{"paper":{"title":"Sharp constructions of eigenfunctions of the magnetic Schr\\\"odinger operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Blair Davey","submitted_at":"2012-12-17T17:59:03Z","abstract_excerpt":"We prove sharpness of quantitative unique continuation results for solutions of $-\\Delta u + W\\cdot \\nabla u + V u = \\la u$, where $\\la \\in \\C$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \\lesssim <x>^{-N}$ and $|W(x)| \\lesssim <x>^{-P}$. For $M(R) = \\inf_{|x_0| = R}||u||_{L^2(B_1(x_0))}$, it was shown in a companion paper that if the solution $u$ is non-zero, bounded, and $u(0) = 1$, then $M(R) \\gtrsim \\exp(-C R^{\\be_0}(\\log R)^{A(R)})$, where $\\be_0 = max{2 - 2P, (4-2N)/3, 1}$. Under certain conditions on $N$, $P$, $\\la$, and the dimension, we construct exampl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.4085","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"}