Localization of the eigenfunctions of a Bloch-Torrey operator on the half-plane
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| Format: | Preprint |
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2025
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| _version_ | 1866912785379622912 |
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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 a non-self adjoint operator of the form $-h^2 Δ+ i(V(x) + α(x)y)$ on the upper half plane $y > 0$ with Dirichlet boundary conditions on $\{y = 0\}$ with $V \geq 0$, $V$ admitting a non-degenerate minimum at $x = 0$ and $α'(0) = 0$. We study its eigenfunctions associated to the smallest eigenvalues in magnitude in the semiclassical limit $h \to 0$. Elementary variational estimates show that these eigenfunctions are localized near the point $(0,0)$ at the scales $O(h^{1/3})$ in $x$ and $O(h^{2/3})$ in $y$. In this paper, we show that the $O(h^{1/3})$ localization in $x$ is not optimal; more precisely, we establish that the eigenfunctions are concentrated in a neighborhood of size $O(h^{1/2})$ of the axis $\{x = 0\}$, and this scale is shown to be sharp. The proof relies on the symbolic calculus of operator-valued pseudodifferential operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_20202 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Localization of the eigenfunctions of a Bloch-Torrey operator on the half-plane Averseng, Martin Frantz, Nicolas Hérau, Frédéric Raymond, Nicolas Mathematical Physics Analysis of PDEs 35P15, 35Q40, 35Sxx, 35J10 We consider a non-self adjoint operator of the form $-h^2 Δ+ i(V(x) + α(x)y)$ on the upper half plane $y > 0$ with Dirichlet boundary conditions on $\{y = 0\}$ with $V \geq 0$, $V$ admitting a non-degenerate minimum at $x = 0$ and $α'(0) = 0$. We study its eigenfunctions associated to the smallest eigenvalues in magnitude in the semiclassical limit $h \to 0$. Elementary variational estimates show that these eigenfunctions are localized near the point $(0,0)$ at the scales $O(h^{1/3})$ in $x$ and $O(h^{2/3})$ in $y$. In this paper, we show that the $O(h^{1/3})$ localization in $x$ is not optimal; more precisely, we establish that the eigenfunctions are concentrated in a neighborhood of size $O(h^{1/2})$ of the axis $\{x = 0\}$, and this scale is shown to be sharp. The proof relies on the symbolic calculus of operator-valued pseudodifferential operators. |
| title | Localization of the eigenfunctions of a Bloch-Torrey operator on the half-plane |
| topic | Mathematical Physics Analysis of PDEs 35P15, 35Q40, 35Sxx, 35J10 |
| url | https://arxiv.org/abs/2512.20202 |