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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.18214 |
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| _version_ | 1866915809589198848 |
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| author | Kämper, Max Schumacher, Christoph Schwarzenberger, Fabian Veselic, Ivan |
| author_facet | Kämper, Max Schumacher, Christoph Schwarzenberger, Fabian Veselic, Ivan |
| contents | The integrated density of states (IDS) is a fundamental spectral quantity for quantum Hamiltonians modeling condensed matter systems, describing how densely energy levels are distributed. It can be interpreted as a volume-averaged spectral distribution. Hence, there are two equivalent definitions of the IDS related by the Pastur-Shubin formula: an operator-theoretic trace formula and a limit of normalized eigenvalue counting functions on finite volumes. We study a discrete random Schrödinger operator with bounded random potentials of finite-range correlations and prove a quantitative concentration inequality ensuring, with explicit high probability, that the empirical IDS (normalized eigenvalue counting function) uniformly approximates the abstract IDS trace formula within a prescribed error, thereby implying confidence regions for the IDS. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18214 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative concentration inequalities for the uniform approximation of the IDS Kämper, Max Schumacher, Christoph Schwarzenberger, Fabian Veselic, Ivan Statistics Theory Mathematical Physics Spectral Theory 47B80, 60B12, 62E20, 82B10 The integrated density of states (IDS) is a fundamental spectral quantity for quantum Hamiltonians modeling condensed matter systems, describing how densely energy levels are distributed. It can be interpreted as a volume-averaged spectral distribution. Hence, there are two equivalent definitions of the IDS related by the Pastur-Shubin formula: an operator-theoretic trace formula and a limit of normalized eigenvalue counting functions on finite volumes. We study a discrete random Schrödinger operator with bounded random potentials of finite-range correlations and prove a quantitative concentration inequality ensuring, with explicit high probability, that the empirical IDS (normalized eigenvalue counting function) uniformly approximates the abstract IDS trace formula within a prescribed error, thereby implying confidence regions for the IDS. |
| title | Quantitative concentration inequalities for the uniform approximation of the IDS |
| topic | Statistics Theory Mathematical Physics Spectral Theory 47B80, 60B12, 62E20, 82B10 |
| url | https://arxiv.org/abs/2602.18214 |