{"paper":{"title":"Scattering, homogenization and interface effects for oscillatory potentials with strong singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Michael I. Weinstein, Vincent Duch\\^ene","submitted_at":"2010-10-13T16:21:19Z","abstract_excerpt":"We study one-dimensional scattering for a decaying potential with rapid periodic oscillations and strong localized singularities. In particular, we consider the Schr\\\"odinger equation \\[ H_\\epsilon \\psi := (-\\partial_x^2 + V_0(x) + q(x,x/\\epsilon)) \\psi = k^2 \\psi, \\] for $k\\in\\RR$ and $\\epsilon << 1$. Here, $q(.,y+1)=q(.,y)$, has mean zero and $|V_0(x)+q(x,.)|$ goes to zero as $|x|$ goes to infinity. The distorted plane waves of $H_\\epsilon$ are solutions of the form: $e^{\\pm ikx}+u^s_\\pm(x;k)$, $u^s_\\pm$ outgoing as $|x|$ goes to infinity. We derive their $\\epsilon$ small asymptotic behavior"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.2694","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1010.2694/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"}