Sharp decay rate for eigenfunctions of perturbed periodic Schrödinger operators
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| Format: | Preprint |
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2024
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| _version_ | 1866909317437849600 |
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| author | Liu, Wencai Matos, Rodrigo Treuer, John N. |
| author_facet | Liu, Wencai Matos, Rodrigo Treuer, John N. |
| contents | This paper investigates uniqueness results for perturbed periodic Schrödinger operators on $\mathbb{Z}^d$. Specifically, we consider operators of the form $H = -Δ+ V + v$, where $Δ$ is the discrete Laplacian, $V: \mathbb{Z}^d \rightarrow \mathbb{R}$ is a periodic potential, and $v: \mathbb{Z}^d \rightarrow \mathbb{C}$ represents a decaying impurity. We establish quantitative conditions under which the equation $-Δu + V u + v u = λu$, for $λ\in \mathbb{C}$, admits only the trivial solution $u \equiv 0$. Key applications include the absence of embedded eigenvalues for operators with impurities decaying faster than any exponential function and the determination of sharp decay rates for eigenfunctions. Our findings extend previous works by providing precise decay conditions for impurities and analyzing different spectral regimes of $λ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_10387 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp decay rate for eigenfunctions of perturbed periodic Schrödinger operators Liu, Wencai Matos, Rodrigo Treuer, John N. Spectral Theory Mathematical Physics Complex Variables This paper investigates uniqueness results for perturbed periodic Schrödinger operators on $\mathbb{Z}^d$. Specifically, we consider operators of the form $H = -Δ+ V + v$, where $Δ$ is the discrete Laplacian, $V: \mathbb{Z}^d \rightarrow \mathbb{R}$ is a periodic potential, and $v: \mathbb{Z}^d \rightarrow \mathbb{C}$ represents a decaying impurity. We establish quantitative conditions under which the equation $-Δu + V u + v u = λu$, for $λ\in \mathbb{C}$, admits only the trivial solution $u \equiv 0$. Key applications include the absence of embedded eigenvalues for operators with impurities decaying faster than any exponential function and the determination of sharp decay rates for eigenfunctions. Our findings extend previous works by providing precise decay conditions for impurities and analyzing different spectral regimes of $λ$. |
| title | Sharp decay rate for eigenfunctions of perturbed periodic Schrödinger operators |
| topic | Spectral Theory Mathematical Physics Complex Variables |
| url | https://arxiv.org/abs/2409.10387 |