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This integral is used to determine the probability for the quantum harmonic oscillator in the $n$th energy eigenstate to tunnel into the classically forbidden region. Numerical results are given to illustrate the accuracy of the expansion."},"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":"1502.03382","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-02-11T17:15:09Z","cross_cats_sorted":[],"title_canon_sha256":"564f093a04c765b38523bc68ea761987eda71c40faeb91e58686dc5ab0def9b0","abstract_canon_sha256":"0b2adfea34db727f00faa816565501d6179d403ccd276f56f56e39835c6096eb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:19.533529Z","signature_b64":"GyyrOe3ruo+0FRCZDigLCUxP2XvSScKzyHOjl9f0roYQ3gsD0ZAWh+PwugMBBP8ISF96bl0Ot/E0Ql5qHak1Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"200ff5af84086d6f04e20a355ce716a9ffc43be7ac1ac72192cd0200eac50a50","last_reissued_at":"2026-05-18T02:27:19.532802Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:19.532802Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic evaluation of an integral arising in quantum harmonic oscillator tunnelling probabilities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"R B Paris","submitted_at":"2015-02-11T17:15:09Z","abstract_excerpt":"We obtain an asymptotic evaluation of the integral \\[\\int_{\\sqrt{2n+1}}^\\infty e^{-x^2} H_n^2(x)\\,dx\\] for $n\\rightarrow\\infty$, where $H_n(x)$ is the Hermite polynomial. 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