Sums of projections with random coefficients
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arXiv
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| Format: | Preprint |
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2025
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| author | Pastur, Leonid Pushnitski, Alexander |
| author_facet | Pastur, Leonid Pushnitski, Alexander |
| contents | We study infinite sums \[ {\mathcal P}_{\varkappa}=\sum_{n=-\infty}^\infty \varkappa_n \langle\cdot, ψ_n\rangleψ_n \] of rank-one projections in a Hilbert space, where $\{ψ_n\}_{n\in\mathbb Z}$ are norm-one vectors, not necessarily orthogonal, and $\{\varkappa_n\}_{n\in\mathbb Z}$ are independent identically distributed positive random variables. Assuming that the Gram matrix $\{\langleψ_n,ψ_m\rangle\}_{n,m\in\mathbb Z}$ defines a bounded operator on $\ell^2(\mathbb Z)$ and that its entries depend only on the difference $n-m$, we analyse ${\mathcal P}_{\varkappa}$ within the framework of spectral theory of ergodic operators. Inspired by the spectral theory of ergodic Schrödinger operators, we define the integrated density of states (IDS) measure $ν_{{\mathcal P}_\varkappa}$ for ${\mathcal P}_{\varkappa}$ and establish results on its continuity and absolute continuity, including Wegner-type estimates and Lifshitz tail behaviour near the spectral edges. In the asymptotic regime of nearly-orthogonal $ψ_n$, we prove the Anderson-type localisation result: the spectrum of ${\mathcal P}_{\varkappa}$ is pure point almost surely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21539 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sums of projections with random coefficients Pastur, Leonid Pushnitski, Alexander Spectral Theory 47B80 We study infinite sums \[ {\mathcal P}_{\varkappa}=\sum_{n=-\infty}^\infty \varkappa_n \langle\cdot, ψ_n\rangleψ_n \] of rank-one projections in a Hilbert space, where $\{ψ_n\}_{n\in\mathbb Z}$ are norm-one vectors, not necessarily orthogonal, and $\{\varkappa_n\}_{n\in\mathbb Z}$ are independent identically distributed positive random variables. Assuming that the Gram matrix $\{\langleψ_n,ψ_m\rangle\}_{n,m\in\mathbb Z}$ defines a bounded operator on $\ell^2(\mathbb Z)$ and that its entries depend only on the difference $n-m$, we analyse ${\mathcal P}_{\varkappa}$ within the framework of spectral theory of ergodic operators. Inspired by the spectral theory of ergodic Schrödinger operators, we define the integrated density of states (IDS) measure $ν_{{\mathcal P}_\varkappa}$ for ${\mathcal P}_{\varkappa}$ and establish results on its continuity and absolute continuity, including Wegner-type estimates and Lifshitz tail behaviour near the spectral edges. In the asymptotic regime of nearly-orthogonal $ψ_n$, we prove the Anderson-type localisation result: the spectrum of ${\mathcal P}_{\varkappa}$ is pure point almost surely. |
| title | Sums of projections with random coefficients |
| topic | Spectral Theory 47B80 |
| url | https://arxiv.org/abs/2509.21539 |