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We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on $M$ related to the non resonance of the linear part, this normal form is integrable when $d=1$ and gives rise to simple (in particular bounded) dynamics when $d\\geq 2$. As a consequence we prove the almost glo"},"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":"0808.0995","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2008-08-07T10:37:16Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"b321f6f01f4fb93185f500045f3ea455ff4385143a6f6e90cbc0ea8f1ffd5feb","abstract_canon_sha256":"1ec3513e6f92b953d946be81876b782257d45149d8ccd335f1d6f6fb801fac76"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:15:43.814632Z","signature_b64":"uC7HOs+e+JyLyJZs8+cZvuCwurp773DHyiZlspyvxjxac7Ec54ApPd1HKoJbCpIm8NW5vV0JxtIvU7s3XfJCBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"00c4ae6b188168527d3de4a332357ee84b01c10a53e9894112fce7053a17407a","last_reissued_at":"2026-05-18T02:15:43.814020Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:15:43.814020Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Normal Forms for Semilinear Quantum Harmonic Oscillators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AP","authors_text":"Benoit Grebert (LMJL), Eric Paturel (LMJL), Rafik Imekraz (LMJL)","submitted_at":"2008-08-07T10:37:16Z","abstract_excerpt":"We consider the semilinear harmonic oscillator $$i\\psi_t=(-\\Delta +\\va{x}^{2} +M)\\psi +\\partial_2 g(\\psi,\\bar \\psi), \\quad x\\in \\R^d, t\\in \\R$$ where $M$ is a Hermite multiplier and $g$ a smooth function globally of order 3 at least. We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on $M$ related to the non resonance of the linear part, this normal form is integrable when $d=1$ and gives rise to simple (in particular bounded) dynamics when $d\\geq 2$. 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