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We consider a Schr\\\" odinger operator $H= -\\Delta + V$ on $\\mathrm{L}^2(\\mathbb{R}^d)$, and let $H_{\\Lambda}$ denote its restriction to a finite box $\\Lambda$ with either Dirichlet or periodic boundary condition. We prove unique continuation estimates of the type $\\chi_I (H_\\Lambda) W \\chi_I (H_\\Lambda) \\ge \\kappa\\, \\chi_I (H_\\Lambda) $ with $\\kappa >0$ for appropriate potentials $W\\ge 0$ and intervals $I$. As an application, we obtain optimal Wegner estimates at all energies for a class of non-ergod"},"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":"1209.4863","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2012-09-21T17:12:30Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"6079ab2d09ed34a13791b7db07434ca8fc53d2d32aa8f6de2d49609bf67e815d","abstract_canon_sha256":"7709fe52631524b1dca1ae05f7be88b7a45bda50dde14b83a9db2d4c7820d11b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:36:56.351529Z","signature_b64":"jDLkaeL3G2sJEmIsBjfFPGpV9ypy94AcULwJEK0KOc/7H0AGIDMgwBWY1oeFjziaRr8hMm1uf2N5yrU9CjJDBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"556ddf5ca760a94c3598ef9e53fad7b4686ce22135073cb35ca333656a5963e0","last_reissued_at":"2026-05-18T03:36:56.350770Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:36:56.350770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unique continuation principle for spectral projections of Schr\\\" odinger operators and optimal Wegner estimates for non-ergodic random Schr\\\" odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Abel Klein","submitted_at":"2012-09-21T17:12:30Z","abstract_excerpt":"We prove a unique continuation principle for spectral projections of Schr\\\" odinger operators. We consider a Schr\\\" odinger operator $H= -\\Delta + V$ on $\\mathrm{L}^2(\\mathbb{R}^d)$, and let $H_{\\Lambda}$ denote its restriction to a finite box $\\Lambda$ with either Dirichlet or periodic boundary condition. We prove unique continuation estimates of the type $\\chi_I (H_\\Lambda) W \\chi_I (H_\\Lambda) \\ge \\kappa\\, \\chi_I (H_\\Lambda) $ with $\\kappa >0$ for appropriate potentials $W\\ge 0$ and intervals $I$. 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