{"paper":{"title":"The log-Sobolev inequality for the ground state of a Schr\\\"odinger operator on bounded convex domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dejun Luo, Huaiqian Li","submitted_at":"2013-01-02T07:07:48Z","abstract_excerpt":"We consider the ground state $\\phi_0$ of the Schr\\\"odinger operator $L=-\\Delta+V$ on the bounded convex domain $\\Omega\\subset\\R^n$, satisfying the Dirichlet boundary condition. Assume that $V\\in C^1(\\Omega)$ and it admits an even function $\\tilde V\\in C^1([-D/2,D/2])$ as its modulus of convexity, where $D$ is the diameter of $\\Omega$. If the first Dirichlet eigenvalue $\\tilde\\lambda_0$ of $-\\frac{\\d^2}{\\d t^2}+\\tilde V$ on the interval $[-D/2,D/2]$ satisfies $\\tilde\\lambda_0>\\tilde V(0)$, then the measure $\\d\\mu=\\phi_0 \\d x$ satisfies the log-Sobolev inequality on $\\Omega$ with the constant $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.0177","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"}