Eigenfunctions with double exponential rate of localization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913666651127808 |
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| author | Krymskii, S. Logunov, A. Pagano, F. |
| author_facet | Krymskii, S. Logunov, A. Pagano, F. |
| contents | We construct a real-valued solution to the eigenvalue problem $-\text{div}(A\nabla u)=λu$, $λ>0,$ in the cylinder $\mathbb{T}^2\times \mathbb{R}$ with a real, uniformly elliptic, and uniformly $C^1$ matrix $A$ such that $|u(x,y,t)|\leq C e^{-c e^{c|t|}}$ for some $c,C>0$. We also construct a complex-valued solution to the heat equation $u_t=Δu + B \nabla u$ in a half-cylinder with continuous and uniformly bounded $B$, which also decays with double exponential speed. Related classical ideas, used in the construction of counterexamples to the unique continuation by Plis and Miller, are reviewed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_15354 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eigenfunctions with double exponential rate of localization Krymskii, S. Logunov, A. Pagano, F. Analysis of PDEs Classical Analysis and ODEs Spectral Theory We construct a real-valued solution to the eigenvalue problem $-\text{div}(A\nabla u)=λu$, $λ>0,$ in the cylinder $\mathbb{T}^2\times \mathbb{R}$ with a real, uniformly elliptic, and uniformly $C^1$ matrix $A$ such that $|u(x,y,t)|\leq C e^{-c e^{c|t|}}$ for some $c,C>0$. We also construct a complex-valued solution to the heat equation $u_t=Δu + B \nabla u$ in a half-cylinder with continuous and uniformly bounded $B$, which also decays with double exponential speed. Related classical ideas, used in the construction of counterexamples to the unique continuation by Plis and Miller, are reviewed. |
| title | Eigenfunctions with double exponential rate of localization |
| topic | Analysis of PDEs Classical Analysis and ODEs Spectral Theory |
| url | https://arxiv.org/abs/2501.15354 |