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By extending a monotonicity formula found by Cabre and Sire \\cite{CaSi-10}, we show that the linear equation $(-\\Delta)^s u+ Vu = 0$ in $\\mathbb{R}^N$ has at most one radial and bounded solution vanishing at infinity, provided that the potential $V$ is a radial and non-decreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schr\\\"odinger operator $H=("},"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":"1302.2652","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-02-11T21:36:32Z","cross_cats_sorted":["math-ph","math.MP","math.SP"],"title_canon_sha256":"f2b4d52c384e5e23fd3ea6bfe34555840c0d866f1576378cca0fe96e668188f6","abstract_canon_sha256":"0233c573df5f17c605a4f609090450eb8d0c32469ccd573cb3cb642a23b913c7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:47.927547Z","signature_b64":"25yyzeqDcJgYsvOfhsHxSG4N71t8nlR8U5UIL/U88OnV4jPJbaZXowEiqUOw6gINKJ9+p0uumk5TsE8eh7lUBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ad9d245658aa73970a35ed3e1531c68281a9934425f94636b03fde6f01c0297","last_reissued_at":"2026-05-18T02:20:47.926781Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:47.926781Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniqueness of radial solutions for the fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.SP"],"primary_cat":"math.AP","authors_text":"Enno Lenzmann, Luis Silvestre, Rupert L. 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