{"paper":{"title":"Perturbation theory for almost-periodic potentials I. One-dimensional case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Leonid Parnovski, Roman Shterenberg","submitted_at":"2017-11-10T18:25:29Z","abstract_excerpt":"We consider the family of operators $H^{(\\epsilon)}:=-\\frac{d^2}{dx^2}+\\epsilon V$ in ${\\mathbb R}$ with almost-periodic potential $V$. We study the behaviour of the integrated density of states (IDS) $N(H^{(\\epsilon)};\\lambda)$ when $\\epsilon\\to 0$ and $\\lambda$ is a fixed energy. When $V$ is quasi-periodic (i.e. is a finite sum of complex exponentials), we prove that for each $\\lambda$ the IDS has a complete asymptotic expansion in powers of $\\epsilon$; these powers are either integer, or in some special cases half-integer. These results are new even for periodic $V$. We also prove that when"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.03950","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}