{"paper":{"title":"Relativistic Comparison Theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","physics.atom-ph","quant-ph"],"primary_cat":"math-ph","authors_text":"Richard L. Hall","submitted_at":"2010-04-07T15:15:39Z","abstract_excerpt":"Comparison theorems are established for the Dirac and Klein--Gordon equations. We suppose that V^{(1)}(r) and V^{(2)}(r) are two real attractive central potentials in  d  dimensions that support discrete Dirac eigenvalues E^{(1)}_{k_d\\nu} and E^{(2)}_{k_d\\nu}. We prove that if V^{(1)}(r) \\leq V^{(2)}(r), then each of the corresponding discrete eigenvalue pairs is ordered  E^{(1)}_{k_d\\nu} \\leq E^{(2)}_{k_d\\nu}. This result generalizes an earlier more restrictive theorem that required the wave functions to be node free. For the  the Klein--Gordon equation, similar reasoning also leads to a comp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.1109","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"}