{"paper":{"title":"Existence of the spectral gap for elliptic operators","license":"","headline":"","cross_cats":["math.SP"],"primary_cat":"math.DG","authors_text":"Feng-Yu Wang","submitted_at":"1998-04-14T00:00:00Z","abstract_excerpt":"Let $M$ be a connected, noncompact, complete Riemannian manifold, consider the operator $L=\\DD +\\nn V$ for some $V\\in C^2(M)$ with $\\exp[V]$ integrable w.r.t. the Riemannian volume element. This paper studies the existence of the spectral gap of $L$. As a consequence of the main result, let $\\rr$ be the distance function from a point $o$, then the spectral gap exists provided $\\lim_{\\rr\\to\\infty}\\sup L\\rr<0$ while the spectral gap does not exist if $o$ is a pole and $\\lim_{\\rr\\to\\infty}\\inf L\\rr\\ge 0.$ Moreover, the elliptic operators on $\\mathbb R^d$ are also studied."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9804151","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"}