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When the harmonic component of $A$ satifies some quantified condition, the spectrum of $-\\Delta_A$ is discrete. In this case we prove that the counting function of the eigenvalues of $-\\Delta_{A}$ satisfies the classical Weyl formula, even when $dA=0. $"},"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":"1004.5291","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2010-04-29T13:58:22Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"33b8f4d11ecac7557e655e72764a381c36f4da8df238ee3e57f17a00c6fda297","abstract_canon_sha256":"49812e42ec3db622bd6ec751ce2e32c3b65f15b0f87e48c04a44d42b7400baa2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:07:45.152494Z","signature_b64":"BTSxP5WkFNtOKvjgNiFELc9B404Ac8UwSewlfuM2iuANQNpfXOgEAPfnX7poQXYp4B8NQ45m8ixLxdr/RX/xAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9e476fc29a4b80b0322f716d6c5430f3ceb931e2d8ad8ca809039ddc2f490f51","last_reissued_at":"2026-05-18T02:07:45.151841Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:07:45.151841Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Eigenvalues of Laplacian with constant magnetic field on non-compact hyperbolic surfaces with finite area","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Abderemane Morame (LMJL), Francoise Truc (IF)","submitted_at":"2010-04-29T13:58:22Z","abstract_excerpt":"We consider a magnetic Laplacian $-\\Delta_A=(id+A)^\\star (id+A)$ on a noncompact hyperbolic surface $\\mM $ with finite area. $A$ is a real one-form and the magnetic field $dA$ is constant in each cusp. 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