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If A=0 and $\\epsilon=\\hbar$ is Planck's constant this problem is equivalent to the Schr\\\"odinger-Newton equations proposed by Penrose in \\cite{pe2}\\ to de"},"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":"1110.4213","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-10-19T09:02:29Z","cross_cats_sorted":[],"title_canon_sha256":"53efb3e16ffc3f346a9c34e201297c6abaa5a3ca796186013bd987b3227c89ac","abstract_canon_sha256":"82357c65ecb2f14756b0213be670102dd841f8cbdfcd7c0084d51c61c12caec1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:36.852094Z","signature_b64":"CXeYWt+mozGBUcI8gRs0kgH0hRo5fZ/9vR3Hu2AEAZrpDLpjJ6b4dU07z8eXfvTRIuBirF3ryiYkFezyhrZaCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aef7e1355acd9a6204a11b5edd0da697fd906a5a96cdc1424a0599304cba2f09","last_reissued_at":"2026-05-18T04:10:36.851368Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:36.851368Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Intertwining semiclassical solutions to a Schr\\\"{o}dinger-Newton system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"M\\'onica Clapp, Silvia Cingolani, Simone Secchi","submitted_at":"2011-10-19T09:02:29Z","abstract_excerpt":"We study the problem (-\\epsilon\\mathrm{i}\\nabla+A(x)) ^{2}u+V(x)u=\\epsilon ^{-2}(\\frac{1}{|x|}\\ast|u|^{2}) u, u\\in L^{2}(\\mathbb{R}^{3},\\mathbb{C}),\\text{\\ \\ \\ \\}\\epsilon\\nabla u+\\mathrm{i}Au\\in L^{2}(\\mathbb{R}^{3},\\mathbb{C}^{3}), where $A\\colon\\mathbb{R}^{3}\\rightarrow\\mathbb{R}^{3}$ is an exterior magnetic potential, $V\\colon\\mathbb{R}^{3}\\rightarrow\\mathbb{R}$ is an exterior electric potential, and $\\epsilon$ is a small positive number. 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