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While most of the latter focuses on the case where $A(t)$ all have purely discrete spectrum, we now particularly study situations permitting essential spectra.\n  Introducing $H_1={D_A}^* D_A$ and $H_2=D_A {D_A}^*$, we c"},"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":"1509.01580","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-09-04T08:39:09Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"6630b63906f1ad56d5edc3432ef909f4628efd66fd4b632b92779b96c64ac2da","abstract_canon_sha256":"5af0213cc385595c4dc228b6fd7b1c0a72b671bd417b7cf6e658a29439c2ab6c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:51.110100Z","signature_b64":"rMgbGYMu3/IFMCcCG8+lI7a0+6/F2DQdrdNyYioW79nKaVi6I+tlYFsts7ZdRxh9/16344ug4OnFpIaN0/3mDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41c754717f4d911cf374bd8792c93ba5e4f43c6a4475975c9f402f4630ca3480","last_reissued_at":"2026-05-18T01:33:51.109601Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:51.109601Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Index of a Non-Fredholm Model Operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"Alan Carey, Fedor Sukochev, Fritz Gesztesy, Galina Levitina","submitted_at":"2015-09-04T08:39:09Z","abstract_excerpt":"Let $\\{A(t)\\}_{t \\in \\mathbb{R}}$ be a path of self-adjoint Fredholm operators in a Hilbert space $\\mathcal{H}$, joining endpoints $A_\\pm$ as $t \\to \\pm \\infty$. 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