{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:XSXRRPLOOC4DWRPXIJXPBUVNIS","short_pith_number":"pith:XSXRRPLO","schema_version":"1.0","canonical_sha256":"bcaf18bd6e70b83b45f7426ef0d2ad44929aa37203be884c18f7899d2dd1627f","source":{"kind":"arxiv","id":"1801.06813","version":2},"attestation_state":"computed","paper":{"title":"Lower bounds on the growth of Sobolev norms in some linear time dependent Schr\\\"odinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alberto Maspero","submitted_at":"2018-01-21T13:19:42Z","abstract_excerpt":"In this paper we consider linear, time dependent Schr\\\"odinger equations of the form $i \\partial_t \\psi = K_0 \\psi + V(t) \\psi $, where $K_0$ is a positive self-adjoint operator with discrete spectrum and whose spectral gaps are asymptotically constant.\n  We give a strategy to construct bounded perturbations $V(t)$ such that the Hamiltonian $K_0 + V(t)$ generates unbounded orbits. We apply our abstract construction to three cases: (i) the Harmonic oscillator on $\\mathbb R$, (ii) the half-wave equation on $\\mathbb T$ and (iii) the Dirac-Schr\\\"odinger equation on the sphere. In each case, $V(t)$"},"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":"1801.06813","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-01-21T13:19:42Z","cross_cats_sorted":[],"title_canon_sha256":"59739d6b1e043b1886fba13eeac37bd540291a10ed66697b48a85980672e3b77","abstract_canon_sha256":"9abf2855f98dc304acced511ef1c724f130efc2dd65ea0f63e3f4992adced0de"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:25:07.111423Z","signature_b64":"18w8NJrBy28Z+ql71pamuISARZjzKMF/Bq7V1I8Yx8Wo3B0vhKaUoTXqskzcnHwoXAuWEXzZLb582ImZGRNlCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bcaf18bd6e70b83b45f7426ef0d2ad44929aa37203be884c18f7899d2dd1627f","last_reissued_at":"2026-05-18T00:25:07.110857Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:25:07.110857Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lower bounds on the growth of Sobolev norms in some linear time dependent Schr\\\"odinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alberto Maspero","submitted_at":"2018-01-21T13:19:42Z","abstract_excerpt":"In this paper we consider linear, time dependent Schr\\\"odinger equations of the form $i \\partial_t \\psi = K_0 \\psi + V(t) \\psi $, where $K_0$ is a positive self-adjoint operator with discrete spectrum and whose spectral gaps are asymptotically constant.\n  We give a strategy to construct bounded perturbations $V(t)$ such that the Hamiltonian $K_0 + V(t)$ generates unbounded orbits. We apply our abstract construction to three cases: (i) the Harmonic oscillator on $\\mathbb R$, (ii) the half-wave equation on $\\mathbb T$ and (iii) the Dirac-Schr\\\"odinger equation on the sphere. In each case, $V(t)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.06813","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1801.06813","created_at":"2026-05-18T00:25:07.110939+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.06813v2","created_at":"2026-05-18T00:25:07.110939+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.06813","created_at":"2026-05-18T00:25:07.110939+00:00"},{"alias_kind":"pith_short_12","alias_value":"XSXRRPLOOC4D","created_at":"2026-05-18T12:33:01.666342+00:00"},{"alias_kind":"pith_short_16","alias_value":"XSXRRPLOOC4DWRPX","created_at":"2026-05-18T12:33:01.666342+00:00"},{"alias_kind":"pith_short_8","alias_value":"XSXRRPLO","created_at":"2026-05-18T12:33:01.666342+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/XSXRRPLOOC4DWRPXIJXPBUVNIS","json":"https://pith.science/pith/XSXRRPLOOC4DWRPXIJXPBUVNIS.json","graph_json":"https://pith.science/api/pith-number/XSXRRPLOOC4DWRPXIJXPBUVNIS/graph.json","events_json":"https://pith.science/api/pith-number/XSXRRPLOOC4DWRPXIJXPBUVNIS/events.json","paper":"https://pith.science/paper/XSXRRPLO"},"agent_actions":{"view_html":"https://pith.science/pith/XSXRRPLOOC4DWRPXIJXPBUVNIS","download_json":"https://pith.science/pith/XSXRRPLOOC4DWRPXIJXPBUVNIS.json","view_paper":"https://pith.science/paper/XSXRRPLO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.06813&json=true","fetch_graph":"https://pith.science/api/pith-number/XSXRRPLOOC4DWRPXIJXPBUVNIS/graph.json","fetch_events":"https://pith.science/api/pith-number/XSXRRPLOOC4DWRPXIJXPBUVNIS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XSXRRPLOOC4DWRPXIJXPBUVNIS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XSXRRPLOOC4DWRPXIJXPBUVNIS/action/storage_attestation","attest_author":"https://pith.science/pith/XSXRRPLOOC4DWRPXIJXPBUVNIS/action/author_attestation","sign_citation":"https://pith.science/pith/XSXRRPLOOC4DWRPXIJXPBUVNIS/action/citation_signature","submit_replication":"https://pith.science/pith/XSXRRPLOOC4DWRPXIJXPBUVNIS/action/replication_record"}},"created_at":"2026-05-18T00:25:07.110939+00:00","updated_at":"2026-05-18T00:25:07.110939+00:00"}