{"paper":{"title":"Schr\\\"odinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues","license":"","headline":"","cross_cats":["hep-th","math-ph","math.MP","quant-ph"],"primary_cat":"math.SP","authors_text":"Kwang C. Shin","submitted_at":"2004-11-07T08:37:59Z","abstract_excerpt":"For integers $m\\geq 3$ and $1\\leq\\ell\\leq m-1$, we study the eigenvalue problem $-u^{\\prime\\prime}(z)+[(-1)^{\\ell}(iz)^m-P(iz)]u(z)=\\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\\arg z=-\\frac{\\pi}{2}\\pm \\frac{(\\ell+1)\\pi}{m+2}$ in the complex plane, where $P(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial. We provide asymptotic expansions of the eigenvalue counting function and the eigenvalues $\\lambda_{n}$. Then we apply these to the inverse spectral problem, reconstructing some coefficients of polynomial potentials fr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0411143","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/math/0411143/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"}