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It is shown that if a solution w(t,x) to the Schr\\\"odinger equation \\partial_t w(t,g)=i Lw(t,g), w(0,g)=f(g), satisfies a suitable Gaussian type estimate at time t= 0 and at some time t=T\\ne 0, then w=0 . The proof is based on Hardy's uncertainty principle and explicit computations within Howe's oscillator semigroup. Our results extend work by Ben Said"},"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":"1207.4652","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-19T13:09:43Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"1665ffd8c3bbc94fe4e6e46b8ca6cbd04f8482789989086cb6ff51d35e5b36cf","abstract_canon_sha256":"e1af6a08626fa84fd7d1191433a745cc4ed6029083d9f02bf05f71bd3f6cba22"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:36.291024Z","signature_b64":"7Pg8r/kKP/5AHEKZU6ix3xhsUIz+L7gg1xr+o58vZB847dygIuqXsxVq9fgLCmH8R+IrACge4VMiI64qidS4Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c268f9913c25b5ba256018790435823bb350c7aaa1da73ecf1ad59fd3edff832","last_reissued_at":"2026-05-18T03:50:36.290197Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:36.290197Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniqueness of solutions to Schr\\\"odinger equations on 2-step nilpotent Lie groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Detlef M\\\"uller, Jean Ludwig","submitted_at":"2012-07-19T13:09:43Z","abstract_excerpt":"Let g=g_1+g_2, [g,g] =g_2, be a nilpotent Lie algebra of step 2, V_1,..., V_m a basis of g_1 and L=\\sum_{j,k} a_{jk} V_j V_k be a left-invariant differential operator on G=exp (g), where the coefficients a_{jk} form a real, symmetric mxm-matrix. It is shown that if a solution w(t,x) to the Schr\\\"odinger equation \\partial_t w(t,g)=i Lw(t,g), w(0,g)=f(g), satisfies a suitable Gaussian type estimate at time t= 0 and at some time t=T\\ne 0, then w=0 . The proof is based on Hardy's uncertainty principle and explicit computations within Howe's oscillator semigroup. 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