{"paper":{"title":"An Inverse Theorem for Partially Symmetric Two-Dimensional Semiclassical Schr\\\"odinger Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.SP"],"primary_cat":"math.DS","authors_text":"Kuo Wang","submitted_at":"2026-08-05T13:06:29Z","abstract_excerpt":"We prove a formal local inverse spectral result for a two-dimensional semiclassical Schr\\\"odinger operator whose potential well possesses a single reflection symmetry. After a harmonic linear normalization, the potential can be written as $V(x_1,x_2)=\\frac{1}{2}(v_1x_1^2+v_2x_2^2)+\\sum_{j+2k\\ge 3}a_{j,2k}\\,x_1^j x_2^{2k}$, with $v_1/v_2\\notin\\mathbb{Q}$. The operator can be brought into a quantum Birkhoff normal form whose Weyl symbol is a formal series $B \\equiv H_2 + \\sum_{2r+k+\\ell \\ge 2} b_{r,k,\\ell}\\, \\hbar^{2r} \\Omega_1^k \\Omega_2^\\ell$. If the coefficient $a_{30}$ of the cubic term $x_1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.04796","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/2608.04796/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"}