{"paper":{"title":"Schr\\\"odinger operators with concentric $\\delta$-shells","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Aleksey Kostenko, Hagen Neidhardt, Mark Malamud, Sergio Albeverio","submitted_at":"2012-11-16T21:56:07Z","abstract_excerpt":"We investigate the spectral properties of the Schr\\\"odinger operators in $L^2(\\mathbb{R}^n)$ with a singular interaction supported by an infinite family of concentric spheres $$ \\mathbf{H}_{R,\\alpha}=-\\Delta+\\sum_{k=1}^\\infty\\alpha_k\\delta(|x|-r_k). $$ We obtain necessary and sufficient conditions for the operator $\\mathbf{H}_{R,\\alpha}$ to be self-adjoint, lower-semibounded. Also we investigate the spectral types of $\\mathbf{H}_{R,\\alpha}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.4048","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":""},"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"}