{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:QPKR2HP2YAFS4SX7X534WGUR7N","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"fdd372d88f342f995772abd67d7d1a9d46720d18b724f3227b2410b57409e12a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-04T16:28:13Z","title_canon_sha256":"2f1efe78f65fc13cbd1cffdd5ce8e3de432ea004b17658ae43639ffba95dc772"},"schema_version":"1.0","source":{"id":"1207.1052","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.1052","created_at":"2026-05-18T03:51:47Z"},{"alias_kind":"arxiv_version","alias_value":"1207.1052v1","created_at":"2026-05-18T03:51:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.1052","created_at":"2026-05-18T03:51:47Z"},{"alias_kind":"pith_short_12","alias_value":"QPKR2HP2YAFS","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QPKR2HP2YAFS4SX7","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QPKR2HP2","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:5f18b876124818842f643d5fde34e74b4ab1198aa53a7059b7e98797a833ff82","target":"graph","created_at":"2026-05-18T03:51:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This paper shows that the nonlinear periodic eigenvalue problem $${cases} -\\Delta u + V(x) u - f(x,u) = \\lambda u, u \\in H^1(\\IR^N), {cases}$$ has a nontrivial branch of solutions emanating from the upper bound of every spectral gap of $-\\Delta + V$. No convexity condition is assumed. The following result of independent interest is also proven: the direct sum $Y \\oplus Z$ in $H^1(\\IR^N)$ associated to a decomposition of the spectrum of $-\\Delta+V$ remains \"topologically direct\" in the $L^p$'s (in the sense that the projections from $Y+Z$ onto $Y$ and $Z$ are $L^p$-continuous).","authors_text":"Christophe Troestler","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-04T16:28:13Z","title":"Bifurcation into spectral gaps for a noncompact semilinear Schr\\\"odinger equation with nonconvex potential"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.1052","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2d6daf6184195cbd218afa6f959475f1e1813c98bdd56352bbe2d5f3a03071c4","target":"record","created_at":"2026-05-18T03:51:47Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"fdd372d88f342f995772abd67d7d1a9d46720d18b724f3227b2410b57409e12a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-04T16:28:13Z","title_canon_sha256":"2f1efe78f65fc13cbd1cffdd5ce8e3de432ea004b17658ae43639ffba95dc772"},"schema_version":"1.0","source":{"id":"1207.1052","kind":"arxiv","version":1}},"canonical_sha256":"83d51d1dfac00b2e4affbf77cb1a91fb48b77020a956e05b790c124d8899641f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"83d51d1dfac00b2e4affbf77cb1a91fb48b77020a956e05b790c124d8899641f","first_computed_at":"2026-05-18T03:51:47.641444Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:51:47.641444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vK0wxKDDkaPD2IoOIo7iYXWshRXKIWjKZtyRkpwgdm7IzdSexNPiMDg11G4m3otWC+EVu3u9GBsd/N6/kfc+AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:51:47.642125Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.1052","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2d6daf6184195cbd218afa6f959475f1e1813c98bdd56352bbe2d5f3a03071c4","sha256:5f18b876124818842f643d5fde34e74b4ab1198aa53a7059b7e98797a833ff82"],"state_sha256":"3c600ef1dcf4a2c169f8f9f991246281d3e0229d2b2bb2285099c0e37071e11b"}