{"paper":{"title":"Gradient Schr\\\"odinger Operators, Manifolds with Density and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jose M. Espinar","submitted_at":"2012-09-27T08:53:17Z","abstract_excerpt":"The aim of this paper is twofold. On the one hand, the study of gradient Schr\\\"{o}dinger operators on manifolds with density $\\phi$. We classify the space of solutions when the underlying manifold is $\\phi-$parabolic. As an application, we extend the Naber-Yau Liouville Theorem, and we will prove that a complete manifold with density is $\\phi -$parabolic if, and only if, it has finite $\\phi-$capacity. Moreover, we show that the linear space given by the kernel of a nonnegative gradient Schr\\\"{o}dinger operators is one dimensional provided there exists a bounded function on it and the underlyin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.6162","kind":"arxiv","version":7},"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"}