Semiclassical tunneling for some 1D Schrödinger operators with complex-valued potentials
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917365206220800 |
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| author | Averseng, Martin Frantz, Nicolas Hérau, Frédéric Raymond, Nicolas |
| author_facet | Averseng, Martin Frantz, Nicolas Hérau, Frédéric Raymond, Nicolas |
| contents | We consider the non-selfadjoint, semiclassical Schrödinger operator $\mathscr{L}(h) := -h^2\partial_x^2+e^{iα}V$, where $α\in (-π,π)$ and $V: \mathbb{R}\to \mathbb{R}_+$ is even and vanishes at exactly two (symmetric) non-degenerate minima. We establish a semiclassical tunneling result: the spectrum of $\mathscr{L}(h)$ near the origin is given by a sequence of algebraically simple eigenvalues which come in exponentially close pairs (within a $\mathscr{O}(e^{-S/h})$ distance where $S > 0$ is explicit), each pair being separated from the others by a distance $\mathscr{O}(h)$. A one-term estimate of the gap between the two smallest eigenvalues in magnitude is derived; it reveals that, when $α\neq 0$, they quickly rotate around each other as $h$ goes to $0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_04296 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Semiclassical tunneling for some 1D Schrödinger operators with complex-valued potentials Averseng, Martin Frantz, Nicolas Hérau, Frédéric Raymond, Nicolas Mathematical Physics Spectral Theory We consider the non-selfadjoint, semiclassical Schrödinger operator $\mathscr{L}(h) := -h^2\partial_x^2+e^{iα}V$, where $α\in (-π,π)$ and $V: \mathbb{R}\to \mathbb{R}_+$ is even and vanishes at exactly two (symmetric) non-degenerate minima. We establish a semiclassical tunneling result: the spectrum of $\mathscr{L}(h)$ near the origin is given by a sequence of algebraically simple eigenvalues which come in exponentially close pairs (within a $\mathscr{O}(e^{-S/h})$ distance where $S > 0$ is explicit), each pair being separated from the others by a distance $\mathscr{O}(h)$. A one-term estimate of the gap between the two smallest eigenvalues in magnitude is derived; it reveals that, when $α\neq 0$, they quickly rotate around each other as $h$ goes to $0$. |
| title | Semiclassical tunneling for some 1D Schrödinger operators with complex-valued potentials |
| topic | Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2510.04296 |