{"paper":{"title":"Metric completion of the Bender--Brody--M\\\"uller Hamiltonian: dilation spectrum and missing eigenstates","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MP","math.SP","quant-ph"],"primary_cat":"math-ph","authors_text":"Kejun Liu","submitted_at":"2026-07-21T12:58:24Z","abstract_excerpt":"The Bender--Brody--M\\\"uller (BBM) Hamiltonian was proposed as a non-Hermitian Hilbert--P\\'olya operator. We analyze the Hilbert completion induced, on the standard half-line $L^2$ core, by BBM's candidate metric $\\hat\\eta=\\sin^2(\\hat p/2)=\\Delta^\\dagger\\Delta/4$. The form $\\eta_0=\\Delta^\\dagger\\Delta$ is positive with trivial kernel but is not coercive. Completing $C_c^\\infty(0,\\infty)$ in the norm $\\|\\psi\\|_{\\eta_0}=\\|\\Delta\\psi\\|$ gives a Hilbert space canonically unitarily equivalent to $L^2(\\mathbb R_+)$. Its free self-adjoint realization is the dilation generator, with simple, purely abso"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.19067","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.19067/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}