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We also assume that $V_1(0) >0$. Then, denoting by $e_1$ the first eigenvalue of $-\\Delta + \\la V_2\"(0)x,x\\ra /2$, and under some ellipticity condition on $r_{1,"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1205.5196","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-05-23T15:06:03Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"f2c1969a55986239e526a5ce78a43da94ecc45201699e847ef26d0ed5623eae7","abstract_canon_sha256":"18f0a5a1da75640860cd55d02135e1323e424d4ccd5c55db9ff0dd7e846a3bf1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:26.616830Z","signature_b64":"r3wSQFLf7QLGlqAhq8TlpAKS1U5gEpAs7H6rvZWl1bNolAyHatuGlIl0T/tGQNEqFv5Pu6ol+M/vQed8NZ/mBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a89c49978a1db3d6e0e1f5f07cd8f51050ce0d5c7ab519b7f7937bdee690ead","last_reissued_at":"2026-05-18T01:22:26.616205Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:26.616205Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Resonance widths for the molecular predissociation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Alain Grigis, Andr\\'e Martinez","submitted_at":"2012-05-23T15:06:03Z","abstract_excerpt":"We consider a semiclassical $2\\times 2$ matrix Schr\\\"odinger operator of the form $P=-h^2\\Delta {\\bf I}_2 + {\\rm diag}(V_1(x), V_2(x)) +hR(x,hD_x)$, where $V_1, V_2$ are real-analytic, $V_2$ admits a non degenerate minimum at 0, $V_1$ is non trapping at energy $V_2(0)=0$, and $R(x,hD_x)=(r_{j,k}(x,hD_x))_{1\\leq j,k\\leq 2}$ is a symmetric off-diagonal $2\\times 2$ matrix of first-order pseudodifferential operators with analytic symbols. We also assume that $V_1(0) >0$. Then, denoting by $e_1$ the first eigenvalue of $-\\Delta + \\la V_2\"(0)x,x\\ra /2$, and under some ellipticity condition on $r_{1,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5196","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1205.5196","created_at":"2026-05-18T01:22:26.616326+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.5196v1","created_at":"2026-05-18T01:22:26.616326+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.5196","created_at":"2026-05-18T01:22:26.616326+00:00"},{"alias_kind":"pith_short_12","alias_value":"JKE4JGLYUHNT","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_16","alias_value":"JKE4JGLYUHNT23QO","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_8","alias_value":"JKE4JGLY","created_at":"2026-05-18T12:27:11.947152+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JKE4JGLYUHNT23QOD5PQPTMPKE","json":"https://pith.science/pith/JKE4JGLYUHNT23QOD5PQPTMPKE.json","graph_json":"https://pith.science/api/pith-number/JKE4JGLYUHNT23QOD5PQPTMPKE/graph.json","events_json":"https://pith.science/api/pith-number/JKE4JGLYUHNT23QOD5PQPTMPKE/events.json","paper":"https://pith.science/paper/JKE4JGLY"},"agent_actions":{"view_html":"https://pith.science/pith/JKE4JGLYUHNT23QOD5PQPTMPKE","download_json":"https://pith.science/pith/JKE4JGLYUHNT23QOD5PQPTMPKE.json","view_paper":"https://pith.science/paper/JKE4JGLY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.5196&json=true","fetch_graph":"https://pith.science/api/pith-number/JKE4JGLYUHNT23QOD5PQPTMPKE/graph.json","fetch_events":"https://pith.science/api/pith-number/JKE4JGLYUHNT23QOD5PQPTMPKE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JKE4JGLYUHNT23QOD5PQPTMPKE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JKE4JGLYUHNT23QOD5PQPTMPKE/action/storage_attestation","attest_author":"https://pith.science/pith/JKE4JGLYUHNT23QOD5PQPTMPKE/action/author_attestation","sign_citation":"https://pith.science/pith/JKE4JGLYUHNT23QOD5PQPTMPKE/action/citation_signature","submit_replication":"https://pith.science/pith/JKE4JGLYUHNT23QOD5PQPTMPKE/action/replication_record"}},"created_at":"2026-05-18T01:22:26.616326+00:00","updated_at":"2026-05-18T01:22:26.616326+00:00"}