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We show that $\\delta$NLS shares many properties in common with those previously established for the focusing autonomous translationally-invariant NLS $$ (\\text{NLS}) \\qquad i\\partial_t \\psi + \\Delta \\psi + |\\psi|^{p-1}\\psi=0 \\,. $$ The critical Sobolev space $\\dot H^{\\sigma_c}$ for $\\delta$NLS is $\\sigma_c=\\frac12-\\frac{1}{p-1}$, whereas"},"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":"1510.03491","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-13T00:41:33Z","cross_cats_sorted":[],"title_canon_sha256":"10363124e17562060775edadc4f418d124a11421d788975561a875e30a30bea0","abstract_canon_sha256":"f65f69140e2748671031c7ed0c1cb0bb70656fecb506ab19724bb4139f6fa1c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:17.087510Z","signature_b64":"Ajwgzy1zNDwXY6sy+jj36x5Iq0Y2Re1gQRrTsWa2Yf/uqsLUsnI89tD1O6rtnGIAfy2eaW81rzBTuIKYzfTDCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b54c90a8d1ff247de5daa656f604102f3b49c0f1b0415070a0f2fef15e7adcde","last_reissued_at":"2026-05-18T01:30:17.086854Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:17.086854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Blow-up for the 1D nonlinear Schr\\\"odinger equation with point nonlinearity I: Basic theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chang Liu, Justin Holmer","submitted_at":"2015-10-13T00:41:33Z","abstract_excerpt":"We consider the 1D nonlinear Schr\\\"odinger equation (NLS) with focusing point nonlinearity, $$ (\\delta\\text{NLS}) \\qquad i\\partial_t\\psi + \\partial_x^2\\psi + \\delta|\\psi|^{p-1}\\psi = 0, $$ where $\\delta=\\delta(x)$ is the delta function supported at the origin. We show that $\\delta$NLS shares many properties in common with those previously established for the focusing autonomous translationally-invariant NLS $$ (\\text{NLS}) \\qquad i\\partial_t \\psi + \\Delta \\psi + |\\psi|^{p-1}\\psi=0 \\,. $$ The critical Sobolev space $\\dot H^{\\sigma_c}$ for $\\delta$NLS is $\\sigma_c=\\frac12-\\frac{1}{p-1}$, whereas"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03491","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":"1510.03491","created_at":"2026-05-18T01:30:17.086960+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.03491v1","created_at":"2026-05-18T01:30:17.086960+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.03491","created_at":"2026-05-18T01:30:17.086960+00:00"},{"alias_kind":"pith_short_12","alias_value":"WVGJBKGR74SH","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_16","alias_value":"WVGJBKGR74SH3ZO2","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_8","alias_value":"WVGJBKGR","created_at":"2026-05-18T12:29:47.479230+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/WVGJBKGR74SH3ZO2UZLPMBAQF4","json":"https://pith.science/pith/WVGJBKGR74SH3ZO2UZLPMBAQF4.json","graph_json":"https://pith.science/api/pith-number/WVGJBKGR74SH3ZO2UZLPMBAQF4/graph.json","events_json":"https://pith.science/api/pith-number/WVGJBKGR74SH3ZO2UZLPMBAQF4/events.json","paper":"https://pith.science/paper/WVGJBKGR"},"agent_actions":{"view_html":"https://pith.science/pith/WVGJBKGR74SH3ZO2UZLPMBAQF4","download_json":"https://pith.science/pith/WVGJBKGR74SH3ZO2UZLPMBAQF4.json","view_paper":"https://pith.science/paper/WVGJBKGR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.03491&json=true","fetch_graph":"https://pith.science/api/pith-number/WVGJBKGR74SH3ZO2UZLPMBAQF4/graph.json","fetch_events":"https://pith.science/api/pith-number/WVGJBKGR74SH3ZO2UZLPMBAQF4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WVGJBKGR74SH3ZO2UZLPMBAQF4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WVGJBKGR74SH3ZO2UZLPMBAQF4/action/storage_attestation","attest_author":"https://pith.science/pith/WVGJBKGR74SH3ZO2UZLPMBAQF4/action/author_attestation","sign_citation":"https://pith.science/pith/WVGJBKGR74SH3ZO2UZLPMBAQF4/action/citation_signature","submit_replication":"https://pith.science/pith/WVGJBKGR74SH3ZO2UZLPMBAQF4/action/replication_record"}},"created_at":"2026-05-18T01:30:17.086960+00:00","updated_at":"2026-05-18T01:30:17.086960+00:00"}