Propagation for Schrödinger operators with potentials singular along a hypersurface
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909098084139008 |
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| author | Galkowski, Jeffrey Wunsch, Jared |
| author_facet | Galkowski, Jeffrey Wunsch, Jared |
| contents | In this article, we study propagation of defect measures for Schrödinger operators, $-h^2Δ_g+V$, on a Riemannian manifold $(M,g)$ of dimension $n$ with $V$ having conormal singularities along a hypersurface $Y$ in the sense that derivatives along vector fields tangent to $Y$ preserve the regularity of $V$. We show that the standard propagation theorem holds for bicharacteristics travelling transversally to the surface $Y$ whenever the potential is absolutely continuous. Furthermore, even when bicharacteristics are tangent to $Y$ at exactly first order, as long as the potential has an absolutely continuous first derivative, standard propagation continues to hold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_08154 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Propagation for Schrödinger operators with potentials singular along a hypersurface Galkowski, Jeffrey Wunsch, Jared Analysis of PDEs Spectral Theory In this article, we study propagation of defect measures for Schrödinger operators, $-h^2Δ_g+V$, on a Riemannian manifold $(M,g)$ of dimension $n$ with $V$ having conormal singularities along a hypersurface $Y$ in the sense that derivatives along vector fields tangent to $Y$ preserve the regularity of $V$. We show that the standard propagation theorem holds for bicharacteristics travelling transversally to the surface $Y$ whenever the potential is absolutely continuous. Furthermore, even when bicharacteristics are tangent to $Y$ at exactly first order, as long as the potential has an absolutely continuous first derivative, standard propagation continues to hold. |
| title | Propagation for Schrödinger operators with potentials singular along a hypersurface |
| topic | Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2302.08154 |