{"paper":{"title":"Band invariants for perturbations of the harmonic oscillator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Alejandro Uribe, Victor Guillemin, Zuoqin Wang","submitted_at":"2011-09-02T22:10:19Z","abstract_excerpt":"We study the direct and inverse spectral problems for semiclassical operators of the form $S = S_0 +\\h^2V$, where $S_0 = \\frac 12 \\Bigl(-\\h^2\\Delta_{\\bbR^n} + |x|^2\\Bigr)$ is the harmonic oscillator and $V:\\bbR^n\\to\\bbR$ is a tempered smooth function. We show that the spectrum of $S$ forms eigenvalue clusters as $\\h$ tends to zero, and compute the first two associated \"band invariants\". We derive several inverse spectral results for $V$, under various assumptions. In particular we prove that, in two dimensions, generic analytic potentials that are even with respect to each variable are spectra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.0567","kind":"arxiv","version":1},"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"}