{"paper":{"title":"A Spectral Gap Estimate and Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP"],"primary_cat":"math.SP","authors_text":"Bogdan Georgiev, Mayukh Mukherjee, Stefan Steinerberger","submitted_at":"2016-12-27T10:38:04Z","abstract_excerpt":"We consider the Schr\\\"odinger operator $$-\\frac{d^2}{d x^2} + V \\qquad \\mbox{on an interval}~~[a,b]~\\mbox{with Dirichlet boundary conditions},$$ where $V$ is bounded from below and prove a lower bound on the first eigenvalue $\\lambda_1$ in terms of sublevel estimates: if $ w_V(y) = |I_y|,\\text{ where } I_y := \\left\\{ x \\in [a,b]: V(x) \\leq y \\right\\},$ then $$ \\lambda_1 \\geq \\frac{1}{250} \\min_{y > \\min V}{\\left(\\frac{1}{w_V(y)^2} + y\\right)}.$$ The result is sharp up to a universal constant if $\\left\\{ x \\in [a,b]: V(x) \\leq y \\right\\}$ is an interval for the value of $y$ solving the minimiza"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08565","kind":"arxiv","version":2},"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"}