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For $0\\le k\\le 2n+1$, let $\\Delta_k$ denote the Hodge Laplacian restricted to $k$-forms.\n  Our first main result shows that $L^2\\Lambda^k(H_n)$ decomposes into finitely many mutually orthogonal subspaces $\\V_\\nu$ with the properties: {itemize} $\\dom \\Delta_k$ splits along the $\\V_\\nu$'s as $\\sum_\\nu(\\dom\\Delta_k\\cap \\V_\\nu)$; $\\Delta_k:(\\dom\\Delta_k\\cap \\V_\\nu)\\longrightarrow \\V_\\nu$ for every $\\nu$; for each $\\nu$, there is a Hilbert space $\\cH_\\nu$ of $L"},"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":"1206.4540","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-06-20T15:37:23Z","cross_cats_sorted":[],"title_canon_sha256":"af3264b8bbe60652a85c0478d32e6227b1f5b7004b4fc8e50a7bd2a296a2bc97","abstract_canon_sha256":"c28f2f7bd4a53bc911f28bba034b4d6a8e5615bb8ca2a53586c28062c0769880"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:53:05.477391Z","signature_b64":"lpj64wXRrRdE1FvJlRlcxXBMYedVKU/XBpBhcCxTFPwm3HeZREyspM++x+eEGmcg6r96Xg7x6lqhA4CcKi4lBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"82de752c96a941a2aab4263fbe7815dd4c1cf579d7b0975d6cdb064d8d473dea","last_reissued_at":"2026-05-18T03:53:05.476624Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:53:05.476624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Analysis of the Hodge Laplacian on the Heisenberg group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Detlef M\\\"uller, Fulvio Ricci, Marco M. 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