Non-harmonic analysis of the wave equation for Schrödinger operators with complex potential
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| Format: | Preprint |
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2024
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| _version_ | 1866912015274999808 |
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| author | Dasgupta, Aparajita Mohan, Lalit Mondal, Shyam Swarup |
| author_facet | Dasgupta, Aparajita Mohan, Lalit Mondal, Shyam Swarup |
| contents | This article investigates the wave equation for the Schrödinger operator on $\mathbb{R}^{n}$, denoted as $\mathcal{H}_0:=-Δ+V$, where $Δ$ is the standard Laplacian and $V$ is a complex-valued multiplication operator. We prove that the operator $\mathcal{H}_0$, with $\operatorname{Re}(V)\geq 0$ and $\operatorname{Re}(V)(x)\to\infty$ as $|x|\to\infty$, has a purely discrete spectrum under certain conditions. In the spirit of Colombini, De Giorgi, and Spagnolo, we also prove that the Cauchy problem with regular coefficients is well-posed in the associated Sobolev spaces, and when the propagation speed is Hölder continuous (or more regular), it is well-posed in Gevrey spaces. Furthermore, we prove that it is very weakly well-posed when the coefficients possess a distributional singularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_03027 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-harmonic analysis of the wave equation for Schrödinger operators with complex potential Dasgupta, Aparajita Mohan, Lalit Mondal, Shyam Swarup Analysis of PDEs Primary 46F05, Secondary 58J40, 22E30 This article investigates the wave equation for the Schrödinger operator on $\mathbb{R}^{n}$, denoted as $\mathcal{H}_0:=-Δ+V$, where $Δ$ is the standard Laplacian and $V$ is a complex-valued multiplication operator. We prove that the operator $\mathcal{H}_0$, with $\operatorname{Re}(V)\geq 0$ and $\operatorname{Re}(V)(x)\to\infty$ as $|x|\to\infty$, has a purely discrete spectrum under certain conditions. In the spirit of Colombini, De Giorgi, and Spagnolo, we also prove that the Cauchy problem with regular coefficients is well-posed in the associated Sobolev spaces, and when the propagation speed is Hölder continuous (or more regular), it is well-posed in Gevrey spaces. Furthermore, we prove that it is very weakly well-posed when the coefficients possess a distributional singularity. |
| title | Non-harmonic analysis of the wave equation for Schrödinger operators with complex potential |
| topic | Analysis of PDEs Primary 46F05, Secondary 58J40, 22E30 |
| url | https://arxiv.org/abs/2409.03027 |