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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.03027 |
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Table of 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.