{"paper":{"title":"Computing Spectra -- On the Solvability Complexity Index Hierarchy and Towers of Algorithms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math-ph","math.LO","math.MP","math.NA","math.SP"],"primary_cat":"cs.CC","authors_text":"Anders C. Hansen, Jonathan Ben-Artzi, Markus Seidel, Matthew J. Colbrook, Olavi Nevanlinna","submitted_at":"2015-08-13T17:42:14Z","abstract_excerpt":"This paper establishes some of the fundamental barriers in the theory of computations and finally settles the long-standing computational spectral problem. That is to determine the existence of algorithms that can compute spectra $\\mathrm{sp}(A)$ of classes of bounded operators $A = \\{a_{ij}\\}_{i,j \\in \\mathbb{N}} \\in \\mathcal{B}(l^2(\\mathbb{N}))$, given the matrix elements $\\{a_{ij}\\}_{i,j \\in \\mathbb{N}}$, that are sharp in the sense that they achieve the boundary of what a digital computer can achieve. Similarly, for a Schr\\\"odinger operator $H = -\\Delta+V$, determine the existence of algor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.03280","kind":"arxiv","version":5},"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/1508.03280/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"}