{"paper":{"title":"Scattering theory for Klein-Gordon equations with non-positive energy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Christian G\\'erard (LM-Orsay)","submitted_at":"2011-01-11T15:45:57Z","abstract_excerpt":"We study the scattering theory for charged Klein-Gordon equations: \\[\\{{array}{l} (\\p_{t}- \\i v(x))^{2}\\phi(t,x) \\epsilon^{2}(x, D_{x})\\phi(t,x)=0,[2mm] \\phi(0, x)= f_{0}, [2mm] \\i^{-1} \\p_{t}\\phi(0, x)= f_{1}, {array}. \\] where: \\[\\epsilon^{2}(x, D_{x})= \\sum_{1\\leq j, k\\leq n}(\\p_{x_{j}} \\i b_{j}(x))A^{jk}(x)(\\p_{x_{k}} \\i b_{k}(x))+ m^{2}(x),\\] describing a Klein-Gordon field minimally coupled to an external electromagnetic field described by the electric potential $v(x)$ and magnetic potential $\\vec{b}(x)$. The flow of the Klein-Gordon equation preserves the energy: \\[ h[f, f]:= \\int_{\\rr^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.2145","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"}