{"paper":{"title":"A Matrix Integral Solution to [P,Q]=P and Matrix Laplace Transforms","license":"","headline":"","cross_cats":[],"primary_cat":"hep-th","authors_text":"A. Morozov, M. Adler, P. van Moerbeke, T. Shiota","submitted_at":"1996-10-17T21:49:44Z","abstract_excerpt":"In this paper we solve the following problems: (i) find two differential operators P and Q satisfying [P,Q]=P, where P flows according to the KP hierarchy \\partial P/\\partial t_n = [(P^{n/p})_+,P], with p := \\ord P\\ge 2; (ii) find a matrix integral representation for the associated $\\t au$-function. First we construct an infinite dimensional space {\\cal W}=\\Span_\\BC \\{\\psi_0(z),\\psi_1(z),... \\} of functions of z\\in\\BC invariant under the action of two operators, multiplication by z^p and A_c:= z \\partial/\\partial z - z + c. This requirement is satisfied, for arbitrary p, if \\psi_0 is a certain"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"hep-th/9610137","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"}