{"paper":{"title":"The Isospectral Torus of Quasi-Periodic Schr\\\"odinger Operators via Periodic Approximations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"David Damanik (Rice University), Michael Goldstein (University of Toronto), Milivoje Lukic (University of Toronto, Rice University)","submitted_at":"2014-09-08T17:23:08Z","abstract_excerpt":"We study the quasi-periodic Schr\\\"odinger operator $$ -\\psi\"(x) + V(x) \\psi(x) = E \\psi(x), \\qquad x \\in \\mathbb{R} $$ in the regime of \"small\" $V(x) = \\sum_{m\\in\\mathbb{Z}^\\nu}c(m)\\exp (2\\pi i m\\omega x)$, $\\omega = (\\omega_1, \\dots, \\omega_\\nu) \\in \\mathbb{R}^\\nu$, $|c(m)| \\le \\varepsilon \\exp(-\\kappa_0|m|)$. We show that the set of reflectionless potentials isospectral with $V$ is homeomorphic to a torus. Moreover, we prove that any reflectionless potential $Q$ isospectral with $V$ has the form $Q (x) = \\sum_{m \\in \\mathbb{Z}^\\nu} d(m) \\exp (2\\pi i m\\omega x)$, with the same $\\omega$ and wi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.2434","kind":"arxiv","version":3},"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"}