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We prove existence of various optimizers and provide an example when different structures generate the same optimal quasi-(normal-)eigenvalue. To show that quasi-eigenvalues locally optimal in various senses are in the spectrum $\\Sigma^{nl}$ of the bang-bang eigenproblem $y\" = - \\omega^2 y [ b_1 + (b_2 - b_1) \\chi_{\\mathbb{C}_+} (y^2 ) ]$, where $\\chi_{\\mathbb{C}_+} (\\cdot)$ is the indicator function of the upper complex hal"},"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":"1508.04706","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-08-19T17:08:38Z","cross_cats_sorted":["math.AP","math.CA","math.SP","physics.optics"],"title_canon_sha256":"96ed81f7311861c7f83be8bb6c55459dcd5b4a0a6b50130c065d2be1c1c7ddbf","abstract_canon_sha256":"ffbc68fd12b61b0866a36410689065e81e34291194b6126ea3cba77b9b7dc968"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:36.938622Z","signature_b64":"xEXaJ7CEAf7hkoEFw9uqcrT6LSO2tGvacPBpe6bpVLjDTd+/lHEQhWdXCtMKzgYDutFOu6NSSuT7lJHzwduzAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b5009eddae357c73997179ac195fb9909f96ac79daf8b5a2a03a392313307172","last_reissued_at":"2026-05-18T00:09:36.938000Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:36.938000Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nonlinear bang-bang eigenproblems and optimization of resonances in layered cavities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CA","math.SP","physics.optics"],"primary_cat":"math.OC","authors_text":"Ievgen V. 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