{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:PV7O425PA6INHXAKNKRXSQ5WG7","short_pith_number":"pith:PV7O425P","canonical_record":{"source":{"id":"1604.06188","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-21T05:53:43Z","cross_cats_sorted":[],"title_canon_sha256":"981468750ce4591349013fc6b72a2e58835019a282c8771c07c7757ad57c1f6a","abstract_canon_sha256":"6ec7e46f6de025fcd2eb563d8ee71abfd355888fe61e9b7457666e3bd53c5e81"},"schema_version":"1.0"},"canonical_sha256":"7d7eee6baf0790d3dc0a6aa37943b637f0a2a3e8353b08f5161724c4381b0e8b","source":{"kind":"arxiv","id":"1604.06188","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.06188","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"arxiv_version","alias_value":"1604.06188v2","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.06188","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"pith_short_12","alias_value":"PV7O425PA6IN","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"PV7O425PA6INHXAK","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"PV7O425P","created_at":"2026-05-18T12:30:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:PV7O425PA6INHXAKNKRXSQ5WG7","target":"record","payload":{"canonical_record":{"source":{"id":"1604.06188","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-21T05:53:43Z","cross_cats_sorted":[],"title_canon_sha256":"981468750ce4591349013fc6b72a2e58835019a282c8771c07c7757ad57c1f6a","abstract_canon_sha256":"6ec7e46f6de025fcd2eb563d8ee71abfd355888fe61e9b7457666e3bd53c5e81"},"schema_version":"1.0"},"canonical_sha256":"7d7eee6baf0790d3dc0a6aa37943b637f0a2a3e8353b08f5161724c4381b0e8b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:34.248753Z","signature_b64":"GwGLpLM1RVYjjVrN6euQwzxBTI0TN8270X5sGCqiJWi2pf1pmq0a2Lq8bvq7NlqiRI+Iu6Dw4Oew872o/CyJAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d7eee6baf0790d3dc0a6aa37943b637f0a2a3e8353b08f5161724c4381b0e8b","last_reissued_at":"2026-05-18T01:12:34.248335Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:34.248335Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.06188","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:12:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tnXAKlSc2YC+D+govtRRwR8EZjOM8RnlzzOVpC7C3960J5CaUWrUmdp54BgBjGajnrfi/0xE52Oe/MW+5QVUDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T11:19:07.373060Z"},"content_sha256":"89dda6293b9a3affc41a545552218f295e1ea04c7bcbe06b0d55ba24419af5db","schema_version":"1.0","event_id":"sha256:89dda6293b9a3affc41a545552218f295e1ea04c7bcbe06b0d55ba24419af5db"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:PV7O425PA6INHXAKNKRXSQ5WG7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multiple complex-valued solutions for nonlinear magnetic Schrodinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kazunaga Tanaka, Louis Jeanjean, Silvia Cingolani","submitted_at":"2016-04-21T05:53:43Z","abstract_excerpt":"We study, in the semiclassical limit, the singularly perturbed nonlinear Schr\\\"odinger equations $$ L^{\\hbar}_{A,V} u = f(|u|^2)u \\quad \\mbox{in } R^N $$ where $N \\geq 3$, $L^{\\hbar}_{A,V}$ is the Schr\\\"odinger operator with a magnetic field having source in a $C^1$ vector potential $A$ and a scalar continuous (electric) potential $V$ defined by \\begin{equation} L^{\\hbar}_{A,V}= -\\hbar^2 \\Delta-\\frac{2\\hbar}{i} A \\cdot \\nabla + |A|^2- \\frac{\\hbar}{i}\\operatorname{div}A + V(x). \\end{equation} Here $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.06188","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:12:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tc9y7XVD/+kQWy4ydbhTlaiPqBWaWX4mo/L3XNH22vbdGu7vTxzXg6dFO5gcMm1YBXqKNpShK7GvAImOSkZ4Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T11:19:07.373593Z"},"content_sha256":"30e7e67655d0a17b5feebfc0789b96f6e5e515a2c269232313cd61e755a0af9e","schema_version":"1.0","event_id":"sha256:30e7e67655d0a17b5feebfc0789b96f6e5e515a2c269232313cd61e755a0af9e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PV7O425PA6INHXAKNKRXSQ5WG7/bundle.json","state_url":"https://pith.science/pith/PV7O425PA6INHXAKNKRXSQ5WG7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PV7O425PA6INHXAKNKRXSQ5WG7/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-02T11:19:07Z","links":{"resolver":"https://pith.science/pith/PV7O425PA6INHXAKNKRXSQ5WG7","bundle":"https://pith.science/pith/PV7O425PA6INHXAKNKRXSQ5WG7/bundle.json","state":"https://pith.science/pith/PV7O425PA6INHXAKNKRXSQ5WG7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PV7O425PA6INHXAKNKRXSQ5WG7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:PV7O425PA6INHXAKNKRXSQ5WG7","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":"6ec7e46f6de025fcd2eb563d8ee71abfd355888fe61e9b7457666e3bd53c5e81","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-21T05:53:43Z","title_canon_sha256":"981468750ce4591349013fc6b72a2e58835019a282c8771c07c7757ad57c1f6a"},"schema_version":"1.0","source":{"id":"1604.06188","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.06188","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"arxiv_version","alias_value":"1604.06188v2","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.06188","created_at":"2026-05-18T01:12:34Z"},{"alias_kind":"pith_short_12","alias_value":"PV7O425PA6IN","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"PV7O425PA6INHXAK","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"PV7O425P","created_at":"2026-05-18T12:30:39Z"}],"graph_snapshots":[{"event_id":"sha256:30e7e67655d0a17b5feebfc0789b96f6e5e515a2c269232313cd61e755a0af9e","target":"graph","created_at":"2026-05-18T01:12:34Z","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":"We study, in the semiclassical limit, the singularly perturbed nonlinear Schr\\\"odinger equations $$ L^{\\hbar}_{A,V} u = f(|u|^2)u \\quad \\mbox{in } R^N $$ where $N \\geq 3$, $L^{\\hbar}_{A,V}$ is the Schr\\\"odinger operator with a magnetic field having source in a $C^1$ vector potential $A$ and a scalar continuous (electric) potential $V$ defined by \\begin{equation} L^{\\hbar}_{A,V}= -\\hbar^2 \\Delta-\\frac{2\\hbar}{i} A \\cdot \\nabla + |A|^2- \\frac{\\hbar}{i}\\operatorname{div}A + V(x). \\end{equation} Here $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that ","authors_text":"Kazunaga Tanaka, Louis Jeanjean, Silvia Cingolani","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-21T05:53:43Z","title":"Multiple complex-valued solutions for nonlinear magnetic Schrodinger equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.06188","kind":"arxiv","version":2},"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":"sha256:89dda6293b9a3affc41a545552218f295e1ea04c7bcbe06b0d55ba24419af5db","target":"record","created_at":"2026-05-18T01:12:34Z","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":"6ec7e46f6de025fcd2eb563d8ee71abfd355888fe61e9b7457666e3bd53c5e81","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-04-21T05:53:43Z","title_canon_sha256":"981468750ce4591349013fc6b72a2e58835019a282c8771c07c7757ad57c1f6a"},"schema_version":"1.0","source":{"id":"1604.06188","kind":"arxiv","version":2}},"canonical_sha256":"7d7eee6baf0790d3dc0a6aa37943b637f0a2a3e8353b08f5161724c4381b0e8b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7d7eee6baf0790d3dc0a6aa37943b637f0a2a3e8353b08f5161724c4381b0e8b","first_computed_at":"2026-05-18T01:12:34.248335Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:34.248335Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GwGLpLM1RVYjjVrN6euQwzxBTI0TN8270X5sGCqiJWi2pf1pmq0a2Lq8bvq7NlqiRI+Iu6Dw4Oew872o/CyJAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:34.248753Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.06188","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89dda6293b9a3affc41a545552218f295e1ea04c7bcbe06b0d55ba24419af5db","sha256:30e7e67655d0a17b5feebfc0789b96f6e5e515a2c269232313cd61e755a0af9e"],"state_sha256":"18de6eb49ac6b280409a1ef0fb14f77cf7b6545cb46f98af08ef18f35cc28302"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f7trX2FSKDMXu1Si1Q7pI5IeV7owfQVHy3O+kcJfu3GBctrBqLavoqOkQ60IABNpGvVZe5359gQ76/wK/M0uDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T11:19:07.375709Z","bundle_sha256":"83460834cf0888f74d734ef27e0d800b378ffe5d43b163a3720aa692e97b3eef"}}