Level spacing and Poisson statistics for continuum random Schrödinger operators
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866914637076758528 |
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| author | Dietlein, Adrian Elgart, Alexander |
| author_facet | Dietlein, Adrian Elgart, Alexander |
| contents | We prove a probabilistic level-spacing estimate at the bottom of the spectrum for continuum alloy-type random Schrödinger operators, assuming sign-definiteness of a single-site bump function and absolutely continuous randomness. More precisely, given a finite-volume restriction of the random operator onto a box of linear size $L$, we prove that with high probability the eigenvalues below some threshold energy $E_{\rm sp}$ keep a distance of at least $e^{-(\log L)^β}$ for sufficiently large $β>1$. This implies simplicity of the spectrum of the infinite-volume operator below $E_{\rm sp}$. Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy $E$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1712_03925 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Level spacing and Poisson statistics for continuum random Schrödinger operators Dietlein, Adrian Elgart, Alexander Mathematical Physics Spectral Theory 82B44, 47B80, 60H25 We prove a probabilistic level-spacing estimate at the bottom of the spectrum for continuum alloy-type random Schrödinger operators, assuming sign-definiteness of a single-site bump function and absolutely continuous randomness. More precisely, given a finite-volume restriction of the random operator onto a box of linear size $L$, we prove that with high probability the eigenvalues below some threshold energy $E_{\rm sp}$ keep a distance of at least $e^{-(\log L)^β}$ for sufficiently large $β>1$. This implies simplicity of the spectrum of the infinite-volume operator below $E_{\rm sp}$. Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy $E$. |
| title | Level spacing and Poisson statistics for continuum random Schrödinger operators |
| topic | Mathematical Physics Spectral Theory 82B44, 47B80, 60H25 |
| url | https://arxiv.org/abs/1712.03925 |