{"paper":{"title":"Complex Eigenvalues of the Parabolic Potential Barrier and Gel'fand Triplet","license":"","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Toshiki Shimbori, Tsunehiro Kobayashi","submitted_at":"1999-10-04T10:49:44Z","abstract_excerpt":"The paper deals with the one-dimensional parabolic potential barrier $V(x)={V_0-m\\gamma^2 x^2/2}$, as a model of an unstable system in quantum mechanics. The time-independent Schr\\\"{o}dinger equation for this model is set up as the eigenvalue problem in Gel'fand triplet and its exact solutions are expressed by generalized eigenfunctions belonging to complex energy eigenvalues ${V_0\\mp i\\Gammav_n}$ whose imaginary parts are quantized as ${\\Gammav_n=(n+1/2)\\hslash\\gamma}$. Under the assumption that time factors of an unstable system are square integrable, we provide a probabilistic interpretatio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/9910009","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/math-ph/9910009/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"}