{"paper":{"title":"A variational approach to dislocation problems for periodic Schr\\\"odinger operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Martin Kohlmann, Rainer Hempel","submitted_at":"2010-09-18T21:00:45Z","abstract_excerpt":"As a simple model for lattice defects like grain boundaries in solid state physics we consider potentials which are obtained from a periodic potential $V = V(x,y)$ on $\\R^2$ with period lattice $\\Z^2$ by setting $W_t(x,y) = V(x+t,y)$ for $x < 0$ and $W_t(x,y) = V(x,y)$ for $x \\ge 0$, for $t \\in [0,1]$. For Lipschitz-continuous $V$ it is shown that the Schr\\\"odinger operators $H_t = -\\Delta + W_t$ have spectrum (surface states) in the spectral gaps of $H_0$, for suitable $t \\in (0,1)$. We also discuss the density of these surface states as compared to the density of the bulk. Our approach is va"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.3581","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"}