{"paper":{"title":"Eigenfunctions with double exponential rate of localization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.SP"],"primary_cat":"math.AP","authors_text":"A.Logunov, F.Pagano, S.Krymskii","submitted_at":"2025-01-26T00:22:30Z","abstract_excerpt":"We construct a real-valued solution to the eigenvalue problem $-\\text{div}(A\\nabla u)=\\lambda u$, $\\lambda>0,$ in the cylinder $\\mathbb{T}^2\\times \\mathbb{R}$ with a real, uniformly elliptic, and uniformly $C^1$ matrix $A$ such that $|u(x,y,t)|\\leq C e^{-c e^{c|t|}}$ for some $c,C>0$. We also construct a complex-valued solution to the heat equation $u_t=\\Delta u + B \\nabla u$ in a half-cylinder with continuous and uniformly bounded $B$, which also decays with double exponential speed. Related classical ideas, used in the construction of counterexamples to the unique continuation by Plis and Mi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.15354","kind":"arxiv","version":1},"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/2501.15354/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"}