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Using Vasy's second 2nd-microlocal approach, \"the Lagrangian approach,\" we analyze -- uniformly, all the way down to $E=0$ -- the output of the limiting resolvent $R(E\\pm i 0) = \\lim_{\\epsilon \\to 0^+} R(E\\pm i \\epsilon)$. The Coulomb potential causes the output of the low-energy resolvent to possess oscillatory asymptotics which differ substantially from the sorts of asymptotics observed in the short-range case by Guillarmo"},"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":"2204.08355","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2022-04-18T14:54:54Z","cross_cats_sorted":["math-ph","math.CA","math.DG","math.MP","math.SP"],"title_canon_sha256":"75fb1387dd7a3656ad285b16912cf04592ec6fbe013954e126a5f0fc0469a525","abstract_canon_sha256":"3bf3c090ae79d8db0b7e3cfa7c34d18a9819554e333b454eade7a6637b288d6a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:14:21.424208Z","signature_b64":"4rLw9iF9dVpFZkAvA/4QNvsqISYEhtz8a4qsejFFHdC5QPVbpH9Yc0ty+MnLM4D2JtvXEr1CXEzKraEPMq57DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50a1bccdf341b92ff4c430e52440f454ba92949f7098001fd8d1709edf19cc6e","last_reissued_at":"2026-07-05T10:14:21.423716Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:14:21.423716Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hydrogen-like Schr\\\"odinger Operators at Low Energies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.CA","math.DG","math.MP","math.SP"],"primary_cat":"math.AP","authors_text":"Ethan Sussman","submitted_at":"2022-04-18T14:54:54Z","abstract_excerpt":"Consider a Schr\\\"odinger operator on an asymptotically Euclidean manifold $X$ of dimension at least two, and suppose that the potential is of attractive Coulomb-like type. 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