{"paper":{"title":"Sharp comparison theorems for the Klein--Gordon equation in $d$ dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","quant-ph"],"primary_cat":"math-ph","authors_text":"Petr Zorin, Richard L. Hall","submitted_at":"2015-06-04T21:05:14Z","abstract_excerpt":"We establish sharp (or `refined') comparison theorems for the Klein--Gordon equation. We show that the condition $V_a\\le V_b$, which leads to $E_a\\le E_b$, can be replaced by the weaker assumption $U_a\\le U_b$ which still implies the spectral ordering $E_a\\le E_b$. In the simplest case, for $d=1$, $U_i(x)=\\int_0^x V_i(t)dt$, $i=a$ or $b$, and for $d>1$, $U_i(r)=\\int_0^r V_i(t) t^{d-1}dt$, $i=a$ or $b$. We also consider sharp comparison theorems in the presence of a scalar potential $S$ (a `variable mass') in addition to the vector term $V$ (the time component of a $4$-vector). The theorems are"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.01728","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"}