Simple Eigenvalues and Non-vanishing Eigenvectors of the Anderson Model
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arXiv
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| Format: | Preprint |
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2025
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| author | Lindblad, Oluyinka Guerrero, Ezra |
| author_facet | Lindblad, Oluyinka Guerrero, Ezra |
| contents | We consider the Anderson model on the finite grid $G = \mathbb Z/L_1\mathbb Z\times\cdots\times\mathbb Z/L_d\mathbb Z$, defined by the random Hamiltonian $H_t=Δ+tV$, where $Δ$ is the discrete Laplacian and $V=\mathrm{diag}(\{ω_{x}\}_{x\in G})$ is a random onsite potential with $ω_x\simμ$ i.i.d. We ask the natural question of when $H_t$ has simple eigenvalues and non-vanishing eigenvectors. We prove that, when $μ$ is a continuous probability distribution, $H_t$ has this property for all but finitely many $t$ values with probability $1$. However, when $μ$ is a Bernoulli distribution, the conditions fail with positive probability, for which we give a lower bound. We also calculate the exact probability of these conditions being met in the Bernoulli case when $d = 1$ and $L = L_1$ is prime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00278 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simple Eigenvalues and Non-vanishing Eigenvectors of the Anderson Model Lindblad, Oluyinka Guerrero, Ezra Mathematical Physics Combinatorics 05C50 (Primary) 15A42, 15B52 (Secondary) We consider the Anderson model on the finite grid $G = \mathbb Z/L_1\mathbb Z\times\cdots\times\mathbb Z/L_d\mathbb Z$, defined by the random Hamiltonian $H_t=Δ+tV$, where $Δ$ is the discrete Laplacian and $V=\mathrm{diag}(\{ω_{x}\}_{x\in G})$ is a random onsite potential with $ω_x\simμ$ i.i.d. We ask the natural question of when $H_t$ has simple eigenvalues and non-vanishing eigenvectors. We prove that, when $μ$ is a continuous probability distribution, $H_t$ has this property for all but finitely many $t$ values with probability $1$. However, when $μ$ is a Bernoulli distribution, the conditions fail with positive probability, for which we give a lower bound. We also calculate the exact probability of these conditions being met in the Bernoulli case when $d = 1$ and $L = L_1$ is prime. |
| title | Simple Eigenvalues and Non-vanishing Eigenvectors of the Anderson Model |
| topic | Mathematical Physics Combinatorics 05C50 (Primary) 15A42, 15B52 (Secondary) |
| url | https://arxiv.org/abs/2512.00278 |