{"paper":{"title":"Spectral estimates for the Schr\\\"odinger operators with sparse potentials on graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Grigori Rozenblum, Michael Solomyak","submitted_at":"2011-04-18T12:30:10Z","abstract_excerpt":"The construction of \"sparse potentials\", suggested in \\cite{RS09} for the lattice $\\Z^d,\\ d>2$, is extended to a wide class of combinatorial and metric graphs whose global dimension is a number $D>2$. For the Schr\\\"odinger operator $-\\D-\\a V$ on such graphs, with a sparse potential $V$, we study the behavior (as $\\a\\to\\infty$) of the number $N_-(-\\D-\\a V)$ of negative eigenvalues of $-\\D-\\a V$. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of $N_-(-\\D-\\a V)$ under very mild regularity assumptions. A similar construction works also for the lattice"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3455","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"}