\partial_x u_\text{loc}(t)$ is localized near the origin uniformly in time. Since decay conditions on the potential do not preclude resonances unless $d \geq 5$, our results can be seen as a natural extension of [Terence Tao. Dynamics of Partial Differential Equations, 5(2), 2008] and [Avy Soffer, Xiaoxu Wu. arXiv:2304.04245] to the lower-dimensional case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_16283 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weakly localized states of one dimensional Schrodinger equations have localized energy Stewart, Gavin Soffer, Avy Analysis of PDEs 35Q40 We study the asymptotics of the Schrödinger equation with time-dependent potential in dimension one. Assuming that the potential decays sufficiently rapidly as $|x| \to \infty$, we prove that the solution can be written as the sum of a free wave $e^{-itΔ} u_+$ and a weakly bound component $u_{\text{wb}}(t)$. Moreover, we show that the weakly bound part decomposes as $u_{\text{wb}}(t) = u_{\text{loc}}(t) + o_{\dot{H}^1}(1)$, where $\partial_x u_\text{loc}(t)$ is localized near the origin uniformly in time. Since decay conditions on the potential do not preclude resonances unless $d \geq 5$, our results can be seen as a natural extension of [Terence Tao. Dynamics of Partial Differential Equations, 5(2), 2008] and [Avy Soffer, Xiaoxu Wu. arXiv:2304.04245] to the lower-dimensional case. |
| title | Weakly localized states of one dimensional Schrodinger equations have localized energy |
| topic | Analysis of PDEs 35Q40 |
| url | https://arxiv.org/abs/2510.16283 |