{"paper":{"title":"Existence and uniqueness of solutions of Schr\\\"odinger type stationary equations with very singular potentials without prescribing boundary conditions and some applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David G\\'omez-Castro, Jean-Michel Rakotoson, Jes\\'us Ildefonso D\\'iaz","submitted_at":"2017-10-18T11:22:02Z","abstract_excerpt":"Motivated mainly by the localization over an open bounded set $\\Omega$ of $\\mathbb R^n$ of solutions of the Schr\\\"odinger equations, we consider the Schr\\\"odinger equation over $\\Omega$ with a very singular potential $V(x) \\ge C d (x, \\partial \\Omega)^{-r}$ with $r\\ge 2$ and a convective flow $\\vec U$. We prove the existence and uniqueness of a very weak solution of the equation, when the right hand side datum $f(x)$ is in $L^1 (\\Omega, d(\\cdot, \\partial \\Omega))$, even if no boundary condition is a priori prescribed. We prove that, in fact, the solution necessarily satisfies (in a suitable wa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.06679","kind":"arxiv","version":3},"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"}