{"paper":{"title":"Constructions of the soluble potentials for the non-relativistic quantum system by means of the Heun functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"quant-ph","authors_text":"C. Mejia Garcia, Guo-Hua Sun, G. Yanez-Navarro, M. A. Mercado Sanchez, Shi-Hai Dong, Shishan Dong","submitted_at":"2017-10-19T15:46:38Z","abstract_excerpt":"The Schr\\\"{o}dinger equation $\\psi\"(x)+\\kappa^2 \\psi(x)=0$ where $\\kappa^2=k^2-V(x)$ is rewritten as a more popular form of a second order differential equation through taking a similarity transformation $\\psi(z)=\\phi(z)u(z)$ with $z=z(x)$. The Schr\\\"{o}dinger invariant $I_{S}(x)$ can be calculated directly by the Schwarzian derivative $\\{z, x\\}$ and the invariant $I(z)$ of the differential equation $u_{zz}+f(z)u_{z}+g(z)u=0$. We find an important relation for moving particle as $\\nabla^2=-I_{S}(x)$ and thus explain the reason why the Schr\\\"{o}dinger invariant $I_{S}(x)$ keeps constant. As an "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.07199","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"}