{"paper":{"title":"Asymptotic velocity of a position-dependent quantum walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.SP","authors_text":"Akito Suzuki","submitted_at":"2015-07-30T16:14:54Z","abstract_excerpt":"We consider a position-dependent coined quantum walk on $\\mathbb{Z}$ and assume that the coin operator $C(x)$ satisfies \\[ \\|C(x) - C_0 \\| \\leq c_1|x|^{-1-\\epsilon},\n  \\quad x \\in \\mathbb{Z} \\] with positive $c_1$ and $\\epsilon$ and $C_0 \\in U(2)$. We show that the Heisenberg operator $\\hat x(t)$ of the position operator converges to the asymptotic velocity operator $\\hat v_+$ so that \\[ \\mbox{s-}\\lim_{t \\to \\infty} {\\rm exp}\\left( i \\xi \\frac{\\hat x(t)}{t} \\right)\n  = \\Pi_{\\rm p}(U) + {\\rm exp}(i \\xi \\hat v_+) \\Pi_{\\rm ac}(U) \\] provided that $U$ has no singular continuous spectrum. Here $\\Pi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08562","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}